RAPID DESIGN OF STEEL MONOSYMMETRIC PLATE AND BOX GIRDERS by MILAD KHORASANI B.A.Sc., UNIVERSITY OF BRITISH COLUMBIA, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2010 © Milad Khorasani, 2010 ii Abstract This thesis deals with the design process for steel plate girders and box girders. The design of plate girders is quite prescriptive. A more fundamental approach is required for the design of box girders. Equations explicitly for the design of plate and box girders are heavily influenced by empirical data. This work considers pure steel box girders only, and does not include the design of composite box girder sections. CAN/CSA-S6-00 “Canadian Highway Bridge Design Code” provides detailed design requirements for these composite girders. The design of plate girders follows mostly the requirements specified in the “Handbook of Steel Construction” CAN/CSA-S16.1. However, the S16.1 clauses relating to bending capacity are not well suited for the design of monosymmetric plate girders. Therefore, the code recommends a rational method of analysis such as methods explained in the Structural Stability Research Council’s Guide to Stability Design Criteria for Metal Structures. In addition, “Canadian Highway Bridge Design Code” CAN/CSA-S6-00 provides additional design information for monosymmetric sections. A steel box girder excluding composite design, hereon simply referred to as a box girder, is a purely steel section that could be designed in accordance with CAN/CSA-S16 “Limit States Design of Steel Structures”. However, this standard focuses on clauses for plate girder design, with little specific reference to box girders. Therefore, additional reference materials such as: 1) Guide to Stability Design Criteria for Metal Structures, 2) Crane Manufacturer’s Association of America (CMAA 74-2) standards, and 3) Canadian Highway Bridge Design Code are used for the design of monosymmeteric box girders. An integrated design and analysis environment in a form of formatted spreadsheet is implemented to ease the design process. The spreadsheet checks for both strength and serviceability requirements according to the applicable codes and standards. Included with this project is a clear procedure manual in chapter 7, so that the spreadsheet can be utilized for commercial design. iii Table of Contents Abstract........................................................................................................................................... ii Table of Contents........................................................................................................................... iii List of Tables .................................................................................................................................. v List of Figures................................................................................................................................ vi Acknowledgments ........................................................................................................................ vii 1.0 Introduction............................................................................................................................... 1 2.0 Buckling of Plates..................................................................................................................... 2 2.1 Buckling of Unstiffened Plates............................................................................................. 2 2.1.1 Uniaxial uniform compression ...................................................................................... 2 2.1.2 Pure bending .................................................................................................................. 5 2.1.3 Pure shear....................................................................................................................... 6 2.1.4 Combined stresses ......................................................................................................... 7 2.2 Buckling of Stiffened Plates ................................................................................................. 9 3.0 Design of Plate Girders (CAN/CSA-S16-01)......................................................................... 10 3.1 Preliminary Sizing .............................................................................................................. 13 3.2 Design of Cross Section for Flexure................................................................................... 14 3.2.1 Lateral torsional buckling ............................................................................................ 16 3.3 Design of Cross Section for Shear...................................................................................... 19 3.3.1 Unstiffened girder webs............................................................................................... 19 3.3.2 Transversely stiffened girder webs.............................................................................. 22 3.4 Design of Cross Section for Combined Flexure and Shear ................................................ 24 3.5 Transverse Stiffeners .......................................................................................................... 25 3.6 Bearing Stiffeners ............................................................................................................... 26 4.0 Design of Plate Girders (CAN/CSA-S6-06)........................................................................... 28 4.1 Moment Resistance............................................................................................................. 28 4.1.1 Class 1 and 2 sections .................................................................................................. 28 4.1.2 Class 3 and 4 sections .................................................................................................. 30 4.1.3 Stiffened plate girders.................................................................................................. 31 4.2 Shear Resistance ................................................................................................................. 31 4.3 Combined Shear and Moment Design................................................................................ 33 4.4 Intermediate Transverse Stiffeners ..................................................................................... 34 4.5 Longitudinal Web Stiffeners .............................................................................................. 35 4.6 Bearing Stiffeners ............................................................................................................... 36 5.0 Comparison of Codes for Plate Girder Design ....................................................................... 38 5.1 Design Requirement for Plate Girders................................................................................ 39 6.0 Box Girder Design.................................................................................................................. 45 6.1 Section Classification ......................................................................................................... 45 Description of Element ..................................................................................................... 46 6.2 Shear Strength.................................................................................................................. 46 6.2.1 Shear strength of box girders based on CAN/CSA-S16-01 ........................................ 46 6.2.2 Shear strength of box sections according to SSRC ..................................................... 47 6.3 Bending Strength of Box Sections...................................................................................... 49 6.4 Design Based on CMAA .................................................................................................... 50 6.4.1 Sectional properties ..................................................................................................... 50 6.4.2 Stresses ........................................................................................................................ 50 6.4.3 Yielding failure............................................................................................................ 52 6.4.4 Buckling failure ........................................................................................................... 52 6.4.5 Stiffeners...................................................................................................................... 52 iv 7.0 Introduction to Formatted Spreadsheet................................................................................... 53 8.0 Design Using Formatted Spreadsheet..................................................................................... 56 8.1 Plate Girder Spreadsheet .................................................................................................... 56 8.1.1 User input..................................................................................................................... 57 8.1.2 Preliminary girder dimension computation ................................................................. 57 8.1.3 Girder resistance calculation........................................................................................ 58 8.1.4 Transverse stiffeners.................................................................................................... 58 8.1.5 Bearing stiffeners......................................................................................................... 58 8.1.6 Weld design and girder weight computation ............................................................... 59 8.2 Box Girder Spreadsheet ...................................................................................................... 60 9.0 Fabrication Considerations ..................................................................................................... 63 9.1 Materials ............................................................................................................................. 63 9.2 Proportioning of Spans ....................................................................................................... 64 9.3 Selection of a Girder Cross Section.................................................................................... 64 9.4 Webs ................................................................................................................................... 64 9.5 Stiffeners............................................................................................................................. 65 9.6 Flanges................................................................................................................................ 66 9.7 Field Splices........................................................................................................................ 67 9.8 Fatigue Details .................................................................................................................... 68 10.0 Erection Considerations........................................................................................................ 69 11.0 Conclusions .......................................................................................................................... 71 References..................................................................................................................................... 72 Appendices ................................................................................................................................... 74 Appendix A: Plate Girder Spreadsheet (CSA-S16-01) ............................................................ 75 Appendix B: Plate Girder Spreadsheet (CSA-S6-06)............................................................... 96 Appendix C: Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) ................................ 118 Appendix D: Box Girder Spreadsheet (CSA-S16-01)............................................................ 140 Appendix E: Box Girder Spreadsheet (CMAA)..................................................................... 152 v List of Tables Table 1 – Section class ................................................................................................................. 14 Table 2 – Differences in the design of plate girders..................................................................... 44 Table 3 – Section classification .................................................................................................... 46 vi List of Figures Figure 1 - Uniform compression buckling coefficients, k, for equation 1 ..................................... 3 Figure 2 - Buckling of a plate with an aspect ratio of 3:1 .............................................................. 4 Figure 3 - Definition of effective width.......................................................................................... 5 Figure 4 - Plate subject to pure bending ......................................................................................... 5 Figure 5 - Plate subject to pure shear.............................................................................................. 6 Figure 6 - Buckling coefficients for combined bending and compression..................................... 7 Figure 7 - Stress states in a longitudinally stiffened plate .............................................................. 9 Figure 8 – Unstiffened and stiffened plate girders ....................................................................... 11 Figure 9 – Monosymmetric cross sections of plate girders .......................................................... 12 Figure 10 – Effective distribution of bending stresses ................................................................. 15 Figure 11 – Lateral-torsional buckling motion............................................................................. 16 Figure 12 – M-∆ relationships for laterally unbraced beams ....................................................... 17 Figure 13 – Beam Failure modes.................................................................................................. 17 Figure 14 – Shear strength versus web slenderness...................................................................... 20 Figure 15 – Web shear strength – unstiffened web ...................................................................... 21 Figure 16 - Tension field in stiffened girder web ......................................................................... 22 Figure 17 - Web shear strength – stiffened web ........................................................................... 24 Figure 18 - Shear-moment interaction diagram............................................................................ 25 Figure 19 - Shear strength versus web slenderness ...................................................................... 32 Figure 20 – Tension field action................................................................................................... 35 Figure 21 – Plate girders in the Oak Street Bridge, Vancouver ................................................... 38 Figure 22 – Plate girders in a building floor system..................................................................... 39 Figure 23 – Transverse and longitudinal stiffeners in plate girders ............................................. 43 Figure 24 – Plate girders with openings ....................................................................................... 43 Figure 25 – Common box sections ............................................................................................... 45 Figure 26 – Shear buckling of web............................................................................................... 48 Figure 27 - Shear flow around the section due to Vf .................................................................... 51 Figure 28 – Description section.................................................................................................... 53 Figure 29 – Input section .............................................................................................................. 54 Figure 30 – Calculation section .................................................................................................... 54 Figure 31 – Macro window........................................................................................................... 54 Figure 32 – Equations and referrences ......................................................................................... 55 Figure 33 – Side view and section of mono-symmetric plate girder............................................ 56 Figure 34 - User input................................................................................................................... 57 Figure 35 - Girder parameters....................................................................................................... 57 Figure 36 - Transverse stiffener parameters ................................................................................. 58 Figure 37 - Bearing stiffener parameters ...................................................................................... 59 Figure 38 - Input weld design....................................................................................................... 59 Figure 39 - Monosymmetric box girder designed in spreadsheet................................................. 60 Figure 40 – Input section-box girder spreadsheet......................................................................... 61 vii Acknowledgments I would like to thank the following who have helped and inspired me during my masters program. I would like to gratefully thank my research supervisor, Prof. Dr.-Ing. S. F. Stiemer, University of British Columbia (UBC), Canada, for his guidance and for the effort he has put into making this research an interesting and enlightening experience for me. His perpetual energy and enthusiasm in research had motivated all his students, including me. In addition, he was always reachable and willing to help and support his students with their research. As a result, research life became smooth and rewarding for me. I would also like to thank the financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the contribution of the industrial sponsor Dynamic Structures Ltd. My deepest gratitude goes to my family for their love and support throughout my life. I would like to thank my mother whose love is boundless, my father who is my role model and my sister for sharing her wit and wisdom. 1 1.0 Introduction Steel girders are typically rolled beams, or welded from steel rolled plates resulting in plate or box girders. A rolled beam is a steel girder which has been formed by hot-rolling. The most common type of rolled beams used are the W-, H-, or I-shapes. These shapes have the advantage of quick erection, straight forward fabrication, and an overall simplicity of design. Rolled beams are sometimes reinforced with a flat plate, or cover plate, at the flange. The advantage or cover plate is to increase flexural resistance of the section without having to use a large size rolled beam or plate girder. A plate girder, like a rolled beam, has two flanges and a central web section. Rather than being hot-rolled as a whole, the girder is welded from steel plate elements. Older designs may be bolted or rivetted. When the designer has the ability to specify the section properties to accommodate the local forces it usually results in greater economy of material usage. Variation in plate sizes may represent a girder with the least weight; however, this may increase the fabrication costs. Plate girders are generally used for larger spans than rolled beams. Bridge superstructures and stadium roofs are common applications for steel plate and box girders. Box girders are more expensive to fabricate than plate girders because of their complexity, however, have a number of significant advantages, particularly for longer spans. Firstly because of the shape of the box, the top flange itself can act as the decking without the need for a concrete or timber decking. Box girders possess excellent torsional stiffness, thus they do not usually require secondary members to provide bracing like plate girders. They can also be designed with an aerodynamic shape, again making them ideal for long spans. A famous example for aero-elastic problems with plate girders of I-shape was the Tacoma Bridge. This report presents the design equations and their background for the design of monosymmetic plate girders according to CAN/CSA-S16 and CSA/CAN-S6 standards. A fundamental first- principle approach is chosen for the design of box girders along with design equations based on CAN/CSA-S16, CMAA and SSRC. 2 2.0 Buckling of Plates Webs and flanges of both steel plate and box girders are comprised of flat steel plates with uniform thicknesses. The design of these girders consists of choosing appropriate cross sections and boundary conditions to each of these plate elements, ensuring that each plate element does not fail due to local buckling, yielding or an interaction of the two and that the girder does not fail due to global buckling. It is desirable for global buckling to be the first mode of failure reached as the loads on a girder are increased past the anticipated service loading. Global buckling is associated with large deformations which give warning of failure, while local buckling is generally brittle and sudden. 2.1 Buckling of Unstiffened Plates The buckling stresses are obtained from the concept of bifurcation of an initially perfect structure. In practice, the response of the structure is continuous, due to the inevitable presence of initial imperfections. For a plate to be considered slender, the in-plane dimensions, “a”, and, “b”, need to be significantly greater than the plate thickness, “t”. The dimension, “b”, is usually taken as the direction transverse to the main direction of in-plane loading named width while “a” is taken as in line with the applied load, commonly named length 2.1.1 Uniaxial uniform compression The elastic critical stress of a long plate segment, σc, is determined by the plate width-to- thickness ratio, b/t, by the restraint conditions along the longitudinal boundaries, and by the elastic material properties. It is expressed as ( )( )22 2 c b/tν112 Eπ kσ − = (1) 3 σ Segment of long plate having thickness t, width b, and various edge conditions as tabulated below σ bCase Description of edge support k 1 2 3 4 5 Both edges simply supported One edge simply supported, the other fixed Both edges fixed One edge simply supported, the other free One edge fixed, the other free 4.00 5.42 6.97 0.425 1.277 Figure 1 - Uniform compression buckling coefficients, k, for equation 1 The buckling coefficient, k, is determined by a theoretical critical load analysis, and is a function of plate geometry and boundary conditions. The values given in figure 1 are the lower bounds, with the actual value depending on the plate aspect ratio, m (a/b). This is because a perfect plate under in-plane compression will buckle into m square half waves if the plate aspect ratio is an integer, as this corresponds to the lowest energy mode and for non integer ratios the plate will in theory have a higher buckling stress. 4 Aspect ratio m = a/b k 4 1 2 3 4 1 2 3 4 a = 3b b 2 6 12 No. of half waves Figure 2 - Buckling of a plate with an aspect ratio of 3:1 When a plate is relatively short in the direction of the compressive stress (i.e. a/b << 1), the critical stress may be conservatively estimated by assuming that a unit width of plate behaves like a column.Local buckling causes a loss of stiffness and a redistribution of stresses. Membrane tensions are set up, which resist the growth of deflection and give the plate postbuckling strength. Uniform edge compression in the longitudinal direction results in a non- uniform stress distribution after buckling, and the plate derives almost all of its stiffness from the longitudinal edge supports. Elastic postbuckling stiffness is measured in terms of the apparent modulus of elasticity E* (the ratio of the average stress carried by the plate to the average strain). For simply supported edges E* = 0.5E. There is a decrease in stress at the center of the panel because of the reduction of in-plane stiffness along the center line of the plate caused by the lateral deflection. This reduction of stress due to buckling action gives rise to a semi-empirical method of estimating the maximum strength of plates by the use of the effective width concept. 5 Displacement controlled loading be is plate buckling effective width Region assumed not to transmit stress because of buckling Actual distribution of stress be 2 be 2 σe σe b t Figure 3 - Definition of effective width It assumed that the maximum edge stress acts uniformly over two ‘strips’ of plate leaving the central region unstressed as shown in Figure 3. This width is evaluated in order that the total force carried by the plate is equal to the actual response. 2.1.2 Pure bending Equation 1 is used to calculate the critical buckling load, with substitution of the appropriate buckling coefficient, k. The buckling coefficient of a plate in bending is significantly influenced by the fact that half (in the linear response range) of the load is applied in tension. b a σ −σ σ −σ Figure 4 - Plate subject to pure bending 6 Pure bending plate buckling coefficients: k = 23.9 edges simply supported k = 39.6 unloaded edges fixed k = 0.85 top edge free, bottom edge simply supported k = 2.15 top edge free, bottom edge fixed 2.1.3 Pure shear In a plate subject to pure shear, there exists tension and compression stresses equal in magnitude to the shear stress and inclined at 45º. The destabilizing influence of compressive stresses is resisted by tensile stresses in the perpendicular direction. The critical stress can be obtained by substituting τc and ks for σc and k in equation 1. Unlike the case of edge compression, the buckling mode is composed of a combination of several waveforms, making the buckling analysis of shear more complex. b a τ τ α = b a Figure 5 - Plate subject to pure shear ( )( )c s 22 2E k 12 1 b / t π τ = − ν (2) Shear buckling coefficients: 1. Plate simply supported on four edges: α ≤ 1: s 2 5.34 k =4.00+ α (3) α ≥ 1: s 2 4.00 k =5.34+ α (4) 2. Plate clamped on four edges: α ≤ 1: s 2 8.98 k =5.60+ α (5) 7 α ≥ 1: s 2 5.60 k =8.98+ α (6) 3. Plate clamped on two opposite edges and simply supported on the other two edges: α ≤ 1: s 2 8.98 k 5.61 1.99= + − α α (7) α ≥ 1: s 2 2 5.61 1.99 k 8.98= + − α α (8) 2.1.4 Combined stresses The Crane Code (CMAA Specifications #70 & #74, revised 2000) gives equations for the buckling coefficient for simply supported plates subject to combined in-plane bending and compression according to the cases shown in Figure 6. Figure 6 - Buckling coefficients for combined bending and compression The critical stress is then calculated using the buckling coefficient, k (obtained using equations 9 – 13), in equation 1. σ 1 σ 1 σ 1 ψσ 1 ψσ 1 σ 1 −σ 1 −σ 1 ψσ 1 ψσ 1 σ 1 σ 1 ψσ 1 ψσ 1 σ 1 σ 1 Loading Case Compressive stresses, varying as a straight line. 0 < ψ < 1 Compressive and tensile stresses; varying as a straight line and with the compression predominating. -1< ψ <0 Compressive and tensile stresses; varying as a straight line, with equal edge values, ψ = -1 or with predominantly tensile stresses. ψ < -1 1 2 3 a= α b b a= α b b a= α b b a= α b b 8 Case 1: 1α ≥ 8.4 k 1.1 = Ψ + (9) 1α < 2 1 2.1 k 1.1 = α + α ψ + (10) Case 2: ( ) ( ) ( )k 1 k ' k '' 10 1= + ψ − ψ + ψ + ψ (11) k’ is the buckling coefficient for ψ = 0 (case 1) k’’ is the buckling coefficient for ψ = -1 (case 3) Case 3: 2 / 3α ≥ k 23.9= (12) 2 / 3α < 2 2 1.87 k 15.87 8.6= + + α α (13) With predominant tension replace the width of the plate, b, by 2 times the width of the compression zone for calculation of α and σc. The crane code also gives interaction failure criteria for plates subject to in-plane bending, compression and shear. First the comparison stress, σ1k, is calculated. 2 2 1k 2 2 c c c 3 1 3 4 4 σ + τ σ = + ψ σ − ψ σ τ + + σ σ τ (14) If the resulting critical stress is below the proportional limit, σp, buckling is elastic. If the resulting value is above the proportional limit, buckling is said to be inelastic. For inelastic buckling the compression stress is reduced to σ1kR. 2 y 1k 1kR 2 2 y 1k0.1836 σ σ σ = σ + σ (15) σp = σy/1.32 (16) 9 The comparison stress is then used to calculate a safety factor, ϑB, which is then compared with allowable design factor values, DFB, for each load combination. Elastic buckling 1k B 2 2 DFB 3 σ ϑ = ≥ σ + τ (17) Inelastic buckling 1kR B 2 2 DFB 3 σ ϑ = ≥ σ + τ (18) Design factor DFB requirements: Case 1 DFB = 1.7 + 0.175(Ψ - 1) (19) Case 2 DFB = 1.5 + 0.125(Ψ - 1) (20) Case 3 DFB = 1.35 + 0.05(Ψ - 1) (21) 2.2 Buckling of Stiffened Plates d1 d2 d3 τσ1 σ2 σ3 σ4 Figure 7 - Stress states in a longitudinally stiffened plate Figure 7 shows the stress state in each sub panel for a stiffened plate subject to bending and shear (compressive stresses can easily included by modifying the ratio of longitudinal stresses at the edge of each panel). Each sub panel can be checked for local buckling subject to these stresses using the appropriate buckling coefficients given. A second mode of failure needs to be checked, too, which is local buckling of the plate as a whole including the stiffeners. The third and final mode of failure involves the local buckling of elements of the stiffeners. This is usually avoided by adhering to slenderness limits, which is a simplified buckling check. 10 3.0 Design of Plate Girders (CAN/CSA-S16-01) Plate girders are built up flexural members with slender webs that are usually used in long spans between 15 to 45 m, exceeding the useful range of available rolled shapes. The webs can fail due to buckling, yielding or a combination of the two in either shear or bending. Web dimensions and stiffener spacing are chosen in order to ensure that there is an adequate safety margin with respect to these failure modes. Flanges are sized to prevent local buckling or yielding. A typical plate girder consists of two flanges and a web welded together to form an I-section. There are several types of plate girders depending on the stiffeners used. There are unstiffened plate girders with no stiffeners, stiffened plate girders with only transverse web stiffeners, and stiffened plate girders with transverse and longitudinal web stiffeners. Figure 8 shows these types of plate girders and various stiffeners generally used in the plate girders. Figure 8d shows a typical section of a plate girder. The stiffeners are used to improve the shear capacity of the webs instead of increasing the overall web thickness. The use of stiffeners is more economical in longer spans under greater loads. Although the plate girder sections are bisymmetric in Figure 8, the monosymmetric plate girders are also common. Figure 9 shows various monosymmetric cross sections of plate girders. 11 Figure 8 – Unstiffened and stiffened plate girders 12 Figure 9 – Monosymmetric cross sections of plate girders 13 3.1 Preliminary Sizing The selection of most economical girder dimensions is dependent on a number of variables which are primarily a function of the weight of steel used and the amount of fabrication. A good value to start with for the good depth in regard of moment resistance according to allowable stress design is: f y 1/ 3 M h 540 F ≈ (22) Fy is the yield strength of steel and Mf is the maximum factored moment along the span. An approximated flange area Af can be calculated by assuming that lateral torsional buckling will not govern the design, and the contribution of the web to the bending resistance of the girder is negligible. This assumption is valid as long as lateral supports are provided at intervals close enough to prevent lateral-torsional buckling. Af which represents area of one flange only can be obtained using: f f y M A F h ≈ (23) The web thickness, w, can be calculated by assuming that the entire shear is carried by the web. Therefore, f w s V A wh F = = φ (24) Vf is the maximum factored shear along the span and Φ is the performance factor equal to 0.9 for structural steel and Fs is the ultimate shear stress of steel. Fs will be discussed later. This parameter depends on the web slenderness ratio and the existence of transverse stiffeners. However, for preliminary design Fs is given by equation 25. s yF 0.66F= (25) Furthermore, the code imposes limits on the maximum and minimum web thickness allowed. The index min and max represents minimum and maximum values respectively. y min F h w 83000 = (26) wmax Fy h⋅ 1900 := (27) 14 3.2 Design of Cross Section for Flexure A plate girder subject primarily to bending moment usually fails by lateral-torsional buckling, local buckling of the compression flange, or yielding of one or both flanges. The code divides the section into 4 classes according to their width to thickness ratios. These limits are summarized in the table 1. Table 1 – Section class Class Plate girder Flange under compression Web under flexural compression Class 1 b0 /t ≤ 145/√Fy h/w ≤1100/√Fy Class 2 b0 /t ≤170/√Fy h/w ≤1700/√Fy Class 3 b0 /t ≤200/√Fy h/w ≤1900/√Fy Class 4 b0 /t >200/√Fy h/w >1900/√Fy The Standard specifies for laterally supported members the moment resistance are: Class 1, 2: Mr = φMp = φZFy (28) Class 3: Mr = φMy = φSFy (29) Where S denotes the section modulus (S = I/ŷ). Z represents the plastic modulus which is the first moment of area of the tension and compression zones about the neutral axis. φ=0.9 is the performance factor of steel and Fy is the yield stress of steel. Class 4 sections are such that they buckle locally at a moment less than My and the moment resistance is a function of the width-to-thickness ratios of the component elements. This class is subdivided into three categories. The first category, (i), contains those sections having both flange and web plates falling within Class 4. The second category, (ii), contains those sections having flanges meeting the requirements of Class 3 but having Class 4 webs. The third category, (iii), contains sections having web plates meeting the Class 3 requirements, but with compression flanges exceeding Class 3 limits. The code recommends using CSA Standard S136 for calculation of the moment resistances of Class 4(i) and 4(iii) sections. However, it allows the use of equation 30 shown below for Class 4(iii) sections as an alternative to CSA S136. 15 Mr = φSeFy (30) Where Se is the effective elastic section modulus determined using an effective flange width. The effective width is 670t/√Fy for flanges supported along two edges parallel to the direction of stress and 200t/√Fy for flanges supported along one edge parallel to the direction of stress. For flanges supported along one edge, in no case shall b/t exceed 60. Plate girder sections that have flanges meeting the requirements of Class 3 but having Class 4 webs (Class 4(ii)) can not attain the full moment resistance from equation 8 due to the local buckling (softening) of the slender web. Most plate girders fall into this class of section. This effect is accounted for in the code through the use of an effective width, by only considering 1/6 of the web area in the compression zone to be effective in resisting lateral buckling as shown in Figure 10. Compression flange of Web Area 1 6 Theoretical Experimental Figure 10 – Effective distribution of bending stresses In the code, it is assumed that the maximum moment that can be carried by the sections is that which causes the extreme fiber in the compression zone to reach yield stress, as the thin web will not permit attainment of the theoretical plastic moment of the section. A linear reduction to this maximum attainable value is then applied, which is a function of web slenderness, the relative proportions of the flange and web, and the buckling load of the web. w r r f f A h 1900 M ' M 1.0 0.0005 A w M / S = − − φ (31) Where Aw and Af are the web and flange area respectively. Mf is the factored moment in the girder due to factored dead and live loads. 16 3.2.1 Lateral torsional buckling Beams subjected to flexure have much greater strength and stiffness in the plane in which the loads are applied (major principal axis) than in the plane of the minor axis. It has been assumed thus far that the strength of the beam is determined by the capacity of its cross-section, and this, in turn, is dependent on the local buckling capacity of its plate elements. However if the beam is laterally unsupported, the strength may be governed instead by lateral- torsional buckling of the complete member, as shown in figure 11. Beams are especially prone to this type of buckling during the construction phase, where lateral bracing are either absent or different in type from their permanent ones. Position before loading Position before buckling Position after buckling ∆ Figure 11 – Lateral-torsional buckling motion At a given stage of loading, the cross-section may twist and bend about its weak axis, reducing its ultimate moment capacity due to large deflections and yielding. The main parameter affecting lateral-torsional buckling strength is the distance between lateral braces. Other influences are: the type and position of the loads, the restraints at the ends and at intermediate locations, the type of cross sections, continuity at supports, the presence or absence of stiffening devices that restrain warping at critical locations, the material properties, the magnitude and distribution of the residual stresses, prestressing forces, initial imperfections of geometry and loading, discontinuities in the cross section, cross-sectional distortion, and interaction between local and overall buckling. 17 M Mp My ∆ A B C D L ∆ M M Figure 12 – M-∆ relationships for laterally unbraced beams Beams can also be classified in terms of the effect of lateral-torsion buckling on the ultimate moment capacity attainable, as shown in figure 13. Local Buckling Inelastic Lateral Buckling Elastic Lateral Buckling Stocky Intermediate Slender Mcr L Figure 13 – Beam Failure modes A stocky beam is defined as a beam which is able to reach its local buckling capacity before lateral buckling occurs. The local buckling capacity of Class 1 or 2 sections is Mp and for Class 3 sections, My. A slender beam buckles laterally before the member yields, and the resistance to lateral-torsion buckling is based on full elastic action. For the intermediate beam, the bending 18 moment at the instant before lateral buckling is sufficient to cause portions of the section to yield, thus the resistance to both lateral and twisting motions is reduced. The Standard provides an equation for calculating the elastic lateral buckling strength of doubly symmetric beams. ( ) wy 2 y 2 u CIπE/LGJEIL πω M += (32) 2 2 1.75 1.05 0.3 2.5ω = + ς + ς ≤ (33) where ζ is the ratio of the smaller bending moment to the larger bending moment at opposite ends of the unbraced length. This equation provides a reasonable estimate of the moment at which lateral buckling will occur, provided that the strains in the member are less than the yield strain at the instant before buckling. Thus, equation 32 is accepted as the basis for the design of slender members. Due to relatively large residual stresses in the flange tips, yielding will occur when the applied moment reaches approximately two-thirds of the buckling capacity of the member, Mp for Class 1 or 2 sections, and My for Class 3 or 4 sections. Equation 32 is thus valid until Mu reaches two- thirds of Mp for Class 1 or 2 sections, and My for Class 3 or 4 sections. Mu ≤ 0.67Mp (Slender members): Mr = φMu for Classes 1- 4 (34) Mu ≥ 0.67Mp (Stocky members): p r p p u 0.28M M 1.15 M 1- M M = φ ≤ φ Class 1 or 2 (35) y r y y u 0.28M M 1.15 M 1 M M = φ − ≤ φ Class 3 or 4 (36) CAN/CSA-S16 does not give equations for the calculation of mono-symmetric sections such as box girders or plate girders with flanges of differing width. The code recommends the use of equations given in ‘Guide to stability Design Criteria for metal structures’; however these are in the general form. CAN/CSA-S6-06 gives worked examples based on the same expressions listed 19 in ‘Guide to Stability Design Criteria for metal structures’ for monosymmetric plate girders and open top box girders. Lateral torsional buckling can be avoided by properly spaced and designed lateral bracing, or by using cross sections which are torsionally stiff, such as box-shaped sections or open-section beam groups connected intermittently by triangulated lacing or by diaphragms or by ensuring that the required design moment does not exceed the lateral-torsional buckling capacity. 3.3 Design of Cross Section for Shear CAN/CSA-S16-01 identifies 3 limiting states for determining the shear capacity of the web; shear buckling, shear yielding or a combination of both. The first mode of failure is dependent on the web slenderness and stiffener spacing. In the following section the shear resistance of the plate girders with and without transverse stiffeners will be considered. 3.3.1 Unstiffened girder webs When steel is subject to a combined stress condition, the yield stress in shear, Fy, is normally approximated by the Von-Misses value, which is increased to allow for the strengthening effects of strain hardening. The shear yielding strength is given by: y s y F F 0.66F 3 = λ → (37) The general equation for buckling of a plate subject to pure shear (equation 2) is used to calculate the resistance to shear buckling after substituting the correct notation for dimensions of plate girder webs. ( )( ) 2 cr 22 k E 12 1 h / w π τ = − ν (38) For a/b ≥ 1.0, for simply supported edges, it is found that: ( )2 4.0 k 5.34 a/h = + (39) 20 The Standard assumes representative values for the terms in equation 38 (E = 200 GPa, υ = 0.3, k = kv, τcr = Fs). With these values equation 38 reduces to: ( ) v s 2 180000k F h / w ≈ (40) Figure 14 below shows equations 38 and 40 plotted on the same chart, and clearly demonstrates that the failure mode is dependent on the slenderness, (h/w), of the web. Fs MPa Web slenderness h/w Fs = 180000 kv (h/w)2 Fs = 0.66Fy Figure 14 – Shear strength versus web slenderness A third mode of failure due to combined shear yielding and buckling creates a transition curve between the curves given by equation 40 shown in Figure 14. The equation for this curve is given in the code and was chosen mainly on the basis of experimental evidence. 1. ( ) v yh / w 439 k / F :≤ s yF 0.66F= (41) 2. ( )v y v y439 k / F h / w 621 k / F :≤ ≥ 21 ( ) y v s cri 290 F k F F h/w = = (42) 3. ( ) v yh / w 621 k / F :≥ ( ) v s cre 2 180000k F F h/w = = (43) The Standard also imposes a limit on slenderness: y h 83000 w F ≤ (44) These equations are presented graphically in figure 15. Fs = 0.66Fy Fs = 290 Fykv (h/w) Fs = 180000 kv (h/w)2 621439 83000 Fy kv Fy kv Web slenderness h/w Fy Fs MPa Figure 15 – Web shear strength – unstiffened web The capacity of the section is calculated by multiplying the ultimate shear stress for the web multiplied by a performance factor φ and the web area Aw. swr FφAV = (45) 22 3.3.2 Transversely stiffened girder webs Stiffened webs may fail due to shear buckling before shear yielding occurs, however, subsequent to buckling the stress distribution in the web changes and significant amount of postbuckling strength may be developed because of the diagonal tension that develops in web panels and compressive forces in the transverse stiffeners that border those panels. This is called tension field action. Figure 16 shows the general distribution of the tension field that develops in the plate girder with transverse stiffeners. This tension field is anchored by the flanges and stiffeners. a a s h T σtaw T + ∆T Tw 2 N.A. F a V Tw 2 2 a 2 2 V 2 θ Figure 16 - Tension field in stiffened girder web The code specifies several equations for the shear resistance of the web based on the web slenderness and stiffener spacing. 1. ( ) v yh / w 439 k / F :≤ s yF 0.66F= (46) 23 2. ( )v y v y439 k / F h / w 502 k / F :≤ ≥ ( ) y v s cri 290 F k F F h/w = = (47) 3. ( )v y v y502 k / F h / w 621 k / F :≤ ≥ s cri tF F F= + (48) y cri t 2 0.50F 0.866F F 1 (a / h) − = + (49) 4. ( ) v yh / w 621 k / F :≥ s cre tF F F= + (50) ( ) v cre 2 180000k F h/w = (51) y cre t 2 0.50F 0.866F F 1 (a / h) − = + (52) where ( )v 2 4.0 a / h 1: k 5.34 a / h > = + (53) ( )v 2 5.34 a / h 1: k 4.0 a / h < = + (54) Where w, h and a are web thickness, web height and stiffeners spacing respectively. Fcre and Fcri are critical elastic and inelastic buckling stress in shear. Ft is the contribution due to tension field action. In the above equations, Fy is in MPa units. The Standard also imposes the same limit on slenderness as for unstiffened webs: y h 83000 w F ≤ (55) These equations are shown diagrammatically in Figure 17. The dotted line shows the shear resistance of the unstiffened web which is shifted upwards by the development of the tension 24 field Ft in the stiffened web. Fs = 0.66Fy Fs = 290 Fykv (h/w) Fs = 180000 kv + Ft (h/w)2 621439 83000 Fy kv Fy kv Web slenderness h/w Fy Fs MPa 502 Fy kv Fs = 290 Fykv + Ft (h/w) Ft Ft Figure 17 - Web shear strength – stiffened web The capacity of the section is calculated by multiplying the ultimate shear stress for the web by a resistance factor φ and the web area Aw. swr FφAV = (56) 3.4 Design of Cross Section for Combined Flexure and Shear The presence of significant shear and moment together can occur at certain locations along the girder span such as at the interior supports of a continuous beam. In such cases, the effect of the interaction between these two forces upon girder strength must be considered. This is shown diagrammatically in figure 18. The diagram shows that if the factored moment is less than 75% of the moment resistance, the full shear shear resistance may be used. Similarly, if the factored shear is less than 60% of the shear resistance, the full moment resistance is used. 25 Figure 18 - Shear-moment interaction diagram The code applies a straight line to the interaction curve in order to simplify the equations. f f r r M V 0.727 0.455 1.0 M V + ≤ (57) 3.5 Transverse Stiffeners Transverse stiffeners are used throughout the web to provide tension field action. The S16-01 Standard does place limits on maximum stiffener spacing. The limits for stiffener spacing are as follows: 2)/( 67500 wh h a ≤ when 150/ >wh or; (58) ha 3≤ when 150/ ≤wh (59) Each stiffener should resist the summation of the vertical components of the tension field action (F) over one panel width. ( )2t 2 a / hhw a F 2 h 1 (a / h) σ = − + (60) 0.6 1.0 1.0 0.75 Mf Mr Vf Vr 0.727 + 0.455 = 1.0 Mf Mr Vf Vr 26 Based on the assumption that the stiffener will yield before buckling, CAN/CSA-S16-01 provides an equation for the stiffener area required. y s 2 ys Faw a / h A 1 C D 2 F1 (a / h) ≥ − + (61) v 2 y 310000K C 1 0.1 F (h / w) = − ≥ (62) The stiffener factor, D, can be found as follows D = 1.0 stiffeners furnished in pairs D = 1.8 stiffeners composed of angles placed on one side of web only D = 2.4 stiffeners composed of plates placed on one side of web only Furthermore, the code imposes a limit on the moment of inertia of the stiffener to prevent lateral displacement of the web. This is given as follows: 4 s h I 50 ≥ (63) To prevent local buckling of the stiffener under the compressive force, F, the slenderness ratio (b/t) should not exceed: y b 200 t F ≤ (64) 3.6 Bearing Stiffeners The application of a concentrated load to the flange of a girder can result in local failure. This can happen either by localized buckling of the web in the region where it connects to the flange or by overall buckling of the web. In a stockier web, the web will fail due to yielding. In a more slender web, crippling, or localized buckling, will govern. Bearing resistance must be calculated for both possible modes of failure, and the smaller value will govern as the bearing resistance. The equations for bearing for an interior location of the girder are given in clause 14.3.2(a): ybir FtNwB )10( += φ (65) 27 EFwB ybir 245.1 φ= (66) Where N= length of bearing w= web thickness t= flange thickness φbi = 0.8 While the equations for bearing for the end reactions are given by: yber FtNwB )4( += φ (67) EFwB yber 260.0 φ= (68) Where φbe = 0.75 28 4.0 Design of Plate Girders (CAN/CSA-S6-06) This section thoroughly covers the design of plate girders according to Canadian Highway Bridge Design Code CAN/CSA-S6-06. 4.1 Moment Resistance The moment resistance calculations for plate girders are divided into two categories: those for Class 1 and 2 sections in 10.10.2 and those for Class 3 and 4 sections in 10.10.3. The fundamental distinction is that Class 1 and 2 sections use the plastic section modulus to calculate the moment resistance, while Class 3 and 4 sections use the elastic modulus. The reasoning for this is described in the previous section of this report. Both 10.10.2 and 10.10.3 include four main subsections. These describe (i) the limiting width- to-thickness ratios for steel sections of the corresponding classes, (ii) moment resistance for laterally supported members, (iii) moment resistance for laterally unsupported members and (iv) moment resistance for bending about the minor axis. Each of these conditions makes use of different geometric properties of the plate girder to calculate the moment resistance. 4.1.1 Class 1 and 2 sections The basic equation for the moment resistance of Class 1 and 2 sections is the factored plastic moment resistance for laterally supported members, defined in clause 10.10.2.2 as: pxsyxsr MFZM φφ == (69) Laterally supported members are expected to attain their full plastic moment strength, and thus the moment resistance varies only with the plastic section modulus and the yield strength of the steel. A laterally unsupported member may fail by lateral torsional buckling or a combination of weak axis buckling and lateral buckling. In order to account for this reduction in bending strength due to buckling, clause 10.10.2.3 (a) introduces equations that reduce the moment resistance of the section based on its geometric properties. These equations are as follows: ps u p psr M M M MM φφ ≤−= ] 28.0 1[15.1 when pu MM 67.0> (70) 29 usr MM φ= when pu MM 67.0≤ (71) The Mu term in these equations is the critical elastic moment, which can be described as the moment that will cause buckling in the unbraced beam. This moment is defined in clause 10.10.2.3 (b) by the equations: )]1([ 2121 2 BBBJGIE L M sysu +++= πω (72) 5.23.005.175.1 22 ≤++= κκω (73) JG IE L B s ysx 21 β π= (74) JGL CE B s ws 2 2 2 π = (75) The coefficient ω2 accounts for the for the increased moment resistance of the beam when subjected to a moment gradient. This coefficient depends on κ, the ratio of the smaller factored moment to the larger factored moment at opposite ends of the beam. The coefficients B1 and B2 are included to account for the monosymmetric nature of the plate girder. These coefficients vary with βx, the coefficient of monosymmetry, as well as J, the St. Venant torsional constant and Cw, the warping torsional constant. These values all take into account the complex torsional buckling tendencies of the unusual box girder cross-section. Equations for these terms are given in the S6 commentary, in section C10.10.2.3, Laterally Unsupported Members. The coefficient of monosymmetry is defined by the equation: ∫ ++= Axx x edAyxy I 2)( 1 22β (76) The closed-form solution to the integral is also provided in the commentary, but is not shown here. Also not shown are the equations for J and Cw. These formulas are complex, and would be beyond the scope of this report to derive from the geometry of the cross-section. The last clause in section 10.10.2 identifies the moment resistance for bending about the minor axis of the box girder. This equation is: 30 pysyysr MFZM φφ == (77) 4.1.2 Class 3 and 4 sections Class 3 sections will not attain the plastic moment capacity, so the moment resistance is based on the yield moment. The equation for laterally supported Class 3 sections in bending is defined in clause 10.10.3.2 as: ysyxsr MFSM φφ == (78) Laterally unsupported plate girder sections may be subject to lateral torsional buckling, thus beams in these conditions have reduced moment resistance equations listed in clause 10.10.3.3: ys u ysr M M My MM φφ ≤−= ] 28.0 1[15.1 when yu MM 67.0> (79) usr MM φ= when yu MM 67.0≤ (80) These equations are very similar to the laterally unbraced bending equations from clause 10.10.2.3, except that the plastic moment capacity has been replace with the elastic moment capacity. Similar to Class 1 and 2 sections, the moment resistance for Class 3 sections is based solely on the section properties about the weak axis. The plastic section modulus has once again been replaced with the elastic section modulus, leading to the following equation in 10.10.2.5: ysyysr MFSM φφ == (81) Section 10.10.3 also provides for the calculation of the moment resistance for certain Class 4 sections. Plate girders must meet certain additional conditions, such as that the compression flange must have continuous lateral support, and the web must still meet Class 3 requirements. In such cases, the moment resistance of the section can be calculated using the equations for Class 3 sections, except that the elastic section modulus is replaced with the effective section modulus, Se. The section modulus has been reduced by using only a portion of the actual flange width. 31 4.1.3 Stiffened plate girders The clauses in section 10.10.4 reduce the moment capacity of the plate girders when the webs are slender. This is due to the susceptibility of slender webs to buckle during flexure, which would seem to be equally applicable to single webbed plate girders or double webbed box girders. This section provides additional limits the width-to-thickness ratios of webs that have transverse stiffeners, which will be further discussed later in this report. For girders that have webs without longitudinal stiffeners and that are more slender than the limit of yc Fwd /1900/2 > , the moment resistance will be reduce by the following factor: SMw d A A sf c w cf φ/ 19002 ( 1200 300 1 0.1[ − + − )] (82) 4.2 Shear Resistance The shear resistance of the plate girder is calculated using the area of the webs, neglecting the contribution of the flanges. The ultimate shear stress Ft, depends on the slenderness of the web. The ultimate shear stress includes two components, the shear buckling stress Fcr, and the tension field component of the post-buckling stress Ft. These components are added together to provide the ultimate shear stress. swsr FAV φ= (83) tcrs FFF += (84) 32 Figure 19 - Shear strength versus web slenderness The buckling stress and post-buckling stress vary depending on the slenderness of the web. The variation of these stresses with the web slenderness is shown in the graph above, from the S16.1 Commentary. Three sets of equations are given to represent the different modes of behavior for the webs in shear. The first set of equations, in clause 10.10.5.1(a), describes the criteria for stockier webs, where y v F k w h 502≤ , and the web will fail in full yielding. The post-buckling stress is neglected, and the buckling stress is derived from the vonMises-Hencky- Huber yield criterion. This value is more conservative than that used in S16.1. ycr FF 577.0= (85) 0=tF (86) As the slenderness of the web increases, the failure mode shifts from full yielding to inelastic buckling. This occurs when y v y v F k w h F k 621502 ≤≤ , corresponding to another set of equations for the ultimate shear stress in 10.10.5.1(b). These equations now include a component for the post-buckling strength due to tension field action. This tension field action is only relevant if the web of the girder is stiffened. 33 wh kF F vy cr / 290 = (87) ) )/(1 1 )(866.05.0( 2ha FFF cryt + −= (88) The equations for the most slender category of webs, when y v F k w h 621> , are defined in clause 10.10.5.1(c). For this case, the web will fail due to elastic buckling. The post-buckling stress, assuming the web is stiffened, remains the same as for the inelastic buckling case. However, the equation describing the buckling stress has been modified. 2)/( 180000 wh k F vcr = (89) ) )/(1 1 )(866.05.0( 2ha FFF cryt + −= (90) The shear buckling stress in both elastic and inelastic buckling varies with the shear buckling coefficient kv. This coefficient depends on the ratio of the stiffener spacing to the height of the web, and is calculated from the following equations. 2)/( 34.5 4 ha kv += when 1/ <ha (91) 2)/( 4 34.5 ha kv += when 1/ ≥ha (92) Once all of the parameters that define the ultimate shearof the web have been evaluated, the factored shear resistance of the plate girder can be determined. This value is compared to the factored shear force in the girder to determine if the shear capacity is adequate. 4.3 Combined Shear and Moment Design Clause 10.10.5.2 provides the check for sections subject to combined shear and moment. The equation involves linear interaction with coefficients for shear and moment based on Basler (1963). The clause only applies to webs of girders that depend on tension field action to carry shear, that is with y v F k w h 502> . The girder does not lose a significant amount of shear strength 34 if Mf/Mr<0.75 because the moment will be carried by the flanges of the girder, and the web will still be able to carry the shear. The combined shear and moment equation is shown below: 0.1455.0727.0 <+ r f r f V V M M (93) 4.4 Intermediate Transverse Stiffeners The first portion of section 10.10.6 is related to intermediate transverse stiffeners. Web stiffeners in a plate girder will not be required if the factored shear load is less than the unstiffened shear resistance, and the slenderness of the web does not exceed the limit of 150/ ≤wh . In such instances, the web does not need tension field action to resist the shear loads on the girder. If these restrictions are not met, then web stiffeners are required. The following paragraph in section 10.10.6 identifies the limits for spacing between web stiffeners. The limits for stiffener spacing are as follows: 2)/( 67500 wh h a ≤ when 150/ >wh or; (94) ha 3≤ when 150/ ≤wh (95) The limit of 3h for stockier webs is necessary to ensure that tension field action is properly developed. Interestingly, the limit for the case of slender webs is provided only to ensure the ease in handling and fabrication the girders. This limit is not related to the strength of the girder, and is based practical limits and experience in the industry. Once the maximum spacing of the stiffeners has been determined, the stiffeners must be sized. Lower limits have been placed on the moment of inertia and the cross-sectional area of the stiffeners. These limits are necessary to ensure that the stiffeners can withstand the compression loads from the tension field action in the web. The limits for stiffener moment of inertia and area are given below: jawI 3≥ where 2)/(5.2 2 −= ahj but not less than 0.5 (96) 0)18] )/(1 / 1[ 2 ( 2 2 ≥− + −= YwCD V V ha haaw A r f s (97) 35 2)/( 310000 1 whF k C y v−= but not less than 0.10 (98) The tension field action in the web develops like a truss, where the segments of web between the stiffeners will transfer the shear loads in tension to the adjacent stiffeners. These stiffeners will then carry the shear in compression, completing the truss. Figure 20 – Tension field action Additionally, limits are placed on the width-to-thickness ratios of the stiffeners to prevent local buckling 4.5 Longitudinal Web Stiffeners The main effect of installing longitudinal web stiffeners is the change in the allowable spacing between intermediate transverse stiffeners. These new spacing requirements are defined in section 10.10.7. This modified spacing may provide an overall reduction in material if the transverse stiffener spacing is increased sufficiently. The maximum transverse stiffener spacing is modified to 1.5hp, where hp is the maximum subpanel depth. However, this spacing limit is not necessarily an increase, since webs without longitudinal stiffeners have a maximum spacing of 3h, if 150/ ≤wh . The creation of subpanels due the longitudinal stiffener reduces the slenderness of the web, and increases the web stiffness. Therefore; slender webs with longitudinal stiffeners are not subjected to the stiffener spacing limit of: 36 2)/( 67500 wh h a ≤ when 150/ >wh . (99) Clause 10.10.7.2 provides limits for the size and thickness of longitudinal stiffeners. The maximum width-to-thickness ratio is yF/200 , and the maximum width of the stiffener is 30t. This clause also specifies minimum values for the moment of inertia and the radius of gyration for the longitudinal stiffeners: ]13.0)/(4.2[ 23 −≥ hahwI (100) 1900 yF ar ≥ (101) These limits ensure that the longitudinal stiffeners will not undergo local buckling. Clause 10.10.7.3 provides a number of adjustments to the parameters used to calculate the properties of transverse stiffeners when longitudinal stiffeners are present. The main difference is that typically the subpanel height is used instead of the full web height when calculating slenderness ratios. In this way, longitudinal stiffeners can significantly improve the effectiveness of transverse stiffeners. 4.6 Bearing Stiffeners Section 10.10.8 provides equations to determine the strength of the web in bearing. Webs can fail in bearing either due to crippling or yielding. The failure mode depends on the slenderness of the web. In a stockier web, the web will fail due to yielding. In a more slender web, crippling, or localized buckling, will govern. Unlike similar clauses, a limiting width-to- thickness ratio is not provided to determine which condition will govern. Instead, a bearing resistance must be calculated for both possible modes of failure, and the smaller value is used as the overall bearing resistance. Two sets of equations are given for bearing at the end of the beam and along the span of the beam. The equations are similar in form, but have slightly different coefficients. Notably, these clauses include a distinct resistance factor for bearing, being smaller than the resistance factor for flexural compression. The equations for bearing within the span of the beam are giving in clause 10.10.8.1(a): ybir FtNwB )10( += φ (102) EFwB ybir 245.1 φ= 37 While the equations for bearing at the end of the beam are given in clause 10.10.8.1(b): yber FtNwB )4( += φ (103) EFwB yber 260.0 φ= These equations are not the same as those given in S16. The S6 equations do not consider the distance from the flange to the toe of the web fillet weld, nor the depth of the web. The equations in S6 are derived from finite element analysis, while those from S16.1 are based on empirical analysis. If the bearing resistance of the web is less than the factored concentrated loads at the point of application, bearing stiffeners are required. S6 also includes equations that determine the bearing resistance of the bearing stiffeners. 38 5.0 Comparison of Codes for Plate Girder Design In this section the requirements for the design of plate girder design will be compared between the “Handbook of Steel Construction CAN/CSA-S16” and the “Canadian Highway Bridge Design Code CAN/CSA-S6-00”. Plate girders can be used in bridges, as crane girders in industrial buildings, and for long floor spans in other buildings. As their cross sections are chosen according to the load demand of the individual structure, they are most economical and efficient. Figure 21 shows plate girders employed in bridge span. Figure 22 shows plate girders as a part of building floor system. Figure 21 – Plate girders in the Oak Street Bridge, Vancouver 39 Figure 22 – Plate girders in a building floor system 5.1 Design Requirement for Plate Girders The building code and bridge code are similar in the major aspects of design of the plate girders. The proportioning of flanges, web transverse stiffeners, and bearing stiffeners, design for resistance of combined shear and moment, and resistance to web crippling and yielding are similar in most parts in both the codes. Minor differences exist between the codes in the proportioning. The bridge code limits the minimum web thickness to 10mm while the building code has no such restriction (S6.06 Clause 10.7.2). This restriction in bridge code is to reduce the susceptibility of web to fatigue and brittle fracture. On the other hand, the building code restricts the web slenderness ratio, h/w, to less that 83000/Fy, (S16.01 Clause 14.3.1) where h = height of the web and clear distance between the flanges w = thickness of the web Fy = yield strength of the steel in MPa. For Fy =350 MPa, the slenderness ratio is limited to 237. This limit is waived if it is proven that the compression flange does not buckle under factored loads. In the case of bridge code, the 40 maximum slenderness ratio is 6000/√Fy, for webs stiffened with both longitudinal and transverse stiffeners (S6.06 Clause 10.10.4.2). This ratio is equal to 321. A comparison of the ratios at first indicates that the bridge code is more lenient with respect to web slenderness. However, it must be noted that the use of longitudinal web stiffener would reduce to the effective height of web and provides increased resistance to web buckling. In the case of building code, although the use of transverse stiffeners is implicit, the limit on slenderness ratio is for webs without longitudinal stiffeners. Hence, the building code is more lenient in limiting the web slenderness. However, the building code specifies that the structures subjected to fatigue must have web slenderness ratio less than 3150/√Fy, for webs with transverse stiffeners (S16.01 Clause 26.4.2). This is similar to the restriction in the bridge code for webs with transverse stiffeners (S6.06 Clause 10.17.2.5). Therefore, the bridge code is more conservative with proportioning of webs due to fatigue considerations. Similarly, for flanges the building code limits the maximum width-to- thickness ratio to 60 (S16.01 Clause 13.5), whereas the bridge code restricts it to 30 (S6.06 Clause 10.10.3.4). Furthermore, the bridge code replaces the web height h, with 2dc where dc is the depth of compression portion of web, for the computation of web slenderness under flexural compression (S6.06 Clause 10.10.3.1). This implies that the depth of compression in the web must be equal to half of web height in order to be comparable to similar restrictions in the building code. Therefore, the bridge code considers the web as slender when (2dc/w) >1900/√Fy, whereas the building code considers it for h/w >1900/√ Fy. The reduction in moment resistance due to slender web, in building code is computed as, (S16.01 Clause 14.3.4), −−= SMw h A A MM ff w rr φ/ 1900 0005.01' (104) where Mr = factored moment resistance of the plate girder Aw = web area Af = flange area Mf = maximum bending moment φ = resistance factor of steel = 0.90 S = elastic section modulus In bridge code, it is computed as (S6.06 Clause 10.10.4.3), 41 − + −= SMw d AA A MM f c cfw w rr φ/ 19002 1200300 1' (105) where φ = resistance factor of steel = 0.95 Acf = area of the compression flange The comparison of equations 104 and 105 yields two main differences. The web height in building code is replaced by 2dc in the bridge code and the factor of 0.0005 applied to the ratio of web area to flange area. The factor of 0.0005 Aw/Af is equivalent to Aw/2000Af. If compression flange area and web area are equivalent in equation 105, ratio of areas becomes Aw/1500Acf. This implies that the bridge code specifies a greater reduction in the moment resistance for plate girders with slender webs compared to the building code with a similar girder design. Moreover, for the web in flexural and axial compression, the slenderness ratio limit in building code includes φ factor applied to Cy, axial compression load at yield stress (S16.01 Table 2, Clause 11.2). The bridge code follows the previous edition of the building code, which does not include the φ factor (S6.06 Table 10.3). Both codes are similar in all other aspects of width-to- thickness ratios. Further major differences between the codes are in two main areas: stiffeners and openings. The building code is more lenient with respect to the stiffeners than the bridge code. In building code, the intermediate transverse stiffeners are waived if the factored shear resistance, Vr, is greater than the shear force under factored load, Vf. In bridge code, the intermediate transverse stiffeners must be designed unless h/w ≤ 150, and Vr > Vf (S6.06 Clause 10.10.6.1). Another minor difference between the codes is in the computation of factored shear resistance. The building code specifies greater shear resistance for h/w ≤ 439√kv/Fy, where kv is the shear buckling coefficient (S16.01 Clause 13.4.1.1(a)). In contrast, the bridge code maintains that all webs with slenderness ratio less than 502√kv/ Fy, have same shear resistance (S6.06 Clause 10.10.5.1). In designing the transverse stiffeners, the building codes specifies the minimum moment of inertia of the stiffener about the web, I, as (h/50)4, whereas the bridge code specifies 42 I ≥ aw3j (106) where, a = spacing between the stiffeners j = 2.5(h/a)2 – 2 ≥ 0.5 Furthermore, the area of the transverse stiffeners is specified in the building code as (S16.01 Clause 14.5.3), + −= CYD ha haaw As 2)/(1 / 1 2 (107) Where 10.0 )/( 310000 1 2 ≥−= whF k C y v (108) D = stiffener factor Y = ratio of specified minimum yield point of web steel to specified minimum yield point of the stiffener steel The area in Eq. (27) can be further decreased by a ratio of Vf / Vr. In the case of bridge code, the area of transverse stiffener is proportioned as (S6.06 Clause 10.10.6.2(b)), 018 )/(1 / 1 2 2 2 ≥ − + −= YwCD V V ha haaw A r f s (109) The comparison of the proportioning of As reveals that the bridge code allows lesser area of stiffeners than the building code. Moreover, the bridge code specifies the width of the plate used as stiffener to be greater than a quarter of the flange width. It should also be greater than (50+h/30) (S6.06 Clause 10.10.6.2). The building code does not have these restrictions. 43 Figure 23 – Transverse and longitudinal stiffeners in plate girders Figure 24 – Plate girders with openings In addition to the above differences, the building code addresses the connections of stiffeners, proportioning of the end panel and cover plates to the flanges. These details are not addressed in the bridge code. Instead, the proportioning of the longitudinal stiffeners and the design of transverse stiffeners in the presence of longitudinal stiffeners are detailed. In contrast, the 44 building code does not stipulate design procedures for longitudinal stiffeners. The plate girders in the buildings do not usually need the longitudinal stiffeners since it is more economical in buildings to employ other structural systems, such as trusses, if greater shear resistance is warranted. Figure 23 shows the transverse and longitudinal stiffeners in a plate girder for bridge. Furthermore, openings in the plate girders are explicitly addressed in the building code. For the plate girders utilized in the buildings, the openings are essential to run the building utilities. The openings are not a concern in case of bridges and hence, these are not addressed in the bridge code. Figure 24 shows the plate girders with openings. Table 2 summarizes the differences between the building and bridge codes in the design of plate girders, along with the relevant clauses. Table 2 – Differences in the design of plate girders Aspect of design Building Code Clause Bridge Code Clause Minimum web thickness Not addressed 10.7.2 Web slenderness ratio 14.3.1 10.10.4.2 Width-to-thickness ratio Table 2 Table 10.3 Class 4 sections 13.5(c) 10.10.3.4 Reduction in moment resistance for slender webs 14.3.4 10.10.4.3 Monosymmetric sections Not Addressed 10.10.2.3 Waiver of transverse stiffeners 13.4.1.1, 14.5 10.10.6.1 Factored shear resistance 13.4.1.1(a) Not addressed Proportioning of moment of inertia of transverse stiffeners 14.5.3 10.10.6.2(a) Proportioning of area of transverse stiffeners 14.5.3 10.10.6.2(b) Proportioning of width of transverse stiffeners Not addressed 10.10.6.2(b) Connections of stiffeners 14.5.4 10.10.6.4 Proportioning of end panel 14.4.1 Not addressed Cover plates of flanges 14.2.4 Not addressed Longitudinal stiffeners Not addressed 10.10.7 Openings 14.3.3 Not addressed 45 6.0 Box Girder Design Steel box girders are used in bridges due to their torsional capacity and good stability during construction. Box girders are more stable and able to span greater distances than plate girders. However the design and construction of box girders are more difficult than plate girders. The shapes of box girders are usually rectangular or trapezoidal in and can be either open or closed, as shown in below. Multi-Spine Multi-Cell "Bathtub" Open top Trapezoidal Rectangular Closed top Figure 25 – Common box sections One component of the overall design procedure is to determine the capacity of the box girder prior to attaining composite action with the concrete deck. This situation arises during construction, when the concrete deck has not yet been poured, and after the concrete has been poured but has not yet hardened. In such cases, the box girder is designed as a non-composite steel section. This report will discuss this non-composite aspect of steel box girder design. CAN/CSA-S16-01 guides the designer to consult alternative reference material such as SSRC’s (Structural Stability Research Council) “Guide to Stability design criteria for metal structures” for the design of box girders. 6.1 Section Classification The maximum width-to- thickness ratios are presented as a constant divided by the square root of the specified yield strength of the steel. The limiting values are given in the table below: 46 Table 3 – Section classification Description of Element Class 1 Class 2 Class 3 Flanges of box girders yFt b 525 ≤ yFt b 525 ≤ yFt b 670 ≤ Webs in flexural compression yFt h 1100 ≤ yFt h 1700 ≤ yFt h 1900 ≤ 6.2 Shear Strength 6.2.1 Shear strength of box girders based on CAN/CSA-S16-01 The shear design of box girders based on CSA-S16-01 is presented in this section. The design is applicable to design of both symmetric and mono-symmetric sections. The factored shear resistance, Vr, developed by the web of the flexural member shall be taken as swr FAV Φ= (110) where wA = Shear Area sF = as follows (a) when y v F k w h 439≤ (111) ys FF 66.0= (b) when y v y v F k w h F K 502439 ≤< (112) cris FF = (c) when y v y v F k w h F K 621502 ≤< (113) )866.050.0( criyacris FFkFF −+= (d) when w h F k y v <621 (114) )866.050.0( criyacres FFkFF −+= 47 vk = shear buckling coefficient (i) when a / h < 1 (115) 2)/( 34.5 4 ha kv += (ii) when 1/ ≥ha (116) 2)/( 4 34.5 ha kv += a = distance between the stiffeners h = web depth )/( 290 wh KF F vy cri = (117) aK = aspect coefficient (118) = 2)/(1 1 ha+ 2)/( 180000 wh k F vcre = (119) 6.2.2 Shear strength of box sections according to SSRC To find the shear strength of box section, the approach defined by SSRC’s (Structural Stability Research Council) “Guide to Stability design criteria for metal structures” is also applicable because in such sections the shear strength is the combination of strength provided by both, the web before buckling and the diagonal tension after the buckling of the web. The shear strength of the box-section can thus be found by the following formula: )1(2 2 αα ++ = = += Tw T vcrwB TBu FDt V FDtV VVV (120) Where =D depth of the web between flanges =0d transverse stiffener separation Ddo /=α =wt web thickness =vcrF critical buckling shear stress =TF tension-field stress 48 There are various models to calculate the post-buckling strength of the web of the box or plate girder. Basler (1963) was first to model the tension field action for the plate girder. Woulchuk and Mayrbourl (1980) suggest the application of the Basler’s model, which is based on the assumption of negligible bending rigidity of the flanges, for the box girders. According to Basler tension field model (1963) tension field stress and critical buckling shear stress are additive, assuming that tension field stress acts at 45 degree and the model uses the resulting combination of the principal stresses in the linear approximation of the Mises yield condition. This results in the following: −= vyw vcr ywT F F FF 1 (121) where =ywF yield stress for web in tension =vywF yield stress for web in shear =vcrF critical buckling shear stress Figure 26 – Shear buckling of web 49 6.3 Bending Strength of Box Sections SSRC guidelines are followed to estimate the flexural capacity of the box-section. According to SSRC guide lines, “The flexural strength of box sections is rarely governed by flexural torsional buckling. Instead the governing criteria is buckling of compression flange or yielding of the tension flange, which ever occurs first”. The bucking of the compression flange can calculated using the basic plate buckling equation: 22 2 )/)(1(12 tb E kc ν π σ − = (122) where E = modulus of elasticity ν = Poisson ratio k= buckling coefficient t = thickness of the compression flange b = width of the plate (distance between the webs for this case) The top flange is in the state of uniform compression. It can be assumed as a plate with simply supported edges. In other words, buckling coefficient can be assumed as k=4.0. If buckling stress is less than the yield stress then ccr SM σΦ= (123) Otherwise ttr SM σΦ= (124) where tσ = yield stress in tension cσ = compression flange buckling stress tS = section modulus for extreme tension fiber cS = section modulus for extreme compression fiber 50 6.4 Design Based on CMAA The analysis uses solid mechanics principles to calculate stresses at each panel in the box girder and then uses the equations in CMAA standard to determe the limiting stresses. 6.4.1 Sectional properties The neutral axis height, ў, and moment of inertia of the box girder section, Ixtot, are easily calculated using equations 121 and 122. i i tot y A y A ∑= (125) 2I I A (y y)xtot xi i i= + −∑ ∑ (126) For the calculation of effective thickness of flanges or webs with the longitudinal stiffeners smeared, equations 123 and 124 are used. 2 xstiffenedplate xplate xstiffeners i stiffenedplateI I I A(y y )= + + −∑ ∑ (127) 1/ 3 xstiffenedplate eff plate 12I t L = (128) 6.4.2 Stresses The factored moment and previously calculated sectional properties allow the calculation of the longitudinal stresses at any point in the section (assuming that plane sections remain plane). M yf Ixtot σ = (129) The factored shear (assumed to be applied along the line of symmetry) gives rise to a shear flow around the section, q, which is calculated using equation 126. The general distribution of shear around the section is given in Figure below: f x o xtot V q D q I = + (130) 51 N. A. VF Figure 27 - Shear flow around the section due to Vf By taking a cut along the vertical line of symmetry of the bottom flange, the constant shear flow term, qo, is zero and the shear flow, q, can be evaluated traveling anti-clockwise around the section by evaluating Dx. s x si sioD tyds y A= − − ∑∫ (131) xy q t τ = (132) Once the longitudinal (normal) and shear stresses are known, the principal stresses, σ1 and σ2, can be calculated (the maximum and minimum normal stresses in a plane, always perpendicular to each other and oriented in directions for which the shear stresses are zero). 2 x y x y 2 1 2 xy, 2 2 σ + σ σ + σ σ σ = ± + τ (133) The principal stresses are calculated at several discrete points; the intersection of the flanges and web, the web stiffener locations, and at the neutral axis. 52 6.4.3 Yielding failure These principal stresses are used to check for local yielding failure of the cross section using the Von Mises failure criterion for plane stress, which is given in equation 130 (this assumes that failure occurs when the energy of distortion reaches the same energy for yield/failure in uniaxial tension). 2 2 2 1 1 2 2 yFσ − σ σ + σ ≤ (134) The von Mises yield criterion was chosen over other yield failure criteria such as the Tresca criterion, as it is more conservative. 6.4.4 Buckling failure Equations presented in section 2 are used to check for local buckling of the sub panels between longitudinal stiffeners and also the wider panels with the longitudinal stiffeners smeared to create a plate with increased thickness. Lateral-torsional buckling is not evaluated for the closed cell box girder, as it is unlikely that this failure mode will dominate for typical box girders. However if the height to width ratio of the box is relatively large, lateral-torsional buckling may dominate and needs to be evaluated. 6.4.5 Stiffeners No provision for design of stiffeners has been made in the box girder formatted spreadsheet, buckling of the longitudinal stiffeners can be avoided through conformance with slenderness limits given in the Standard. 53 7.0 Introduction to Formatted Spreadsheet The formatted spreadsheet applies macros in Microsoft Excel to perform the required calculations. Therefore, the user should have enabled macros in Excel for the spreadsheet to perform properly. The formatted spreadsheet starts with a diagram or an image describing the problem in the description area. This is shown in the following figure: Figure 28 – Description section The next section is the input section. The user is required to enter the value for the parameters in column G. Column H is the unit of the parameter, Column B represents the full name of variable or parameter. Column C represents the short variable name which is used in the equations. This is shown in the figure below: 54 Figure 29 – Input section The last section is the calculations section. The user does not require to enter any inputs here. Column E represents the equations used. The references for the equations or formulas can be found in column I. This is shown in the figure below: Figure 30 – Calculation section To perform the calculations, the user is required to press Alt/F8 simultaneously, to bring up the macro table FormatSheet is already selected, therefore only the Enter key needs to be pressed. Figure 31 – Macro window 55 The macro fills in all equal signs, and then parses the equations from text to working formulae in column G with relative referencing. Figure 32 – Equations and referrences If the user wants to change something in the input, as long as it is the numerical value in column G is changed, the entire spreadsheet will immediately change the results. Therefore, it is not required to press Alt/F8 again. 56 8.0 Design Using Formatted Spreadsheet Formatted spreadsheets are developed for ease of design. The spreadsheets follow the same approach given by the Standard and presented in the preceding sections. 8.1 Plate Girder Spreadsheet The plate girder spreadsheet consists of three sub sheets. The first sheet is based on CAN/CSA- S16 standard. The second sheet is design based on CAN/CSA-S6-00 standard. The third sheet is design of plate girder based on the combination of both codes. The spreadsheets are applicable to both doubly-symmetric mono-symmetric sections. The figure below illustrates the section and side view of the mono-symmetric plate girder. Figure 33 – Side view and section of mono-symmetric plate girder The following sections provide complete user guide for the plate girder spreadsheet. 57 8.1.1 User input The spreadsheet requires the user to enter the parameters such as material strength properties, length of span, maximum factored shear and moment in the span and section geometry (single/double symmetry). Figure below illustrates the input parameters. INPUT LOAD PARAMETERS span L = 32.0[m] factored moment Mf = 6200[kNm] factored shear Vf = 2200[kN] max. (space for) girder depth d_max = 3200[mm] INPUT MATERIAL PROPERTIES specified material yield strength Fy = 300[MPa] ultimate material yield strength Fu = 450[MPa] weld metal strength Xu = 490[MPa] material Shear Modulus G = 77000[MPa] material Young's Modulus Est = 200000[MPa] performance factor φ = 0.9 performance factor for welds φw = 0.67 Figure 34 - User input 8.1.2 Preliminary girder dimension computation The most economical girder dimensions are computed based on the user input in the previous section. However, the user is also given the option to enter his/her desirable dimension for further calculation. This is shown in the following figure Figure 35 - Girder parameters INPUT GIRDER DIMENSIONS Select Girder Parameters web thickness w = 18[mm] web depth h = 1600[mm] compression flange width b_com = 500[mm] compression flange thickness t_com = 32[mm] tension flange width b_ten = 500[mm] tension flange thickness t_ten = 32[mm] 58 8.1.3 Girder resistance calculation The spreadsheet calculates the moment resistance of the section based on girder dimensions given. This is dependent on the class of web and flanges. Next, the spreadsheet calculates the shear resistance of the girder. The shear resistance depends on the thickness of the web, the presence of transverse stiffeners and their spacing. The analysis also checks if transverse stiffeners are required. Then the combined action of the applied shear and moment is checked to ensure that it has sufficient capacity. 8.1.4 Transverse stiffeners The spreadsheet allows the user to enter the required data such as number of stiffeners, shape and dimension of the stiffeners. This is shown in the following figure: INPUT TRANSVERSE STIFFENERS stiffener type stiff_type = plate angle/plate stiffener furnishing stiff_furn = pair pair/single stiffener yield strength Fy_stiff = 350 [MPa] stiffener thickness primary leg ts_a = 6 [mm] stiffener width primary leg bs_a = 200 [mm] stiffener thickness secondary leg ts_b = 12 [mm] stiffener width secondary leg bs_b = 100 [mm] stiffener effective length factor K = 0.75 >=0.75 Figure 36 - Transverse stiffener parameters Furthermore, the spreadsheet checks to see if the stiffeners have the adequate slenderness and capacity. 8.1.5 Bearing stiffeners The spreadsheet allows the user to enter the required data for the check of the bearing stiffeners. The spreadsheet accounts for both interior and end bearing stiffeners. This is shown in following figure: 59 Figure 37 - Bearing stiffener parameters Bearing resistance must be calculated for both possible modes of failure, namely localized buckling and yielding. The smaller value is noted as the bearing resistance by the spreadsheet. 8.1.6 Weld design and girder weight computation The spreadsheet checks the adequacy of weld between flanges and web (two fillet welds each). The required user inputs for this check are the weld size, length and spacing. The figure below illustrates the user input; Figure 38 - Input weld design The spreadsheet calculates the shear flow per length of the girder and checks to see if the weld shear resistance is adequate. Finally, the spreadsheet computes the total weight per girder. The spreadsheet requires the user to enter the density of steel in kg/mm3. INPUT BEARING STIFFENERS factored load Cf = 1600 [kN] performance factor φbi = 0.8 weld performance factor φω = 0.7 performance factor φbe = 0.75 length of the bearing plate N = 300 [mm] flange to web weld depth d_weld = 6 [mm] bearing stiffeners under load bea_stiff_i = exist exist/none end bearing stiffeners bea_stiff_e = exist exist/none end stiffener width bs_e = 125 [mm] end stiffener thickness ts_e = 16 [mm] intermediate stiffener width bs_i = 125 intermediate stiffener thickness ts_i = 12 stiffener contact length cpl = 100 [mm] parameter n = 1.34 INPUT WELD DESIGN Select Weld Between Flanges and Web (two fillet welds each) weld size w_size = 8 [mm] intermittend weld length w_length = 200 [mm] spacing on centre w_spacing = 400 [mm] 60 8.2 Box Girder Spreadsheet For the design of closed box girders, a formatted spreadsheet was developed which checks for local buckling or yielding of a monosymmetric box girder stiffened longitudinally and transversely, with webs and flanges each stiffened longitudinally with two stiffeners. The spreadsheet contains of two sub spreadsheets. The first spreadsheet is based on CAN/CSA-S16 standard and SSRC guidelines. The second spreadsheet is based on basic solid mechanics principles and CMAA standard. The spreadsheets require the user to enter the complete dimensions of the girder according to the figure provided below. LTF LBF LW ITFS,ATFS IBFS,ABFS IWS,AWS tBF,IBF,ABF tTF,ITF,ATF yBFS yWS1 yWS2 yTFS hws XTFS1 XTFS2 XBFS2 XBFS1 Figure 39 - Monosymmetric box girder designed in spreadsheet 61 The input section is shown in the figure below. Figure 40 – Input section-box girder spreadsheet INPUT Factored Loads and Moments Factored Moment Mf = 3500.00[kNm] Factored Shear Vf = 1750.00[kN] Material Properties Material Yield Strength Fy = 350.00[MPa] Material Young's Modulus E = 200000[MPa] Material poisson's ratio v = 0.30 Performance Factor φ = 0.90 Girder Dimensions web thickness t_w = 20.00[mm] web length L_w = 900.00[mm] Top flange width L_tf = 500.00[mm] Top flange thickness t_tf = 35.00[mm] Bottom flange width L_bf = 300.00[mm] Bottom flange thickness t_bf = 25.00[mm] Web Longitudinal stiffener moment of Inertia I_ws = 131000[mm^4] Web longitudinal stiffener area A_ws = 524.00[mm^2] web longitudinal stiffener 1 height y_ws1 = 250.00[mm] web longitudinal stiffener 2 height y_ws2 = 875.00[mm] web longitudinal stiffener offset h_ws = 31.90[mm] Top flange Longitudinal stiffener moment of Inertia I_tfs = 131000[mm^4] Top flange Longitudinal stiffener area A_tfs = 524.00[mm^2] Top flange longitudinal stiffener height y_tfs = 913.60[mm] Top flange stiffener 1 offset x_tfs1 = 166.67[mm] Top flange stiffener 2 offset x_tfs2 = 333.33[mm] Bottom flange Longitudinal stiffener moment of Inertia I_bfs = 131000[mm^4] Bottom flange Longitudinal stiffener area A_bfs = 524.00[mm^2] Bottom flange longitudinal stiffener height y_bfs = 41.90[mm] Bottom flange stiffener 1 offset x_bfs1 = 100.00[mm] Bottom flange stiffener 2 offset x_bfs2 = 200.00[mm] Stiffeners Transverse stiffener/internal diaphragm spacing a = 2000.00[mm] Transverse Stiffeners stf = exist(exist/none) Longitudinal Web Stiffeners stf_lw = exist(exist/none) Longitudinal Flange Stiffeners stf_lf = exist(exist/none) Number of Longitudinal stiffeners n = 2.00 62 The spreadsheet is designed in a way that it calculates the flexural resistance and shear resistance of the section with and without longitudinal stiffeners. If there are no stiffeners, the user has to enter “none” in the input section for stiffeners The spreadsheet checks for the web-crippling and compression flange buckling and calculates the ultimate moment resistance of the box girder using SSRC guidelines. Furthermore, the shear resistance of the girder is calculated using both the CSA-S16 code and SSRC guidelines as described in sections 6.1 to 6.3. The second spreadsheet follows the exact procedure and equations explained in section 6.4. 63 9.0 Fabrication Considerations Evaluation of the economics of design often includes a perception that least weight and least cost are synonymous. Although cost of a structure is related to the weight of steel material, there are numerous other considerations in purchasing, fabricating, shipping, and erection and effective use of material locally which may override the decision to aim for a least weight structure. Some of these considerations require familiarity with purchasing, fabricating and erecting processes. Various fabricators have their own processes, and it is difficult for a designer to produce a design to satisfy everyone. Fabricators should be allowed flexibility in detailing, with designer approval, to make adjustments to the number and location of splices. Material content is only one element in the cost equation and will represent about 20% to 30% of the total ‘in place’ cost in fairly standard bridges. The total rate per tonne (metric = 1000 kg) of steel depends on several factors, including: Complexity of details Quality control requirements Amount of welding, including grinding, type and amount of inspection etc… The amount of repetition and reuse of assembly jigs Size and number of individual pieces to be fabricated Other demands on shop space, particularly when large box girders are involved The access for erection Number of girder field splices The allowable fabrication tolerances are defined in W59, clauses 5.8 and 5.9. The tolerances in the individual pieces that make up a continuous span will be additive. 9.1 Materials Weathering steel is now the norm for bridges in Canada. Painted steels are used in environments not considered acceptable for the weathering process, such as continued wetness due to climate and precipitation, proximity to airborne chlorides e.g. near the sea coast or above a high traffic volume expressway, and exposure to harsh industrial environments. In many cases weathering steel is selected even when a paint system is to be applied due its lower strength to cost ratio and its ability to form a superior base for paint systems. A designer needs to be aware of the plate sizes available so that spices in webs and flanges are kept to a minimum, particularly longitudinal splices which should be avoided. The maximum 64 length of plate that may be supplied is dependent on the thickness of the plate and the material type, and will vary from mill to mill and hence local fabricators should be consulted. The designer should also be aware of other factors which influence material cost such as: There is a small premium on plates longer than 18 meters (about 4%) Plates less than 9 mm thick and more than 25mm thick attract a premium of from 4% - 8% Small orders also incur mill extras and small quantities of any one plate thickness should be avoided. As a general guide the maximum piece-weight for handling in the shop is of the order of 50 tonnes, and the optimum length for the shop is about 27m although these values are increasing. 9.2 Proportioning of Spans When there is choice in the positioning of piers for a continuous bridge. The end spans should be approximately 75% of the length of the main span, this will permit balancing of dead and live load moments, reduce the potential for uplift at the abutments, and permit the most economical design when proportioning the girder. 9.3 Selection of a Girder Cross Section For compositely designed continuous spans, the designer should start with a main span to girder depth ration of approximately 28 for box girders and 26 for I girders. On bridges where there are no pedestrians the bridge may be made more slender due to the reduced deflection requirements, ratios of 30 to 34 may be used successfully. 9.4 Webs The optimum web thickness and subsequent number of transverse stiffeners depends on the depth of the web and should be considered. For example, it is economically advantageous to have an unstiffened web if the girder is 1200 mm deep or less. The economics of unstiffened webs decreases as web depth increases. Sometimes the minimum web thickness is dictated by the method of construction such as launching, in this case thicker webs will usually be the economical solution because of local bearing, buckling, and crippling considerations, as well as overall stability. 65 Changes in web thickness should coincide with either a field splice or a maximum length of mill material available for the thickness and depth of web being considered. In addition, it will usually be found satisfactory to avoid grinding or to use only nominal grinding to touch up the profile of full penetration butt welds in the web when using the submerged arc process. In many cases it will be found economical to maintain a constant thickness of web throughout the girder, varying the spacing of intermediate transverse stiffeners according to the shear diagram and possible eliminating transverse stiffeners in the areas of low shear. The recommended minimum web thickness is ½ inch as thinner plate is subject to excessive distortion from welding. Web thickness increments should be 1/16 inch up to a plate thickness of ¾ inch, use 1/8 increments up to 1 inch, if the web plate needs to be thicker than 1 inch, use ¼ inch increments. For web splices use the submerged arc process and avoid grinding if possible, or use only nominal grinding to touch up the profile of full penetration butt welds in the web. Web shop splices should be at least 10 feet apart and at least 6 inches away from a flange splice or transverse stiffener, in order to facilitate testing of the weld. 9.5 Stiffeners Welding of bearing stiffeners to the bottom flange should be specified as fillet welds, use of full penetration welds is costly and can cause distortion of the bottom flange, thus making it difficult to achieve the desired flatness for the sole plate or bearing. For composite bridge girders, stiffeners welded to the top flange throughout do not alter the fatigue category of the flange (already Class ‘C’ because of the studs). The use of both transverse and longitudinal stiffeners is difficult to avoid on deeper girders. However every effort should be made to place longitudinal stiffeners on one side of the web, with transverse stiffeners on the other so that interferences occur only where the longitudinal stiffener meets the double sided web stiffeners used on I girders for connection of cross frames. Fabricators have indicated that flat bars are typically more economical than plates for stiffeners. The clear distance between longitudinal stiffeners should be no less than 24 inches, to accommodate automated welding equipment. 66 It is highly preferable not to have several stiffener sizes for a girder. Bearing stiffener thickness that matches the flange thickness is suggested. Bearing stiffeners should be thick enough to preclude the need for multiple bearing stiffeners at any given bearing, as multiple stiffeners present fabrication difficulties and usually are not needed. It is very important that the width be sufficient to provide clearance for field welding of diaphragm members to the stiffener. Four inches or more of clearance between the web face and a vertical weld on a gusset plate/diaphragm member is required for good welding access. Three inches or more of clearance is needed between a gusset plate/diaphragm member and a flange. For box girders, the current trend is to longitudinally stiffen the webs and flanges and use internal diaphragms, without the requirement for transverse stiffeners. 9.6 Flanges When deciding how to fit the flange sizes of a girder to the moment envelope, the designer must consider the cost implications as well as technical factors. The trade-off to be considered here is the cost of the material saved by reducing the flange size versus the cost of the full penetration but welded splice in the material including; material preparation, fitting up, welding, gouging, grinding, inspection and possibly repairs and so can involve a considerable number of man- hours. In I girders and open top box girders, the designer may change the flange width or thickness or both. It is usually more economical to produce several flange splices simultaneously, this process involves butting two thicknesses of plate, wide enough to produce 2,4 or 6 flanges, producing one butt weld across them and then flame cutting (stripping or ripping) the flanges longitudinally. Thus by making the flanges a constant width between field splices, the costly procedure of butt welding individual plates is avoided, although this is not always possible e.g. a constant width may require a flange which is locally beyond a practical thickness. If flange widths are varied, it is best to change the width at field splices only Top and bottom flanges should be the same width. Girders in positive bending that are composite with a slab can have a top flange narrower than the bottom flange, but the weight savings achieved are typically not worth the reduced lateral stability prior to hardening of the deck. Also, if continuous construction is used, the top flange width would normally have to be increased for the negative moment sections, which creates slab-forming difficulties. 67 The desirable maximum flange thickness is 3 inches. Grade 50 and HPS70W steels are not available in thicknesses greater than 4 inches. Weld time is disproportionately increased when splicing plates thicker than 3 inches. A 10-foot minimum length should be used for any given flange segment on a girder. It is only economical to introduce a flange splice if it is possible to save about 800 – 1000 pounds, these numbers are approximate and are a function of the current cost of steel plate. Flange thickness increments should be 1/8 inch for thicknesses from ¾ to 1 inch, ¼ inch from 1 to 3 inches, and ½ inch from 3 to 4 inches. A change in thickness should be made at a slope of 1 in 2½. Flange thicknesses should be sufficient to preclude the need for lateral bracing. Lateral bracing is to be avoided because it creates fatigue-sensitive details and is costly to fabricate and install. Flange splices should be located at least 6 inches away from a web splice or transverse stiffener, in order to facilitate testing of the weld. Splices should be at least 10 feet apart. Field splices are good locations to change flange sizes. Top flanges for open box girders should follow the suggestions for plate girder flanges, except for the stability criteria. Top and bottom flanges of closed box girders and bottom flanges of open box girders should extend past the centerline of each web a minimum of 2 inches to allow for automated welding equipment. Flange width is somewhat dependent on the need for enough room inside the box girder to allow the passage of inspection personnel. Provision must be made for entrance to the box girder by inspection personnel, typically a hatch-type, lockable door at each end of the box is sufficient. For wide bottom flanges of box girders, plate distortion during fabrication and erection can be a problem. Designer should check with fabricators when using bottom tension flange plates of less than 1” thickness in order to determine whether practical stiffness needs are met. In no case should bottom tension flanges be less than ½” thick. 9.7 Field Splices The gap between girder ends should be made large enough to accommodate normal shop tolerances. A dimension of 10mm is commonly used, smaller values would be difficult to work to, and unnecessarily expensive. 68 Designer should use one bolt diameter throughout a structure, if practical, and ensure that it is physically possible to install bolts in their specified locations. When welded field splices are specified, usually it is because aesthetics are paramount and a bolted splice is deemed unattractive. They have several disadvantages compared with bolted and are rarely seen. It can be difficult to detail an all welded splice to have acceptable fatigue and fracture performance, not to mention the problems of welding (including possible repairs), grinding and inspecting the welds in the field. Temporary connections are required to hold the parts in alignment during welding, and the accuracy of fabrication and fit up is more critical than with a bolted splice. 9.8 Fatigue Details Flanges with welded shear studs and a web with welded transverse stiffeners both fall into Category ‘C’. Grinding is expensive and if carried out improperly can be detrimental to the fatigue life of the structure. Each tension flange butt weld should be radiographed, compression flange butt splices should be radio-graphed randomly (form 10% to 25%) and only butt splices in webs in critical tensile areas (e.g. 20% of the web adjacent to a tension flange) should be radio- graphed. A radius should be provided at the end of the gusset to eliminate a sharp notch, reduce the stiffness at the tip and minimize longitudinal stresses at the tip of the attachment. 69 10.0 Erection Considerations A well conceived economical steel bridge requires consideration of its erection at two stages in the design process. Firstly, erection must be considered at the concept stage because it typically represents about 30% of the superstructure cost and therefore the most economical arrangement cannot evolve without its consideration. Truss versus girder, curved versus parallel chords and flanges, continuity, main member dimensions, drop in spans, pier arrangements, etc… all have significance at this stage. Secondly, erection must be considered at the detail stage. Details of splices, diaphragms, bracing and pier members are very significant contributors to erection cost. Those elements which are in the control of the designer should be designed to facilitate construction wherever possible. Field labor is very expensive, therefore keep things simple. Realistic tolerances must be built into the system wherever shop fabricated elements meet field construction. Access to splices, anchor bolts and bearings and adequate space to install jacks is very necessary for proper installation, inspection and future maintenance. Constant depth or Curved chords: Strictly from an erection point of view, constant depth girders have the advantage. Pier sections of haunched girders frequently require extraordinary effort in shipping, handling and turning because of their increased bulk. Constant depth girders are much easier to ship, to turn and to lift and block. Plate girder or box girder:From an erection point of view, the box girders are usually preferable to the plate girders because there are fewer pieces of girder and less bracing. Box girders are reasonably stable in shipping, handling and free cantilever, whereas plate girders, particularly if slender in flange width, can pose stability problems in shipping, and handling and frequently require top chord stiffening trusses in cantilever erection. Plate girders can often be nested during shipping whereas the internal diaphragms present in common box girders prevent nesting. The particular configuration of box selected has a very significant effect on the erection cost of the bridge. Unless circumstances dictate boxes larger than about 3.5 m in width should be avoided because they will cause excessive shipping and handling problems and, in the limit, will require a longitudinal splice. Box girders having more than two webs should be avoided except for special situations such as an axial girder cable stayed bridge. 70 Flange width has an impact on the stability of the girder during handling and erection. According to an industry rule of thumb, I-girders will be stable if their length is less than or equal to 60 times the flange width. If this is exceeded the erector and fabricator may need to use temporary bracing to handle and erect the girder. The maximum economy will result if the fabricator/erector is permitted freedom to choose the splice locations that best suit his equipment. If the strength requirements of the splices are spelled out in general terms in the drawings and specifications, then the Contractor can detail the bridge with his preferred splice locations for the Engineer’s approval. It is common to have all holes drilled or punched sub-size and then reamed to full size in full girder assembly of not fewer than three girder sections, laid on blocking corresponding to the cambered shape. If this method is performed accurately, all components should fit precisely in the field and the required bridge geometry will be attained. This method has the advantage of minimum time spent on field fitting and rework, as well as optimum quality in the connection. However it should be noted that the large assemblies tie up a lot of shop space and reaming is very time consuming. While angle bracing and diaphragms are very cheap to fabricate, they are generally very expensive to erect, due to the cost of the crane and labor for erection. In order to minimize these costs, the designer should not use bracing and diaphragms indiscriminately, but only were strictly necessary. 71 11.0 Conclusions This report has outlined the CAN/CSA S16.1 approach to the design of doubly symmetric plate girders. The Standard gives a prescriptive method which was easily transferable to formatted spreadsheets, this report expands on the clause equations to give some explanation of their derivation and why they are applied. In order to expand on the applicability of the spreadsheet, equations for analyzing the lateral-torsional buckling resistance of monosymmetric plate girders was obtained from CAN/CSA-S6-06 (Canadian Highway Bridge Design Code). There is a high level of confidence in using the plate girder spreadsheet, as it is based entirely on CSA standards equations and clauses. Design of steel box girders is based on a more fundamental approach was required. This approach is influenced by equations given in the Crane Code (CMAA Specifications #70 & #74), which allows plate elements of box girders subject to combined linearly varying normal edge stress and shear stress to be checked for buckling. A formatted spreadsheet was created for the design of closed cell box girders, however unlike the spreadsheets for the design of plate girders which are based on limit states design, this spreadsheet is based on allowable or working stress design. The second spreadsheet is also developed for the design of box girders based on CAN/CSA S16.1 Standard and the equations provided in SSRC’s Guide to Stability Design Criteria for Metal Structures. To ease the use of spreadsheet, a complete manual is provided in the report. This report also includes a discussion of economical and practical aspects associated with the design, fabrication and erection of steel plate and box girders. The general considerations given are true across North America, and will continue to be true for the foreseeable future. However the reader should bear in mind that the exact values given will vary from one geographical location to the next and also with time. In any case it is of paramount importance to collaborate with and receive input from local fabricators when designing plate and box girders, in order to ensure a practical and economical design. 72 References [1] Basler, K., Thurlimann,B. Trans., “Strength of Plate Girders in Bending”, ASCE, Vol. 128, Part II, p.655, 1963 [2] Bridge Design Specifications, 2nd Edition Washington D.C, American Association of Highway and Transportation Officials (AASHTO), 1997 [3] Bridge Design Specifications, 3rd Edition, Washington D.C, American Association of Highway and Transportation Officials (AASHTO), 2004 [4] CAN/CSA-S6-06. Canadian Highway Bridge Design Code (CHBDC) A National Standard of Canad , CSA International, Toronto, Ontario, November 2006 [5] Commentary to the Canadian Highway Bridge Design Code – CAN/CSA-S6-00 CSA International, Toronto, Ontario, 2006 [6] Concrete Design Handbook, 3rd Edition, Cement Association of Canada, Ottawa, Ontario, Canada, January 2006 [7] Crane Code (CMAA Specifications #70 & #74, revised 2000, Material Handling Industry) [8] Handbook of Steel Construction, 8th Edition, third printing, Toronto, Ontario, Canadian Institute of Steel Construction,, December 2005 [9] Kulak Grondin, “Limit States Design in Structural Steel”, 8th Edition, Willowdale, Ontario, CISC, 2006 [10] Preferred Practices for Steel Bridge Design, Fabrication and Erection, Texas, Texas Steel Quality Council, Texas Department of Transportation (TxDOT), November, 2000 [11] Steel Bridges, Design, Fabrication, Construction, ‘Notes and References’, Canadian Institute of Steel Construction 73 [12] Taylor, “Bridge Erection – The designer’s Role”, Canadian Structural Engineering Conference, 1982 [13] Theodore V. Galambos, “Guide to Stability Design Criteria for Metal Structures”, New York, John Wiley & Sons, 1998 [14] Wolchuk, R., Mayrbourl, “Proposed Design Specification for Steel Box Girder Bridges”, Washington, D.C, U.S. Department of Transportation, Federal Highway Administration, 1980 74 Appendices 75 Appendix A: Plate Girder Spreadsheet (CSA-S16-01) Plate Girder Spreadsheet (CSA-S16-01) 76 DESIGN OF PLATE GIRDERS REFERENCES INPUT GENERAL PARAMETERS specified material yield strength Fy = 300[MPa] ultimate material yield strength Fu = 450[MPa] weld metal strength Xu = 490[MPa] performance factor Φ = 0.90 performance factor for welds Φw = 0.67 Plate Girder Spreadsheet (CSA-S16-01) 77 span L = 25.0[m] factored moment Mf = 9000[kNm] factored shear Vf = 3000[kN] max. (space for) girder depth d_max = 4000[mm] COMPUTATIONS GENERAL PARAMETERS Slenderness Limits min.web SL for red. moment SL_wmin = 1900/SQRT(Fy) = 110 $14.3.4 max. web slenderness SL_wmax = 83000/Fy = 277 $14.3.1 max. flange slenderness SL_fmax = 200/SQRT(Fy) = 12 $11.2 Table 2 Preliminary Sizing height for max bending efficiency h_a = 540*(Mf/Fy)^(1/3) = 1678[mm] maximum shear strength Fs = 0.66*Fy = 198.00[MPa] $13.4.1.1(a) minimum web area Aw_min = Vf*1000/(φ*Fs) = 16835[mm2] min web thickness w_a = Aw_min/h_rec = 10.0[mm] recommended web slenderness SL_recw = h_rec/ w_a = 167 ratio of web slendernesses r_SLw = SL_recw/SL_wmax = 0.60 minimum flange area Af_min = Mf * 1000000 / ( Fy * h_rec) = 17879[mm2] recommended web thickness w_rec = IF(r_SLw>1,w_a*r_SLw,w_a) = 10[mm] recommended web depth h_rec = IF(h_a>d_max,d_max,h_a) = 1678[mm] recommended flange thickness t_rec = SQRT((Af_min/SL_fmax)/2) = 28[mm] recommended flange width b_rec = Af_min/t_rec = 643[mm] total girder depth d_rec = 2*t_rec+h_rec = 1734[mm] Plate Girder Spreadsheet (CSA-S16-01) 78 INPUT GIRDER PARAMETERS Select Girder Parameters web thickness W = 18[mm] web depth H = 1600[mm] flange thickness T = 32[mm] flange width B = 500[mm] CHECKS CONCEPTUAL DESIGN web slenderness check s_w_chec = IF(h/w>SL_wmax,"reduce slenderness of web","web slenderness OK") = web slenderness OK efficiency web slenderness wb_eff = (h/w)/SL_wmax = 0.32 flange slenderness check sL_fsel_chec = IF(b/(2*t)>SL_fmax,"reduce flange slenderness ","flange slenderness OK") = flange slenderness OK efficiency flange slenderness fl_eff = (b/(2*t))/SL_fmax = 0.68 girder depth check d_sel_chec = IF(d_sel>d_max,"reduce recommended girder depth","girder depth OK") = girder depth OK efficiency flange slenderness d_eff = d_sel/d_max = 0.42 Fs = 0.66Fy Fs = 290 Fykv (h/w) Fs = 180000 kv + Ft (h/w)2 621439 83000 Fy kv Fy kv Web slenderness h/w Fy Fs MPa 502 Fy kv Fs = 290 Fykv + Ft (h/w) Ft Ft Plate Girder Spreadsheet (CSA-S16-01) 79 INPUT TO CHECK SHEAR RESISTANCE transverse stiffeners exist? Stiffeners = exist(exist/none) intermediate trans. stiff. spacing A = 2000[mm] panel location p_loc = interiorinterior/end anchorage for end panel Anch = yesyes/no $15.7.1 factored shear in end panel Vf_e = 3000[kN] CALCULATIONS OF SHEAR RESISTANCE web area Aw = h*w = 28800[mm2] panel Ratio a_h = a/h = 1.25 factored shear stress Ff_i = (Vf/Aw)*1000 = 104.17[MPa] factored shear stress at end panel Ff_e = (Vf_e/Aw)*1000 = 104.17[MPa] Panel Ratio Check max panel ratio one a_hmax_a = 67500/(s_w)^2 = 8.54 $14.5.2 Table 5 max panel ratio two a_hmax_b = 3 = 3.00 $14.5.2 Table 5 panel ratio check a_hcheck = IF(stiffeners="exist",IF(s_w>150,IF(a_h<=a _hmax_a,"OK","decrease stiffener spacing"),IF(a_h<=a_hmax_b,"OK","decrea se stiffener spacing")),"NA") = OK $14.5.2 Table 5 Web Slenderness Check web maximum allowable slenderness s_wmax = 83000/Fy = 277 $14.3.1 web slenderness s_w = h/w = 89 web slenderness check s_wcheck = IF(s_w>s_wmax,"reduce slenderness","slenderness OK") = slenderness OK $14.3.1 Plate Girder Spreadsheet (CSA-S16-01) 80 Ultimate Shear Stress (Fs) (a) yielding in shear Fs_a = 0.66 * Fy = 198.00[MPa] $13.4.1.1(a) (b) elasto-plastic action Fs_b = F_cri = 158.83[MPa] $13.4.1.1(b) (c) tension field action Fs_c = F_cri+ka*(0.5*Fy -0.866*F_cri) = 166.61[MPa] $13.4.1.1(c) (d) elastic buckling Fs_d = F_cre+ka*(0.5*Fy-0.866*F_cre) = 176.31[MPa] $13.4.1.1(d) tension field contribution Ft = ka*(0.5*Fy-0.866*F_cre) = -3.66[MPa] coefficient for no stiffeners kv_n = 5.34 = 5.34 coefficient case a/h<1 kv_s1 = 4+5.34/(a/h)^2 = 7.42 $13.4.1.1 coefficient case a/h>=1 kv_s2 = 5.34+4/(a/h)^2 = 7.90 $13.4.1.1 kv_s = IF(a/h<1,kv_s1,kv_s2) = 7.90 $13.4.1.1 Kv = IF(stiffeners = "none",kv_n,kv_s) = 7.90 $13.4.1.1 = critical shear stress inelastic F_cri = 290 * (((Fy * kv)^.5)/(h / w)) = 158.83[MPa] aspect coefficient Ka = 1/SQRT(1+(a/h)^2) = 0.62 critical shear stress elastic F_cre = 180000*kv/(s_w)^2 = 179.97[MPa] coefficient factor cF = SQRT(kv/Fy) = 0.16[MPa-0.5] slenderness case h/w Case = IF(s_w<=439*cF,"c_a",IF(s_w<=502*cF,"c_ b",IF(s_w<=621*cF,"c_c","c_d"))) = c_c Fs_stiff = IF(case="c_a",Fs_a,IF(case="c_b",Fs_b,IF( case="c_c",Fs_c,Fs_d))) = 166.61[MPa] $13.4.1.1 (a-d) Fs_unstiff = IF(case="c_a",Fs_a,IF(case="c_b",Fs_b,IF( case="c_c",F_cri,F_cre))) = 158.83[MPa] $13.4.1.1 (a-d) = = stiffener check st_check IF(Vf>(Fs_unstiff*Aw/1000),"transverse stiffener required","transverse stiffener not required") transverse stiffener not required F_s = IF(p_loc="interior",IF(stiffeners="exist",Fs_ stiff,Fs_unstiff),IF(anch="yes",Fs_stiff,Fs_u nstiff)) = 166.61[MPa] $13.4.1.1 (a-d) Plate Girder Spreadsheet (CSA-S16-01) 81 End Panel Calculation minimum shear buckling coefficient kv_min = (Ff_e*(s_w)^2)/(180000*0.9) = 5.08 minimum end panel spacing a_e = SQRT(MAX((4*h^2)/(kv_min-5.34),0)) = 0[mm] FINAL CHECKS SHEAR RESISTANCE shear resistance Vr = φ * Aw * F_s / 1000 = 4318[kN] $13.4.1.1 efficiency shear resistance Vf_r = Vf_e/Vr = 0.69 check shear resistance Vcheck = IF(Vf_r<1,"OK","increase shear resistance") = OK INPUT 3: GIRDER PARAMETERS - UNEQUAL FLANGES compression flange width b_com = 500[mm] compression flange t_com = 32[mm] Plate Girder Spreadsheet (CSA-S16-01) 82 thickness tension flange width b_ten = 500[mm] tension flange thickness t_ten = 32[mm] material Shear Modulus G = 77000[MPa] material Young's Modulus Est = 200000[MPa] unsupported Length Lu = 5000[mm] girder loading: Uniform …. Load = load (load/moment ) longitudinal stiffener at 0.2d Long = noneexist/none longitudinal stiffener area A_stiff = 2100[mm 2] longitudinal stiffener weak axis I Ix_stiff = 1.51E+06 [mm4] longitudinal stiffener strong axis I Iy_stiff = 1.51E+06 [mm4] longitudinal strong neutral axis height x_stiffna = 27[mm] long. stiff. strong axis plastic modulus Z_stiff = 0.00E+00[mm3] point of application of transverse loading as a fraction of girder depth d g'' = 1 moment distribution factor ω2 = 1 $13.6(a) CALCULATIONS OF MOMENT RESISTANCE Section Element Slenderness and Class flange max. allowable slenderness s_fmax = = 60.0 $13.5 (c) flange (in compression) slenderness s_f = b_com/2/t_com = 7.8 flange slenderness check f_check = IF(s_f>s_fmax,"reduce flange width","flange width OK") = flange width OK Plate Girder Spreadsheet (CSA-S16-01) 83 flange class f_class = IF(s_f<=145/SQRT(Fy),"1",IF(s_f<=170/SQ RT(Fy),"2",IF(s_f<=200/SQRT(Fy),"3","4"))) = 1 $11.2 Table 2 web class w_class = IF(s_w<=1100/SQRT(Fy),"1",IF(s_w<=170 0/SQRT(Fy),"2",IF(s_w<=1900/SQRT(Fy)," 3","4"))) = 2 $11.2 Table 2 class_a = IF(f_class>w_class,f_class,w_class) = 2 section class Class = IF(class_a="4",IF(w_class="4",IF(f_class=" 4","4(i)","4(ii)"),"4(iii)"),class_a) = 2 $13.5 section class check cl_check = IF(class="4(i)","consult CSA S136","OK") = OK $13.5 Sectional Properites without Longitudinal Stiffener compression flange area Af_com = b_com*t_com = 16000.00[mm 2] tension flange area Af_ten = b_ten*t_ten = 16000.00[mm 2] total area A_tot = Aw+Af_com+Af_ten = 60800.00[mm 2] overall girder depth Gd = h+t_ten+t_com = 1664[mm] neutral axis height y_na = (Af_com*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_ten))/ A_tot = 832[mm] neutral axis height x_na = b_com/2 = 250[mm] outermost fibre distance y_max = MAX(gd-y_na,y_na) = 832[mm] strong axis second moment of area Ix = 1/12*(b_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_com*(gd- t_com/2-y_na)^2+Af_ten*(y_na- t_ten/2)^2+Aw*(h/2+t_ten-y_na)^2 = 2.745.E+ 10 [mm4] effective elastic compression flange half width be_a = 200*t_com/SQRT(Fy) = 370[mm] $13.5(c)iii compression flange half width Bo = b_com/2 = 250[mm] be_com = MIN(bo,be_a) = 250[mm] $13.5(c)iii effective compression flange area Af_come = 2*be_com*t_com = 16000 [mm2] Plate Girder Spreadsheet (CSA-S16-01) 84 total area A_tote = Aw+Af_come+Af_ten = 60800.00[mm 2] effective neutral axis height y_nae = (Af_come*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_ten))/ A_tote = 832[mm] effective outermost fibre distance y_maxe = MAX((gd-y_nae),y_nae) = 832[mm] effective strong axis second moment of area Ixe = 1/12*(2*be_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_come*(gd- t_com/2-y_nae)^2+Af_ten*(y_nae- t_ten/2)^2+Aw*(h/2+t_ten-y_nae)^2 = 2.745.E+ 10 [mm4] elastic section modulus S = Ix/y_max = 3.29977E +07 [mm3] effective elastic section modulus Se = Ixe/y_maxe = 3.29977E +07 [mm3] plastic section modulus Z = 1/4*(b_com*t_com^2+w*h^2+b_ten*t_ten^2 )+Af_com*((gd-t_com/2- y_na)^2)^0.5+Af_ten*((y_na- t_ten/2)^2)^0.5+Aw*((h/2+t_ten- y_na)^2)^0.5 = 3.78880E +07 [mm3] st. venant torsion constant J = 1/3*(b_com*t_com^3+b_ten*t_ten^3+h*w^3 ) = 1.40E+07 [mm4] weak axis second moment of area Iy = 1/12*(t_com*b_com^3+h*w^3+t_ten*b_ten^ 3) = 6.674E+0 8 [mm4] compression flange Iy Iyc = 1/12*(t_com*b_com^3) = 3.333E+0 8 [mm4] Modified Sectional Properites with Longitudinal Stiffener modified neutral axis height y_na' = (A_tot*y_na+A_stiff*0.2*gd)/(A_tot+A_stiff) = 815.3 modified strong axis second moment of area Ix' = Ix + A_tot*(y_na-y_na')^2+A_stiff*(0.2*gd- y_na')^2+Ix_stiff = 2.796E+1 0 modified outermost fibre distance y_max' = MAX(y_na',gd-y_na' ) = 848.67 modifed effective neutral axis height y_nae' = (A_tote*y_nae+A_stiff*0.2*gd)/(A_tote+A_s tiff) = 815.33 Plate Girder Spreadsheet (CSA-S16-01) 85 modified effective strong axis second moment of area Ixe' = Ixe + A_tote*(y_nae- y_na')^2+A_stiff*(0.2*gd-y_na')^2+Ix_stiff = 2.796E+1 0 modified effective neutral axis height y_maxe' = MAX(y_nae',gd-y_nae' ) = 8.487E+0 2 modified weak axis neutral axis height x_na' = (A_tot*x_na+A_stiff*(x_stiffna+x_na+w/2))/( A_tot+A_stiff) = 251.20 modified weak axis second moment of area Iy' = Iy+A_tot*(x_na- x_na')^2+A_stiff*(x_stiffna+x_na+w/2- x_na')^2 = 6.701E+0 8 elastic section modulus S' = Ix'/y_max' = 3.29E+07[mm3] effective elastic section modulus Se' = Ixe'/y_maxe' = 3.29E+07[mm3] plastic section modulus Z' = Z + Z_stiff+A_stiff*((0.2*gd- y_na')^2)^0.5+A_tot*((y_na-y_na')^2)^0.5 = 3.991E+0 7 [mm3] Moment Resistances: plastic moment Mp = Z*Fy/1000000 = 11366[kNm] class 1,2 moment resistance Ma_unstiff = φ*Z*Fy/1000000 = 10230[kNm] $13.5(a) Ma_stiff = φ*Z'*Fy/1000000 = 10777[kNm] $13.5(a) class 3 moment resistance Mb_unstiff = φ*S*Fy/1000000 = 8909[kNm] $13.5(b) Mb_stiff = φ*S' *Fy/1000000 = 8896[kNm] class 4 (ii) moment resistance Mc_unstiff = Mb_unstiff*(1-(0.0005*Aw/Af_com)*((s_w)- (1900/SQRT(Mf*1000000/(φ*S))))) = 9.07180E +03 [kNm] $14.3.4 Mc_stiff = Mb_stiff = 8896[kNm] class 4 (iii) moment resistance Md_unstiff = φ*Se*Fy/1000000 = 8909[kNm] $13.5(c) Md_stiff = φ*Se'*Fy/1000000 = 8896[kNm] $13.5(c) Ma = IF(long="none",Ma_unstiff,Ma_stiff) = 10230[kNm] Mb = IF(long="none",Mb_unstiff,Mb_stiff) = 8909[kNm] Mc = IF(long="none",Mc_unstiff,Mc_stiff) = 9072[kNm] Md = IF(long="none",Md_unstiff,Md_stiff) = 8909[kNm] Plate Girder Spreadsheet (CSA-S16-01) 86 Lateral Torsional Buckling: d' = h+(t_com+t_ten)/2 = 1632 distance from centroid to tension flange midline y_2 = y_na-t_ten/2 = 816 E = (d'*(b_com^3*t_com)/(b_com^3*t_com+b_t en^3*t_ten))-y_2 = 0.0 warping constant Cw = d'^2/12*(b_com^3*t_com*b_ten^3*t_ten)/(b _com^3*t_com+b_ten^3*t_ten) = 4.44.E+1 4 [mm6] $13.6 B2 = (PI()*PI()*Est*Cw)/((Lu)^2*G*J) = 32.44 $13.6 elastic lateral torsional buckling resistance Mu = IF(Lu=0, "NA!",( ω2*PI())/(Lu)*(SQRT(Est*Iy*G*J)*(SQRT(1 +B2)))/1000000) = 43635[kNm] $13.6 Actual Moment Resistance lateral buckling ratio Mratio = IF(Lu=0,"NA!",Mu/Mp) = 3.84 $13.6 Mr_a = IF(class="1",Ma,IF(class="2",Ma,IF(class=" 3",Mb,IF(class="4(i)","Consult CSA S136!",IF(class="4(ii)",Mc,Md))))) = 10230[kNm] $13.6 Mr_b' = IF(Lu=0,"NA!",1.15*Mr_a*(1- 0.28/φ*Mr_a/Mu)) = 10906[kNm] $13.6 Mr_b = MIN(Mr_a,Mr_b') = 10230[kNm] $13.6 Mr_c = IF(L=0,"NA!",φ*Mu) = 39272[kNm] FINAL CHECKS MOMENT RESISTANCE moment resistance Mr = IF(class="4(i)","Consult CSA S136!",IF(L=0,Mr_a,IF(Mratio>0.67,Mr_b,M r_c))) = 10230[kNm] $13.6,$13.5 efficiency moment resistance Mf_r = Mf/Mr = 0.88 check moment resistance Mcheck = IF(Mf_r<1,"OK","increase moment resistance") = OK Plate Girder Spreadsheet (CSA-S16-01) 87 FINAL CHECKS COMBINED SHEAR AND MOMENT RESISTANCE check if interaction critical IF(Vf>=0.6*Vr, "interaction is critical",""interaction not critical") = interaction is critical interaction efficiency Inter = 0.727*Mf_r+0.455*Vf_r = 0.96 $14.6. check resistance against shear and moment C_check = IF(stiffeners="exist",IF(Inter>=1,"increase moment or shear resistance","combined shear and moment capacity OK"),"NA") = combined shear and moment capacity OK Plate Girder Spreadsheet (CSA-S16-01) 88 RECOMMENDATIONS FOR TRANSVERSE STIFFENERS Required Stiffener Properties kv_a = 4+ 5.34/(a/h)^2 = 7.42 $13.4.1.1 kv_b = 5.34 + 4/(a/h)^2 = 7.90 $13.4.1.1 shear buckling coefficient Kvs = IF(a/h<1,kv_a,kv_b) = 7.90 $13.4.1.1 C_a = (1-(310000*kvs)/(Fy*(h/w)^2)) = -0.03 $13.4.1.1 C = IF(C_a<0.1,0.1,C_a) = 0.10 $13.4.1.1 Y = Fy/Fy_stiff = 0.86 $13.4.1.1 Plate Girder Spreadsheet (CSA-S16-01) 89 stiffener factor D = IF(stiff_furn="pair",1,IF(stiff_type="angle",1. 8,2.4)) = 1.00 $13.4.1.1 total required area of stiffener(s) Asr = a*w/2*(1-(a/h)/SQRT(1+(a/h)^2))*C*Y*D = 338[mm2] $14.5.3 required moment of inertia I_req'd = (h / 50)^4 = 1.05.E+0 6 [mm4] $14.5.3 INPUT TRANSVERSE STIFFENERS stiffener type stiff_type = plateangle/plate stiffener furnishing stiff_furn = pairpair/single stiffener yield strength Fy_stiff = 350[MPa] stiffener thickness primary leg ts_a = 6[mm] stiffener width primary leg bs_a = 200[mm] stiffener thickness secondary leg ts_b = 12[mm] stiffener width secondary leg bs_b = 100[mm] stiffener effective length factor K = 0.75>=0.75 $10.2.1 compute section properties or input? Comp = computeinput/comput e stiffener area As_input = 995[mm2] stiffener moment of inertia I_input = 8.50E+05[mm4] CALCULATIONS TRANSVERSE STIFFENERS Stiffener Properties area of stiffener angle As_a = ts_a*bs_a+ts_b*bs_b = 2400[mm2] area of stiffener plate As_b = ts_a*bs_a = 1200[mm2] Plate Girder Spreadsheet (CSA-S16-01) 90 computed stiffener area As_comp = IF(stiff_furn="pair",IF(stiff_type="angle",2*A s_a,2*As_b),IF(stiff_type="angle",As_a,As_ b)) = 2400[mm2] inertia of stiffener angle plus web y_a = (ts_a*bs_a*(bs_a/2+ts_b+w)+bs_b*(ts_b+w )^2/2)/(bs_a*ts_a+bs_b*(ts_b+w)) = 48[mm] I_a = (ts_a*bs_a^3)/12+bs_a*ts_a*(bs_a/2+ts_b +w- y_a)^2+(bs_b*(ts_b+w)^3)/12+bs_b*(ts_b+ w)*((ts_b+w)/2-y_a) = 1.22E+07[mm4] I_atwo = 1/12*(ts_a*(bs_a*2+w+2*ts_b)^3+(bs_b- ts_a)*(2*ts_b+w)^3) = 4.38E+07[mm4] inertia of stiffener plate I_b = 1/12*(ts_a*(w+bs_a)^3) = 5.18E+06[mm4] I_btwo = 1/12*(ts_a*(w+2*bs_a)^3) = 3.65E+07[mm4] computed stiffener Inertia I_comp = IF(stiff_furn="pair",IF(stiff_type="angle",I_at wo,I_btwo),IF(stiff_type="angle",I_a,I_b)) = 3.65E+07[mm4] stiffener area A_stift = IF(comp="compute",As_comp,As_input) = 2400[mm2] stiffener inertia I_stiff = IF(comp="compute",I_comp,I_input) = 3.65E+07[mm4] CHECKS TRANSVERSE STIFFENERS Check Against Requirements: area ratio A_ratio = A_stift/Asr = 7.10[mm2] inertia ratio I_ratio = I_stiff/I_req'd = 34.83 check meeting requirements Check = IF(MIN(A_ratio,I_ratio)<=1.0,"increase size of stiffener","OK") = OK Stiffener Slenderness Check stiffener readius of gyration R = SQRT(I_stiff/A_stift) = 123 stiffener slenderness ratio sr_stiff = K*h/r = 10 stiffener slenderness ratio check sr_stiff_check = IF(sr_stiff>200,"stiffener to slender, increase size","stiffener slenderness OK") = stiffener slenderness OK $10.4.2.1 Plate Girder Spreadsheet (CSA-S16-01) 91 stiffener bs/ts b_t_a = bs_a/ts_a = 33 b_t_b = bs_b/ts_b = 8 b_t_max = 200/SQRT(Fy_stiff) = 11 efficiency stiffener slenderness stiff_st = MIN(b_t_a,b_t_b)/b_t_max = 0.78 check stiffener slenderness check2 = IF(stiff_st<1,"OK","choose stockier stiff.") = OK $11.2 Table 1 INPUT BEARING STIFFENERS factored load Cf = 1600[kN] performance factor φbi = 0.8 weld performance factor φω = 0.7 performance factor φbe = 0.75 length of the bearing plate N = 300[mm] flange to web weld depth d_weld = 6[mm] bearing stiffeners under load bea_stiff_i = existexist/none end bearing stiffeners bea_stiff_e = existexist/none end stiffener width bs_e = 125[mm] end stiffener thickness ts_e = 16[mm] intermediate stiffener width bs_i = 125 intermediate stiffener thickness ts_i = 12 stiffener contact length Cpl = 100[mm] parameter N = 1.34 fillet weld size S = 6[mm] CALCULATIONS AND CHECKS BEARING STIFFENERS stiffener requirement at unframed ends check stiff_check_en = IF(s_w>1100/SQRT(Fy),"bearing stiff.required at unframed ends","no bearing stiff. required at unframed ends") = bearing stiff.required at unframed ends $14.4.1 flange thickness plus weld K = t + d_weld = 38 Plate Girder Spreadsheet (CSA-S16-01) 92 stiffener area Abs_e = bs_e*ts_e = 2000 stiffener moment of inertia I_e = 1/12*(ts_e*bs_e^3) = 2604167[mm4] Stiffener Slenderness Check stiffener readius of gyration r_e = SQRT(I_e/Abs_e) = 36 stiffener slenderness ratio sr_stiff_e = K*h/r_e = 33 stiffener slenderness ratio check chksl_e = IF(bea_stiff_e="exist",IF(sr_stiff_e>200,"stif fener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK $10.4.2.1 stiffener dimension check chkdim_e = IF(bea_stiff_e="exist",IF(bs_e/ts_e<200/SQ RT(Fy),"OK","increase stiffener thickness"),"NA") = OK Unstiffened End Bearing Resistance web crippling Bre_a = φbe*w*(N+4*t)*Fy/1000 = 1733[kN] $14.3.2b(i) web yielding Bre_b = 0.6*φbe*w^2*SQRT(Fy/1000*Est/1000) = 1129[kN] $14.3.2b(ii) bearing resistance Br_end = MIN( Bre_a, Bre_b) = 1129[kN] Stiffened Exterior Compression Resistance resisting area Ae = 12*w^2+2*Abs_e = 7888[mm2] $14.4.2 moment of inertia 1 Ie = 1/12*(ts_e*(2*bs_e+w)^3) = 25665109[mm4] radius of gyration Re = SQRT(Ie/Ae) = 57[mm] lambda λe = K*h/r_e*SQRT(Fy/(PI()^2*Est)) = 0.41 $13.3.1 axial compression resistance Cre = Ae*φ*Fy*(1+λe^(2*n))^(-1/n)/1000 = 1995[kN] $13.3.1 efficiency bearing stiffener br_sfe = Cf/Cre = 0.80 stiffener bearing resistance Bstiff_e = φ*1.5*Fy*(cpl*ts_e*2)/1000 = 1296[kN] $13.10(a) efficiency stiffener bearing st_sfe = Cf/Bstiff_e = 1.23 min. stiffener efficiency sf_ext = MIN(br_sfe,st_sfe) = 0.80 Plate Girder Spreadsheet (CSA-S16-01) 93 capacity check chk_cap_ext = IF(bea_stiff_e="exist",IF(sf_ext>1,"increase stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = stiffener OK Stiffener Under Concerated Loads Unstiffened Interior Bearing Resistance web crippling Bri_a = φbi*w*(N+10*t)*Fy/1000 = 2678[kN] $14.3.2a(i) web yielding Bri_b = 1.45*φbi*w^2*SQRT(Fy/1000*Est/1000) = 2911[kN] $14.3.2a(ii) bearing resistance Br_int = MIN( Bri_a, Bri_b) = 2678[kN] stiffener requirement under concentrated load stiff_check_in = IF(Cf>Br_int,"intermediate bearing stiff. required","intermediate bearing stiff. not required") = intermediate bearing stiff. not required stiffener area Abs_i = bs_i*ts_i = 1500 stiffener moment of Inertia I_i = 1/12*(ts_e*bs_e^3) = 2604167 Stiffener Slenderness Check stiffener readius of gyration r_i = SQRT(I_i/Abs_i) = 42 stiffener slenderness ratio sr_stiff_i = K*h/r_i = 29 stiffener slenderness ratio check chksl_i = IF(bea_stiff_i="exist",IF(sr_stiff_i>200,"stiff ener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK stiffener dimension check chkdim_i = IF(bea_stiff_i="exist",IF(bs_i/ts_i<200/SQR T(Fy),"OK","Increase Stiffener's thickness"),"NA") = OK Stiffened Interior Compression Resistance resisting area Ai = 25*w^2+2*Abs_i = 11100[mm2] $14.4.2 Plate Girder Spreadsheet (CSA-S16-01) 94 moment of inertia 1 Ii = 1/12*(ts_i*(2*bs_i+w)^3) = 19248832[mm4] radius of gyration Ri = SQRT(Ii/Ai) = 42[mm] lambda λ = K*h/ri*SQRT(Fy/(PI()^2*Est)) = 0.36 $13.3.1 axial compression resistance Cri = Ai*φ*Fy*(1+λ^(2*n))^(-1/n)/1000 = 2865[kN] $13.3.1 efficiency bearing br_sfi = Cf/Cri = 0.56 stiffener bearing resistance Bstiff_i = 1.5*φ*Fy*(cpl*ts_i*2)/1000 = 972[kN] $13.10(a) efficiency stiffener st_sfi = Cf/Bstiff_i = 1.65 min. stiffener efficiency sf_int = MIN(br_sfi,st_sfi) = 0.56 capacity check chk_cap_int = IF(bea_stiff_i="exist",IF(sf_int>1,"increase stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = stiffener OK Welding of Bearing Stiffener strength of base metal Vr_b = (0.67∗φω*s*Fu)/1000 = 1.21[kN/mm] $13.13.2.2(a) strength of weld Vr_w = (0.67∗φω*s*0.7071*Xu)/1000 = 0.93[kN/mm] $13.13.2.2(b) governing strength Vr_gov = Min(Vr_b,Vr_w) = 0.93[kN/mm] total weld length L_w = Cf/Vr_gov = 1715[mm] INPUT WELD DESIGN Select Weld Between Flanges and Web (two fillet welds each) weld size w_size = 8[mm] intermittend weld length w_length = 200[mm] spacing on centre w_spacing = 400[mm] COMPUTATIONS WELD DESIGN girder depth d_sel = h+2*t_com = 1664[mm] moment of inertia Ig = ((b*d_sel^3)-(b-w)*h^3)/12 = 2.75E+10[mm4] Qs = t_com*b_com*h/2 = 1.28E+07[mm3] shear flow per mm length Q = (Vf*Qs)/Ig*1000 = 1399[N/mm] Plate Girder Spreadsheet (CSA-S16-01) 95 weld resistance (two weld Lines) vr_base = 2*0.67*φw*w_size*Fu = 3232[N/mm] w_throat = 2*0.707*w_size = 11[mm] vr_throat = 2*0.67*φw*w_throat*Xu = 4976[N/mm] vr_min = MIN(vr_base,vr_throat) = 3232[N/mm] shear resistance per mm length v_r = vr_min*(w_length/w_spacing) = 1616[N/mm] weld check w_check = IF(v_r>=q,"weld flange to web OK","increase weld amount") = weld flange to web OK efficiency weld we_eff = q/v_r = 0.87 INPUT Weight Computation Steel Density S_den = 0.000007 8500[Kg/mm^3] COMPUTATIONS OF TOTAL WEIGHT PER GIRDER mass of compression flange M_com_Fl = b_com*t_com*(L*1000)*S_den = 3140.00[kg] mass of tension Flange M_ten_Fl = b_ten*t_ten*(L*1000)*S_den = 3140[kg] mass of web M_Web = if(stiffeners="exist",(L*1000 / a - 1) * bs_a * ts_a * h * S_den * if(stiff_furn="pair",2,1),0) = 346.66[kg] mass of transverse stiffeners M_t_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 175.84[kg] mass of bearing stiffener M_b_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 175.84[kg] total mass T_Mass = M_com_Fl+M_ten_Fl+M_Web+M_t_stiff+M _b_stiff = 6978.34[kg] total weight per girder T_Weight = T_Mass*9.8/1000 = 68.4[kN] Handbook of Steel Construction CAN/CSA- S16-01 - 9th Edition 96 Appendix B: Plate Girder Spreadsheet (CSA-S6-06) Plate Girder Spreadsheet (CSA-S6-06) 97 DESIGN OF PLATE GIRDERS REFERENCES INPUT GENERAL PARAMETERS specified material yield strength Fy = 350[MPa] ultimate material yield strength Fu = 450[MPa] weld metal strength Xu = 490[MPa] performance factor φ = 0.95 Plate Girder Spreadsheet (CSA-S6-06) 98 performance factor for welds φw = 0.67 span L = 30.0[m] factored moment Mf = 9500[kNm] factored shear Vf = 4000[kN] max. (space for) girder depth d_max = 3200[mm] presence of stiffeners (No stiffeners = 0, Transverse stiffeners = 1, Longitudinal stiffeners = 2) Stiff = = 1 COMPUTATIONS GENERAL PARAMETERS Slenderness Limits min.web SL for red. moment SL_wmin = 1900/SQRT(Fy) = 102 $10.10.4.3 max. web slenderness SL_wmax = IF(Stiff = 2, 6000/sqrt(Fy), 3150/sqrt(Fy)) = 168 $10.10.4.2 max. flange slenderness SL_fmax = 200/SQRT(Fy) = 11 $10.9.2 TabLe 10.3 Preliminary Sizing height for max bending efficiency h_a = 540*(Mf/Fy)^(1/3) = 1623[mm] maximum shear strength Fs = 0.577*Fy = 201.95[MPa] $10.10.5.1 minimum web area Aw_min = Vf*1000/(φ*Fs) = 20849[mm2] min web thickness w_a = MAX(Aw_min/h_rec, 10) = 12.8[mm] $10.7.2 recommended web slenderness SL_recw = h_rec/ w_a = 126 ratio of web slendernesses r_SLw = SL_recw/SL_wmax = 0.75 minimum flange area Af_min = Mf * 1000000 / ( Fy * h_rec) = 16725[mm2] recommended web thickness w_rec = IF(r_SLw>1,w_a*r_SLw,w_a) = 13[mm] recommended web depth h_rec = IF(h_a>d_max,d_max,h_a) = 1623[mm] recommended flange thickness t_rec = SQRT((Af_min/SL_fmax)/2) = 28[mm] recommended flange width b_rec = Af_min/t_rec = 598[mm] total girder depth d_rec = 2*t_rec+h_rec = 1679[mm] INPUT GIRDER PARAMETERS Select Girder Parameters web thickness w = 15[mm] web depth h = 1800[mm] Plate Girder Spreadsheet (CSA-S6-06) 99 flange thickness t = 30[mm] flange width b = 500[mm] CHECKS CONCEPTUAL DESIGN web slenderness check s_w_chec = IF(h/w>SL_wmax,"reduce slenderness of web","web slenderness OK") = web slenderness OK efficiency web slenderness wb_eff = (h/w)/SL_wmax = 0.71 web thickness check w_ch = IF(w<10,"Increase web thickness", "web thickness OK") = web thickness OK flange slenderness check sL_fsel_che c = IF(b/(2*t)>SL_fmax,"reduce flange slenderness ","flange slenderness OK") = flange slenderness OK efficiency flange slenderness fl_eff = (b/(2*t))/SL_fmax = 0.78 girder depth check d_sel_chec = IF(d_sel>d_max,"reduce recommended girder depth","girder depth OK") = girder depth OK efficiency flange slenderness d_eff = d_sel/d_max = 0.58 Plate Girder Spreadsheet (CSA-S6-06) 100 INPUT TO CHECK SHEAR RESISTANCE transverse stiffeners exist? stiffeners = exist(exist/non e) intermediate trans. stiff. spacing a = 4500[mm] panel location p_loc = interiorinterior/en d anchorage for end panel anch = yesyes/no factored shear in end panel Vf_e = 2500[kN] CALCULATIONS OF SHEAR RESISTANCE web area Aw = h*w = 27000[mm2] panel Ratio a_h = a/h = 2.50 factored shear stress Ff_i = (Vf/Aw)*1000 = 148.15[MPa] Plate Girder Spreadsheet (CSA-S6-06) 101 factored shear stress at end panel Ff_e = (Vf_e/Aw)*1000 = 92.59[MPa] Panel Ratio Check max panel ratio one a_hmax_a = 67500/(s_w)^2 = 4.69 $10.10.6.1 max panel ratio two a_hmax_b = 3 = 3.00 $10.10.6.1 max panel ratio three a_hmax_c = 1.5 = 1.50 $10.10.7.1 panel ratio check for transverse stiffeners only a_hcheck = (IF(stiffeners="exist",IF(s_w>150,IF(a_h<= a_hmax_a,"OK","decrease stiffener spacing"),IF(a_h<=a_hmax_b,"OK","decre ase stiffener spacing")),"NA")) = OK $10.10.6.1 panel ratio check if longitudinal stiffeners present a_hcheck_ 2 = IF(Stiff = 2, IF(a/hp < a_hmax_c, "OK", "decrease stiffener spacing"),a_hcheck) = OK Web Slenderness Check web maximum allowable slenderness s_wmax = 3150/sqrt(Fy) = 168 $10.10.4.2 web slenderness s_w = h/w = 120 web slenderness check s_wcheck = IF(s_w>s_wmax,"reduce slenderness","slenderness OK") = slenderness OK $10.10.4.2 Ultimate Shear Stress (Fs) (a) yielding in shear Fs_a = 0.577* Fy = 201.95[MPa] $10.10.5.1(a) (b) tension field action Fs_b = F_cri+ka*(0.5*Fy -0.866*F_cri) = 140.00[MPa] $10.10.5.1(b) (c) elastic buckling Fs_c = F_cre+ka*(0.5*Fy-0.866*F_cre) = 115.70[MPa] $10.10.5.1(c) tension field contribution Ft = ka*(0.5*Fy-0.866*F_cre) = 40.95[MPa] coefficient for no stiffeners kv_n = 5.34 = 5.34 coefficient case a/h<1 kv_s1 = 4+5.34/(a/h)^2 = 4.85 $10.10.5.1 coefficient case a/h>=1 kv_s2 = 5.34+4/(a/h)^2 = 5.98 $10.10.5.1 kv_s = IF(a/h<1,kv_s1,kv_s2) = 5.98 $10.10.5.1 kv = IF(stiffeners = "none",kv_n,kv_s) = 5.98 $10.10.5.1 = critical shear stress inelastic F_cri = 290 * (((Fy * kv)^.5)/(h / w)) = 110.56[MPa] aspect coefficient ka = 1/SQRT(1+(a/h)^2) = 0.37 critical shear stress elastic F_cre = 180000*kv/(s_w)^2 = 74.75[MPa] Plate Girder Spreadsheet (CSA-S6-06) 102 coefficient factor cF = SQRT(kv/Fy) = 0.13[MPa-0.5] slenderness case h/w case = IF(s_w<=502*cF,"c_a",IF(s_w<=621*cF,"c _b","c_c")) = c_c Fs_stiff = IF(case="c_a",Fs_a,IF(case="c_b",Fs_b,Fs _c)) = 115.70[MPa] $10.10.5.1 (a-c) Fs_unstiff = IF(case="c_a",Fs_a,IF(case="c_b",F_cri,F_ cre))) = 74.75[MPa] $10.10.5.1 (a-c) = = stiffener check st_check IF(Vf>(Fs_unstiff*Aw/1000),"transverse stiffener required", IF(h/w>150, "transverse stiffener required", "transverse stiffener not required")) transverse stiffener required F_s = IF(p_loc="interior",IF(stiffeners="exist",Fs_ stiff,Fs_unstiff),IF(anch="yes",Fs_stiff,Fs_u nstiff)) = 115.70[MPa] $10.10.5.1 (a-c) End Panel Calculation minimum shear buckling coefficient kv_min = (Ff_e*(s_w)^2)/(180000*0.9) = 8.23 minimum end panel spacing a_e = SQRT((4*h^2)/(kv_min-5.34)) = 2117[mm] FINAL CHECKS SHEAR RESISTANCE shear resistance Vr = φ * Aw * F_s / 1000 = 2968[kN] $10.10.5 efficiency shear resistance Vf_r = Vf_e/Vr = 0.84 check shear resistance Vcheck = IF(Vf_r<1,"OK","increase shear resistance") = OK Plate Girder Spreadsheet (CSA-S6-06) 103 INPUT 3: GIRDER PARAMETERS - UNEQUAL FLANGES compression flange width b_com = 500[mm] compression flange thickness t_com = 30[mm] tension flange width b_ten = 500[mm] tension flange thickness t_ten = 30[mm] material Shear Modulus G = 77000[MPa] material Young's Modulus Est = 200000[MPa] unsupported Length Lu = 5000[mm] girder loading: Uniform …. Load = load (load/mo ment) longitudinal stiffener at 0.2d long = none exist/non e longitudinal stiffener area A_stiff = 2100[mm 2] Plate Girder Spreadsheet (CSA-S6-06) 104 longitudinal stiffener width d_stiff = 150[mm] longitudinal stiffener thickness t_stiff = 10[mm] longitudinal stiffener weak axis I Ix_stiff = 2.80E+0 6 [mm4] longitudinal stiffener strong axis I Iy_stiff = 2.80E+0 6 [mm4] longitudinal strong neutral axis height x_stiffna = 27[mm] long. stiff. strong axis plastic modulus Z_stiff = 0.00E+0 0 [mm3] longitudinal stiffener section modulus about base Sh = 1.00E+0 2 [mm3] longitudinal stiffener subpanel length hp = 10.00[mm] point of application of transverse loading as a fraction of girder depth d g'' = 1 moment distribution factor ω2 = 1 $10.10.2.3 CHECKS FOR LONGITUDINAL STIFFENER SECTION longitudinal stiffener effective width de_Stiff = IF((d_stiff/t_stiff)<(200/sqrt(Fy)), IF((d_stiff/t_stiff)<30, d_stiff, "reduce width- to-thickness ratio"), 200/sqrt(Fy)*t_stiff) = 107[mm] $10.10.7.2(a-b) check for stiffener width d_stiff_ch = IF(de_Stiff < d_stiff, "use effective width", "OK") = use effectiv e width minimum stiffener moment of inertia I_limit = h*w^3*(2.4*(a/h)^2-0.13) = 9033525 0 [mm4] $10.10.7.2 (c ) moment of interia of stiffener section I_sw = Ix_stiff+(w*(2*10*w)^3/12)+(2*10*w^2*x_sti ffna^2) = 3980624 5 [mm4] check for moment of interia I_ch = IF(I_sw>=I_limit, "OK", "increase stiffener section") = increas e stiffener section $10.10.7.2 (c ) minimum radius of gyration r_limit = a*sqrt(Fy)/1900 = 44[mm] check for radius of gyration r_ch = IF(sqrt(I_sw/(A_stiff+20*w^2))>=r_limit, "OK", "increase longitudinal stiffener section") = OK $10.10.7.2(d) Plate Girder Spreadsheet (CSA-S6-06) 105 CALCULATIONS OF MOMENT RESISTANCE Section Element Slenderness and Class flange max. allowable slenderness s_fmax = = 30.0 $10.10.3.4(b) flange (in compression) slenderness s_f = b_com/2/t_com = 8.3 flange slenderness check f_check = IF(s_f>s_fmax,"reduce flange width","flange width OK") = flange width OK flange class f_class = IF(s_f<=145/SQRT(Fy),"1",IF(s_f<=170/SQ RT(Fy),"2",IF(s_f<=200/SQRT(Fy),"3","4")) ) = 2 $10.9.2, Table 10.3 web slenderness for monosymmetric section s_wm = IF(B1 = 0, s_w, 2*dc/w) = 120.000 000 web class w_class = IF(s_wm<=1100/SQRT(Fy),"1",IF(s_wm<= 1700/SQRT(Fy),"2",IF(s_wm<=1900/SQRT (Fy),"3","4"))) = 4 $10.9.2, Table 10.3 class_a = IF(f_class>w_class,f_class,w_class) = 4 section class class = IF(class_a="4",IF(w_class="4",IF(f_class=" 4","4(i)","4(ii)"),"4(iii)"),class_a) = 4(ii) $10.10.3.4 section class check cl_check = IF(class="4(i)","change girder dimensions","OK") = OK $10.10.3.4 Sectional Properites without Longitudinal Stiffener compression flange area Af_com = b_com*t_com = 15000.0 0 [mm2] tension flange area Af_ten = b_ten*t_ten = 15000.0 0 [mm2] total area A_tot = Aw+Af_com+Af_ten = 57000.0 0 [mm2] overall girder depth gd = h+t_ten+t_com = 1860[mm] neutral axis height y_na = (Af_com*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_ten))/ A_tot = 930[mm] neutral axis height x_na = b_com/2 = 250[mm] outermost fibre distance y_max = MAX(gd-y_na,y_na) = 930[mm] Plate Girder Spreadsheet (CSA-S6-06) 106 strong axis second moment of area Ix = 1/12*(b_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_com*(gd- t_com/2-y_na)^2+Af_ten*(y_na- t_ten/2)^2+Aw*(h/2+t_ten-y_na)^2 = 3.241.E +10 [mm4] effective elastic compression flange half width be_a = 200*t_com/SQRT(Fy) = 321[mm] $10.10.3.4(b) compression flange half width bo = b_com/2 = 250[mm] be_com = MIN(bo,be_a) = 250[mm] $10.10.3.4(b) effective compression flange area Af_come = 2*be_com*t_com = 15000[mm 2] total area A_tote = Aw+Af_come+Af_ten = 57000.0 0 [mm2] effective neutral axis height y_nae = (Af_come*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_ten))/ A_tote = 930[mm] effective outermost fibre distance y_maxe = MAX((gd-y_nae),y_nae) = 930[mm] effective strong axis second moment of area Ixe = 1/12*(2*be_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_come*(gd- t_com/2-y_nae)^2+Af_ten*(y_nae- t_ten/2)^2+Aw*(h/2+t_ten-y_nae)^2 = 3.241.E +10 [mm4] elastic section modulus S = Ix/y_max = 3.48484 E+07 [mm3] effective elastic section modulus Se = Ixe/y_maxe = 3.48484 E+07 [mm3] plastic section modulus Z = 1/4*(b_com*t_com^2+w*h^2+b_ten*t_ten^ 2)+Af_com*((gd-t_com/2- y_na)^2)^0.5+Af_ten*((y_na- t_ten/2)^2)^0.5+Aw*((h/2+t_ten- y_na)^2)^0.5 = 3.98250 E+07 [mm3] st. venant torsion constant J = 1/3*(b_com*t_com^3+b_ten*t_ten^3+h*w^ 3) = 1.10E+0 7 [mm4] weak axis second moment of area Iy = 1/12*(t_com*b_com^3+h*w^3+t_ten*b_ten ^3) = 6.255E+ 08 [mm4] compression flange Iy Iyc = 1/12*(t_com*b_com^3) = 3.125E+ 08 [mm4] Plate Girder Spreadsheet (CSA-S6-06) 107 Modified Sectional Properites with Longitudinal Stiffener modified neutral axis height y_na' = (A_tot*y_na+A_stiff*0.2*gd)/(A_tot+A_stiff) = 910.2 modified strong axis second moment of area Ix' = Ix + A_tot*(y_na-y_na')^2+A_stiff*(0.2*gd- y_na')^2+Ix_stiff = 3.304E+ 10 modified outermost fibre distance y_max' = MAX(y_na',gd-y_na' ) = 949.83 modifed effective neutral axis height y_nae' = (A_tote*y_nae+A_stiff*0.2*gd)/(A_tote+A_s tiff) = 910.17 modified effective strong axis second moment of area Ixe' = Ixe + A_tote*(y_nae- y_na')^2+A_stiff*(0.2*gd-y_na')^2+Ix_stiff = 3.304E+ 10 modified effective neutral axis height y_maxe' = MAX(y_nae',gd-y_nae' ) = 9.498E+ 02 modified weak axis neutral axis height x_na' = (A_tot*x_na+A_stiff*(x_stiffna+x_na+w/2))/( A_tot+A_stiff) = 251.22 modified weak axis second moment of area Iy' = Iy+A_tot*(x_na- x_na')^2+A_stiff*(x_stiffna+x_na+w/2- x_na')^2 = 6.279E+ 08 elastic section modulus S' = Ix'/y_max' = 3.48E+0 7 [mm3] effective elastic section modulus Se' = Ixe'/y_maxe' = 3.48E+0 7 [mm3] plastic section modulus Z' = Z + Z_stiff+A_stiff*((0.2*gd- y_na')^2)^0.5+A_tot*((y_na-y_na')^2)^0.5 = 4.209E+ 07 [mm3] depth of compression portion of web dc = IF(Stiff = 2,gd- y_na'-t_com, gd-y_na- t_com) = 900.00[mm] Moment Resistances: plastic moment Mp = Z*Fy/1000000 = 13939[kNm] class 1,2 moment resistance Ma_unstiff = φ*Z*Fy/1000000 = 13242[kNm] $10.10.2.2 Ma_stiff = φ*Z'*Fy/1000000 = 13993[kNm] $10.10.2.2 class 3 moment resistance Mb_unstiff = φ*S*Fy/1000000 = 11587[kNm] $10.10.3.2 Mb_stiff = φ*S' *Fy/1000000 = 11567[kNm] $10.10.3.2 class 4 (ii) moment resistance Mc_unstiff = Mb_unstiff*(1- (Aw/(300*Aw+1200*Af_com))*((2*dc/w)- (1900/SQRT(Mf*1000000/(φ*S))))) = 11493[kNm] $10.10.3.4 Plate Girder Spreadsheet (CSA-S6-06) 108 Mc_stiff = Mb_stiff = 11567[kNm] class 4 (iii) moment resistance Md_unstiff = φ*Se*Fy/1000000 = 11587[kNm] $10.10.3.4 Md_stiff = φ*Se'*Fy/1000000 = 11567[kNm] $10.10.3.4 Moment reduction factor M_red = 1-(Aw/(300*Aw+1200*Af_com))*((2*dc/w)- 1900/sqrt(SQRT(Mf*1000000/(φ*S)))) = 1.35342 $10.10.4.3 Ma = IF(long="none", Ma_unstiff,Ma_stiff) = 13242[kNm] Mb = IF(long="none",IF((2*dc/w)>1900/sqrt(Fy), Mc_unstiff,Mb_unstiff),Mb_stiff) = 11493[kNm] Mc = IF(long="none",IF(stiffeners = "none", IF(s_w<=150, Mc_unstiff, "stiffeners required"), Mb),Mc_stiff) = 11493[kNm] Md = IF(long="none",IF((2*dc/w)>1900/sqrt(Fy), Md_unstiff*M_red, Md_unstiff),Md_stiff) = 15682[kNm] Lateral Torsional Buckling: $10.10.2.3 d' = h+(t_com+t_ten)/2 = 1830 $10.10.2.3 distance from centroid to tension flange midline y_2 = y_na-t_ten/2 = 915 $10.10.2.3 e = (d'*(b_com^3*t_com)/(b_com^3*t_com+b_t en^3*t_ten))-y_2 = 0.0 $10.10.2.3 coefficient of monosymmetry bx = 0.9*d'*(2*Iyc/Iy-1)*(1-(Iy/Ix)^2) = -1.3325 $10.10.2.3 warping constant Cw = d'^2/12*(b_com^3*t_com*b_ten^3*t_ten)/(b _com^3*t_com+b_ten^3*t_ten) = 5.23.E+ 14 [mm6] $10.10.2.3 B1 = ((PI()*bx)/(2*Lu))*SQRT((Est*Iy)/(G*J)) = -0.01 $10.10.2.3 B2 = (PI()*PI()*Est*Cw)/((Lu)^2*G*J) = 48.67 $10.10.2.3 elastic lateral torsional buckling resistance Mu = IF(Lu=0, "NA!",( ω2*PI())/(Lu)*(SQRT(Est*Iy*G*J)*(B1+SQR T(1+B2+B1^2)))/1000000) = 45601[kNm] $10.10.2.3 Actual Moment Resistance lateral buckling ratio Mratio = IF(Lu=0,"NA!",Mu/Mp) = 3.27 $10.10.2.3 Mr_a = IF(class="1",Ma,IF(class="2",Ma,IF(class=" 3",Mb,IF(class="4(i)","change girder dimensions",IF(class="4(ii)",Mc,Md))))) = 11493[kNm] $10.10.2.3 Plate Girder Spreadsheet (CSA-S6-06) 109 Mr_b' = IF(Lu=0,"NA!",1.15*Mr_a*(1- 0.28/φ*Mr_a/Mu)) = 12235[kNm] $10.10.3.3 Mr_b = MIN(Mr_a,Mr_b') = 11493[kNm] $10.10.3.3 Mr_c = IF(L=0,"NA!",φ*Mu) = 43321[kNm] FINAL CHECKS MOMENT RESISTANCE moment resistance Mr = IF(class="4(i)","change girder dimensions",IF(L=0,Mr_a,IF(Mratio>0.67,M r_b,Mr_c))) = 11493[kNm] $10.10.2.3, $10.10.3.3 efficiency moment resistance Mf_r = Mf/Mr = 0.83 check moment resistance Mcheck = IF(Mf_r<1,"OK","increase moment resistance") = OK FINAL CHECKS COMBINED SHEAR AND MOMENT RESISTANCE check if interaction critical IF(Vf>=0.6*Vr, "interaction is critical",""interaction not critical") = shear and moment is critical interaction efficiency Inter = 0.727*Mf_r+0.455*Vf_r = 0.98 $10.10.5.2 check resistance against shear and moment C_check = IF(stiffeners="exist",IF(Inter>=1,"increase moment or shear resistance","combined shear and moment capacity OK"),"NA") = combined shear and moment capacity OK Plate Girder Spreadsheet (CSA-S6-06) 110 RECOMMENDATIONS FOR TRANSVERSE STIFFENERS Required Stiffener Properties kv_a = 4+ 5.34/(a/h)^2 = 4.85 $10.10.5.1 kv_b = 5.34 + 4/(a/h)^2 = 5.98 $10.10.5.1 shear buckling coefficient kvs = IF(a/h<1,kv_a,kv_b) = 5.98 $10.10.5.1 C_a = (1-(310000*kvs)/(Fy*(h/w)^2)) = 0.63 $10.10.6.2(b) C = IF(C_a<0.1,0.1,C_a) = 0.63 $10.10.6.2(b) Y = Fy/Fy_stiff = 1.00 $10.10.6.2(b) stiffener factor D = IF(stiff_furn="pair",1,IF(stiff_type="angle",1. 8,2.4)) = 1.00 $10.10.6.2(b) minimum stiffener width b_tsl = Max(50+h/30,Min(b_com,b_ten)/4) = 125.00[mm] $10.10.6.2(b) Plate Girder Spreadsheet (CSA-S6-06) 111 If longitudinal stiffener present, required area Asr_l = (a*w/2*(1- (a/hp)/SQRT(1+(a/hp)^2))*Vf_r*C*D- 18*w^2)*Y = -4049.96[mm2] $10.10.7.3 If longitudinal stiffener present, moment of inertia I_lreq = a*w^3*MAX(2.5*(hp/a)^2-2,0.5) = 7.59E+0 6 [mm4] $10.10.7.3 If longitudinal stiffener present, section modulus required S_t = h*Sh/(3*a) = 13.33[mm3] $10.10.7.3 total required area of stiffener(s) Asr = MAX(IF(long = "none",(a*w/2*(1- (a/h)/SQRT(1+(a/h)^2))*Vf_r*C*D- 18*w^2)*Y, Asr_l),0.01) = 0[mm2] $10.10.6.2(b) required moment of inertia I_req'd = IF(long = "none",a*w^3*MAX(2.5*(h/a)^2- 2,0.5), I_lreq) = 7.59.E+ 06 [mm4] $10.10.6.2(b) INPUT TRANSVERSE STIFFENERS stiffener type stiff_type = plateangle/plat e stiffener furnishing stiff_furn = pairpair/singl e stiffener yield strength Fy_stiff = 350[MPa] stiffener thickness primary leg ts_a = 6[mm] stiffener width primary leg bs_a = 200[mm] stiffener thickness secondary leg ts_b = 12[mm] stiffener width secondary leg bs_b = 126[mm] stiffener effective length factor K = 0.75>=0.75 $10.5.9.2.1 stiffener section modulus with I at base Sbase = 100.00[mm3] compute section properties or input? comp = comput e input/com pute stiffener area As_input = 995[mm2] stiffener moment of inertia I_input = 8.50E+0 5 [mm4] Plate Girder Spreadsheet (CSA-S6-06) 112 CALCULATIONS TRANSVERSE STIFFENERS Stiffener Properties area of stiffener angle As_a = ts_a*bs_a+ts_b*bs_b = 2712[mm2] area of stiffener plate As_b = ts_a*bs_a = 1200[mm2] computed stiffener area As_comp = IF(stiff_furn="pair",IF(stiff_type="angle",2*A s_a,2*As_b),IF(stiff_type="angle",As_a,As _b)) = 2400[mm2] inertia of stiffener angle plus web y_a = (ts_a*bs_a*(bs_a/2+ts_b+w)+bs_b*(ts_b+ w)^2/2)/(bs_a*ts_a+bs_b*(ts_b+w)) = 43[mm] I_a = (ts_a*bs_a^3)/12+bs_a*ts_a*(bs_a/2+ts_b +w- y_a)^2+(bs_b*(ts_b+w)^3)/12+bs_b*(ts_b+ w)*((ts_b+w)/2-y_a) = 1.26E+0 7 [mm4] I_atwo = 1/12*(ts_a*(bs_a*2+w+2*ts_b)^3+(bs_b- ts_a)*(2*ts_b+w)^3) = 4.29E+0 7 [mm4] inertia of stiffener plate I_b = 1/12*(ts_a*(w+bs_a)^3) = 4.97E+0 6 [mm4] I_btwo = 1/12*(ts_a*(w+2*bs_a)^3) = 3.57E+0 7 [mm4] computed stiffener Inertia I_comp = IF(stiff_furn="pair",IF(stiff_type="angle",I_at wo,I_btwo),IF(stiff_type="angle",I_a,I_b)) = 3.57E+0 7 [mm4] stiffener area A_stift = IF(comp="compute",As_comp,As_input) = 2400[mm2] stiffener inertia I_stiff = IF(comp="compute",I_comp,I_input) = 3.57E+0 7 [mm4] CHECKS TRANSVERSE STIFFENERS Check Against Requirements: area ratio A_ratio = A_stift/Asr = 240000[mm2] inertia ratio I_ratio = I_stiff/I_req'd = 4.71 section modulus ratio S_ratio = IF(long = "none",1.1, Sbase/S_t) = 1.10 width ratio b_ratio = MIN(bs_a,bs_b)/b_tsl = 1.01 check meeting requirements check = IF(MIN(A_ratio,I_ratio,b_ratio,S_ratio)<=1. = OK Plate Girder Spreadsheet (CSA-S6-06) 113 0,"increase size of stiffener","OK") Stiffener Slenderness Check stiffener readius of gyration r = SQRT(I_stiff/A_stift) = 122 stiffener slenderness ratio sr_stiff = K*h/r = 11 stiffener slenderness ratio check sr_stiff_che ck = IF(sr_stiff>200,"stiffener to slender, increase size","stiffener slenderness OK") = stiffener slenderness OK stiffener bs/ts b_t_a = bs_a/ts_a = 33 b_t_b = bs_b/ts_b = 11 b_t_max = MIN(200/SQRT(Fy_stiff),30) = 11 $10.10.6.2 (b) efficiency stiffener slenderness stiff_st = MIN(b_t_a,b_t_b)/b_t_max = 0.98 check stiffener slenderness check2 = IF(stiff_st<1,"OK","choose stockier stiff.") = OK INPUT BEARING STIFFENERS factored load Cf = 2000[kN] performance factor φbi = 0.80 weld performance factor φω = 0.67 performance factor φbe = 0.75 length of the bearing plate N = 300[mm] flange to web weld depth d_weld = 6[mm] bearing stiffeners under load bea_stiff_i = existexist/non e end bearing stiffeners bea_stiff_e = existexist/non e end stiffener width bs_e = 125[mm] end stiffener thickness ts_e = 16[mm] intermediate stiffener width bs_i = 125 intermediate stiffener thickness ts_i = 12 stiffener contact length cpl = 100[mm] parameter n = 1.34 fillet weld size s = 6[mm] Plate Girder Spreadsheet (CSA-S6-06) 114 CALCULATIONS AND CHECKS BEARING STIFFENERS stiffener requirement at unframed ends check stiff_check_ en = IF(s_w>1100/SQRT(Fy),"bearing stiff.required at unframed ends","no bearing stiff. required at unframed ends") = bearing stiff.required at unframed ends $10.10.8 flange thickness plus weld k = t + d_weld = 36 stiffener area Abs_e = bs_e*ts_e = 2000 stiffener moment of inertia I_e = 1/12*(ts_e*bs_e^3) = 2604167[mm4] Stiffener Slenderness Check stiffener readius of gyration r_e = SQRT(I_e/Abs_e) = 36 stiffener slenderness ratio sr_stiff_e = K*h/r_e = 37 stiffener slenderness ratio check chksl_e = IF(bea_stiff_e="exist",IF(sr_stiff_e>200,"stif fener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK $10.10.8.2 stiffener dimension check chkdim_e = IF(bea_stiff_e="exist",IF(bs_e/ts_e<200/S QRT(Fy),"OK","increase stiffener thickness"),"NA") = OK Unstiffened End Bearing Resistance web crippling Bre_a = φbe*w*(N+4*t)*Fy/1000 = 1654[kN] $10.10.8.1(b)(i) web yielding Bre_b = 0.6*φbe*w^2*SQRT(Fy/1000*Est/1000) = 847[kN] $10.10.8.1(b)(ii) bearing resistance Br_end = MIN( Bre_a, Bre_b) = 847[kN] Stiffened Exterior Compression Resistance resisting area Ae = 12*w^2+2*Abs_e = 6700[mm2] $10.10.8.3 moment of inertia 1 Ie = 1/12*(ts_e*(2*bs_e+w)^3) = 2481283 3 [mm4] radius of gyration re = SQRT(Ie/Ae) = 61[mm] lambda λe = K*h/r_e*SQRT(Fy/(PI()^2*Est)) = 0.50 $10.9.3.1 axial compression resistance Cre = Ae*φ*Fy*(1+λe^(2*n))^(-1/n)/1000 = 2001[kN] $10.9.3.1 efficiency bearing stiffener br_sfe = Cf/Cre = 1.00 Plate Girder Spreadsheet (CSA-S6-06) 115 stiffener bearing resistance Bstiff_e = φ*1.5*Fy*(cpl*ts_e*2)/1000 = 1596[kN] $10.10.8.2 efficiency stiffener bearing st_sfe = Cf/Bstiff_e = 1.25 min. stiffener efficiency sf_ext = MIN(br_sfe,st_sfe) = 1.00 capacity check chk_cap_ex t = IF(bea_stiff_e="exist",IF(sf_ext>1,"increase stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = stiffener OK Stiffener Under Concerated Loads Unstiffened Interior Bearing Resistance web crippling Bri_a = φbi*w*(N+10*t)*Fy/1000 = 2520[kN] $10.10.8.1(a)(i) web yielding Bri_b = 1.45*φbi*w^2*SQRT(Fy/1000*Est/1000) = 2184[kN] $10.10.8.1(a)(i) bearing resistance Br_int = MIN( Bri_a, Bri_b) = 2184[kN] stiffener requirement under concentrated load stiff_check_ in = IF(Cf>Br_int,"intermediate bearing stiff. required","intermediate bearing stiff. not required") = intermediate bearing stiff. not required stiffener area Abs_i = bs_i*ts_i = 1500 stiffener moment of Inertia I_i = 1/12*(ts_e*bs_e^3) = 2604167 Stiffener Slenderness Check stiffener readius of gyration r_i = SQRT(I_i/Abs_i) = 42 stiffener slenderness ratio sr_stiff_i = K*h/r_i = 32 stiffener slenderness ratio check chksl_i = IF(bea_stiff_i="exist",IF(sr_stiff_i>200,"stiff ener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK stiffener dimension check chkdim_i = IF(bea_stiff_i="exist",IF(bs_i/ts_i<200/SQR T(Fy),"OK","Increase Stiffener's thickness"),"NA") = OK Stiffened Interior Compression Resistance resisting area Ai = 24*w^2+2*Abs_i = 8400[mm2] $10.10.8.3 Plate Girder Spreadsheet (CSA-S6-06) 116 moment of inertia 1 Ii = 1/12*(ts_i*(2*bs_i+w)^3) = 1860962 5 [mm4] radius of gyration ri = SQRT(Ii/Ai) = 47[mm] lambda λ = K*h/ri*SQRT(Fy/(PI()^2*Est)) = 0.38 $10.9.3.1 axial compression resistance Cri = Ai*φ*Fy*(1+λ^(2*n))^(-1/n)/1000 = 2645[kN] $10.9.3.1 efficiency bearing br_sfi = Cf/Cri = 0.76 stiffener bearing resistance Bstiff_i = 1.5*φ*Fy*(cpl*ts_i*2)/1000 = 1197[kN] $10.10.8.2 efficiency stiffener st_sfi = Cf/Bstiff_i = 1.67 min. stiffener efficiency sf_int = MIN(br_sfi,st_sfi) = 0.76 capacity check chk_cap_int = IF(bea_stiff_i="exist",IF(sf_int>1,"increase stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = stiffener OK Welding of Bearing Stiffener strength of base metal Vr_b = (0.67∗φω*s*Fu)/1000 = 1.21[kN/mm] $10.18.3.2.2(a) strength of weld Vr_w = (0.67∗φω*s*0.7071*Xu)/1000 = 0.93[kN/mm] $10.18.3.2.2(b) governing strength Vr_gov = Min(Vr_b,Vr_w) = 0.93[kN/mm] total weld length L_w = Cf/Vr_gov = 2143[mm] INPUT WELD DESIGN Select Weld Between Flanges and Web (two fillet welds each) weld size w_size = 8[mm] intermittend weld length w_length = 300[mm] spacing on centre w_spacing = 400[mm] COMPUTATIONS WELD DESIGN girder depth d_sel = h+2*t_com = 1860[mm] moment of inertia Ig = ((b*d_sel^3)-(b-w)*h^3)/12 = 3.24E+1 0[mm4] Qs = t_com*b_com*h/2 = 1.35E+0 7[mm3] Plate Girder Spreadsheet (CSA-S6-06) 117 shear flow per mm length q = (Vf*Qs)/Ig*1000 = 1666[N/mm] weld resistance (two weld Lines) vr_base = 2*0.67*φw*w_size*Fu = 3232[N/mm] w_throat = 2*0.707*w_size = 11[mm] vr_throat = 2*0.67*φw*w_throat*Xu = 4976[N/mm] vr_min = MIN(vr_base,vr_throat) = 3232[N/mm] shear resistance per mm length v_r = vr_min*(w_length/w_spacing) = 2424[N/mm] weld check w_check = IF(v_r>=q,"weld flange to web OK","increase weld amount") = weld flange to web OK efficiency weld we_eff = q/v_r = 0.69 INPUT Weight Computation Steel Density S_den = 0.00000 78500 [Kg/mm^ 3] COMPUTATIONS OF TOTAL WEIGHT PER GIRDER mass of compression flange M_com_Fl = b_com*t_com*(L*1000)*S_den = 3532.50[kg] mass of tension Flange M_ten_Fl = b_ten*t_ten*(L*1000)*S_den = 3533[kg] mass of web M_Web = if(stiffeners="exist",(L*1000 / a - 1) * bs_a * ts_a * h * S_den * if(stiff_furn="pair",2,1),0) = 192.17[kg] mass of transverse stiffeners M_t_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 197.82[kg] mass of bearing stiffener M_b_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 197.82[kg] total mass T_Mass = M_com_Fl+M_ten_Fl+M_Web+M_t_stiff+ M_b_stiff = 7652.81[kg] total weight per girder T_Weight = T_Mass*9.8/1000 = 75.0[kN] Canadian Highway Bridge Design Code CAN/CSA-S6- 06 118 Appendix C: Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 119 DESIGN OF PLATE GIRDERS REFERENCES Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 120 INPUT GENERAL PARAMETERS specified material yield strength Fy = 300 [MPa] ultimate material yield strength Fu = 450 [MPa] weld metal strength Xu = 490 [MPa] performance factor φ = 0.9 performance factor for welds φw = 0.67 span L = 32.0 [m] factored moment Mf = 6200 [kNm] factored shear Vf = 2200 [kN] max. (space for) girder depth d_max = 3200 [mm] design for single/double symmetry symmetry = Single (Single/Dou ble) COMPUTATIONS GENERAL PARAMETERS Slenderness Limits min.web SL for red. moment SL_wmin = 1900/SQRT(Fy) = 110 $14.3.4 max. web slenderness SL_wmax = 83000/Fy = 277 $14.3.1 max. flange slenderness SL_fmax = 200/SQRT(Fy) = 12 $11.2 TabLe 2 Preliminary Sizing for Equal Flange Sizes height for max bending efficiency h_a = 540*(Mf/Fy)^(1/3) = 1482 [mm] maximum shear strength Fs = 0.66*Fy = 198.00 [MPa] $13.4.1.1(a) minimum web area Aw_min = Vf*1000/(φ*Fs) = 12346 [mm2] min web thickness w_a = Aw_min/h_rec = 8.3 [mm] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 121 recommended web slenderness SL_recw = h_rec/ w_a = 178 ratio of web slendernesses r_SLw = SL_recw/SL_wmax = 0.64 minimum flange area Af_min = Mf * 1000000 / ( Fy * h_rec) = 13946 [mm2] recommended web thickness w_rec = IF(r_SLw>1,w_a*r_SLw,w_a) = 8 [mm] recommended web depth h_rec = IF(h_a>0.9*d_max,d_max,h_a) = 1482 [mm] recommended flange thickness t_rec = SQRT(Af_min/(2*SL_fmax)) = 25 [mm] recommended flange width b_rec = Af_min/t_rec = 568 [mm] total girder depth d_rec = 2*t_rec+h_rec = 1531 [mm] INPUT GIRDER PARAMETERS Select Girder Parameters web thickness w = 12 [mm] web depth h = 1600 [mm] flange thickness t = 24 [mm] flange width b = 490 [mm] CHECKS CONCEPTUAL DESIGN FOR EQUAL FLANGE SIZES web slenderness check s_w_chec = IF(h/w>SL_wmax,"reduce slenderness of web","web slenderness OK") = web slenderness OK efficiency web slenderness wb_eff = (h/w)/SL_wmax = 0.48 flange slenderness check sL_fsel_che c = IF(b/(2*t)>SL_fmax,"reduce flange slenderness ","flange slenderness OK") = flange slenderness OK efficiency flange slenderness fl_eff = (b/(2*t))/SL_fmax = 0.88 girder depth check d_sel_chec = IF(d_sel>d_max,"reduce recommended girder depth","girder depth OK") = girder depth OK efficiency flange slenderness d_eff = d_sel/d_max = 0.52 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 122 INPUT TO CHECK SHEAR RESISTANCE transverse stiffeners exist? stiffeners = exist (exist/none) intermediate trans. stiff. spacing a = 3000 [mm] panel location p_loc = interior interior/end anchorage for end panel anch = yes yes/no $15.7.1 factored shear in end panel Vf_e = 2500 [kN] Fs = 0.66Fy Fs = 290 Fykv (h/w) Fs = 180000 kv + Ft (h/w)2 621439 83000 Fy kv Fy kv Web slenderness h/w Fy Fs MPa 502 Fy kv Fs = 290 Fykv + Ft (h/w) Ft Ft Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 123 CALCULATIONS OF SHEAR RESISTANCE web area Aw = h*w = 19200 [mm2] panel Ratio a_h = a/h = 1.88 factored shear stress Ff_i = (Vf/Aw)*1000 = 114.58 [MPa] factored shear stress at end panel Ff_e = (Vf_e/Aw)*1000 = 130.21 [MPa] Panel Ratio Check max panel spacing a_hmax_a = 67500/(s_w)^2 = 4 $14.5.2 Table 5 max panel spacing a_hmax_b = 3 = 3 $14.5.2 Table 5 panel ratio check a_hcheck = IF(stiffeners="exist",IF(s_w>150,IF(a_ h<=a_hmax_a,"OK","decrease stiffener spacing"),IF(a_h<=a_hmax_b,"OK","d ecrease stiffener spacing")),"NA") = OK $14.5.2 Table 5 Web Slenderness Check web maximum allowable slenderness s_wmax = 83000/Fy = 277 $14.3.1 web slenderness s_w = h/w = 133 web slenderness check s_wcheck = IF(s_w>s_wmax,"reduce slenderness","slenderness OK") = slenderness OK $14.3.1 Ultimate Shear Stress (Fs) (a) yielding in shear Fs_a = 0.66 * Fy = 198.00 [MPa] $13.4.1.1(a) (b) elasto-plastic action Fs_b = F_cri = 95.88 [MPa] $13.4.1.1(b) (c) tension field action Fs_c = F_cri+ka*(0.5*Fy -0.866*F_cri) = 127.39 [MPa] $13.4.1.1(c) (d) elastic buckling Fs_d = F_cre+ka*(0.5*Fy-0.866*F_cre) = 109.45 [MPa] $13.4.1.1(d) tension field contribution Ft = ka*(0.5*Fy-0.866*F_cre) = 43.86 [MPa] coefficient for no stiffeners kv_n = 5.34 = 5.34 coefficient case a/h<1 kv_s1 = 4+5.34/(a/h)^2 = 5.52 $13.4.1.1 coefficient case a/h>=1 kv_s2 = 5.34+4/(a/h)^2 = 6.48 $13.4.1.1 kv_s = IF(a/h<1,kv_s1,kv_s2) = 6.48 $13.4.1.1 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 124 kv = IF(stiffeners = "none",kv_n,kv_s) = 6.48 $13.4.1.1 critical shear stress inelastic F_cri = 290 * (((Fy * kv)^.5)/(h / w)) = 95.88 [MPa] aspect coefficient ka = 1/SQRT(1+(a/h)^2) = 0.47 critical shear stress elastic F_cre = 180000*kv/(s_w)^2 = 65.59 [MPa] coefficient factor cF = SQRT(kv/Fy) = 0.15 [MPa-0.5] slenderness case h/w case = IF(s_w<=439*cF,"c_a",IF(s_w<=502*c F,"c_b",IF(s_w<=621*cF,"c_c","c_d"))) = c_d Fs_stiff = IF(case="c_a",Fs_a,IF(case="c_b",Fs _b,IF(case="c_c",Fs_c,Fs_d))) = 109.45 [MPa] $13.4.1.1 (a-d) Fs_unstiff = IF(case="c_a",Fs_a,IF(case="c_b",Fs _b,IF(case="c_c",F_cri,F_cre))) = 65.59 [MPa] $13.4.1.1 (a-d) = = stiffener check st_check IF(Vf>(Fs_unstiff*Aw/1000),"transvers e stiffener required","transverse stiffener not required") transverse stiffener required F_s = IF(p_loc="interior",IF(stiffeners="exist", Fs_stiff,Fs_unstiff),IF(anch="yes",Fs_s tiff,Fs_unstiff)) = 109.45 [MPa] $13.4.1.1 (a-d) End Panel Calculation minimum shear buckling coefficient kv_min = (Ff_e*(s_w)^2)/(180000*0.9) = 14.29 minimum end panel spacing a_e = SQRT((4*h^2)/(kv_min-5.34)) = 1070 [mm] FINAL CHECKS SHEAR RESISTANCE shear resistance Vr = φ * Aw * F_s / 1000 = 1891 [kN] $13.4.1.1 efficiency shear resistance Vf_r = Vf_e/Vr = 1.32 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 125 check shear resistance Vcheck = IF(Vf_r<1,"OK","increase shear resistance") = increase shear resistance INPUT 3: GIRDER PARAMETERS - UNEQUAL FLANGE SIZES compression flange width b_com = 500 [mm] compression flange thickness t_com = 32 [mm] tension flange width b_ten = 500 [mm] tension flange thickness t_ten = 32 [mm] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 126 material Shear Modulus G = 77000 [MPa] material Young's Modulus Est = 200000 [MPa] unsupported Length Lu = 5000 [mm] girder loading: Uniform …. Load = load (load/mome nt) longitudinal stiffener at 0.2d long = none exist/none longitudinal stiffener area A_stiff = 2100 [mm 2] longitudinal stiffener weak axis I Ix_stiff = 1.51E+06 [mm 4] longitudinal stiffener strong axis I Iy_stiff = 1.51E+06 [mm 4] longitudinal strong neutral axis height x_stiffna = 27 [mm] long. stiff. strong axis plastic modulus Z_stiff = 0.00E+00 [mm3] point of application of transverse loading as a fraction of girder depth d g'' = 1 moment distribution factor ω2 = 1 $13.6(a) CALCULATIONS OF MOMENT RESISTANCE Section Element Slenderness and Class flange max. allowable slenderness s_fmax = = 60.0 $13.5 (c) flange (in compression) slenderness s_f = b_com/2/t_com = 7.8 flange slenderness check f_check = IF(s_f>s_fmax,"reduce flange width","flange width OK") = flange width OK flange class f_class = IF(s_f<=145/SQRT(Fy),"1",IF(s_f<=17 0/SQRT(Fy),"2",IF(s_f<=200/SQRT(Fy ),"3","4"))) = 1 $11.2 Table 2 web class w_class = IF(s_w<=1100/SQRT(Fy),"1",IF(s_w< =1700/SQRT(Fy),"2",IF(s_w<=1900/S QRT(Fy),"3","4"))) = 4 $11.2 Table 2 class_a = IF(f_class>w_class,f_class,w_class) = 4 section class class = IF(class_a="4",IF(w_class="4",IF(f_cla ss="4","4(i)","4(ii)"),"4(iii)"),class_a) = 4(ii) $13.5 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 127 section class check cl_check = IF(class="4(i)","consult CSA S136","OK") = OK $13.5 Sectional Properites without Longitudinal Stiffener compression flange area Af_com = b_com*t_com = 16000.00 [mm 2] tension flange area Af_ten = b_ten*t_ten = 16000.00 [mm 2] total area A_tot = Aw+Af_com+Af_ten = 51200.00 [mm 2] overall girder depth gd = h+t_ten+t_com = 1664 [mm] neutral axis height y_na = (Af_com*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_te n))/A_tot = 832 [mm] neutral axis height x_na = b_com/2 = 250 [mm] outermost fibre distance y_max = MAX(gd-y_na,y_na) = 832 [mm] strong axis second moment of area Ix = 1/12*(b_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_com*(gd- t_com/2-y_na)^2+Af_ten*(y_na- t_ten/2)^2+Aw*(h/2+t_ten-y_na)^2 = 2.541.E+10 [mm4] effective elastic compression flange half width be_a = 200*t_com/SQRT(Fy) = 370 [mm] $13.5(c)iii compression flange half width bo = b_com/2 = 250 [mm] be_com = MIN(bo,be_a) = 250 [mm] $13.5(c)iii effective compression flange area Af_come = 2*be_com*t_com = 16000 [mm 2] total area A_tote = Aw+Af_come+Af_ten = 51200.00 [mm 2] effective neutral axis height y_nae = (Af_come*(gd- t_com/2)+Af_ten*t_ten/2+Aw*(h/2+t_te n))/A_tote = 832 [mm] effective outermost fibre distance y_maxe = MAX((gd-y_nae),y_nae) = 832 [mm] effective strong axis second moment of area Ixe = 1/12*(2*be_com*t_com^3 + w*h^3+b_ten*t_ten^3)+Af_come*(gd- t_com/2-y_nae)^2+Af_ten*(y_nae- t_ten/2)^2+Aw*(h/2+t_ten-y_nae)^2 = 2.541.E+10 [mm4] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 128 elastic section modulus S = Ix/y_max = 3.05362E+0 7 [mm3] effective elastic section modulus Se = Ixe/y_maxe = 3.05362E+0 7 [mm3] plastic section modulus Z = 1/4*(b_com*t_com^2+w*h^2+b_ten*t_t en^2)+Af_com*((gd-t_com/2- y_na)^2)^0.5+Af_ten*((y_na- t_ten/2)^2)^0.5+Aw*((h/2+t_ten- y_na)^2)^0.5 = 3.40480E+0 7 [mm3] st. venant torsion constant J = 1/3*(b_com*t_com^3+b_ten*t_ten^3+h *w^3) = 1.18E+07 [mm4] weak axis second moment of area Iy = 1/12*(t_com*b_com^3+h*w^3+t_ten*b _ten^3) = 6.669E+08 [mm4] compression flange Iy Iyc = 1/12*(t_com*b_com^3) = 3.333E+08 [mm 4] Modified Sectional Properites with Longitudinal Stiffener modified neutral axis height y_na' = (A_tot*y_na+A_stiff*0.2*gd)/(A_tot+A_ stiff) = 812.3 modified strong axis second moment of area Ix' = Ix + A_tot*(y_na- y_na')^2+A_stiff*(0.2*gd- y_na')^2+Ix_stiff = 2.591E+10 modified outermost fibre distance y_max' = MAX(y_na',gd-y_na' ) = 851.67 modifed effective neutral axis height y_nae' = (A_tote*y_nae+A_stiff*0.2*gd)/(A_tote +A_stiff) = 812.33 modified effective strong axis second moment of area Ixe' = Ixe + A_tote*(y_nae- y_na')^2+A_stiff*(0.2*gd- y_na')^2+Ix_stiff = 2.591E+10 modified effective neutral axis height y_maxe' = MAX(y_nae',gd-y_nae' ) = 8.517E+02 modified weak axis neutral axis height x_na' = (A_tot*x_na+A_stiff*(x_stiffna+x_na+w /2))/(A_tot+A_stiff) = 251.30 modified weak axis second moment of area Iy' = Iy+A_tot*(x_na- x_na')^2+A_stiff*(x_stiffna+x_na+w/2- x_na')^2 = 6.691E+08 elastic section modulus S' = Ix'/y_max' = 3.04E+07 [mm3] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 129 effective elastic section modulus Se' = Ixe'/y_maxe' = 3.04E+07 [mm3] plastic section modulus Z' = Z + Z_stiff+A_stiff*((0.2*gd- y_na')^2)^0.5+A_tot*((y_na- y_na')^2)^0.5 = 3.606E+07 [mm3] Moment Resistances: plastic moment Mp = Z*Fy/1000000 = 10214 [kNm] class 1,2 moment resistance Ma_unstiff = φ*Z*Fy/1000000 = 9193 [kNm] $13.5(a) Ma_stiff = φ*Z'*Fy/1000000 = 9737 [kNm] $13.5(a) class 3 moment resistance Mb_unstiff = φ*S*Fy/1000000 = 8245 [kNm] $13.5(b) Mb_stiff = φ*S' *Fy/1000000 = 8214 [kNm] class 4 (ii) moment resistance Mc_unstiff = Mb_unstiff*(1- (0.0005*Aw/Af_com)*((s_w)- (1900/SQRT(Mf*1000000/(φ*S))))) = 8.21097E+0 3 [kNm] $14.3.4 Mc_stiff = Mb_stiff = 8214 [kNm] class 4 (iii) moment resistance Md_unstiff = φ*Se*Fy/1000000 = 8245 [kNm] $13.5(c) Md_stiff = φ*Se'*Fy/1000000 = 8214 [kNm] $13.5(c) Ma = IF(long="none",Ma_unstiff,Ma_stiff) = 9193 [kNm] Mb = IF(long="none",Mb_unstiff,Mb_stiff) = 8245 [kNm] Mc = IF(long="none",Mc_unstiff,Mc_stiff) = 8211 [kNm] Md = IF(long="none",Md_unstiff,Md_stiff) = 8245 [kNm] Lateral Torsional Buckling: %10.10.2.3 d' = h+(t_com+t_ten)/2 = 1632 %10.10.2.3 distance from centroid to tension flange midline y_2 = y_na-t_ten/2 = 816 %10.10.2.3 e = (d'*(b_com^3*t_com)/(b_com^3*t_com +b_ten^3*t_ten))-y_2 = 0.0 %10.10.2.3 coefficient of monosymmetry bx = IF(symmetry="Single",0.9*d'*(2*Iyc/Iy- 1)*(1-(Iy/Ix)^2),"0") = -0.5071 %10.10.2.3 warping constant Cw = d'^2/12*(b_com^3*t_com*b_ten^3*t_te n)/(b_com^3*t_com+b_ten^3*t_ten) = 4.44.E+14 [mm6] %10.10.2.3 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 130 B1 = ((PI()*bx)/(2*Lu))*SQRT((Est*Iy)/(G*J)) = 0.00 %10.10.2.3 B2 = (PI()*PI()*Est*Cw)/((Lu)^2*G*J) = 38.43 %10.10.2.3 elastic lateral torsional buckling resistance Mu = IF(Lu=0, "NA!",( ω2*PI())/(Lu)*(SQRT(Est*Iy*G*J)*(B1+ SQRT(1+B2+B1^2)))/1000000) = 43502 [kNm] %10.10.2.3 Actual Moment Resistance lateral buckling ratio Mratio = IF(Lu=0,"NA!",Mu/Mp) = 4.26 $13.6 Mr_a = IF(class="1",Ma,IF(class="2",Ma,IF(cla ss="3",Mb,IF(class="4(i)","Consult CSA S136!",IF(class="4(ii)",Mc,Md))))) = 8211 [kNm] $13.6 Mr_b' = IF(Lu=0,"NA!",1.15*Mr_a*(1- 0.28/φ*Mr_a/Mu)) = 8888 [kNm] $13.6 Mr_b = MIN(Mr_a,Mr_b') = 8211 [kNm] $13.6 Mr_c = IF(L=0,"NA!",φ*Mu) = 39152 [kNm] FINAL CHECKS MOMENT RESISTANCE moment resistance Mr = IF(class="4(i)","Consult CSA S136!",IF(L=0,Mr_a,IF(Mratio>0.67,Mr _b,Mr_c))) = 8211 [kNm] $13.6,$13.5 efficiency moment resistance Mf_r = Mf/Mr = 0.76 check moment resistance Mcheck = IF(Mf_r<1,"OK","increase moment resistance") = OK FINAL CHECKS COMBINED SHEAR AND MOMENT RESISTANCE INPUT COMBINED SHEAR AND BENDING CHECK Factored Moment at 0.6 Vr Mf_c = = 4800.00 [kNm] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 131 CALCULATIONS Factored shear at 0.6Vr Vf_c = 0.6*Vr = 1134.75 [kNm] Moment ratio Mf_rc = Mf_c/Mr = 0.58 Shear ratio Vf_rc = Vf_c/Vr = 0.60 check if interaction critical IF(Vf>=0.6*Vr, "interaction is critical",""interaction not critical") = shear and moment is critical $14.6. interaction efficiency Inter = 0.727*Mf_rc+0.455*Vf_rc = 0.70 check resistance against shear and moment C_check = IF(stiffeners="exist",IF(Inter>=1,"incre ase moment or shear resistance","combined shear and moment capacity OK"),"NA") = combined shear and moment capacity OK RECOMMENDATIONS FOR TRANSVERSE STIFFENERS Required Stiffener Properties kv_a = 4+ 5.34/(a/h)^2 = 5.52 $13.4.1.1 kv_b = 5.34 + 4/(a/h)^2 = 6.48 $13.4.1.1 shear buckling coefficient kvs = IF(a/h<1,kv_a,kv_b) = 6.48 $13.4.1.1 C_a = (1-(310000*kvs)/(Fy*(h/w)^2)) = 0.62 $13.4.1.1 C = IF(C_a<0.1,0.1,C_a) = 0.62 $13.4.1.1 Y = Fy/Fy_stiff = 0.86 $13.4.1.1 stiffener factor D = IF(stiff_furn="pair",1,IF(stiff_type="angl e",1.8,2.4)) = 1.00 $13.4.1.1 total required area of stiffener(s) Asr = a*w/2*(1- (a/h)/SQRT(1+(a/h)^2))*C*Y*D = 1132 [mm2] $14.5.3 required moment of inertia I_req'd = (h / 50)^4 = 1.05.E+06 [mm4] $14.5.3 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 132 INPUT TRANSVERSE STIFFENERS stiffener type stiff_type = plate angle/plate stiffener furnishing stiff_furn = pair pair/single stiffener yield strength Fy_stiff = 350 [MPa] stiffener thickness primary leg ts_a = 6 [mm] stiffener width primary leg bs_a = 200 [mm] stiffener thickness secondary leg ts_b = 12 [mm] stiffener width secondary leg bs_b = 100 [mm] stiffener effective length factor K = 0.75 >=0.75 $10.2.1 compute section properties or input? comp = compute input/comp ute stiffener area As_input = 995 [mm2] stiffener moment of inertia I_input = 8.50E+05 [mm4] CALCULATIONS TRANSVERSE STIFFENERS Stiffener Properties area of stiffener angle As_a = ts_a*bs_a+ts_b*bs_b = 2400 [mm2] area of stiffener plate As_b = ts_a*bs_a = 1200 [mm2] computed stiffener area As_comp = IF(stiff_furn="pair",IF(stiff_type="angle ",2*As_a,2*As_b),IF(stiff_type="angle" ,As_a,As_b)) = 2400 [mm2] inertia of stiffener angle plus web y_a = (ts_a*bs_a*(bs_a/2+ts_b+w)+bs_b*(ts _b+w)^2/2)/(bs_a*ts_a+bs_b*(ts_b+w) ) = 49 [mm] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 133 I_a = (ts_a*bs_a^3)/12+bs_a*ts_a*(bs_a/2+t s_b+w- y_a)^2+(bs_b*(ts_b+w)^3)/12+bs_b*(t s_b+w)*((ts_b+w)/2-y_a) = 1.07E+07 [mm4] I_atwo = 1/12*(ts_a*(bs_a*2+w+2*ts_b)^3+(bs_ b-ts_a)*(2*ts_b+w)^3) = 4.18E+07 [mm4] inertia of stiffener plate I_b = 1/12*(ts_a*(w+bs_a)^3) = 4.76E+06 [mm4] I_btwo = 1/12*(ts_a*(w+2*bs_a)^3) = 3.50E+07 [mm4] computed stiffener Inertia I_comp = IF(stiff_furn="pair",IF(stiff_type="angle ",I_atwo,I_btwo),IF(stiff_type="angle",I _a,I_b)) = 3.50E+07 [mm4] stiffener area A_stift = IF(comp="compute",As_comp,As_inpu t) = 2400 [mm2] stiffener inertia I_stiff = IF(comp="compute",I_comp,I_input) = 3.50E+07 [mm4] CHECKS TRANSVERSE STIFFENERS Check Against Requirements: area ratio A_ratio = A_stift/Asr = 2.12 [mm2] inertia ratio I_ratio = I_stiff/I_req'd = 33.35 check meeting requirements check = IF(MIN(A_ratio,I_ratio)<=1.0,"increase size of stiffener","OK") = OK Stiffener Slenderness Check stiffener readius of gyration r = SQRT(I_stiff/A_stift) = 121 stiffener slenderness ratio sr_stiff = K*h/r = 10 stiffener slenderness ratio check sr_stiff_che ck = IF(sr_stiff>200,"stiffener to slender, increase size","stiffener slenderness OK") = stiffener slenderness OK $10.4.2.1 stiffener bs/ts b_t_a = bs_a/ts_a = 33 b_t_b = bs_b/ts_b = 8 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 134 b_t_max = 200/SQRT(Fy_stiff) = 11 efficiency stiffener slenderness stiff_st = MIN(b_t_a,b_t_b)/b_t_max = 0.78 check stiffener slenderness check2 = IF(stiff_st<1,"OK","choose stockier stiff.") = OK $11.2 Table 1 INPUT BEARING STIFFENERS factored load Cf = 1600 [kN] performance factor φbi = 0.8 weld performance factor φω = 0.7 performance factor φbe = 0.75 length of the bearing plate N = 300 [mm] flange to web weld depth d_weld = 6 [mm] bearing stiffeners under load bea_stiff_i = exist exist/none end bearing stiffeners bea_stiff_e = exist exist/none end stiffener width bs_e = 125 [mm] end stiffener thickness ts_e = 16 [mm] intermediate stiffener width bs_i = 125 intermediate stiffener thickness ts_i = 12 stiffener contact length cpl = 100 [mm] parameter n = 1.34 fillet weld size s = 6 [mm] CALCULATIONS AND CHECKS BEARING STIFFENERS stiffener requirement at unframed ends check stiff_check_ en = IF(s_w>1100/SQRT(Fy),"bearing stiff.required at unframed ends","no = bearing stiff.required at unframed ends $14.4.1 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 135 bearing stiff. required at unframed ends") flange thickness plus weld k = t + d_weld = 30 stiffener area Abs_e = bs_e*ts_e = 2000 stiffener moment of inertia I_e = 1/12*(ts_e*bs_e^3) = 2604167 [mm4] Stiffener Slenderness Check stiffener readius of gyration r_e = SQRT(I_e/Abs_e) = 36 stiffener slenderness ratio sr_stiff_e = K*h/r_e = 33 stiffener slenderness ratio check chksl_e = IF(bea_stiff_e="exist",IF(sr_stiff_e>20 0,"stiffener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK $10.4.2.1 stiffener dimension check chkdim_e = IF(bea_stiff_e="exist",IF(bs_e/ts_e<20 0/SQRT(Fy),"OK","increase stiffener thickness"),"NA") = OK Unstiffened End Bearing Resistance web crippling Bre_a = φbe*w*(N+4*t)*Fy/1000 = 1069 [kN] $14.3.2b(i) web yielding Bre_b = 0.6*φbe*w^2*SQRT(Fy/1000*Est/1000) = 502 [kN] $14.3.2b(ii) bearing resistance Br_end = MIN( Bre_a, Bre_b) = 502 [kN] Stiffened Exterior Compression Resistance resisting area Ae = 12*w^2+2*Abs_e = 5728 [mm2] $14.4.2 moment of inertia 1 Ie = 1/12*(ts_e*(2*bs_e+w)^3) = 23979637 [mm4] radius of gyration re = SQRT(Ie/Ae) = 65 [mm] lambda λe = K*h/r_e*SQRT(Fy/(PI()^2*Est)) = 0.41 $13.3.1 axial compression resistance Cre = Ae*φ*Fy*(1+λe^(2*n))^(-1/n)/1000 = 1449 [kN] $13.3.1 efficiency bearing stiffener br_sfe = Cf/Cre = 1.10 Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 136 stiffener bearing resistance Bstiff_e = φ*1.5*Fy*(cpl*ts_e*2)/1000 = 1296 [kN] $13.10(a) efficiency stiffener bearing st_sfe = Cf/Bstiff_e = 1.23 min. stiffener efficiency sf_ext = MIN(br_sfe,st_sfe) = 1.10 capacity check chk_cap_ex t = IF(bea_stiff_e="exist",IF(sf_ext>1,"incr ease stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = increase stiffener thickness or increase bearing seat length Stiffener Under Concerated Loads Unstiffened Interior Bearing Resistance web crippling Bri_a = φbi*w*(N+10*t)*Fy/1000 = 1555 [kN] $14.3.2a(i) web yielding Bri_b = 1.45*φbi*w^2*SQRT(Fy/1000*Est/1000 ) = 1294 [kN] $14.3.2a(ii) bearing resistance Br_int = MIN( Bri_a, Bri_b) = 1294 [kN] stiffener requirement under concentrated load stiff_check_ in = IF(Cf>Br_int,"intermediate bearing stiff. required","intermediate bearing stiff. not required") = intermediate bearing stiff. required stiffener area Abs_i = bs_i*ts_i = 1500 stiffener moment of Inertia I_i = 1/12*(ts_e*bs_e^3) = 2604167 Stiffener Slenderness Check stiffener readius of gyration r_i = SQRT(I_i/Abs_i) = 42 stiffener slenderness ratio sr_stiff_i = K*h/r_i = 29 stiffener slenderness ratio check chksl_i = IF(bea_stiff_i="exist",IF(sr_stiff_i>200, "stiffener too slender, increase size","stiffener slenderness OK"),"NA") = stiffener slenderness OK Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 137 stiffener dimension check chkdim_i = IF(bea_stiff_i="exist",IF(bs_i/ts_i<200/ SQRT(Fy),"OK","Increase Stiffener's thickness"),"NA") = OK Stiffened Interior Compression Resistance resisting area Ai = 25*w^2+2*Abs_i = 6600 [mm2] $14.4.2 moment of inertia 1 Ii = 1/12*(ts_i*(2*bs_i+w)^3) = 17984728 [mm4] radius of gyration ri = SQRT(Ii/Ai) = 52 [mm] lambda λ = K*h/ri*SQRT(Fy/(PI()^2*Est)) = 0.28 $13.3.1 axial compression resistance Cri = Ai*φ*Fy*(1+λ^(2*n))^(-1/n)/1000 = 1738 [kN] $13.3.1 efficiency bearing br_sfi = Cf/Cri = 0.92 stiffener bearing resistance Bstiff_i = 1.5*φ*Fy*(cpl*ts_i*2)/1000 = 972 [kN] $13.10(a) efficiency stiffener st_sfi = Cf/Bstiff_i = 1.65 min. stiffener efficiency sf_int = MIN(br_sfi,st_sfi) = 0.92 capacity check chk_cap_int = IF(bea_stiff_i="exist",IF(sf_int>1,"incre ase stiffener thickness or increase bearing seat length","stiffener OK"),"NA") = stiffener OK Welding of Bearing Stiffener strength of base metal Vr_b = (0.67∗φω*s*Fu)/1000 = 1.21 [kN/mm] $13.13.2.2(a) strength of weld Vr_w = (0.67∗φω*s*0.7071*Xu)/1000 = 0.93 [kN/mm] $13.13.2.2(b) governing strength Vr_gov = Min(Vr_b,Vr_w) = 0.93 [kN/mm] total weld length L_w = Cf/Vr_gov = 1715 [mm] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 138 INPUT WELD DESIGN Select Weld Between Flanges and Web (two fillet welds each) weld size w_size = 8 [mm] intermittend weld length w_length = 200 [mm] spacing on centre w_spacing = 400 [mm] COMPUTATIONS WELD DESIGN girder depth d_sel = h+2*t_com = 1664 [mm] moment of inertia Ig = ((b*d_sel^3)-(b-w)*h^3)/12 = 2.50E+10 [mm4] Qs = t_com*b_com*h/2 = 1.28E+07 [mm3] shear flow per mm length q = (Vf*Qs)/Ig*1000 = 1127 [N/mm] weld resistance (two weld Lines) vr_base = 2*0.67*φw*w_size*Fu = 3232 [N/mm] w_throat = 2*0.707*w_size = 11 [mm] vr_throat = 2*0.67*φw*w_throat*Xu = 4976 [N/mm] vr_min = MIN(vr_base,vr_throat) = 3232 [N/mm] shear resistance per mm length v_r = vr_min*(w_length/w_spacing) = 1616 [N/mm] weld check w_check = IF(v_r>=q,"weld flange to web OK","increase weld amount") = weld flange to web OK efficiency weld we_eff = q/v_r = 0.70 INPUT Weight Computation Steel Density S_den = 0.00000785 00 [Kg/mm^3] COMPUTATIONS OF TOTAL WEIGHT PER GIRDER mass of compression flange M_com_Fl = b_com*t_com*(L*1000)*S_den = 4019.20 [kg] mass of tension Flange M_ten_Fl = b_ten*t_ten*(L*1000)*S_den = 4019 [kg] Plate Girder Spreadsheet (CSA-S16-01 and CSA-S6-06) 139 mass of web M_Web = if(stiffeners="exist",(L*1000 / a - 1) * bs_a * ts_a * h * S_den * if(stiff_furn="pair",2,1),0) = 291.39 [kg] mass of transverse stiffeners M_t_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 175.84 [kg] mass of bearing stiffener M_b_stiff = (if(bea_stiff_e="exist",(bs_e * ts_e * h)*2,0) + if(bea_stiff_i="exist",(bs_i * ts_i * h)*2,0))*S_den*2 = 175.84 [kg] total mass T_Mass = M_com_Fl+M_ten_Fl+M_Web+M_t_st iff+M_b_stiff = 8681.47 [kg] total weight per girder T_Weight = T_Mass*9.8/1000 = 85.1 [kN] Handbook of Steel Construction CAN/CSA-S16-01 - 9th Edition Canadian Highway Bridge Design Code CAN/C S6-00 140 Appendix D: Box Girder Spreadsheet (CSA-S16-01) Box Girder Spreadsheet (CSA-S16-01) 141 DESIGN OF BOX GIRDERS DESCRIPTION LTF LBF LW ITFS,ATFS IBFS,ABFS IWS,AWS tBF,IBF,ABF tTF,ITF,ATF yBFS yWS1 yWS2 yTFS hws XTFS1 XTFS2 XBFS2 XBFS1 Box Girder Spreadsheet (CSA-S16-01) 142 INPUT Factored Loads and Moments Factored Moment Mf = 3500.00[kNm] Factored Shear Vf = 1750.00[kN] Material Properties Material Yield Strength Fy = 350.00[MPa] Material Young's Modulus E = 200000[MPa] Material poisson's ratio v = 0.30 Performance Factor φ = 0.90 Girder Dimensions web thickness t_w = 20.00[mm] web length L_w = 900.00[mm] Top flange width L_tf = 500.00[mm] Top flange thickness t_tf = 35.00[mm] Bottom flange width L_bf = 300.00[mm] Bottom flange thickness t_bf = 25.00[mm] Web Longitudinal stiffener moment of Inertia I_ws = 131000[mm^4] Web longitudinal stiffener area A_ws = 524.00[mm^2] web longitudinal stiffener 1 height y_ws1 = 250.00[mm] web longitudinal stiffener 2 height y_ws2 = 875.00[mm] web longitudinal stiffener offset h_ws = 31.90[mm] Top flange Longitudinal stiffener moment of Inertia I_tfs = 131000[mm^4] Top flange Longitudinal stiffener area A_tfs = 524.00[mm^2] Top flange longitudinal stiffener height y_tfs = 913.60[mm] Top flange stiffener 1 offset x_tfs1 = 166.67[mm] Top flange stiffener 2 offset x_tfs2 = 333.33[mm] Bottom flange Longitudinal stiffener moment I_bfs = 131000[mm^4] Box Girder Spreadsheet (CSA-S16-01) 143 of Inertia Bottom flange Longitudinal stiffener area A_bfs = 524.00[mm^2] Bottom flange longitudinal stiffener height y_bfs = 41.90[mm] Bottom flange stiffener 1 offset x_bfs1 = 100.00[mm] Bottom flange stiffener 2 offset x_bfs2 = 200.00[mm] Stiffeners Transverse stiffener/internal diaphragm spacing a = 2000.00[mm] Transverse Stiffeners stf = exist(exist/non e) Longitudinal Web Stiffeners stf_lw = exist (exist/non e) Longitudinal Flange Stiffeners stf_lf = exist (exist/non e) Number of Longitudinal stiffeners n = 2.00 SECTIONAL PROPERTIES CALCULATED FOR THE BOX SECTION References Section Properties without Longitudinal Stiffeners Girder Depth D = t_bf+L_w*COS(w_incl)+t_tf = 954.43[mm] web inclination w_incl = ASIN(((L_tf-L_bf)/2)/L_w) = 0.111[rad] Top flange area A_tf = L_tf*t_tf = 17500[mm^2] Bottom flange area A_bf = L_bf*t_bf = 7500.0[mm^2] Web area A_w1 = L_w*t_w = 18000.0[mm^2] Total area A_unstf = A_tf+A_bf+2*A_w1 = 61000.0[mm^2] Top flange centroid height y_tf = t_bf+L_w*COS(w_incl)+t_tf/2 = 936.92719 1[mm] Bottom flange centroid height y_bf = t_bf/2 = 12.50[mm] Web centroid height y_w = L_w*COS(w_incl)/2+t_bf = 472.21[mm] Box Girder Spreadsheet (CSA-S16-01) 144 Top flange Ix Ix_tf = L_tf*(t_tf^3)/12 = 1.79E+06[mm^4] Bottom flange Ix Ix_bf = L_bf*(t_bf^3)/12 = 3.91E+05[mm^4] Web Ix Ix_w = t_w*(COS(w_incl))^2*(L_w^3)/12 = 1.20E+09[mm^4] Total section neutral axis height y_unstf = (A_tf*y_tf+A_bf*y_bf+2*A_w1*y_w)/(A _unstf) = 549.01[mm] Total section Ix Ix_unst = Ix_tf+Ix_bf+2*Ix_w+A_tf*(y_tf- y_tot)^2+A_bf*(y_bf- y_tot)^2+2*A_w1*(y_w-y_tot)^2 = 7.41E+09[mm^4] Section Modulus in Compression S_c = Ix_unst/(D-y_unstf) = 1.8.E+07[mm^3] Section Modulus in Tension S_t = Ix_unst/y_unstf = 1.35E+07[mm^3] Section Properties with Longitudinal stiffeners smeared into plate elements Total area A_tot = A_tf+A_bf+2*A_w1+2*(A_bfs+A_tfs+2* A_ws) = 65192.0[mm^2] Combined stiffened top flange plate neutral axis height y_tfc = (2*A_tfs*y_tfs+A_tf*y_tf)/(A_tf+2*A_tfs ) = 935.61[mm] Combined stiffened top flange plate Ix Ix_tfc = 2*I_tfs+L_tf*t_tf^3/12+2*A_tfs*(y_tfs- y_tfc)^2+A_tf*(y_tf-y_tfc)^2 = 2.59E+06[mm^4] Effective top flange flat plate thickness t_tf_sm = (12*Ix_tfc/L_tf)^(1/3) = 39.60[mm] Combined stiffened bottom flange plate neutral axis height y_bfc = (2*A_bfs*y_bfs+A_bf*y_bf)/(A_bf+2*A _bfs) = 16[mm] Combined stiffened bottom flange plate Ix Ix_bfc = 2*I_bfs+L_bf*t_bf^3/12+2*A_bfs*(y_bf s-y_tfc)^2+A_tf*(y_bf-y_tfc)^2 = 1.57E+10[mm^4] Effective bottom flange flat plate thickness t_bf_sm = (12*Ix_bfc/L_bf)^(1/3) = 857.26[mm] Combined stiffened web plate neutral axis height y_tot_w = (2*A_ws*h_ws+A_w1*t_w/2)/(A_w1+2 *A_ws) = 11.20[mm] Combined stiffened web plate Ix Ix_wc = 2*I_ws+L_w*t_w^3/12+2*A_ws*(h_ws- y_tot_w)^2+A_w1*(t_w/2-y_tot_w)^2 = 1.34E+06[mm^4] Effective web plate thickness t_w_sm = (12*Ix_wc/L_w)^(1/3) = 26[mm] Total section neutral axis height y_tot = (A_tf*y_tf+A_bf*y_bf+2*A_w1*y_w+2*( A_tfs*y_tfs+A_bfs*y_bfs+A_ws*y_ws1 +A_ws*y_ws2))/(A_tot) = 547.15[mm] Box Girder Spreadsheet (CSA-S16-01) 145 Total section Ix Ix_tot = Ix_tf+Ix_bf+2*Ix_w+A_tf*(y_tf- y_tot)^2+A_bf*(y_bf- y_tot)^2+2*A_w1*(y_w- y_tot)^2+2*I_bfs+2*I_tfs+I_ws*4+2*A_t fs*(y_tfs-y_tot)^2+2*A_bfs*(y_bfs- y_tot)^2+2*A_ws*(y_ws1- y_tot)^2+2*A_ws*(y_ws2-y_tot)^2 = 8.02E+09[mm^4] Section Modulus in Compression Sc_stf = Ix_tot/(D-y_tot) = 2.0.E+07[mm^3] Section Modulus in Tension St_stf = Ix_tot/y_tot = 1.47E+07[mm^3] First web panel height h_wp1 = (y_ws1-t_bf)/(COS(w_incl)) = 2.26E+02 Second web panel height h_wp2 = (y_ws2-y_ws1)/(COS(w_incl)) = 6.29E+02 Third web panel height h_wp3 = L_w-(h_wp1+h_wp2) = 4.47E+01 SECTION STRENGTH CHECK FOR FLEXURAL RESISTANCE (SSRC) Excluding Longitudinal Stiffeners COMPRESSION FLANGE CLASS Max Slenderness for Box Flanges slf_max = 670/(sqrt(Fy)) = 35.81 $11.2 Table 2 Compression flange maximum slenderness ratio sl_tfus = L_tf/(2*t_tf) = 7.14 Flange Slenderness check slf_chu = if(sl_tfus<slf_max,"OK","NOT OK") = OK COMPRESSION FLANGE BUCKLIING Top flange longitudinal stress σ_tf_us = Mf*(y_tf-y_unstf)/Ix_unst*1000000 = 183.30[Mpa] Critical Buckling Stress σcr_tfu = kb_tf*PI()^2*E/(12*(1-v^2)*(L_tf/t_tf)^2) = 3542.93[Mpa] Galambos 4.2.1 Safety Check = if(σ_tf_us<σcr_tfu, "OK", "NOT OK") = OK Box Girder Spreadsheet (CSA-S16-01) 146 FLEXURAL STRENGTH Flexural Moment Resistance Mr_uns = min(0.9*(σcr_tfu*S_c),0.9*(Fy*S_t))/10 ^6 = 4249.79[kN.m] Galambos 7.4 Flexural Strength check if(Mr_uns<Mf, "NOT OK", "OK") = OK WEB CRIPPLING Limiting web depth to web thickness Ratio to control Flexural Buckling of the web with Longitudinal stiffener R1_uns = 6.04*sqrt(E/Fy) = 144.38 Galambos 6.2 maximum web panel slenderness slw_uns = L_w/t_w = 45.00 IF(R1_uns>slw_uns, "OK","Not OK") = OK Including Longitudinal Stiffeners COMPRESSION FLANGE CLASS Compression flange maximum slenderness ratio sl_tf = b_tfm/(2*t_tf) = 2.38 Flange Slenderness check slf_chk = if(sl_tf<slf_max,"OK","NOT OK") = OK COMPRESSION FLANGE BUCKLIING Top flange longitudinal stress σ_tf = Mf*(y_tfc-y_tot)/Ix_tot*1000000 = 169.50[Mpa] Buckling Co-efficient for top flange kb_tf = 4 Galambos Figure 4.8 Top flange subpanel plate width 1 b_tf_a = x_tfs1 = 166.7[mm] Top flange subpanel plate width 2 b_tf_b = x_tfs2-x_tfs1 = 166.7[mm] Top flange subpanel plate width 3 b_tf_c = L_tf-x_tfs2 = 166.7[mm] Top flange maximum plate width b_tfm = MAX(b_tf_a,b_tf_b,b_tf_c) = 166.7[mm] Critical Buckling Stress σcr_tf = kb_tf*PI()^2*E/(12*(1- v^2)*(b_tfm/t_tf)^2) = 31886.4[MPa] Galambos 4.2.1 Box Girder Spreadsheet (CSA-S16-01) 147 Safety Check = if(σ_tf<σcr_tf, "OK", "NOT OK") = OK FLEXURAL STRENGTH Flexural Moment Resistance Mr = min(0.9*(σcr_tf*Sc_stf),0.9*(Fy*St_stf)) /10^6 = 4618.0kNm Galambos 7.4 Flexural Strength check if(Mr<Mf, "NOT OK", "OK") = OK Overall Flexural Resistance Flexural resistance Mr_com = IF(stf_lf="exist",Mr,Mr_uns) = 4617.98[kN] WEB CRIPPLING Limiting web depth to web thickness Ratio to control Flexural Buckling of the web with Longitudinal stiffener R1 = 6.04*sqrt(E/Fy) = 144.4 Galambos 6.2 Maximum web panel height h_max = MAX(h_wp1,h_wp2,h_wp3) = 628.89 Maximum web panel slenderness Sl_maxp = h_max/t_w = 31.44 IF(R1>Sl_maxp, "OK","Not OK") = OK SHEAR STRENGTH SSRC (without longitudinal stiffeners) Tension field Stress ft = Fy*(1-Fs_eb/Fs_sy) = -1232.00[MPa] Galambos 6.3 Shear Strength contribution due to diagonal tension Vt = (L_w*t_w*ft/(2*sqrt(1+(L_w/t_w)^2)+(L _w/t_w)))/1000 = -164.24[kN] Galambos 7.5.1 Tension field check Vt_c = IF(Vt<0, 0, Vt) = 0 Shear Strength contribution due to web before buckling Vb = (L_w*t_w*Fs)/1000 = 4158.000[kN] Galambos 7.5.1 Final Shear Resistance Vu = 0.9*(Vb+Vt_c) = 3742.2[kN] Galambos 7.5.1 if(Vu>Vf_d, "OK", "NOT OK") = OK Box Girder Spreadsheet (CSA-S16-01) 148 SHEAR STRENGTH CSA Excluding Longitudinal Stiffeners Factored Design Shear for each Web Plate Vf_d = Vf/(2*COS(w_incl)) = 880.45 Web Area Aw = L_w*t_w = 18000.00[mm^2] Web Slenderness s_w = L_w/t_w = 45.00 Panel Ratio a_h = a/L_w = 2.22 Factored shear stress Ff_d = (Vf_d/Aw)*1000 = 48.91[Mpa] Panel ratio check Max panel ratio one ahmax_a = 67500/(s_w)^2 = 33.33 $14.5.2 Table 5 Max panel ratio two ahmax_b = 3 = 3 " Panel ratio check a_check = IF(stf="exist",IF(s_w>150,IF(a_h<=ah max_a,"OK!","Decrease stiffener spacing"),IF(a_h<=ahmax_b,"OK!","De crease stiffener spacing")),"NA") = OK! " Web slenderness check Maximum allowable slenderness s_wmax = 83000/Fy = 237.14 $14.3.1 web slenderness check s_checx = IF(slw_uns>s_wmax,"Reduce Slenderness!","Slenderness OK!") = Slenderne ss OK! $14.3.1 Calculation of Shear Stress Resistance: Shear Buckling Coefficient kv_nx = 5.34 = 5.34 $13.4.1.1 kv_s1x = 4+5.34/(a/L_w)^2 = 5.08 " kv_s2x = 5.34+4/(a/L_w)^2 = 6.15 " kv_sx = IF(a/L_w<1,kv_s1x,kv_s2x) = 6.15 " kvx = IF(stf = "none",kv_nx,kv_sx) = 6.15 " Ultimate Shear Stress (Fs) (a) Shear Yielding Fs_syx = 0.66 * Fy = 231[MPa] $13.4.1.1(a) (b) Inelastic Buckling Fs_ibx = 290 * (((Fy * kvx)^.5)/(L_w/ t_w)) = 299.0[MPa] $13.4.1.1(b) Box Girder Spreadsheet (CSA-S16-01) 149 tension field contribution Fs_ibtx = (0.5 * Fy - 0.866*Fs_ibx)*sqrt(1/(1+(a/L_w)^2)) = -34.4 [MPa] $13.4.1.1(c) (d) Elastic Buckling Fs_ebx = 180000*kvx/(slw_uns)^2 = 546.7[MPa] $13.4.1.1(d) tension field contribution Fs_ebtx = (0.5*Fy- 0.866*Fs_ebx)*sqrt(1/(1+(a_h)^2)) = -122.5 [MPa] " K Factor KFx = (kvx/Fy)^0.5 = 0.13 (MPa^- 0.5] slenderness case h/w casex = IF(slw_uns<=439*KF,"i",IF(slw_uns<= 502*KF,"ii",IF(slw_uns<=621*KF,"iii","i v"))) = i [MPa] Fs_stx = IF(casex="i",Fs_syx,IF(casex="ii",Fs_i bx,IF(casex="iii",Fs_ibx+Fs_ibtx,Fs_eb x+Fs_ebtx))) = 231 [MPa] $13.4.1.1 (a- d) Fs_unsx = IF(casex="i",Fs_syx,IF(casex="ii",Fs_i bx,IF(casex="iii",Fs_ibx,Fs_ebx))) = 231 [MPa] " Stiffener check st_chx = IF(Ff_d>Fs_unsx,"Transverse Stiffener Required!","Transverse Stiffener not Required!") = Transvers e Stiffener not Required! Fsx = IF(stf="exist",Fs_st,Fs_unst) = 231[MPa] " SHEAR RESISTANCE Vr_uns = φ * Aw * Fs / 1000 = 3742.2[kN] $13.4.1.1 Shear ratio Vratiox = Vf_d/Vr_uns = 0.235 Shear ratio check Vcheckx = IF(Vratiox<1,"OK!!","Increase shear resistance") = OK!! Including Longitudinal Stiffeners web slenderness check s_chek = IF(Sl_maxp>s_wmax,"Reduce Slenderness!","Slenderness OK!") = Slenderne ss OK! Calculation of Shear Stress Resistance: Shear Buckling Coefficient kv_n = 5.34 = 5.34 $13.4.1.1 kv_s1 = 4+5.34/(a/h_max)^2 = 4.53 " Box Girder Spreadsheet (CSA-S16-01) 150 kv_s2 = 5.34+4/(a/h_max)^2 = 5.74 " kv_s = IF(a/h_max<1,kv_s1,kv_s2) = 5.74 " kv = IF(stf = "none",kv_n,kv_s) = 5.74 " Ultimate Shear Stress (Fs) (a) Shear Yielding Fs_sy = 0.66 * Fy = 231[MPa] $13.4.1.1(a) (b) Inelastic Buckling Fs_ib = 290 * (((Fy * kv)^.5)/(h_max/ t_w)) = 413.2[MPa] $13.4.1.1(b) tension field contribution Fs_ibt = (0.5 * Fy - 0.866*Fs_ib)*sqrt(1/(1+(a/h_max)^2)) = -54.8 [MPa] $13.4.1.1(c) (d) Elastic Buckling Fs_eb = 180000*kv/(Sl_maxp)^2 = 1044.1[MPa] $13.4.1.1(d) tension field contribution Fs_ebt = (0.5*Fy- 0.866*Fs_eb)*sqrt(1/(1+(a/h_max)^2)) = -218.7 [MPa] " K Factor KF = (kv/Fy)^0.5 = 0.13 (MPa^- 0.5] slenderness case h/w case = IF(Sl_maxp<=439*KF,"i",IF(Sl_maxp< =502*KF,"ii",IF(Sl_maxp<=621*KF,"iii", "iv"))) = i [MPa] Fs_st = IF(case="i",Fs_sy,IF(case="ii",Fs_ib,IF (case="iii",Fs_ib+Fs_ibt,Fs_eb+Fs_ebt ))) = 231 [MPa] $13.4.1.1 (a- d) Fs_unst = IF(case="i",Fs_sy,IF(case="ii",Fs_ib,IF (case="iii",Fs_ib,Fs_eb))) = 231 [MPa] " Stiffener check st_ch = IF(Ff_d>Fs_unst,"Transverse Stiffener Required!","Transverse Stiffener not Required!") = Transvers e Stiffener not Required! Fs = IF(stf="exist",Fs_st,Fs_unst) = 231[MPa] " SHEAR RESISTANCE Vr = φ * Aw * Fs / 1000 = 3742.2[kN] $13.4.1.1 Shear ratio Vratio = Vf_d/Vr = 0.235 Shear ratio check Vcheck = IF(Vratio<1,"OK!!","Increase shear resistance") = OK!! Overall Shear Resistance overall shear resistance Vr_com = IF(stf_lw="exist",Vr,Vr_uns) = 3742.2[kN] Box Girder Spreadsheet (CSA-S16-01) 151 Combined Flexure and Shear Check (CSA) Excluding Longitudinal Stiffeners Factored Moment at 0.6 Vr Mf_c = 3000[kN.m] Factored shear at 0.6Vr Vf_c = 0.6*Vr = 2245[kN] Moment ratio Mf_rcx = Mf_c/Mr_uns = 0.7059 Shear ratio Vf_rcx = Vf_c/Vr_uns = 0.6 check if interaction critical chkx = IF(Vf>=0.6*Vr_uns, "Interaction is critical","Interaction not critical") = Interactio n not critical interaction efficiency Interx = 0.727*Mf_rc+0.455*Vf_rc = 0.745 $14.6. check resistance against shear and moment C_checx = IF(stf="exist",IF(Inter>=1,"increase moment or shear resistance","combined shear and moment capacity OK"),"NA") = combined shear and moment capacity OK Including Longitudinal Stiffeners Moment ratio Mf_rc = Mf_c/Mr = 0.6496 Shear ratio Vf_rc = Vf_c/Vr = 0.6 check if interaction critical chk = IF(Vf>=0.6*Vr, "Interaction is critical","Interaction not critical") = Interactio n not critical interaction efficiency Inter = 0.727*Mf_rc+0.455*Vf_rc = 0.745 $14.6. check resistance against shear and moment C_check = IF(stf="exist",IF(Inter>=1,"increase moment or shear resistance","combined shear and moment capacity OK"),"NA") = combined shear and moment capacity OK 152 Appendix E: Box Girder Spreadsheet (CMAA) Box Girder Spreadsheet (CMAA) 153 DESIGN OF BOX GIRDERS DESCRIPTION REFERENCE Crane Code (CMAA Specifications #70 & #74 INPUT Factored Loads and Moments Factored Moment Mf = 3500.00[kNm] Factored Shear Vf = 1750.00[kN] LTF LBF LW ITFS,ATFS IBFS,ABFS IWS,AWS tBF,IBF,ABF tTF,ITF,ATF yBFS yWS1 yWS2 yTFS hws XTFS1 XTFS2 XBFS2 XBFS1 Box Girder Spreadsheet (CMAA) 154 Material Properties Material Yield Strength Fy = 350.00[MPa] Material Young's Modulus E = 200000[MPa] Material poisson's ratio v = 0.30 Performance Factor φ = 0.90 Girder Dimensions web thickness t_w = 20.00[mm] web length L_w = 900.00[mm] Top flange width L_tf = 500.00[mm] Top flange thickness t_tf = 35.00[mm] Bottom flange width L_bf = 300.00[mm] Bottom flange thickness t_bf = 25.00[mm] Web Longitudinal stiffener moment of Inertia I_ws = 131000[mm^4] Web longitudinal stiffener area A_ws = 524.00[mm^2] web longitudinal stiffener 1 height y_ws1 = 250.00[mm] web longitudinal stiffener 2 height y_ws2 = 875.00[mm] web longitudinal stiffener offset h_ws = 31.90[mm] Top flange Longitudinal stiffener moment of Inertia I_tfs = 131000[mm^4] Top flange Longitudinal stiffener area A_tfs = 524.00[mm^2] Top flange longitudinal stiffener height y_tfs = 913.60[mm] Top flange stiffener 1 offset x_tfs1 = 166.67[mm] Top flange stiffener 2 offset x_tfs2 = 333.33[mm] Bottom flange Longitudinal stiffener moment of Inertia I_bfs = 131000[mm^4] Bottom flange Longitudinal stiffener area A_bfs = 524.00[mm^2] Bottom flange longitudinal stiffener height y_bfs = 41.90[mm] Bottom flange stiffener 1 offset x_bfs1 = 100.00[mm] Bottom flange stiffener 2 offset x_bfs2 = 200.00[mm] Transverse stiffener/internal diaphragm spacing a = 2000.00[mm] Box Girder Spreadsheet (CMAA) 155 SECTIONAL PROPERTIES CALCULATED FOR THE BOX SECTION Section Properties without Longitudinal Stiffeners web inclination w_incl = ASIN(((L_tf-L_bf)/2)/L_w) = 0.11[rad] Top flange area A_tf = L_tf*t_tf = 17500[mm^2] Bottom flange area A_bf = L_bf*t_bf = 7500[mm^2] Web area A_w1 = L_w*t_w = 18000[mm^2] Total area A_tot = A_tf+A_bf+2*A_w1+2*(A_bfs+A_tfs +2*A_ws) = 65192[mm^2] Top flange centroid height y_tf = t_bf+L_w*COS(w_incl)+t_tf/2 = 936.9[mm] Bottom flange centroid height y_bf = t_bf/2 = 12.5[mm] Web centroid height y_w = L_w*COS(w_incl)/2+t_bf = 472.2[mm] Total section neutral axis height y_tot = (A_tf*y_tf+A_bf*y_bf+2*A_w1*y_w+ 2*(A_tfs*y_tfs+A_bfs*y_bfs+A_ws*Y _ws1+A_ws*Y_ws2))/(A_tot) = 547.15[mm] Top flange Ix Ix_tf = L_tf*(t_tf^3)/12 = 1.79E+0 6[mm^4] Bottom flange Ix Ix_bf = L_bf*(t_bf^3)/12 = 3.91E+0 5[mm^4] Web Ix Ix_w = t_w*(COS(w_incl))^2*(L_w^3)/12 = 1.20E+0 9[mm^4] Total section Ix Ix_tot = Ix_tf+Ix_bf+2*Ix_w+A_tf*(y_tf- y_tot)^2+A_bf*(y_bf- y_tot)^2+2*A_w1*(y_w- y_tot)^2+2*I_bfs+2*I_tfs+I_ws*4+2* A_tfs*(y_tfs- y_tot)^2+2*A_bfs*(y_bfs- y_tot)^2+2*A_ws*(y_ws1- y_tot)^2+2*A_ws*(y_ws2-y_tot)^2 = 8.02E+0 9[mm^4] Section Properties with Longitudinal stiffeners smeared into plate elements Combined stiffened top flange plate neutral axis height y_tot_tf = (2*A_tfs*y_tfs+A_tf*y_tf)/(A_tf+2*A_t fs) = 935.61 Combined stiffened top flange plate Ix Ix_tot_tf = 2*I_tfs+L_tf*t_tf^3/12+2*A_tfs*(y_tfs = 2586514 Box Girder Spreadsheet (CMAA) 156 -y_tot_tf)^2+A_tf*(y_tf-y_tot_tf)^2 Effective top flange flat plate thickness t_tf_sm = (12*Ix_tot_tf/L_tf)^(1/3) = 39.60[mm] Combined stiffened top flange plate neutral axis height y_tot_bf = (2*A_bfs*y_bfs+A_bf*y_bf)/(A_bf+2* A_bfs) = 16.10 Combined stiffened top flange plate Ix Ix_tot_bf = 2*I_bfs+L_bf*t_bf^3/12+2*A_bfs*(y_ bfs-y_tot_bf)^2+A_tf*(y_bf- y_tot_bf)^2 = 1577339 Effective top flange flat plate thickness t_bf_sm = (12*Ix_tot_bf/L_bf)^(1/3) = 39.81 Combined stiffened top flange plate neutral axis height y_tot_w = (2*A_ws*h_ws+A_w1*t_w/2)/(A_w1 +2*A_ws) = 11.20 Combined stiffened top flange plate Ix Ix_tot_w = 2*I_ws+L_w*t_w^3/12+2*A_ws*(h_ ws-y_tot_w)^2+A_w1*(t_w/2- y_tot_w)^2 = 1336977 Effective top flange flat plate thickness t_w_sm = (12*Ix_tot_w/L_w)^(1/3) = 26.12 Longitudinal stress distribution, panel sizes and loading cases Top flange longitudinal stress σ_tf = Mf*(y_tot_tf-y_tot)/Ix_tot*1000000 = 169.50[MPa] Top flange loading ratio Ψ_tf = 1 = 1 % Table 3.4.8.2-1 Top flange loading case case_tf = 1 = 1 % Table 3.4.8.2-1 Top flange subpanel plate width 1 b_tf_a = x_tfs1 = 167[mm] Top flange subpanel plate width 2 b_tf_b = x_tfs2-x_tfs1 = 167[mm] Top flange subpanel plate width 3 b_tf_c = L_tf-x_tfs2 = 167[mm] Top flange maximum plate width b_tf_max = MAX(b_tf_a,b_tf_b,b_tf_c) = 167[mm] Top flange maximum subpanel aspect ratio α_tf = a/b_tf_max = 12.00 Longitudinal stress in bottom flange σ_bf = Mf*(y_tot_bf-y_tot)/Ix_tot*1000000 = -231.71[MPa] Bottom flange loading ratio Ψ_bf = 1 = 1 % Table 3.4.8.2-1 Bottom flange loading case case_bf = 1 = 1 % Table 3.4.8.2-1 Bottom flange subpanel plate width 1 b_bf_a = x_bfs1 = 100[mm] Bottom flange subpanel plate width 2 b_bf_b = x_bfs2-x_bfs1 = 100[mm] Bottom flange subpanel plate width 3 b_bf_c = L_bf-x_bfs2 = 100[mm] Bottom flange maximum plate width b_bf_max = MAX(b_bf_a,b_bf_b,b_bf_c) = 100[mm] Bottom flange maximum subpanel aspect ratio α_bf = a/b_bf_max = 20.00 Box Girder Spreadsheet (CMAA) 157 Web longitudinal stress at first stiffener σ_w_s1 = Mf*(y_ws1-y_tot)/Ix_tot*1000000 = -129.66[MPa] Web longitudinal stress at second stiffener σ_w_s2 = Mf*(y_ws2-y_tot)/Ix_tot*1000000 = 143.05[MPa] Web subpanel 1 loading ratio Ψ_w1 = σ_w_s1/σ_bf = 0.56 % Table 3.4.8.2-1 Web subpanel 2 loading ratio Ψ_w2 = σ_w_s2/σ_w_s1 = -1.10 % Table 3.4.8.2-1 Web subpanel 3 loading ratio Ψ_w3 = σ_tf/σ_w_s2 = 1.18 % Table 3.4.8.2-1 Web subpanel 1 loading case case_w1 = IF(σ_bf<0,IF(σ_w_s1<0,"Tension!!",I F(Ψ_w1<- 1,3,IF(Ψ_w1<0,2,1))),IF(Ψ_w3<- 1,3,IF(Ψ_w1<0,2,1))) = Tension !! % Table 3.4.8.2-1 Web subpanel 2 loading case case_w2 = IF(σ_w_s1<0,IF(σ_w_s2<0,"Tensio n!!",IF(Ψ_w2<- 1,3,IF(Ψ_w1<0,2,1))),IF(Ψ_w3<- 1,3,IF(Ψ_w1<0,2,1))) = 3 % Table 3.4.8.2-1 Web subpanel 3 loading case case_w3 = IF(σ_tf<0,IF(σ_w_s2<0,"Tension!!!",I F(Ψ_w3<- 1,3,IF(Ψ_w1<0,2,1))),IF(Ψ_w3<- 1,3,IF(Ψ_w1<0,2,1))) = 1 % Table 3.4.8.2-1 Web subpanel plate width 1 b_w1 = (y_ws1-t_bf)/COS(w_incl) = 226.4[mm] Web subpanel plate width 2 b_w2 = (y_ws2-y_ws1)/COS(w_incl) = 628.9[mm] Web subpanel plate width 3 b_w3 = (L_w-y_ws2)/COS(w_incl) = 25.2[mm] Web subpanel aspect ratio 1 α_w1 = a/b_w1 = 8.83 Web subpanel aspect ratio 2 α_w2 = a/b_w2 = 3.18 Web subpanel aspect ratio 3 α_w3 = a/b_w3 = 79.50 Web loading ratio Ψ_w = σ_tf/σ_bf = -0.73 % Table 3.4.8.2-1 Web loading case case_w = IF(σ_bf<0,IF(σ_tf<0,"Tension!!",IF(Ψ _w<-1,3,IF(Ψ_w<0,2,1))),IF(Ψ_w3<- 1,3,IF(Ψ_w<0,2,1))) = 2 % Table 3.4.8.2-1 Shear flow and shear stress in each panel Dx1 = 0 = 0.000 Dx2 = t_bf*(y_bf-y_tot)*(x_bfs2-L_bf/2)*-1 = 668317 Box Girder Spreadsheet (CMAA) 158 Dx2' = Dx2-(y_bfs-y_tot)*A_bfs = 933070 Dx3 = t_bf*(y_bf-y_tot)*(L_bf-x_bfs2)*- 1+Dx2-(y_bfs-y_tot)*A_bfs = 2269703 Dx4 = t_w*((y_bf-y_tot)*(y_ws1- y_bf)+0.5*(y_ws1- y_bf)^2*COS(w_incl))*-1+Dx3 = 4248738 s4 = (y_bf-y_tot)/COS(w_incl)*-1 = 538 Dx_w1max' = t_w*((y_bf- y_tot)*s4+0.5*s4^2*COS(w_incl))*- 1+Dx3 = 5146057 Dx_w1max = IF(s4>0,IF(s4>b_w1,MAX(Dx4,Dx3), Dx_w1max'),MAX(Dx4,Dx3)) = 4248738 Dx4' = Dx4-(y_ws1-y_tot)*A_ws = 4404446 Dx5 = t_w*((y_ws1-y_tot)*(y_ws2- y_ws1)+0.5*(y_ws2- y_ws1)^2*COS(w_incl))*-1+Dx4- (y_ws1-y_tot)*A_ws = 4236802 s5 = (y_ws1-y_tot)/COS(w_incl)*-1 = 299 Dx_w2max' = t_w*((y_ws1- y_tot)*s5+0.5*s5^2*COS(w_incl))*- 1+Dx4-(y_ws1-y_tot)*A_ws = 5292949 Dx_w2max = IF(s5>0,IF(s5>b_w2,MAX(Dx5,Dx4') ,Dx_w2max'),MAX(Dx5,Dx4')) = 5292949 Dx5' = Dx5-(y_ws2-y_tot)*A_ws = 4065010 Dx6 = t_w*((y_ws2-y_tot)*(y_tf- y_ws2)+0.5*(y_tf- y_ws2)^2*COS(w_incl))*-1+Dx5- (y_ws2-y_tot)*A_ws = 3620846 s6 = (y_ws2-y_tot)/COS(w_incl)*-1 = -330 Dx_w3max' = t_w*((y_ws2- y_tot)*s6+0.5*s6^2*COS(w_incl))*- 1+Dx5-(y_ws2-y_tot)*A_ws = 5146541 Dx_w3max = IF(s6>0,IF(s6>b_w2,MAX(Dx6,Dx5') ,Dx_w3max'),MAX(Dx6,Dx5')) = 4065010 Dx7 = t_tf*(y_tf-y_tot)*(L_tf-x_tfs2)*-1+Dx6 = 1347166 Box Girder Spreadsheet (CMAA) 159 Dx7' = Dx7-(y_tfs-y_tot)*A_tfs = 1155148 Dx8 = t_tf*(y_tf-y_tot)*(x_tfs2-L_tf/2)*- 1+Dx7-(y_tfs-y_tot)*A_tfs = 18308 Maximum shear stress in bottom flange centre panel τ'_bf1 = Vf/Ix_tot*MAX(Dx1,Dx2)/t_bf*1000 = 5.83[MPa] Maximum shear stress in bottom flange outer panels τ'_bf2 = Vf/Ix_tot*Dx3/t_bf*1000 = 19.81[MPa] Maximum shear stress in bottom web panel τ'_w1max = Vf/Ix_tot*Dx_w1max/t_w*1000 = 46.35[MPa] Shear stress at bottom of the web τ'_wbot = Vf/Ix_tot*Dx3/t_w*1000 = 24.76[MPa] Maximum shear stress in mid web panel τ'_w2max = Vf/Ix_tot*Dx_w2max/t_w*1000 = 57.74[MPa] Shear stress at first web stiffener τ'_ws1 = Vf/Ix_tot*Dx4'/t_w*1000 = 48.05[MPa] Maximum shear stress in top web panel τ'_w3max = Vf/Ix_tot*Dx_w3max/t_w*1000 = 44.34[MPa] Shear stress at second web stiffener τ'_ws2 = Vf/Ix_tot*Dx5'/t_w*1000 = 44.34[MPa] Shear stress at the top of the web τ'_wtop = Vf/Ix_tot*Dx6/t_w*1000 = 39.50[MPa] Shear stress in top flange outer panels τ'_tf2 = Vf/Ix_tot*MAX(Dx6,Dx7)/t_tf*1000 = 22.57[MPa] Shear stress in top flange centre panel τ'_tf1 = Vf/Ix_tot*MAX(Dx7,Dx8)/t_tf*1000 = 8.40[MPa] Maximum shear stress in top flange τ'_tfmax = MAX(τ'_tf1,τ'_tf2) = 22.57[MPa] Maximum shear stress in bottom flange τ'_bfmax = MAX(τ'_bf1,τ'_bf2) = 19.81[MPa] Maximum shear stress in web τ'_wmax = MAX(τ'_w1max,τ'_w2max,τ'_w3max ) = 57.74[MPa] Yield Check Allowable Von Misses effective stress σ_al_vm = Fy^2 = 122500. 000 Top flange principal stress 1 σ_tf_p1 = σ_tf/2+SQRT((σ_tf/2)^2+τ'_tfmax^2) = 172.450 Top flange principal stress 2 σ_tf_p2 = σ_tf/2-SQRT((σ_tf/2)^2+τ'_tfmax^2) = -2.954 Top flange Von Misses effective stress σ_tf_vm = σ_tf_p1^2- σ_tf_p1*σ_tf_p2+σ_tf_p2^2 = 30256.9 73 Top flange yield safety factor SF_tf_y = σ_tf_vm/σ_al_vm = 0.247 Top flange yielding check SF_check_tf_y = IF(SF_tf_y>1,"Top flange failed in yielding!, Reduce stresses","Top flange OK!") = Top flange OK! Bottom flange principal stress 1 σ_bf_p1 = σ_bf/2+SQRT((σ_bf/2)^2+τ'_bfmax^ 2) = 1.681 Box Girder Spreadsheet (CMAA) 160 Bottom flange principal stress 2 σ_bf_p2 = σ_bf/2- SQRT((σ_bf/2)^2+τ'_bfmax^2) = - 233.395 Bottom flange Von Misses effective stress σ_bf_vm = σ_bf_p1^2-σ_bf_p1*σ_bf_p2 + σ_bf_p2 ^2 = 54868.1 65 Bottom flange yield safety factor SF_bf_y = σ_bf_vm/σ_al_vm = 0.448 Bottom flange yielding check SF_check_bf_y = IF(SF_bf_y>1,"Bottom flange failed in yielding!, Reduce stresses","Bottom flange OK!") = Bottom flange OK! Bottom web panel principal stress 1 σ_wa_p1 = σ_bf/2+SQRT((σ_bf/2)^2+τ'_wbot^2 ) = 2.616 Bottom web panel principal stress 2 σ_wa_p2 = σ_bf/2-SQRT((σ_bf/2)^2+τ'_wbot^2) = - 234.330 Bottom web panel Von Misses effective stress σ_wa_vm = σ_wa_p1^2-σ_wa_p1*σ_wa_p2 + σ_wa_p2 ^2 = 55530.1 92 Bottom web panel yield safety factor SF_wa_y = σ_wa_vm/σ_al_vm = 0.453 Web panel principal stress 1 at first stiffener σ_wb_p1 = σ_w_s1/2+SQRT((σ_w_s1/2)^2+τ'_ ws1^2) = 15.863 Web panel principal stress 2 at first stiffener σ_wb_p2 = σ_w_s1/2- SQRT((σ_w_s1/2)^2+τ'_ws1^2) = - 145.520 Web panel Von Misses effective stress at first stiffener σ_wb_vm = σ_wb_p1^2-σ_wb_p1*σ_wb_p2 + σ_wb_p2 ^2 = 23736.0 58 Web panel yield safety factor at first stiffener SF_wb_y = σ_wb_vm/σ_al_vm = 0.194 Web panel principal stress 1 at second stiffener σ_wc_p1 = σ_w_s2/2+SQRT((σ_w_s2/2)^2+τ'_ ws2^2) = 155.680 Web panel principal stress 2 at second stiffener σ_wc_p2 = σ_w_s2/2- SQRT((σ_w_s2/2)^2+τ'_ws2^2) = -12.630 Web panel Von Misses effective stress at second stiffener σ_wc_vm = σ_wc_p1^2-σ_wc_p1*σ_wc_p2 + σ_wc_p2 ^2 = 26362.0 17 Web panel yield safety factor at second stiffener SF_wc_y = σ_wb_vm/σ_al_vm = 0.194 Web panel yield safety factor at NA SF_wd_y = τ'_wmax^2/σ_al_vm = 0.027 Web panel yielding check SF_check_w_y = IF(MAX(SF_wa_y,SF_wb_y,SF_wc _y,SF_wd_y)>1,"Web panel failed in yielding!, Reduce stresses","Web OK!") = Web OK! Box Girder Spreadsheet (CMAA) 161 Buckling check of maximum aspect ratio top flange sub panel Longitudinal plate buckling coefficient 1 k_tf1 = 8.4/(Y_tf+1.1) = 4.00 % Table 3.4.8.2-1 Longitudinal plate buckling coefficient 2 k_tf2 = (a_tf+1/a_tf)^2*(2.1/(Y_tf+1.1)) = 146.01 % Table 3.4.8.2-1 Longitudinal plate buckling coefficient k_tf = IF(a_tf<1,k_tf2,k_tf1) = 4 Critical longitudinal buckling stress σcr_tf = k_tf*PI()^2*E/(12*(1- v^2)*(b_tf_max/t_tf)^2) = 31886[MPa] Shear plate buckling coefficient 1 ks_tf1 = 4+5.34/(a_tf^2) = 4 $13.4.1.1 Shear plate buckling coefficient 2 ks_tf2 = 5.34+4/(a_tf^2) = 5 $13.4.1.1 Shear plate buckling coefficient ks_tf = IF(a_tf<1,ks_tf1,ks_tf2) = 5 Critical shear buckling stress τcr_tf = ks_tf*PI()^2*E/(12*(1- v^2)*(b_tf_max/t_tf)^2) = 42790[MPa] elastic comparison stress factor 1 F1_tf = (1+Y_tf)/4*(σ_tf/σcr_tf) = 0.00265 78 elastic comparison stress factor 2 F2_tf = SQRT(((3- Ψ_tf)*σ_tf/(4*σcr_tf))^2+(τ'_tfmax/τc r_tf)^2) = 0.00270 96 elastic comparison stress σ_1k_tf = SQRT(σ_tf^2+3*τ'_tfmax^2)/(F1_tf+ F2_tf) = 32407 % 3.4.8.2 Reduced comparison stress σ_1kR_tf = Fy*σ_1k_tf^2/(0.1836*Fy^2+σ_1k_tf ^2) = 349.992 50 % 3.4.8.2 Proportional limit σ_p_tf = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_tf = IF(σ_1k_tf<σ_p_tf,σ_1k_tf,σ_1kR_tf ) = 349.99 buckling safety factor SF_tf = σ_comp_tf/(SQRT(σ_tf^2+3*τ'_tfma x^2)) = 2.01 Case 1 design factor requirement DFB_1_tf = 1.7+0.175*(Ψ_tf-1) = 1.70 Table 3.4.8.3-1 Case 2 design factor requirement DFB_2_tf = 1.5+0.125*(Ψ_tf-1) = 1.50 Table 3.4.8.3-1 Case 3 design factor requirement DFB_3_tf = 1.35+0.05*(Ψ_tf-1) = 1.35 Table 3.4.8.3-1 Design factor requirement DFB_tf = IF(case_tf=1,DFB_1_tf,IF(case_tf=2 ,DFB_2_tf,DFB_3_tf)) = 1.70 Safety factor ratio SF_ratio_tf = DFB_tf/SF_tf = 0.84 Box Girder Spreadsheet (CMAA) 162 Safety factor ratio check SF_check_tf = IF(SF_ratio_tf<1,"Top flange sub panels OK!", "Increase top flange thickness or decrease maximum stiffener spacing!") = Top flange sub panels OK! Buckling check of top flange with smeared longitudinal stiffeners Aspect ratio α_tf_sm = a/L_tf = 4.00 Longitudinal plate buckling coefficient 1 k_tf1_sm = 8.4/(Ψ_tf+1.1) = 4.00 Table 3.4.8.3-1 Longitudinal plate buckling coefficient 2 k_tf2_sm = (1/α_tf_sm + α_tf_sm)^2*(2.1/(Ψ_tf+1.1)) = 18.06 Table 3.4.8.3-1 Longitudinal plate buckling coefficient k_tf_sm = IF(α_tf_sm<1,k_tf2_sm,k_tf1_sm) = 4 Critical longitudinal buckling stress σcr_tf_sm = k_tf_sm*PI()^2*E/(12*(1- v^2)*(L_tf/t_tf_sm)^2) = 4534[MPa] Shear plate buckling coefficient 1 ks_tf1_sm = 4+5.34/(α_tf_sm^2) = 4 $13.4.1.1 Shear plate buckling coefficient 2 ks_tf2_sm = 5.34+4/(α_tf_sm^2) = 6 $13.4.1.1 Shear plate buckling coefficient ks_tf_sm = IF(α_tf_sm<1,ks_tf1_sm,ks_tf2_sm) = 6 Critical shear buckling stress τcr_tf_sm = ks_tf_sm*PI()^2*E/(12*(1- v^2)*(L_tf/t_tf_sm)^2) = 6337[MPa] elastic comparison stress factor 1 F1_tf_sm = (1+Ψ_tf)/4*(σ_tf/σcr_tf_sm) = 0.01869 04 elastic comparison stress factor 2 F2_tf_sm = SQRT(((3- Ψ_tf)*σ_tf/(4*σcr_tf_sm))^2+(τ'_tfma x/τcr_tf_sm)^2) = 0.01902 67 elastic comparison stress σ_1k_tf_sm = SQRT(σ_tf^2+3*τ'_tfmax^2)/(F1_tf_ sm+F2_tf_sm) = 4612 % 3.4.8.2 Reduced comparison stress σ_1kR_tf_sm = Fy*σ_1k_tf_sm^2/(0.1836*Fy^2+σ_ 1k_tf_sm^2) = 349.630 28 % 3.4.8.2 Proportional limit σ_p_tf_sm = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_tf_sm = IF(σ_1k_tf_sm<σ_p_tf_sm,σ_1k_tf_ sm,σ_1kR_tf_sm) = 349.63 buckling safety factor SF_tf_sm = σ_comp_tf_sm/(SQRT(σ_tf^2+3*τ'_t fmax^2)) = 2.01 Case 1 design factor requirement DFB_1_tf_sm = 1.7+0.175*(Ψ_tf-1) = 1.70 % Table 3.4.8.3-1 Case 2 design factor requirement DFB_2_tf_sm = 1.5+0.125*(Ψ_tf-1) = 1.50 % Table 3.4.8.3-1 Box Girder Spreadsheet (CMAA) 163 Case 3 design factor requirement DFB_3_tf_sm = 1.35+0.05*(Ψ_tf-1) = 1.35 % Table 3.4.8.3-1 Design factor requirement DFB_tf_sm = IF(case_tf=1,DFB_1_tf_sm,IF(case_ tf=2,DFB_2_tf_sm,DFB_3_tf_sm)) = 1.70 Safety factor ratio SF_ratio_tf_sm = DFB_tf_sm/SF_tf_sm = 0.85 Safety factor ratio check SF_check_tf_sm = IF(SF_ratio_tf_sm<1,"Smeared top flange OK!", "Increase top flange thickness or number/size of stiffeners") = Smeared top flange OK! Buckling check of bottom web subpanel Longitudinal plate buckling coefficient 1 k_w1a = 8.4/(Ψ_w1+1.1) = 5.06 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 2 k_w1b = (α_w1+1/α_w1)^2*(2.1/(Ψ_w1+1.1)) = 101.29 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 3 k_w1a' = 8.4/1.1 = 7.64 Longitudinal plate buckling coefficient 4 k_w1b' = (α_w1+1/α_w1)^2*(2.1/1.1) = 152.82 Longitudinal plate buckling coefficient 5 k_w1c' = IF(α_w1<1,k_w1b',k_w1a') = 7.64 Longitudinal plate buckling coefficient 6 k_w1d = 23.9 = 23.90 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 7 k_w1e = 15.87+1.87/(α_w1)^2+8.6*α_w1^2 = 687.01 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 8 k_w1c'' = IF(α_w1<2/3,k_w1e,k_w1d) = 23.90 Longitudinal plate buckling coefficient 9 k_w1c = (1+Y_w1)*k_w1c'- (Y_w1*k_w1c'')+10*Y_w1*(1+Y_w1) = 7.26 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient k_w1 = IF(case_w1=1,IF(α_w1<1,k_w1b,k_ w1a),IF(case_w1=2,k_w1c,IF(case_ w1=3,IF(a_w1<2/3,k_w1e,k_w1d)," Tension!!"))) = Tension! ! Critical longitudinal buckling stress σcr_w1 = IF(k_w1="Tension!!","Tension!!",k_w 1*PI()^2*E/(12*(1- v^2)*(b_w1/t_w)^2)) = Tension! ! [MPa] Shear plate buckling coefficient 1 ks_w1a = 4+5.34/(α_w1^2) = 4.1 $13.4.1.1 Shear plate buckling coefficient 2 ks_w1b = 5.34+4/(α_w1^2) = 5.4 $13.4.1.1 Shear plate buckling coefficient ks_w1 = IF(α_w1<1,ks_w1a,ks_w1b) = 5.4 Critical shear buckling stress τcr_w1 = ks_w1*PI()^2*E/(12*(1- v^2)*(b_w1/t_w)^2) = 7605[MPa] elastic comparison stress factor 1 F1_w1 = IF(k_w1="Tension!!","Tension!!",(1+ Ψ_w1)/4*(σ_w_s1/σcr_w1)) = Tension! ! Box Girder Spreadsheet (CMAA) 164 elastic comparison stress factor 2 F2_w1 = IF(k_w1="Tension!!","Tension!!",SQ RT(((3- Ψ_w1)*σ_w_s1/(4*σcr_w1))^2+(τ'_ w1max/τcr_w1)^2)) = Tension! ! elastic comparison stress σ_1k_w1 = IF(k_w1="Tension!!","Tension!!",SQ RT(σ_w_s1^2+3*τ'_w1max^2)/(F1_ w1+F2_w1)) = Tension! ! % 3.4.8.2 Reduced comparison stress σ_1kR_w1 = IF(k_w1="Tension!!","Tension!!",Fy* σ_1k_w1^2/(0.1836*Fy^2+σ_1k_w1 ^2)) = Tension! ! % 3.4.8.2 Proportional limit σ_p_w1 = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_w1 = IF(σ_1k_w1<σ_p_w1,σ_1k_w1,σ_1 kR_w1) = Tension! ! buckling safety factor SF_w1 = IF(k_w1="Tension!!","Tension!!",s_c omp_w1/(SQRT(s_w_s1^2+3*t'_w1 max^2))) = Tension! ! Case 1 design factor requirement DFB_1_w1 = 1.7+0.175*(Ψ_w1-1) = 1.62 % Table 3.4.8.3-1 Case 2 design factor requirement DFB_2_w1 = 1.5+0.125*(Ψ_w1-1) = 1.44 % Table 3.4.8.3-1 Case 3 design factor requirement DFB_3_w1 = 1.35+0.05*(Ψ_w1-1) = 1.33 % Table 3.4.8.3-1 Design factor requirement DFB_w1 = IF(case_w1="Tension!!","Tension!!", IF(case_w1=1,DFB_1_w1,IF(case_ w1=2,DFB_2_w1,DFB_3_w1))) = Tension !! Safety factor ratio SF_ratio_w1 = IF(k_w1="Tension!!","Tension!!",DF B_w1/SF_w1) = Tension !! Safety factor ratio check SF_check_w1 = IF(case_w1="Tension!!","Tension!!", IF(SF_ratio_w1<1,"Bottom web sub panel OK!", "Increase web thickness or decrease stiffener spacing!")) = Tension !! Buckling check of mid web subpanel Longitudinal plate buckling coefficient 1 k_w2a = 8.4/(Ψ_w2+1.1) = - 2553.01 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 2 k_w2b = (α_w2+1/α_w2)^2*(2.1/(Ψ_w2+1.1)) = - 7794.63 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 3 k_w2a' = 8.4/1.1 = 7.64 Box Girder Spreadsheet (CMAA) 165 Longitudinal plate buckling coefficient 4 k_w2b' = (α_w2+1/α_w2)^2*(2.1/1.1) = 23.31 Longitudinal plate buckling coefficient 5 k_w2c' = IF(α_w2<1,k_w2b',k_w2a') = 7.64 Longitudinal plate buckling coefficient 6 k_w2d = 23.9 = 23.90 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 7 k_w2e = 15.87+1.87/(α_w2)^2+8.6*α_w2^2 = 103.03 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 8 k_w2c'' = IF(α_w2<2/3,k_w2e,k_w2d) = 23.90 Longitudinal plate buckling coefficient 9 k_w2c = (1+Y_w2)*k_w2c'- (Y_w2*k_w2c'')+10*Y_w2*(1+Y_w2) = 26.72 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient k_w2 = IF(case_w2=1,IF(α_w2<1,k_w2b,k_ w2a),IF(case_w2=2,k_w2c,IF(case_ w2=3,IF(a_w2<2/3,k_w2e,k_w2d)," Tension!!"))) = 23.90 Critical longitudinal buckling stress σcr_w2 = k_w2*PI()^2*E/(12*(1- v^2)*(b_w2/t_w)^2) = 4369[MPa] Shear plate buckling coefficient 1 ks_w2a = 4+5.34/(α_w2^2) = 4.5 $13.4.1.1 Shear plate buckling coefficient 2 ks_w2b = 5.34+4/(α_w2^2) = 5.7 $13.4.1.1 Shear plate buckling coefficient ks_w2 = IF(α_w2<1,ks_w2a,ks_w2b) = 5.7 Critical shear buckling stress τcr_w2 = ks_w2*PI()^2*E/(12*(1- v^2)*(b_w2/t_w)^2) = 1049[MPa] elastic comparison stress factor 1 F1_w2 = (1+Ψ_w2)/4*(σ_w_s2/σcr_w2) = - 0.00084 54 elastic comparison stress factor 2 F2_w2 = SQRT(((3- Ψ_w2)*σ_w_s2/(4*σcr_w2))^2+(τ'_ w2max/τcr_w2)^2) = 0.06449 86 elastic comparison stress σ_1k_w2 = SQRT(σ_w_s2^2+3*τ'_w2max^2)/(F 1_w2+F2_w2) = 2742 % 3.4.8.2 Reduced comparison stress σ_1kR_w2 = Fy*σ_1k_w2^2/(0.1836*Fy^2+σ_1k_ w2^2) = 348.956 16 % 3.4.8.2 Proportional limit σ_p_w2 = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_w2 = IF(σ_1k_w2<σ_p_w2,σ_1k_w2,σ_1 kR_w2) = 348.96 buckling safety factor SF_w2 = σ_comp_w2/(SQRT(σ_w_s2^2+3*τ' _w2max^2)) = 2.00 Case 1 design factor requirement DFB_1_w2 = 1.7+0.175*(Ψ_w2-1) = 1.33 % Table 3.4.8.3-1 Box Girder Spreadsheet (CMAA) 166 Case 2 design factor requirement DFB_2_w2 = 1.5+0.125*(Ψ_w2-1) = 1.24 % Table 3.4.8.3-1 Case 3 design factor requirement DFB_3_w2 = 1.35+0.05*(Ψ_w2-1) = 1.24 % Table 3.4.8.3-1 Design factor requirement DFB_w2 = IF(case_w2="Tension!!","Tension!!", IF(case_w2=1,DFB_1_w2,IF(case_ w2=2,DFB_2_w2,DFB_3_w2))) = 1.24 Safety factor ratio SF_ratio_w2 = DFB_w2/SF_w2 = 0.62 Safety factor ratio check SF_check_w2 = IF(case_w2="Tension!!","Tension!!", IF(SF_ratio_w2<1,"Mid web sub panel OK!", "Increase web thickness or decrease stiffener spacing!")) = Mid web sub panel OK! Buckling check of top web subpanel Longitudinal plate buckling coefficient 1 k_w3a = 8.4/(Ψ_w3+1.1) = 3.68 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 2 k_w3b = (α_w3+1/α_w3)^2*(2.1/(Ψ_w3+1.1)) = 5811.39 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 3 k_w3a' = 8.4/1.1 = 7.64 Longitudinal plate buckling coefficient 4 k_w3b' = (α_w3+1/α_w3)^2*(2.1/1.1) = 12071.1 6 Longitudinal plate buckling coefficient 5 k_w3c' = IF(α_w3<1,k_w3b',k_w3a') = 7.64 Longitudinal plate buckling coefficient 6 k_w3d = 23.9 = 23.90 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 7 k_w3e = 15.87+1.87/(α_w3)^2+8.6*α_w3^2 = 54376.3 6 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 8 k_w3c'' = IF(α_w3<2/3,k_w3e,k_w3d) = 23.90 Longitudinal plate buckling coefficient 9 k_w3c = (1+Y_w3)*k_w3c'- (Y_w3*k_w3c'')+10*Y_w3*(1+Y_w3) = 14.25 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient k_w3 = IF(case_w3=1,IF(α_w3<1,k_w2b,k_ w2a),IF(case_w3=2,k_w2c,IF(case_ w3=3,IF(a_w2<2/3,k_w2e,k_w2d)," Tension!!"))) = - 2553.01 Critical longitudinal buckling stress σcr_w3 = k_w3*PI()^2*E/(12*(1- v^2)*(b_w3/t_w)^2) = - 2917054 44[MPa] Shear plate buckling coefficient 1 ks_w3a = 4+5.34/(α_w3^2) = 4.0 $13.4.1.1 Shear plate buckling coefficient 2 ks_w3b = 5.34+4/(α_w3^2) = 5.3 $13.4.1.1 Shear plate buckling coefficient ks_w3 = IF(α_w3<1,ks_w3a,ks_w3b) = 5.3 Box Girder Spreadsheet (CMAA) 167 Critical shear buckling stress τcr_w3 = ks_w3*PI()^2*E/(12*(1- v^2)*(b_w3/t_w)^2) = 610218[MPa] elastic comparison stress factor 1 F1_w3 = (1+Ψ_w3)/4*(σ_tf/σcr_w3) = - 0.00000 03 elastic comparison stress factor 2 F2_w3 = SQRT(((3- Ψ_w3)*σ_tf/(4*σcr_w3))^2+(τ'_w3m ax/τcr_w3)^2) = 0.00007 27 elastic comparison stress σ_1k_w3 = SQRT(σ_tf^2+3*τ'_w3max^2)/(F1_w 3+F2_w3) = 2572023 % 3.4.8.2 Reduced comparison stress σ_1kR_w3 = Fy*σ_1k_w3^2/(0.1836*Fy^2+σ_1k_ w3^2) = 350.000 00 % 3.4.8.2 Proportional limit σ_p_w3 = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_w3 = IF(σ_1k_w3<σ_p_w3,σ_1k_w3,σ_1 kR_w3) = 350.00 buckling safety factor SF_w3 = σ_comp_w3/(SQRT(σ_tf^2+3*τ'_w3 max^2)) = 1.88 Case 1 design factor requirement DFB_1_w3 = 1.7+0.175*(Ψ_w3-1) = 1.73 % Table 3.4.8.3-1 Case 2 design factor requirement DFB_2_w3 = 1.5+0.125*(Ψ_w3-1) = 1.52 % Table 3.4.8.3-1 Case 3 design factor requirement DFB_3_w3 = 1.35+0.05*(Ψ_w3-1) = 1.36 % Table 3.4.8.3-1 Design factor requirement DFB_w3 = IF(case_w3="Tension!!","Tension!!", IF(case_w3=1,DFB_1_w3,IF(case_ w3=2,DFB_2_w3,DFB_3_w3))) = 1.73 Safety factor ratio SF_ratio_w3 = DFB_w3/SF_w3 = 0.92 Safety factor ratio check SF_check_w3 = IF(case_w3="Tension!!","Tension!!", IF(SF_ratio_w3<1,"Top web sub panel OK!", "Increase web thickness or decrease stiffener spacing!")) = Top web sub panel OK! Buckling check of web with smeared longitudinal stiffeners Aspect ratio α_w = a/L_w = 2.22 Longitudinal plate buckling coefficient 1 k_wa = 8.4/(Ψ_w+1.1) = 22.79 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 2 k_wb = (α_w+1/α_w)^2*(2.1/(Ψ_w+1.1)) = 40.69 % Table 3.4.8.3-1 Box Girder Spreadsheet (CMAA) 168 Longitudinal plate buckling coefficient 3 k_wa' = 8.4/1.1 = 7.64 Longitudinal plate buckling coefficient 4 k_wb' = (α_w+1/α_w)^2*(2.1/1.1) = 13.63 Longitudinal plate buckling coefficient 5 k_wc' = IF(α_w<1,k_wb',k_wa') = 7.64 Longitudinal plate buckling coefficient 6 k_wd = 23.9 = 23.90 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 7 k_we = 15.87+1.87/(α_w)^2+8.6*α_w^2 = 58.72 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient 8 k_wc'' = IF(α_w<2/3,k_we,k_wd) = 23.90 Longitudinal plate buckling coefficient 9 k_wc = (1+Y_w)*k_wc'- (Y_w*k_wc'')+10*Y_w*(1+Y_w) = 17.57 % Table 3.4.8.3-1 Longitudinal plate buckling coefficient k_w_sm = IF(case_w=1,IF(α_w<1,k_wb,k_wa), IF(case_w=2,k_wc,IF(case_w=3,IF( a_w<2/3,k_we,k_wd),"Tension!!"))) = 17.57 Critical longitudinal buckling stress σcr_w_sm = k_w_sm*PI()^2*E/(12*(1- v^2)*(L_w/t_w_sm)^2) = 2676[MPa] Shear plate buckling coefficient 1 ks_wa = 4+5.34/(α_w^2) = 5 $13.4.1.1 Shear plate buckling coefficient 2 ks_wb = 5.34+4/(α_w^2) = 6 $13.4.1.1 Shear plate buckling coefficient ks_w_sm = IF(α_w<1,ks_wa,ks_wb) = 6 Critical shear buckling stress τcr_w_sm = ks_w_sm*PI()^2*E/(12*(1- v^2)*(L_w/t_w_sm)^2) = 937[MPa] elastic comparison stress factor 1 F1_w_sm = (1+Ψ_w)/4*(σ_tf/σcr_w_sm) = 0.00425 26 elastic comparison stress factor 2 F2_w_sm = SQRT(((3- Ψ_tf)*σ_tf/(4*σcr_w_sm))^2+(τ'_wm ax/τcr_w_sm)^2) = 0.06930 91 elastic comparison stress σ_1k_w_sm = SQRT(σ_tf^2+3*τ'_wmax^2)/(F1_w_ sm+F2_w_sm) = 2675 % 3.4.8.2 Reduced comparison stress σ_1kR_w_sm = Fy*σ_1k_w_sm^2/(0.1836*Fy^2+σ_ 1k_w_sm^2) = 348.903 58 % 3.4.8.2 Proportional limit σ_p_w_sm = Fy/1.32 = 265.15 % 3.4.8.2 comparison stress σ_comp_w_sm = IF(σ_1k_w_sm<σ_p_w_sm,σ_1k_w _sm,σ_1kR_w_sm) = 348.90 buckling safety factor SF_w_sm = σ_comp_w_sm/(SQRT(σ_tf^2+3*τ'_ wmax^2)) = 1.77 Case 1 design factor requirement DFB_1_w_sm = 1.7+0.175*(Ψ_w-1) = 1.40 % Table 3.4.8.3-1 Case 2 design factor requirement DFB_2_w_sm = 1.5+0.125*(Ψ_w-1) = 1.28 % able 3.4.8.3-1 Box Girder Spreadsheet (CMAA) 169 Case 3 design factor requirement DFB_3_w_sm = 1.35+0.05*(Ψ_w-1) = 1.26 % Table 3.4.8.3-1 Design factor requirement DFB_w_sm = IF(case_w=1,DFB_1_w_sm,IF(case _w=2,DFB_2_w_sm,DFB_3_w_sm) ) = 1.28 Safety factor ratio SF_ratio_w_sm = DFB_w_sm/SF_w_sm = 0.72 Safety factor ratio check SF_check_w_sm = IF(SF_ratio_w_sm<1,"Smeared web OK!", "Increase web thickness or number/size of stiffeners") = Smeared web OK! % Crane Code (CMAA Specifications #70 & #74)
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Rapid design of steel monosymmetric plate and box girders Khorasani, Milad 2010
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Title | Rapid design of steel monosymmetric plate and box girders |
Creator |
Khorasani, Milad |
Publisher | University of British Columbia |
Date Issued | 2010 |
Description | This thesis deals with the design process for steel plate girders and box girders. The design of plate girders is quite prescriptive. A more fundamental approach is required for the design of box girders. Equations explicitly for the design of plate and box girders are heavily influenced by empirical data. This work considers pure steel box girders only, and does not include the design of composite box girder sections. CAN/CSA-S6-00 “Canadian Highway Bridge Design Code” provides detailed design requirements for these composite girders. The design of plate girders follows mostly the requirements specified in the “Handbook of Steel Construction” CAN/CSA-S16.1. However, the S16.1 clauses relating to bending capacity are not well suited for the design of monosymmetric plate girders. Therefore, the code recommends a rational method of analysis such as methods explained in the Structural Stability Research Council’s Guide to Stability Design Criteria for Metal Structures. In addition, “Canadian Highway Bridge Design Code” CAN/CSA-S6-00 provides additional design information for monosymmetric sections. A steel box girder excluding composite design, hereon simply referred to as a box girder, is a purely steel section that could be designed in accordance with CAN/CSA-S16 “Limit States Design of Steel Structures”. However, this standard focuses on clauses for plate girder design, with little specific reference to box girders. Therefore, additional reference materials such as: 1) Guide to Stability Design Criteria for Metal Structures, 2) Crane Manufacturer’s Association of America (CMAA 74-2) standards, and 3) Canadian Highway Bridge Design Code are used for the design of monosymmeteric box girders. An integrated design and analysis environment in a form of formatted spreadsheet is implemented to ease the design process. The spreadsheet checks for both strength and serviceability requirements according to the applicable codes and standards. Included with this project is a clear procedure manual in chapter 7, so that the spreadsheet can be utilized for commercial design. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-08-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0062610 |
URI | http://hdl.handle.net/2429/28007 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2010-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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