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Lateral stability of two-and three-hinged glulam arches Egerup, Arne Ryden 1972

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LATERAL STABILITY OF TWO- AND THREE-HINGED GLULAM ARCHES by ARNE RYDEN EGERUP Bachelor in C i v i l Engineering, The Technical University of Copenhagen 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE ' i n the Department of CIVIL ENGINEERING We accept this thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA April 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t i s h Columbia, I agree that the Library snail make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this I'.. thesis for financial gain shall not be allowed without my written permis-sion. A.R. Egerup Department of C i v i l Engineering The University of Br i t i s h Columbia Vancouver 8, B.C. Canada Date: April 1972. i LATERAL STABILITY OP TWO- AND THREE-HINGED GLUIAM ARCHES , ABSTRACT This t h e s i s presents the r e s u l t s of a t h e o r e t i c a l and experimental s t u -dy of the l a t e r a l buckling of two- and three-hinged arches of rectangu-l a r c r o s s - s e c t i o n with l a t e r a l l y r e s t r a i n e d top edges. The structure i s analysed with and without a l i n e a r t o r s i o n a l r e s t r a i n t along the top ed-ge. The problem i s formulated using the s t i f f n e s s method. A s t i f f n e s s ma-t r i x i n c l u d i n g the e f f e c t s of l a t e r a l bending and t o r s i o n i s used. The buckling load i s defined as the smallest load at which the structure s t i f f -ness matrix becomes si n g u l a r . The method of s o l u t i o n of the t h e o r e t i c a l l a t e r a l buckling i s i t e r a t i o n (eigen value problem) and determinant p l o t . T h i s t h e o r e t i c a l approach i s v e r i f i e d by model t e s t s with two- and three-hinged parabolic glulam arches i n the laboratory. The method of so-l u t i o n f o r model t e s t i s the Southwell p l o t . The r e s u l t s of the t e s t s are presented and are shown to be s a t i s f a c t o r y . A set of numerical r e s u l t s are given f o r a range of arches with t o r s i o -n a l r e s t r a i n t at the top edge and f o r various load d i s t r i b u t i o n s . A sample of c a l c u l a t i o n s of a p r a c t i c a l arch shows that, although the arch i s safe according to the e x i s t i n g code, i t i s only safe considering l a t e r a l buck-l i n g i n c l u d i n g a t o r s i o n a l r e s t r a i n t at the top edge. i i TABLE OF CONTENTS PAGE NO. ABSTRACT i TABLE OF CONTENTS ' _ _ . _ i i LIST OF TABLES ' . i v LIST OF FIGURES v LIST OF PLATES v i i NOTATION v i i i ACKNOWLEDGEMENTS - x l ' CHAPTER I INTRODUCTION 1 I I STIFFNESS MATRICES 2.1 D e s c r i p t i o n of the structure 3; 2.2 Assumptions 3 2.3 S t i f f n e s s Matrix of Arch Segment 6 2.4 T o r s i o n a l Spring S t i f f n e s s 9 2.5 Geometric Transformation 10 2.6 E f f e c t of Top Edge Loading 13 2.7 End- and Top p i n Conditions 15 I I I SOLUTION TECHNIQUES 3.1 Introduction 16 3.2 Existence of Solutions ' 20 3.3 The I t e r a t i v e Methods of Solutions 24 3.4 The Determinant Plot Method 27 i i i TABLE OP CONTENTS (Cont'd) CHAPTER Page No. ... IV EXPERIMENTAL MODEL TEST 4.1 Introduction 28 4.2 Apparatus 28 4.3: .Testing Procedure 39 4.4 Experimental Results .44 4.5 Discussion of Experimental Results 46 V THEORETICAL RESULTS 5.1 Introduction 48 5«2 Structure Parameters 48 5-3 T h e o r e t i c a l r e s u l t s - 50 VI NUMERICAL DESIGN EXAMPLE ' 6.1 Introduction 77 6.2 A Sample Design Calculation 77 6.3 Method of connecting p u r l i n s to the arch r i b 80 VII CONCLUSIONS , 83 LIST OF FffiPERENCES BIBLIOGRAPHY iv LIST OF TABLES Page no. I Arch co-ordinates < 40 I I . Test re s u l t s 45 I I I Theoretical r e s u l t s 55 TV Co values 75 V LIST OF FIGURES Page no. 2.1 Two-hinged Arch 5 2.2 Three-hinged Arch 5 2.3 Forces and Degrees of Freedom on Segment 6 2.4 T o r s i o n a l spring force on Segment 9 2.5 ' L o c a l co-ordinates and g l o b a l co-ordinates 10 2.6 Load Point on Arch 13 2.7 E q u i l i b r i u m .of J o i n t i n General Displaced p o s i t i o n 14 2.8 End conditions and Degrees of Freedom f o r Top p i n 15 3.1 Degrees of freedom f o r Top p i n members 10 and 11 20 3.2 Unsymmetric non-linear member s t i f f n e s s matrix 21 3.3 Midportion of non-linear'structure matrix t^j^j] 22 4.1 General Arrangement of Test Rig 29 4.2 Bearing Assembly 31 4.3 Top p i n 32 4.4 T o r s i o n a l r e s t r a i n t 33 4.5 A p p l i c a t i o n of Load to top of arch 34 4.14 Load D i s t r i b u t i o n f o r Model Test 41 4.15 Model Southwell P l o t 42 4.16 Determination of spring constant 43 4.17 Load-deflection graph f o r a two hinged-arch 48 5.* Load d i s t r i b u t i o n s 54 5-2 o Graphs of CTcr/E VS. kL /EI f o r two-hinged arches, Loadcase 1, f / L =0.39 70 5.3 Graphs of o VS. kL /EI f o r three-hinged arches, Loadcase 1, f/L = 0.39 71 VI LIST OF FIGURES (cont'd) 2 5-4 Graphs of a c r/E VS. kL/EI for two-hinged arches, Loadcase 2, f/L=0.39 72 2 • 5.5 Graphs of o^ /^E VS. kL /EI for three-hinged arches, Loadcase 2, f/L =-0.39 73 2 -5.6 Graph of o^ /E VS. b /Ld for two-hinged arches for varying Loadcases, f/L = 0.39 74 5-7 Graph of C q VS. C b / 0 T f o r two-hinged arches 76 6.01 Determination of spring constant i n a real structure 80 6.02 Moment connection with the purlin on the top of the arch r i b 82 6.03 Moment connection with the purlin down beside the arch rib 82 v i i LIST OF PHOTOS Page No. 4.6 General Arrangement of Test Rig f o r a two-hinged arch 35 4.7 • General Arrangement of Test Rig f o r a three-hinged arch 35. 4.8 End Support Assembly 36 4.9 Top-pin'in the three hinged-arch 36 4.10 Load Hangers 37 4.11 P u r l i n arrangement 37 4.12 T o r s i o n a l R e s t r a i n t Arrangement 38 4.13 D e f l e c t i o n Measurement Device 38 v i i i NOTATION Symbol Definition [A] General symmetric matrix [B] General symmetric positive definite matrix E Young's modulus of e l a s t i c i t y for arch CP). Nodal forces associated with [S] P a Allowable axial stress Fb Allowable bending stress G Shear modulus for arch I Moment of inertia about weak axis, b d/12 J Torsion constant, (b^d/3)(1-O.63 b/d) [K Q] Linear part of [S] from structure constants [ KK ] Linear part of [S] from the spring [ K L ] Non linear part of [S] from P, M and V Non linear part of [S] from load on top edge [ KOK ] Total linear part of [S] [ V Total non-linear part of [S] L Span of arch (L/d) Slenderness ratio of equivalent column M Primary bending moment at fc segments M1,Mr Primary moments on ends of segment N r N 2 Applied external loads on top edge of arch P Primary axial force on segment [s] Structure, stiffness matrix NOTATION (cont'd) Symbol Definition S.J j S ^ S ^ S ^ , Stability functions for axial load P [T] Geometric transformation matrix for segment V Primary shear force on segment X,Y,Z Global co-ordinates axes b Width of arch cross-section C Q Slope of curve CTCX/E VS. b 2/Ld d Depth of arch cross-section e Distance from gravity axis to top edge of segment e' . Vertical distance from load point to joint f Rise of arch f • Computed axial stress f ^ Computed maximum bending stress (f} Nodal forces associated with [k] i General joint number [k] Total stiffness matrix for segment [k ] Linear part of [k] from structure constants o [k^l Linear part of [k] from the spring [kjj Non-linear part of [k] from P, M and V [k^j] Non-linear part of [k] from load on top edge k10(l,1 ),k10(l,2) Elements of 1^(10) k . S p r i n g constant (kips inch/inch/radian) s 1 ; Length of segment m,n Typical segment number X NOTATION (Cont'd) X Increase i n X co-ordinate over length of m'th segment, m Y Increase in Y co-ordinate over length of m'th segment, m {A} Nodal displacement associated with [S] T r i a l vector for iteration cycle. {A^) Resultant vector after iteration cycle. AP,AM,AV Resultant primary forces at a joint. {>&0} Intermediate vector i n iteration cycle, a Load level factor. 6 Nodal displacements associated with [k]. 6 • Slope of m'th segment relative to structure axes. X . Eigen value o Direct compressive stress at point of maximum bending com_-3, pressive stress at buckling load. °k Maximum compressive bending stress at buckling load. o" C r i t i c a l compressive stress i n bottom fibres of arch at ocr point of maximum bending compressive stress at buckling load without spring effect. o". Increase i n c r i t i c a l stress from the spring effect alone, ker o" Total c r i t i c a l compressive stress i n bottom fibres of arch cr at point of maximum bending compressive stress at buckling : load. . [ ] A square matrix —1 [ ] Inverse of a matrix { } A column matrix { }* Transpose matrix x i ACKNOWLEnDGEMENTS The author wishes to thank h i s supervisor Dr. R.P. Hooley, f o r h i s i n v a l u -able assistance during the research and preparation of t h i s t h e s i s . I should l i k e to thank the National Research Council of Canada f o r t h e i r f i n a n c i a l support, and the U n i v e r s i t y of B r i t i s h Columbia Computing Cen-t r e f o r use of t h e i r f a c i l i t i e s . Gratitude i s also expressed to Danmarks Ingeni^rakademi of Copenhagen, and Treebranchens Oplysningsrad and Otto M0nsted Fonden f o r making the opportunity of t h i s work p o s s i b l e . A p r i l , 1972 Copenhagen, Denmark. LATERAL STABILITY OP TWO. AND THREE-HINGED GLULAM ARCHES. 0 CHAPTER I : INTRODUCTION Structures composed of two- or three-hinged glulam arches with a roof deck are frequently used i n prefabricated b u i l d i n g systems. Economy i n design of glulam arches favours deep, narrow sections. The t r a d i t i o n a l method of de-sign i s to consider the structure as a plane frame, which buckles i n the p l a -ne, assuming the roof deck prevents l a t e r a l t o r s i o n a l buckling of the arch. The top edge i s supported against l a t e r a l t r a n s l a t i o n by the roof deck, but the lower edge i s l e f t f r e e . Under loads a r i s i n g from wind, snow and dead loads c e r t a i n portions of t h i s lower edge w i l l be i n compression due to ben-ding and a x i a l stresses. Such a system can become unstable and buckle s i d e -ways i n a t o r s i o n a l - f l e x u r a l mode about the constrained centre of r o t a t i o n at the top. An i n v e s t i g a t i o n of two-hinged arches by R.G.Charlwood [2] showed that the c r i t i c a l s t r e s s i n the t r a d i t i o n a l l y designed arch wherein the l a t e r a l buck-l i n g was neglected was sometimes l e s s than the allowable • Since, as f a r as I am aware, no f a i l u r e i n r e a l structures has occured although some structures ; must have received t h e i r f u l l load,' the roof deck must have increased the c r i -t i c a l s t r e s s thus producing a f a c t o r of safety greater than one. The e f f e c t of the roof deck i s to provide a t o r s i o n a l r e s t r a i n t bn the top.edge, by v i r t u e of connection of the p u r l i n s to the arch r i b . The object of t h i s t h e s i s i s to i n -a. Numbers i n square parenthesis r e f e r to the bibliography. vestigate the e f f e c t t h i s t o r s i o n a l r e s t r a i n t has on the c r i t i c a l s t r e s s i n two- and three-hinged arches under some common loadings. A s t i f f n e s s matrix developed by R.G.Charlwood and modified herein, i s used as the basis of the t h e o r e t i c a l study. T h i s i s checked against two glulam models to e s t a b l i s h i t s v a l i d i t y , and an e l e c t r o n i c d i g i t a l computer i s used to produce a set of nu-merical r e s u l t s from which some graphs are obtained to show the behavior of the st r u c t u r e s . F i n a l l y , a sample of c a l c u l a t i o n s of a p r a c t i c a l arch, shows that, although the arch i s safe according to the e x i s t i n g code, i t i s not safe con-s i d e r i n g l a t e r a l t o r s i o n a l buckling. However, taking the p u r l i n e f f e c t i n t o ac-count w i l l ensure a s u f f i c i e n t f a c t o r of safety. I t i s a major conclusion of t h i s t h e s i s then, that care must be taken i n de-t a i n i n g the p u r l i n ; to r i b connection f o r these structures, so that s u f f i c i e n t r e s t r a i n t i s provided to prevent l a t e r a l buckling. A s t i f f n e s s formulation i s used as i t o f f e r s a convenient method of i n c l u d i n g boundary conditions and f i n -ding buckling loads under any load d i s t r i b u t i o n f o r any shaped arch. Timoshenko [3] gives s o l u t i o n s f o r c i r c u l a r two- and three-hinged arches without l a t e r a l r e s t r a i n t , using the d i f f e r e n t i a l equations of equilibrium. However, the d i f -f e r e n t i a l equations Including l a t e r a l r e s t r a i n t s are too cumbersome to permit any p r a c t i c a l s o l u t i o n . T h i s work i s an i n v e s t i g a t i o n of two- and three-hinged glulam arches with and without a t o r s i o n a l r e s t r a i n t at the top edge, whereas Charlwood i n v e s t i g a t e d twc hinged arches without t o r s i o n a l r e s t r a i n t . 3. CHAPTER II  STIFFNESS MATRICES 2.1 D e s c r i p t i o n of the structure The structures i n v e s t i g a t e d are two- and three-hinged parabolic arches with a narrow rectangular cross-section as shown i n F i g ( 2 . l ) and Fig(2.2). To f a -c i l i t a t e d e s c r i p t i o n a system of g l o b a l coordinate axes i s introduced, as shown i n the f i g u r e s . The support conditions are such that the ends are r e -s t r a i n e d against t r a n s l a t i o n and t o r s i o n a l r o t a t i o n . Rotation at the ends a-bout the major and minor p r i n c i p a l axes i s permitted. T r a n s l a t i o n of the top edge i n the z d i r e c t i o n i s prevented over the whole span, and a l i n e a r t o r s i o -n a l r e s t r a i n t i s provided along t h i s edge. In the three-hinged-arch a top hinge i s added i n the centre of the arch. T h i s top hinge w i l l allow d i f f e r e n t r o t a -t i o n about the major and minor axes on both sides of the hinge, but w i l l give equal t o r s i o n a l r o t a t i o n on e i t h e r side of the hinge because t r a n s l a t i o n i s prevented i n the z - d i r e c t i o n at the top edge. 2.2 Assumptions' I t i s assumed that the arch i s composed of a number of small segments which are loaded and constrained only at the j o i n t s between segments. The s t r u c -ture s t i f f n e s s matrix i s generated from the s t i f f n e s s matrix of these seg-ments . The c r i t i c a l buckling load i s defined as the smallest p o s i t i v e load at which the s t r u c t u r e s t i f f n e s s matrix becomes si n g u l a r . In the d e r i v a t i o n of the member s t i f f n e s s matrix and i n the f o l l o w i n g theo-ry i t has been assumed that: 4. (a) The material remains e l a s t i c . (b) D e f l e c t i o n s remain small. (c) Shear deformations are neglected. (d) The spring i s l i n e a r . (e) Applied loads do not change d i r e c t i o n during deformation. (Conserva-t i v e f o r c e s ) . ( f ) Arch segments are s t r a i g h t and of rectangular c r o s s - s e c t i o n . (g) The width of the arch c r o s s - s e c t i o n i s small i n comparison with i t s depth. D e t a i l s of the s t i f f n e s s matrix and transformations of i t to s u i t the boundary conditions are given i n the f o l l o w i n g sections of t h i s chapter. Top edge laterally fixed and torsionally Figure (2.1) General Arrangement of a Two Hinged Arch Y Figure (2e2) General Arrangement of a Three Hinged Aroh Stiffness Matrix of Arch Segment Mi •M=ML +v|=Mr - V ^ Primary Forces on Segment d I I— —1 M, V, 1 gravity ax is Secondary Forces on Segment. (Degrees of freedom.) Figure (2,3) / Forces and Degrees of Freedom of Segment Fig(2.3) shows a general segment from the arch, when the element i s i n i t s undeformed position. Each end of such a segment i s under the action of three primary in-plane forces P, M and V. These primary forces are ge-nerated by the action of the external load on the system. When the prima-ry forces reach a c r i t i c a l value the arch deforms into the shape shown above i n fig(2.3)• With the element i n i t s deformed position, each end i s under the ac t i o n of two a d d i t i o n a l secondary forces f ^ , f ^ , f ^ and f ^ . These secondary forces are generated by the l a t e r a l motion and hold the structure i n equilibrium i n i t s deformed p o s i t i o n . The elements d e f l e c t e d shape i s then.described by the four d e f l e c t i o n s ^ , ^ and The four degrees of freedom are the r o t a t i o n along and perpendicular to the gr a v i t y axis at each end. For deep narrow, cross-sections the primary forces can be computed i n -dependently of secondary e f f e c t s by any s u i t a b l e method of plane frame a-n a l y s i s . Charlwood has given a s t i f f n e s s matrix l i n k i n g the four d e f l e c -t i o n s of Fig(2.3) to the corresponding four forces such that k& = f (2.01) where o= (5^, 5^, 5^, ^ ) * Is a column vector of nodal displacements, f=(f1» f g , f y f]).)* i s & column vector of conjugate nodal forces and k .is a 4x4 ..member s t i f f n e s s matrix i n member coordinates as follo w s : 12EIe 2S„ T3. + L -(!-1) 6EIeS„ VL IT 12EIe 2S. T_ • 3 L + If 2Me 6EIeS 2 VL 6EIeS, VL EIS. 6EIeS M 2 VL 2EIS, + -12EIe S l_JGx 6EIeS If 2Me 2 VL + T 12EIe 2S. + . , M V -2e V L + 2 6EIeS„ VL ~6~ 6EIeS 2 , VL 2EIS,, 6EIeS, 4EIS. + M VL 8. Here the moment of inertia I about the weak axis i s given by T b^d 12 The torsion constant J may be calculated from the approximate formula J = ^ (1 - 0.63 | ) The segment length i s L, d i s the depth, b is. the width of the cross-sec-tion and e i s the distance between the top edge and the gravity axis of the cross-section. The functions S. and x which account for the effect of I axial load P are defined as 5 1 = u - 5 sin W/12H . ^ - 1 + PL2/10 5 2 = u 2 (1 - cos u)/6 |i - 1 + PL2/60 = a) (sinw - oicos u )/k \i — 1 + PL /50 = u ( G J - sinw )/2 p. ^ 1 - PL2/60 X = 1 + Pd2/12 JG (2.02) 2 2 where u = - PL /EI and \i = (2 - 2 cos u> - u s i n u ). The functions account for the change in bending stiffness due to axial load and are the same functions as given by Gere and Weaver [4]. The func -tion x gives the change i n torsional stiffness due to axial load. It i s a basic assumption in the development of k that the primary loads (P, M, and V) on the segment are much less than the c r i t i c a l primary loads of the segment. In this analysis the structure i s s p l i t into twenty segments, so this basic assumption i s satisfied. Since a large number of segments (20) are being used i n this analysis the c r i t i c a l load of an individual segment as a unit i s far greater than the c r i t i c a l load of the whole structure, hence the value of u defined above i s relatively small. In this case there i s no need to use the exact transcendental equations for S^ Instead, the f i r s t two terms of their series expansion may be used as shown on the right-hand side of equation (2.02). When lateral motion at the top edge i s prevented, this matrix does not contain the effect of a torsional restraint at the top edge and i s not suitable for work with a three-hinged arch. The following sections of this chapter w i l l modify Charlwood's matrix to contain these two effects. Torsional spring stiffness Figure (2.4) Torsional Spring Foroe on Segment The effect of the purlins along the top edge i s a uniformly distributed linear torsional spring. The spring constant has dimension of moment/radian/length of the top edge. I f this spring is.considered to be concentrated half-at each end Pig (2.4) of the segment then the forces f„ and f_ w i l l be given by • 1 3 k L f 1 = -#~ 61 ( 2 - 0 3 ) and 10. The stiffness matrix considering the spring effect alone then becomes s k L s 2 0 0 • 0 0 0 0 0 0 0 k L s 2 0 o 0 0 0 Geometric Transformation Figure (2*5) Segment Co-ordinates and Global Co-ordinates The stiffness matrices i n 2.3 and 2A are given i n a local co-ordinate system. In order to obtain continuity at joints between inclined members a set of glo-bal co-ordinates must be used. Fig(2.5) shows the two systems. An orthogonal transformation between the local and the global co-ordinates i s 6 = T S" . . . . (2.05) f = T f (2.06) where & and f refer to local coordinates, while 5 and f refer to global co-ordinates, The transformation matrix T i s given below 11. cos9 m -sin9 m 0 0 s i n 9 m cos 9m .0 0 0 0 cos 9m - sin 9m 0 0 sin 9m cos 9m If (2.05) and (2.06) are substituted into (2.01), then we obtain: TkT * & = f (2.07) -1 * by using T = T The force-deflection relationship in global co-ordinates k "o = f (2.08) when compared to equation (2.08) shows that-k = T k T* (2.09) Evaluation of equation (2.09) for k gives the Charlwood matrix as follows: 2 2 Bx + Dy -(Q+S)xy 2 2 Ox - Sy^ +(B-D) xy 2 2 Ax + Hy +(Q-R) xy 2 2 Rx + Qy +(A-H) xy 2 2 +.(B-D) xy 2 2 Dx + By +(S+Q) xy 2 2 -Qx - Ry +(A-H) xy 2 2 Hx '+ Ay +(R-Q) xy 2 2 Ax + Hy +(Q-R) xy 2 2 -Q* - Ryr +(A-H) xy 2 2 Cx + Dy +(R-T) xy 2 2 -Rx - Ty +(C-D) xy 2 2 Rx + Qy* +(A-H) xy 2 2 Hx + Ay + (R-QJ xy 2 2 Tx + Ry +(C-D) xy 2 2 Dx + Cy +(T-R) xy where 12. A = -e 212 E l / L 5 - JG/L - e 21 . 5 3 P/L + 2eM/L B = e 212 El/J? + JG/L + e 21 . 5 3 P/L - 2e(M/L - V/2) C = e 212 EI/L 5•+ JG/L + e 21 . 5 3 P/L - 2e(M/L + V/2) D = 4 EI/L + 2PL/15 H = 2 EI/L - PL/30 Q = e 6 E I / L 2 + e P/10 - VL/6 R = e 6 E I / L 2 + e P/10 + VL/6 S = e 6 E J / L 2 + e P/10 - M + VL/3 T = -e 6 E J / L 2 - e P/10 + M + VL/3 (2.10) A s i m i l a r evaluation of equation (2.09) f ° r "the matrix i n 2.4 containing the spring e f f e c t y i e l d s : 2L 2 0 X 0 xy 0 0 0 0 0 2 X xy 0 0 xy 0 13-E f f e c t of Top Edge Loading Figure (2 06) Load Point on Arch The external loads are applied to the structure as point loads, N 1 as horizon-t a l and Ng as v e r t i c a l , to the top edge of the arch at j o i n t s between segments. Loads applied i n t h i s manner are e c c e n t r i c and account must be taken of the mo-ments which are induced. The evaluation of these moments and a method f o r i n -cluding them as terms i n the member s t i f f n e s s matrix are given i n t h i s s e c t i o n . The point of a p p l i c a t i o n of the point loads. N and Ng on the arch i s shown as point 'a' i n F i g ( 2 . 6 ) . Here 'a' i s v e r t i c a l l y above the p o s i t i o n of point 'g which i s the point i n the axis of g r a v i t y between two segments. The distance be tween load point 'a' and point 'g' i s e'. Fig(2.7) shows two adjacent segments taken apart from each other i n a gene-r a l d i s p l a c e d p o s i t i o n . The j o i n t i s held i n equilibrium by i n t e r n a l forces from the end of adjacent segments. The r e s u l t a n t i n t e r n a l forces are f f g , f ^ — 1 and fj, a c t i n g on each segment. A f a i r approximation would be to apply -5- and -g- to the end of each segment. A d d i t i o n a l forces generated by these e c c e n t r i -c i t i e s are 14. Figure (2o?) Equilibrium of Joint in General Displaced Position ? = [ -g- e/cos9 m ] 6 N . f 2 = [ - -± e/cos 9 m ] ^ N f _ = [ e/cos 9m-1 ] \ 3 2 3 N f^ = [ - e/cos 6 m-1 ] I .... (2.11) The contribution to the stiffness matrix from top-edge loading becomes then, when the o i s taken as 1. N2 -g- e/cos 9 m 0 0 0 N1 — 2 " e/cos 9m 0 0 0 0 0 N - T p e/cos 9 m-1 0 0 0 N — r / e/cos 9 m-1 0 Having obtained the stiffness matrices for segments i n terms of global co-or-dinates, the complete structure stiffness matrix can be b u i l t by linear super-position of segment matrices. 15. At the top pin for a 3-Hinged arch an ext ra joint is added, such that the horizontal rotations are equal, but the vertical rotations 10 and 11 are not equal. Rotation of both degrees of freedom prevented at this joint. Rotation of both degrees of freedom permitted at al l internal joints. True Span Section properties of hypothetical segment set as I=o ] = 30 J (0) Figure (2.8) End Conditions and Degrees of Freedom for Top-pin 2 . 7 End- and Top-pin Conditions To obtain the end conditions of no torsional rotation but free rotation about any axis perpendicular to the longitudinal centroidal axis, an additional, hy-pothetical segment i s introduced as shown in Fig ( 2 . 8 ) . For the three-hinged arch an additional, hypothetical joint i s introduced in the top hinge as shown i n Fig ( 2 . 8 ) . 16. CHAPTER I I I SOLUTION TECHNIQUES Introduction The s t i f f n e s s matrices presented i n Chapter I I contain two types of parameters. One type i s the s t r u c t u r e l constants, EJ, JG, k etc., the other type i s the s qu a n t i t i e s P, M and V which depend l i n e a r l y on the magnitude and d i s t r i b u t i o n of the external loads N^ and N^. I t i s convenient to define P, M and V as i n i t i a l values of primary i n t e r n a l forces corresponding to an i n i t i a l set of external loads N^ and N^. Since these i n t e r n a l forces vary l i n e a r l y with external loads, the i n t e r n a l forces can be expressed as ccP.» <xM and aV corresponding to exter-n a l loads of-magnitude and aN^. Here the f a c t o r a describes a v a r i a b l e load l e v e l r e l a t i v e to the i n i t i a l load l e v e l . In t h i s way a l l loads increase at the same rate u n t i l at some c r i t i c a l value of a , the structure buckles. Some arches with a large L/d w i l l buckle i n plane at about the same value of a as they buckle " l a t e r a l l y . In t h i s s i t u a t i o n the i n t e r n a l primary f o r c e s do not increase l i n e a r l y with the external loads. Such a s i t u a t i o n was i n v e s t i -gated by Charlwood and tested by him i n mode! work, but i t w i l l not be considered here. I f such a s i t u a t i o n does a r i s e the same technique as developed by Charlwood w i l l be appl i c a b l e whether or not the top edge i s e l a s t i c a l l y r e s t r a i n e d . The segment s t i f f n e s s matrix w i l l now become at the load l e v e l cc: (3.01) 17. The matrix [k n] of equation (3«01) i s symmetric and l i n e a r as follo w s : -A0x2+ DO y 2 -2Q0 xy 2 2 QO x - QO y +(-AO-DO)xy 2 2 DO x - AO y +2Q0 xy AO x2+H0 y 2 +2Q0 xy -QO x 2-Q0 y 2 +(A0-H0) xy -AO x2+D0 y 2 +2Q0 xy QO x2+Q0 y 2 +(A0-H0) xy HO x 2+A0 y 2 +2Q0 xy QO x2+Q0 y 2 +(-AO-DO) xy 2 2 DO x -AO y +2Q0 xy where: AO = -e 212 E J / L 5 - JG/L DO = 4 EJ/L HO = 2 EJ/L QO = e6 E J / L 2 . The spring matrix [k,,] i s a symmetric l i n e a r matrix as follo w s : k x 2 s k XY s 0 0 0 k x 2 s 0 0 k XY s 0 The unsymmetric matrix [ k l i s given below: 2 2 B1x +D1y -(Q1+Q1)xy 2 2 Q1x -S1y +(B1-D1)xy 2 2 A1x +H1y + (01-R1 )xy 2 2 R1x +01y . +(A1-H1)xy 2 2 Six -01 y +(B1-D1)xy 2 2 D1x +B1y + (S1+Q1)xy 2 2 -01x -R1y + (A1-H1 )xy 2 2 H1x +A1y + (R1 -01 )xy 2 2 A1x +H1y + (Q1-R1 )xy 2 2 -0,1 x -R1y + (A1-H1 )xy 2 2 C1x +D1y +(E1-T1)xy 2 2 -R1x -T1y +(C1-D1)xy 2 2 R1x -H3,1y + (A1-H1 )xy 2 2 H1x +A1y +(R1-Q1)xy 2 2 Tlx +R1y + (C1-D1 )xy 2 2 D1x +C1y + (T.1-R1 )xy where: A1 = -e 2 1.53 P/L + 2 e MA B1 = e 2 1.53 P/L - 2 e (M/L - V/2) C1 = e 2 1.53 P A - 2 e (M/L + V/2) D1 = 2 PL/15 H1 = -PL/30 Q1 = e P/10 - VL/6 R1 = e P/10 + VL/6 S1 = e P/10 - M + VL/3 T i = -e P/10+M+ VL/3 19-The matrix [k^] i s unsymrnetric and shown below-: IV-N 2 e 2 cos^ n 1 e 2 cos 9 n N 2 e 2 cos 9 m 1 e 2 cos Q m The structure stiffness [S] can now be expressed as: [S] = [ K Q + K K] + « [ ^ + . (3-02) where [KQ .+ K^l i s the linear portion of the structure matrix generated by the appropriate addition of [K^] and [K^] for each segment. [K^ -fr K^] are the ap-propriate sums of [K^] and [K^] for each segment. Denoting the sum [K^ + as K and [K^ + K^] as the structure s t i f f -ness matrix becomes [ S ] = K 0 K + a K L N • The structure stiffness equation becomes C K 0 K + < X K L N ] A = P = ° (3-03) . . (3.04) P i s taken as zero only because the eigenvalue i s required and no load corre* sponding to the buckling degrees of freedom need to be applied. The solution to the buckling problem i s to find the smallest value of tx at which the ma-t r i x [S] i s singular. The value of the products aP, aM, av", aN^ and aN^ w i l l then yield the c r i t i c a l value of the forces i n the structure. Three methods of solution were used, two types of iteration processes and, when the iter a ^ 20. t i o n processes f a i l e d , a determinant p l o t . Existence of Solutions The i t e r a t i v e method of determination of the c r i t i c a l load i s an eigenvalue problem K^^X = - a K j ^ X . A s u f f i c i e n t condition f o r the existence of r e a l so-l u t i o n s f o r a i s that and are symmetric, and i s p o s i t i v e d e f i n i t e . These requirements are obviously not s a t i s f i e d . Since i s the l i n e a r portion of the structure s t i f f n e s s matrix i t i s . UK symmetric and p o s i t i v e d e f i n i t e . For a two-hinged arch Charlwood showed that i s symmetric. In order to i n v e s t i g a t e the symmetry of f o r a three-hinged arch, i t i s s u f f i c i e n t to consider only terms of the structure s t i f f n e s s matrix, i n which members connected to the top p i n contribute, because other terms are s i m i l a r to terms from a two-hinged structure. For convenience the degrees of freedom f o r members connected to the top p i n are numbered as shown i n Fig ( 3.l). 20 22 23 21 25 Figure (3.1) Degrees of Freedom for Top-pin Members 10 and 11 Fig(3-2) shows the unsymmetric non-linear member s t i f f n e s s matrices f o r member 10 and 11 with a s p e c i a l notation f o r the elements. 21 W 1 0 ) s 19 20 21 22 klO(1,1) k l 0 ( 1 , 2 ) k i 0 ( 1 , 3 ) k l 0 ( 1 , 4 ) 19 k10.(2,1 ) k 1 0 ( 2 , 2 ) k l 0 ( 2 , 3 ) k l 0 ( 2 , 4 ) 20 k 1 0 ( 3,0 k i 0 ( 3 , 2 ) k l 0 ( 3 , 3 ) k l 0 . ( 3 , 4 ) 21 klO(4,l) k l 0 ( 4 , 2 ) k l 0 ( 4 , 3 ) k l 0 ( 4 , 4 ) 22 21 2 3 24 25 V 1 1 ) = k 1 l ( l , 1 ) •k11 ( 1 ,2) k-11(1,3) kll ( 1,4) 21 k11(2 ,1) k 1 1 ( 2 , 2 ) k 1 l ( 2 , 3 ) kll ( 2,4) 23 kl 1 ( 3 , 1 ) kl 1 ( 3 , 2 ) ki 1 ( 3 , 3 ) ki 1 ( 3 , 4 ) 24 • k l l ( 4 , 1 ) k11(4 ,2) k n ( 4 , 3 ) ki1(4,4) 25 • Pig '(3«2). unsymmetric non-linear member • stiffness matrix. Pig ( 3 . 3 ) shows that portion of the non-linear structure stiffness matrix t'Kjjj] in which the elements of member matrix 10 and 11 are located. It i s only necessary to check the symmetry for off-diagonal terms. Of the off-diagonal terms i t i s only necessary to check terms containing the elements ( 1 , 2 ) , ( 2 , 2 ) , (3,4.)' and ( 4 , 3 ) from the member stiffness matrix as a l l other off-diagonal terms w i l l be symmetric because the corresponding terms i n the member stiffness matrix are symmetric. Symmetry can thus be checked by writing the difference between corresponding elements i n the structure stiffness matrix. Therefore the following equations must hold: k 1 0 ( 4 , 3 ) - k 1 0 ( 3 , 4 ) = 0 k1.1 ( 2 , 1 ) - k11 ( 1 , 2 ) = 0 • k l l ( 4 , 3 ) + k l 2 ( 2 , l ) - kl 1 ( 3 , 4 ) k l 2 ( 4 , 3 ) + k i 3 ( 2 , i ) - k l 2 ( 3 , 4 ) k 1 2 ( l , 2 ) = 0 k 1 3 d , 2 ) = 0 . . . . (3-05) . . . . ( 3 . 0 6 ) . . . . ( 3 . 0 7 ) (3-08) 22. o o 1^- OJ OJ n n T— KA KA ^- -=f OJ "*» » v ^ V. S V. s OJ OJ OJ OJ KA OJ KA .X. X + AJ X AJ A; + • N s—s s. y—N v-KA KA KA r-^t- OJ VO OJ KA ^ V - v • OJ V s OJ KA OJ OJ OJ KA X X —^ X X X X + .—, .—. .—•% • ,—^ ,—. s—^ •=t- OJ • ^t- OJ OJ OJ •V i n OJ KA v- -sf OJ KA V *• •v s V ' ^ ^ V ' • ^ ' OJ T — O r— •<- OJ i - OJ OJ OJ T — T- ^ —^ Ai X X X + X X A! . -^^  -^^  ^-^ KA KA KA ^-•» •» T — OJ K\ T - J ^ - OJ KA .=*-. »—* o v -* V -« ^ • N S V • *—' OJ r— T— ^- OJ •c- OJ OJ OJ —^ -r — X A! A! •+ Ai Ai + Ai Ai OJ OJ OJ OJ KA T- OJ ^ " V ' S V ' OJ o T— T— T— X X -^^  .—, ' — X •* * n OJ OJ. V—^  o o o OJ o Ai o 5 X o X KA KA KA v- KA ^ * • * •» T- OJ KA T- -=t OJ KA V ' v. S OJ o o . o — o T~ T— T— A! X X + r — ^— Ai X X X Ai OJ s— .—s .—^  ^ OJ OJ OJ •=1- -~ o •> OJ K\ OJ KA O ^r- ^ v-^ o OA *-X X o s CD 5 N ' . T — T— r<^  KA . •> -> •V •> OJ CA ^t- — ' W O + O OA r-X X + O X o X OA o T— OJ KA m VO t~ OJ OJ OJ OJ OJ OJ OJ OJ ROWS Pig. J>.2 midportion of non-linear structure stiffness matrix [K,.M] 23. Substitution of the values of the elements from the matrices k^ and i n equa-tion (3-05) gives: T 1i o 4/L?o + R 1 i o 4 / L?o. + ( c 1 i o - D 1 i o } x i o yic/ L?o -- [ - H 1 i o x ? c / L ? o + T 1 i o y ? c / L i o + ( G 1 i o - D 1 i o } xio yic/ L?o ] = 0 ( 3 ' ° 9 ) 2 2 2 By noting that x + y = L and substitution of the values for T1^Q and R1^Q the above equation becomes: V L M 1 Q + - g - = 0 . . . . (3-10) In equation (3.10) + V ^  ^  i s the primary moment at the right hand end of the 10th element, which i s the top pin. The primary moment i n the top pin i s zero, thus equation (3.10) i s satisfied. Therefore k10(4,3) i s equal to k10(3*4). Similarly equation (3-06) can be shown to be satisfied. Substitution of the values of the elements from the matrices k^ and k^ i n equation (3«07) gives: ( Q 1 1 2 - S 1 1 2 ) x 2 ^ - ( R l ^ + T 1 1 l } x 2 / ^ - ( s i 1 2 - Q I 1 2 ) y?2/ L? 2 ~ ( T 1n + ^ 1 1 ^ y ? / L i i " N n e ' = 0 ' 2 2 2 By noting that x + y = L and substitution of the values for Q l ^ ' ^ 1 2 ' R1^^,T1 the above equation becomes: ( M12 " V12 ~T ) ' ( M11 + V11 ~T ) - N n e ' = 0 . . . . ( 3 . 1 2 ) In equation (3.12) the term i n the f i r s t bracket i s the primary moment at the l e f t end of the 12th segment and the second bracket i s the primary moment at the right hand end of the 11th segment (see Pig(2 . 3 ) ) . Hence equation (3.12) i s the equation of rotational equilibrium for the joint about the z-axis and w i l l be automatically satisfied i n the primary for-2 4 . ce a n a l y s i s . S i m i l a r l y equation (3.08) can be shown to be s a t i s f i e d . Therefo-re f o r a three-hinged arch i s symmetric. To summarize t h i s section, i t has been shown that i s symmetric and p o s i t i v e d e f i n i t e and i s symmetric f o r both two- and three-hinged arches; thus r e a l values of a e x i s t . 3.3 The I t e r a t i v e Methods of S o l u t i o n The problem i s formulated as an eigenvalue problem and the " Power Method of i t e r a t i o n " i s used i n modified form. This method converges to the value of a having the smallest absolute value and consequently i s only a p p l i c a b l e when the sign of « so obtained corresponds to the desired d i r e c t i o n of loading. The sign of the products "N^ and aN^ represents the d i r e c t i o n of e x t e r n a l l y a p p l i -ed loads. When the de s i r e d sign of a was obtained, the method proved to be the most e f f i c i e n t with a high degree of accuracy. The basic " Power Method" i s described by C r a n d a l l [5] and requires that the problem equation (3-04) be w r i t t e n : *ut*- -I K O K A .....(3.13-) and rewritten i n the form [" K0K V A = «-A •••• T h i s i s always po s s i b l e since Knir i s p o s i t i v e d e f i n i t e . The procedure i s then OK -1 to i t e r a t e [ - K Q K ^ ^ N ^ i n equation (3 .14) . However, the method i n t h i s form s u f f e r s from the disadvantage that i t de-stroys the narrow band c h a r a c t e r i s t i c s of and when i s in v e r t e d . [-KQ^ ^£N-I b e c o m e s a f u l l matrix and i t e r a t i o n on i t f o r a large structure can be very time-consuming. Therefore an a l t e r n a t i v e method was used which keeps the narrow band c h a r a c t e r i s t i c s of the matrices and was found to be more 25. e f f i c i e n t . In t h i s method a t r i a l vector A > normalized to a maximum value of o 1.0, i s s u b s t i t u t e d into the l e f t hand side of the equation ^ Ao = 4 K0K A1 . . . . 05.15) g i v i n g a vector % = - *LN A O Then solve the equation K Q K A = e .... .(3.16) f o r A„. I f the vector A, i s a s c a l a r multiple of A then A i s an eigenvector 1 1 ^ o o and the r a t i o A^/ A q i s the eigenvalue - . I f 4 i s not a s c a l a r m u l t i p l e of A , then use A„ as a new t r i a l vector, normalise i t , and repeat the procedu-o 1 r e . C r a n d a l l [5] shows that t h i s procedure converges to the eigen vector cor-responding to the eigen value 1/cx of l a r g e s t absolute value when i s sym-metric and p o s i t i v e d e f i n i t e and i s symmetric which has been shown i n sec-t i o n (3.2). . T h i s means that the method w i l l give the value of a ., with the smallest c r i t absolute value. Then the product aN^, °cN2, aM etc. w i l l give the smallest c r i t i c a l values of forces and stresses i n the s t r u c t u r e . The advantage of t h i s modified procedure l i e s i n the f a c t that the product - K y j A Q requires m u l t i p l i c a t i o n over a narrow band and also that i n the s o l u -t i o n of K^ „ A = e , KT i s symmetric and therefore the Choleski t r i a r i g u l a r i -s a t i o n method can be used. In t h i s method, a f t e r the f i r s t cycle when Knv i s t r i a n g u l a r i s e d , a l l successive solutions f o r e simply require m u l t i p l i c a t i o n o of "© onto a narrow band t r i a n g u l a r matrix. The solutions were found by a computer program using the U.B.C. L i b r a r y Subroutine BAND. The c a l c u l a t i o n s 26. were done on the I.B.M. 3 6 0 at the U n i v e r s i t y of B r i t i s h Columbia Compu-t e r Centre. Convergence i n the i t e r a t i o n was tested by i n v e s t i g a t i o n of the values of three d e f l e c t i o n s a f t e r each c y c l e . Three degrees of f r e e -dom were chosen i n which the d e f l e c t i o n s i n the expected mode shape would be strong. The r a t i o s of each term from the t r i a l vector to i t s correspon-ding term i n the s o l u t i o n vector gave three approximate «crit values a f t e r each c y c l e . The c r i t e r i a f o r acceptance of these as the required a c r i t va-lue were: a) The v a r i a t i o n of any one from t h e i r mean, f o r a p a r t i c u l a r cycle had to be l e s s than 0.1 0 / 0 . b) The v a r i a t i o n of the mean of the values from one cycle from the mean of the previous cycle had to be l e s s than 0.1 0 / 0 . The c r i t e r i a b) was introduced because i n cases where convergence was poor, due to the proximity of another eigenvalue, c r i t e r i o n a) would appear to be s a t i s f i e d to a high degree of accuracy but the mean value would jump about from c y c l e to c y c l e . With about h0 degrees of freedom the number of cycles required f o r the above standard of convergence was gene r a l l y about 8 f o r cases when moment dominated and the spring constant was small, and increased to about 8 0 when a x i a l f o r c e dominated and the spring constant was la r g e . The disadvantage i s t h a t the system converges to the lowest absolute value of a and i n some structures t h i s may'yield a negative & i n d i c a t i n g that the external load must be i n the opposite d i r e c t i o n to those assumed and N 2- While t h i s i s mathematically precise, i t i s p h y s i c a l l y i m p r a c t i c a l as the loads gene-r a l l y are g r a v i t y loads. When the above method of i t e r a t i o n f a i l e d to give a c r i t i c a l value o f a of the d e s i r e d s i g n because the reverse load on the arch would given the smal l e s t eigenvalue, another i t e r a t i v e method using the U.B.C. L i b r a r y Subrou-27. t i n e EVPOWR was applied. EVPOWR i s based on Householder's method d e s c r i -bed by Wilkinson [6 ] and requires the problem equation K 0 K A = " M K L N A ' ( 5 - l 6 ) Here i s a banded symmetric p o s i t i v e d e f i n i t e matrix and i s a ban-ded symmetric matrix. The advantage of t h i s procedure i s that a l l eigenva-lues can be determined. The disadvantage of the method i s that convergen-ce may be slower f o r close but not equal eigenvalues. The Determinant P l o t Method When the i t e r a t i o n s technique f a i l e d to give a c r i t i c a l value of « of suf-f i c i e n t accuracy, the determinant p l o t method was used. The determinant p l o t method i s simply to choose a s e r i e s of t r i a l a va-lues of ascending p o s i t i v e values, s u b s t i t u t e them i n t o S matrix and c a l -culate the value of the determinant f o r each a . The c r i t i c a l value of a i s the value at which the determinant i s zero. The method gives any value of « p o s i t i v e or negative. Generally the i -t e r a t i v e method had been t r i e d on the system before the determinant p l o t was used so the order of the eigenvalue was known and used i n the deter-minant p l o t . This was often the case f o r structures with high rise-span r a t i o , l a r ge spring constant and unsymmetric load. In chapter V numerical c a l c u l a t i o n s on various arches using the presen-ted theory to determine the c r i t i c a l load are done. In chapter 17 the theo-ry i s checked against model t e s t s . 28. . CHAPTER IV  EXPERIMENTAL MODEL TESTS 4.1 Introduction A closed form s o l u t i o n f o r the l a t e r a l buckling problem of an arch i s not av a i l a b l e to v e r i f y the t h e o r e t i c a l p r e d i c t i o n of the buckling load by s t i f f -ness matrices. Therefore model,tests were c a r r i e d out on two- and three-hinged parabolic arches with a span of 12 f t , r i s e of 2', depth of 5-3" and width 0.35" • The arches were made of glued laminated Douglas F i r with ^ r " lamination. The three hinged arch was made of the two hinged arch by c u t t i n g i t i n h a l f e s "and adding a hinge at the top. The arches were tested with and without spring e f f e c t along the top edge-4.2 Apparatus The general arrangement of the t e s t r i g i s shown i n Fig.(4.1) and the photos i n Fig.(4.6)-(4.7)• The arch i s supported on a s t i f f bench of timber. A timber frame with plywood as a backboard was r i g i d l y fastened and served to l a t e r a l l y brace the top edge of the arch. The arch was pinned to allow r o t a t i o n about the major and minor axes. T o r s i o n a l r o t a t i o n was prevented by sid e p l a t e s with small steel-rubber pieces i n the bearings, which supported the end of the arch. Trans-l a t i o n s were prevented by having a s p h e r i c a l s t e e l b a l l embedded i n sockets. D e t a i l s of arch support at the ends are shown i n Fig.(4.2). In the three-hinged arch a top p i n was placed at the top point and i n the c e n t e r l i n e of the arch. The top p i n allowed r o t a t i o n about the major and minor axes, When the l a t e r a l motion of the top edge i s constrained the t o r s i o n a l r o t a t i o n of the two cross-sections at each side of the top p i n w i l l be equal. A 5/8" p i n with a sharp t t -YL ply-backboarc 2 xZ, 'framing behind backboard .Top Pin (Figure (43)) 18 Load Points (Figure(4.5)) (Figure(£. ro vo Figure ( 4 o l ) General Arrangement of Test Rig point i n one h a l f of the arch i s f i t t e d i n t o a c o n i c a l hole i n a s t e e l p late i n the other h a l f . D e t a i l s of the top p i n are shown i n Fig.(4.3). L a t e r a l motion of the top edge of the arch was r e s t r a i n e d by aluminium s t r i p s with small pieces of rubber at the ends. These were h o r i z o n t a l l y f a s t e -ned to the back board and pinned to the arch by f i n i s h i n g n a i l s driven i n t o the top edge of i t . The n a i l s passed through holes and rubber i n the s t r i p s . The rubber prevented slippage i n the holes of the s t r i p s and ensured that no t o r -s i o n a l r e s t r a i n t was applied to the arch from the s t r i p s . Furthermore, an ar-rangement at the end of the s t r i p made i t possible to adjust the arch i n the v e r t i c a l d i r e c t i o n . These s t r i p s were placed at 8" i n t e r v a l s over the e n t i r e span. The t o r s i o n a l moment along the top edge of the arch was established by r a -d i a l 1/8" x 3/4" aluminium s t r i p s shown i n Fig.(4.4). One end of each s t r i p was fastened to the arch with a f i x e d connection, and the other end to a rod by a rubber cord i n a pinned connection. The rod was turned to sharp points at each end to f i t i n t o c o n i c a l holes d r i l l e d i n t o the s t r i p and i n t o a s t e e l p l a t e on the backboard. Therefore the s t r i p would not apply any t o r s i o n a l r e s t r a i n t perpendicular to the ce n t e r l i n e of the arch, or any in-plane bending moment. The t o r s i o n a l moment p a r a l l e l to the c e n t e r l i n e of the arch was thus obtained by the s t r i p as a pinned-fixed beam. The rod was made adjustable i n length to ensure that no spring e f f e c t was applied, when there was no t o r s i o n a l r o t a t i o n of the arch. The t o r s i o n a l s t r i p s were placed at 8" i n t e r v a l s over the e n t i r e span. Loads were applied to the arch by 18 hangers, as shown i n Fig.(4.5). Plates weighing 3«75* 1.87 and .94 l b each were used. The hangers were applied to the arch by a p i n as shown i n Fig.(4.5) so that no t o r s i o n a l r e s t r a i n t was provided from the hangers. The p i n had a sharp point and f i t t e d i n t o a c o n i c a l hole i n the s t e e l p late fastened to the top of the arch. rA /Arch Packing Rubber-steel Shearplate 2x2 x yc L 6 x 7 1 x 0 - 6 PL u-Vu 1 0 Steel ball seen from above Packing Section A-A, Front View Figure (4«2) Bearing Assembly 32 Seen from above o o .0 Side View 71 ! r" i—9--Figure (4.3) Top Pin Adjustable Rod A t- — i 1/" 3/ '8 x /U Aluminium S t r ip I y Backboard I - ~l i Arch X T1 Aluminium S t r i p Section A-A Rubberband / B a c k b o a r d N§B&= Figure (4.4) Torsional Restraint 34 Figure (4*5) Application of Load to Top of Arch 35. F i g . 4.7 General arrangement of t e s t r i g f o r three-hinged arch 36, 37. Fig. 4.11 P u r l i n arrangement 38. Fig. 4.12 Torsional restraint arrangement Fig. 4.13 Deflection measurement device 59-The d e f l e c t i o n s were measured using the equipment shown i n F i g ( 4 . 1 5 ) . The approximate t w i s t of the arch was measured by two d i a l gauges placed on each side of a bar clamped to the arch i n a point where the l a r g e s t r o t a t i o n s were expected. The a c t u a l angle of t w i s t was not required, only the r e l a t i v e values, during loading, i n order to f i n d the c r i t i c a l load from a Southwell p l o t . The d i a l gauges read d i r e c t l y to 0.001 M . Testing Procedure When s e t t i n g the s t a t i c a l l y indeterminant two-hinged arch i n the t e s t r i g great care was taken to avoid f o r c i n g the arch out of shape. Before mounting i n the t e s t bench t h i s arch was hung up h o r i z o n t a l l y i n three points, so the arch was unstressed. The center of the bearing's s t e e l b a l l was projected down i n the corresponding point of the support on the t e s t bench. The s t a t i c a l l y determi-nant three-hinged arch would always be unstressed i n the t e s t r i g . V e r t i c a l p o s i t i o n of the arch was obtained by the, adjustable aluminium s t r i p s . The arch was i d e a l i z e d as 19 segments. The loads were applied v e r t i c a l l y to the j o i n t between two segments, but the springs and the p u r l i n s were placed at each side of t h i s j o i n t as near as p o s s i b l e . Co-ordinates of the j o i n t s and section properties of the segments were c a r e f u l l y measured. Table (I) shows the d i f f e r e n c e between the co-ordinates of the J o i n t s f o r an exact p a r a b o l i c -arch and the t e s t - a r c h . The co-ordinates x,y r e f e r to a co-ordinate system : shown i n f i g ( 2 . l ) and f i g ( 2 . 2 ) . The hangers were placed at the top edge of the arch with a constant h o r i z o n t a l distance;thus the f u n i c u l a r curve from the dead load was close to a parabola. In t h i s case the i n i t i a l moments i n the arch were minimised. y for exact y for test X parabola parabola inch inch inch - 72.0 0.0 0.0 - 68.0 2.8 2.7 - 60,0 7.7 7.8 - 52,0 12.0 12.2 - 44.0 15.8 16.0 - 36.O 18.8 19.1 - 28.0 21.2 21.4 - 20.0 23.1 23-3 - 12.0 24.3 24.4 - 4.0 24.9 25.0 0.0 25.0 25.0 4-0 24,9 25.0 12.0 24.3 24.3 20.0 23.1 23.1 28.0 21.2 21.3 36.0 18.8 18.9 44.0 15.8 15.7 52.0 12.0 12.0 60.0 7-7 7.6 68.0 2.8 2.6 72.0 0.0 0.0 TABLE I Arch co-ordinates Three different load distributions were used as shown i n F i g ( 4 . l 4 ) . These were chosen so that axial force would dominate the problem i n loadcase A, moment M would dominate in loadcase B, and loadcase C would be an intermediate case. In loadcase B the deadload, liveload - ratio was approximately 0 . 2 , and in load-case C the ratio was approximately 0 ,14 , dependent on the spring.effect and the number of hinges. Loadcase A Loadcase B Loadcase C Figure (4.14) Load Distribution for Model Test The loads were applied stepwise with a magnitude of approximately 1/10 of the computed buckling load and increased up to approximately 60 0/0 of the buck-lin g load. For each load level deformation was measured after approximately 3 min.duration of the load. Increase in deformation after 3 min.loading was neg-l i g i b l e . The arches were tested with the various load distributions. The c r i t i c a l load was found using Southwell plot as discussed i n Timoshenko [J>] by plotting rotation 6 of the arch against 5/P. The c r i t i c a l value of P i s the cotangents of the slope of the straight line through the points i n the plot. Timoshenko shows that i n the testing of the buckling load of thin plates the Southwell plot can be used. A typical model Southwell plot of an arch i s shown in Fig ( 4 . 1 5 ) . The values of. EI and JG. were found by R.G. Charlwood [2] by clamping the central portion of the arch to a r i g i d bench, so that the end 4 ' - 0 " was a 02 01 s p © ^ 2 Hinged arch 5 . 2 5 x 0 . 3 x 2 . 0 Loadcase 1 k s = 0 © r c r = iy i* Per - -22.3 Lbs ^ © S 0 0.1 0.2 0.3 ( U 4=-Fig.(4.15) Model Southwell plot of a two-hinged arch, Loadcase 1, f/L = O.39 ^ 43. c a n t i l e v e r . A p o i n t l o a d was a p p l i e d t o the f r e e end and the r e s u l t i n g v e r t i -c a l d e f l e c t i o n and t w i s t were measured. Then a couple was a p p l i e d to the f r e e end and a prop used to prevent v e r t i c a l d e f l e c t i o n s , so th a t the t w i s t could be measured. EI and JG (see t a b l e I I ) were c a l c u l a t e d from these r e s u l t s simultaneously with-the h e l p of a plane g r i d computer program to i n c l u d e the e f f e c t of curva-t u r e of the c a n t i l e v e r . T h i s procedure was c a r r i e d out f o r each end of the arch and the average r e s u l t s were used. EI was 0.0398 k * i n c h and JG was 0.00885 2 k - i n c h . The t o r s i o n a l s p r i n g constant of the aluminium s t r i p s was found by f a s t e n i n g a pi e c e of arch w i t h a s t r i p t o a r i g i d bench, so the s t r i p was a c a n t i l e v e r beam. A p o i n t l o a d was a p p l i e d t o the f r e e end and the r e s u l t i n g v e r t i c a l d e f l e c t i o n was measured, from which the s p r i n g constant was c a l c u l a -t e d , (see f i g u r e (4.16)) .The s p r i n g constant k was 0.012 k i p s . s 3 E I lllllllllllllllll Figure (4.16) Determination of Spring Constant 44. 4.4 Experimental Results The arch data and t e s t r e s u l t s are summarized i n Table I I . The c r i t i c a l load i s expressed as a c r i t i c a l value of the maximum compressive s t r e s s . i n the bottom f i b r e s of each arch, °" ... The compressive s t r e s s i n the c r i t bottom f i b r e s was the sum of the compressive bending-and a x i a l s t r e s s . The percentage e r r o r between experimental and t h e o r e t i c a l r e s u l t s i s expres-sed as Model o - Theory a o/o E r r o r = - ° r ^ - x 100 o/o .... (4.01) ' Model a c r i t 2-Hinged arch. k a a Test Load s c r i t c r i t Error no. case [kips] model theory [0/0] [ksi] [ksi] 1 A 0.0 0.192 0.193 + 0.4 2 B . 0.0 0.204 0.207 + 1.3 3 C 0.0 0.210 0.214 + 1.9 4 A 0.012 0.>38 0.410 + 5.8 5 B 0.012 0.318 0.328 + 3.4 6 C 0.012 0.400 0.442 + 7.1 3-Hinged arch. Test Load k 0 a Error s c r i t c r i t no. case [kips]' model theory [0/0] [ksi] [ksi] 7 A 0.0 0.184 0.161 - 12.7 8 B 0.0 0.203 O.196 " 3.6 9 C 0.0 0.341 ^0.370 + 8.0 10 A 0.012 0.299 0.307 + 0.2 11 B 0.012 0.282 O.328 + 14.1 12 C 0.012 0.485 O.56O + 13.4 Arch data: Span L =12.0* Rise f = 2.0* Depth d = 5 - 3 " Breadth b = . 3 5 " E = 1990 ksi, G = 136 ksi TABLE II Test results 46. Discussion of Experimental Results The results were considered to be an acceptable verification of theory ta-king the complexity of the structure into account. The errors between the test results and the theoretical predictions are acceptable considering the determination of the structure parameters EI and GJ, the f r i c t i o n i n the pins and the eccentricities of the loads. When moment dominated the experimental results tended to be slig h t l y lower than the theoretical. The same was found in experiments with spring effect. The Southwell plots were linear up to about 60 o/o of the c r i t i c a l load for the two-hinged arches and then tended to curl downward. The downward curl of the test results was attributed to the effect of a hinge. When the two*"hinged arch tends to buckle sideways a redistribution of stresses i s possible. That portion which buckles sideways tends to lose i t s bending stiffness and create a hinge. This effect would decrease the stiffness of the arch and i n some cases could increase the buckling load. This i s seen from the experimental results (Table I I ) . In loadcase C the buckling load for a three hinged arch i s higher than the buckling load for a two hinged arch. Timoshenko [j>] shows the same effect of a hinge for in-plane buck-li n g of arches. The influence of an extra hinge i n a two-hinged arch i s shown i n Fig (4.17)* which i s a plot of the rotation o against the load N^. Where the curve breaks a hinge i s gradually developed. This effect was minimized by keeping the deflections as small as pos-s i b l e and only using the deflections before the break-point i n the South-well plot to determine the c r i t i c a l load. Also the amount of error obtai-ned by drawing a straight line through the experimental points i n a South-w e l l p l o t must be considered, when comparing the d i f f e r e n c e between theo-r e t i c a l and experimental r e s u l t s i n ta b l e ( I I ) . The experiment c a r r i e d out v e r i f i e d that the computer program, prepa-red i n accordance with the presented theory, was i n f a c t i n close agree-ment with the actu a l p h y s i c a l conditions. Therefore no f u r t h e r t e s t i n g was deemed necessary and the program could be used to check various arch s t r u c -tures to produce some design c r i t e r i a . N2 lbs 2 Hinged arch 5 . 2 5 x 0 . 3 x 2 . 0 Loadcase 1 k s =0 3^* 0 4 / / / 0 / / r 4 / / © / / © / f © / / / / / © © 5 4^ / 1 i - r-, \ 48. CHAPTER V  THEORETICAL RESULTS 5.1 Introduction In this chapter numerical results using the computer, programs are presen-ted to show the behavior of two- and three-hinged glulam arch structures of a parabolic shape under varying load conditions and structure parame-ters . 5.2 Structure Parameters The c r i t i c a l load condition was evaluated as a c r i t i c a l value of the com-pressive stress i n the bottom fibres, due to primary forces at the point of maximum compressive bending stress in bottom of the cross-section. In cases where the moments were zero everywhere, the stress at the quarter span point was used for both two- and three-hinged arches. The c r i t i c a l stress, o" can be found as: cr a = o + a (5.01) cr b a v ' where o i s the maximum compressive bending stress and, o the direct com-pressive stress at the point of maximum compressive bending stress. The c r i t i c a l stress must be a function of the following parameters: a = f [ L , f, d, b, E, G, k , loadcase] (5-02) cr s In the above expression L, f, d and b are defined in Fig(2.1). E and G are Young's modulus and the shear modulus respectively, k i s the spring s 49. c o n s t a n t . Loadcase i s a parameter d e f i n i n g the l o a d d i s t r i b u t i o n on the a r c h . U s i n g the Buckingham u theorem the n i n e parameters i n e q u a t i o n (5 .02), w h i c h c o n t a i n s two p r i m a r y dimensions ( f o r c e and l e n g t h ) , can be r e l a t e d i n seven d i m e n s i o n l e s s g r o u p s . These d i m e n s i o n l e s s groups w i l l then d e -f i n e the problem c o m p l e t e l y . I n the f o l l o w i n g i n v e s t i g a t i o n s these were chosen as ; • 2 (5.03) a c r = f E (£•)' (lO* (z)' (l)' (ifr X a A )_, b where oi/( a + a, ) i s used to d e f i n e d i f f e r e n t l o a d c a s e s . T h i s r a t i o i s . V v a b ' a convenient parameter t o g i v e an approximate r e l a t i o n s h i p between the l o a d c a s e s , but i s not s u f f i c i e n t : t o : d i s t i n g u i s h between i n d i v i d u a l l o a d -c a s e s . The b e h a v i o r of C c r / E was found by v a r y i n g the s i x parameters on the r i g h t - h a n d s i d e o f e q u a t i o n (5.03) w i t h i n the l i m i t s of p r a c t i c a l d e s i g n u s i n g g l u e d - l a m i n a t e d t i m b e r . These are v a r i e d as f o l l o w s : j- = 0.17, 0.30, 0.39 ^ = 3-0, 4 . 0 , 5-0, 7 .0, 10.0 ^ = 28.8, . 41.1, 72.0, 100 E -= =• 14 ( c o n s t a n t ) l i k L 2  S = 0 - 1000 E I a b = o - 1.09 a. + a b E / G was s e t t o 14 which corresponds t o E = 1600 k s i and G .= 1140 k s i f o r l a m i n a t e d Douglas F i r . The v a r i a t i o n of cr^ / a& + f o r the. d i f f e r e n t 50. load cases i s shown later. Theoretical Results It i s impractical to determine an equation linking the c r i t i c a l stress with a l l the parameters defined herein. For this reason the c r i t i c a l stress was found as a function of k L /EI for various values of f/L,d/b, & L/d and different load distributions. The load distributions investigated are shown i n Fig(5.l). The c r i t i c a l stress for the arch was found by considering the'arch as 20 segments and using the methods discussed in chapter 3« Primary forces were found using a linear plane frame program; non-linear effects were not included. The results of the analysis are given i n table (III).(pp.55-69] 2 Fig(5.2) - Fig(5.5) show the function between a /E and k L /EI for CA S various parameters and for two common loadcases, loadcase 1 and 2. It i s seen from the graphs that a c r / E varies from a certain i n i t i a l value corre-sponding to the spring constant equal zero to i n f i n i t y for the spring; con-stant approaching i n f i n i t y . For a practical designed arch cr /E w i l l not exceed a proximately 5.4 • 10 . If ° /E exce ds 5.4 • 10 "''buckling stress cr fc -"•^  i s much greater than the allowable stress i n the material, then failure w i l l occur by tearing of the fibres by in-plane bending or failure i n the material. I t i s not necessary to provide spring, constants higher than a le stress i s set at f, = 3.300 ksi and E = 1800 ksi and a factor of safe-D ty about 3-0. In an economically designed arch the service stress w i l l be approxima-tely 2.400 ksi corresponding to °"/E = 1.5 * 10 . It i s seen from the value corresponding to approximately three times the allowable stress. -3 The upper l i m i t of a /E w i l l then be 5-4 • 10 when the highest allowab-51. graphs that l a t e r a l buck l ing w i l l occur f o r arches where a c r / E i s about - 3 1.5'10 f o r smal l spr ing .cons tants .By comparing the graphs f o r two- and three-hinged arches i t can be seen as expected, that f o r h igh values of the spr ing constant the behaviour of two- and three-hinged arches i s the same, when the primary moment d i s t r i b u t i o n i s the same, because the i n -f luence of the top-hinge i s n e g l i g i b l e f o r modeshapes with many waves. a as a f unc t i on of the spr ing constant k can be w r i t t e n : c r -s a a a c r ocr ker /c nc\ — = — + - E - . •••• < 5 - 0 5 ) a a a. ocr c r ker where —=— i s the value of —=r- , when k i s zero and —=— i s the c o n t r i -ii Jl S Jl but ion from the spr ing, e f f e c t ° k k B L i s a f unc t i on of E E I  T 2 E Thus o ker = f "k L _s EI (5.06) o o c r ocr „ — = — + f •k L' s EI 2 ... (5.0?) Because of the mult i tude of parameters invo lved i t was not poss ib l e to f i n d a r e l a t i v e l y simple func t ion between the c r i t i c a l s t r e ss and the por t i o n conta in ing the spr ing constant ,but i t was poss ib l e to f i n d a simple r e l a t i o n s h i p between the c r i t i c a l s t r e ss and the po r t i on which does not conta in the spr ing, constant . By p l o t t i n g ° c r / E versus b /Ld and mainta in ing a l l other parameters : constant a l i n e a r p l o t was obta ined. Th i s r e s u l t can be expressed i n the fo l l ow ing equat ion: ^ ~ = c £ . . . . (5.08) E o Ld x ' 52. Here the constant v a r i e s w i t h l o a d cases and r i s e / s p a n r a t i o and number 2 of h inges. Some t y p i c a l p l o t s of a o c r / E versus b /Ld are gi v e n i n P i g (5.6), where the slope of the s t r a i g h t l i n e s i s C q . Major v a r i a t i o n s i n c can be seen t o e x i s t between l o a d cases. 0 . . . . • I n order t o o b t a i n a r e l a t i o n which w i l l g i v e o f o r a l l loadcases c r % °h the dimensionless parameter — — = — was intr o d u c e d . The terms °_ V G a °T b and 0" are d e f i n e d i n s e c t i o n 5-2. T h i s r a t i o was chosen because i t w i l l a represent a l l common loadcases i n the range 0-1.05. The a x i a l s t r e s s G 3. and the bending s t r e s s o can be expressed as a = + JL a - bd 6M a _ b " b d 2 . . . . (5.11) where P and M are a x i a l . f o r c e and moment. The r a t i o w i l l then be: •6M o 2 b bd 1 f c - A O \ ° + a ~ 6M + P + P d . . . . KO.i*) h a 7 3 - bd 1 - M ' Z bd For a two- or three-hinged p a r a b o l i c arch the maximum bending .moment oc-curs f o r loadcase 2 and i s * T 2 max 5 5 The corresponding a x i a l f o r c e w i l l be approximately 2. • ' P « 774-- T ^ r (5.14) max O.066 1of . Using the expre s s i o n f o r M and P i n equation (5.12) the r a t i o w i l l max max be o . 53-The depth/span r a t i o f o r p r a c t i c a l arches w i l l be of the order 0.01-0.04. Equation (5.14) w i l l then be CT . CT b b " 1 = 0.98 .... (5.16) ° +°" a -i + 0.022 b a J-For loadcase 1 where the bending s t r e s s i n a parabolic arch i s zero, °\/ a^ i s zero and f o r loadcase 4 (upwards load) where the s t r e s s from a x i a l f o r -ce can be negative a-^/a^ > 1.0. To f i n d how C q depends on the loadcase, C q was p l o t t e d against a /o^ . The values of c and a^/ a r T, f o r various load combinations and f / L arid L/d o b' T r a t i o s are given i n table IV. The graph i s shown i n fig(5-8) where c i s seen to be a function of o a-^/arj, £°r constant f/L value. The r a t i o L/d i s seen from table IV to have n e g l i g i b l e influence on C q i n the range of p r a c t i c a l arches. For small values of ^ / ^ * c 0 v a r i e s l i t t l e with respect to rise/span r a t i o . 54 Load Case Stress Condition tfb Ob -KJa Loading Distribution. 1. o" « max at ends a % - 0 0 2. <^  . max at 3/8 span .88-.98 ••i i i i i i i in i i ! 1 i 3. . max at midspan .92 -.98 • • • I • • • I 4. Wind loading as per A.S.C.E. Report [7] 1.05-1.IC p = l p s " i P = - i 5. Point Load 0.98 6. Combination of Loadcase 1 and 2 (variable) 0 - 1.0 IIIIUillllllllllHIIIIIIIilllLm mm ,„, t—' — 1 — t Figure (5.1) Load Distributions. Table I I I . T h e o r e t i c a l R e s u l t s . Hinges Load L f d b f/L d/b a cr c r c r k L 2 case in. in. i n . in. [kip] [kips] s EI 2 1 1 4 4 . 0 5 6 . 0 1.5 0 . 5 0 . 3 9 3 .0 0 . 0 1.468 0. 916 0. 0 0.002 1.581 0. 989 1.6 • 0. 004 1.695 1.060 3.2 0. 006 1.806 1 .129 4 . 9 0. 008 1.901 1.191 6 .5 0.01 2.005 1. 252 8.1 0. 02 2. 395 1 .499 1 6 . 0 0. 03 2. 685 1. 680 24-0 0 .04 2 .915 1. 820 3 2 . 0 0 .05 3 . 3 2 0 1. 950 40 .4 0. 06 3. 300 2.065 4 9 . 0 0. 08 3. 600 2. 250 65 .0 0. 10 3. 860 2 . 4 1 5 8 1 . 0 0. 12 4. 090 2 . 5 5 5 9 7 . 0 3 1 1 4 4 . 0 5 6 . 0 1.5 0 . 5 0 . 3 9 3 .0 0. 0 1.236 0.776 0. 0 0. 002 1.381 0. 865 1.6 0.004 1.528 • 0 . 9 5 5 3.2 0.006 1.642 1.029 4 . 9 0. 008 1.762 1.101 6 .5 0. 01 1 .890 1. 182 8.1 0. 02 2. 350 1.470 16 . 0 0. 03 2. 680 1.680 2 4 . 0 0.04 2. 930 1.831 32 . 0 0 .05 3 . 1 3 0 1.956 40 .4 0 .06 3. 300 2.070 4 9 . 0 0 .08 3.610 2. 260 65 .0 U l U l T h e o r e t i c a l R e s u l t s H i n g e s Load L f d b f / L d/b k a c r a c r k L 2 c a s e i n . i n . i n . i n . s [ k i p ] [ k i p s ] — 1 0 s E I 3 1 144.0 56.0 1.5 0.5 , 0. 39 3-0 0. 10 3. 380 2.425 81 . 0 0. 12 4. 090 2. 560 97 . 0 0.14 4. 320 2.700 113 . 0 0. 16 4. 530 2. 840 129 . 0 0.18 4.720 2. 950 145 .5 0. 26 4. 900 3.060 209. 0 0. 28 4.950 3.100 226. 0 0. 30 4. 960 3. 120 242. 0 0. 32 5. 020 "3.140 258 . 0 0. 34 5. 060 3.160 276 . 0 0. 36 5. 110 3.195 290. 0 • 0. 38 5. 160 3.215- 306. 0 2 2 144.0 56.0 1.5 0.5 0.39 3.0 0.0 4.459 2.795 0.0 0. 002 4.680 2.925 1.6 0. 004 4. 860 3. 040 3.2 0. 006 5.030 3.145 4.9 0.008 5.180 3.240 6.5 0.01 5- 340 3. 330 8.1 0. 02 6.010 3.770 16. 0 3 2 144.0 56.0 1.5 0.5 0 .39 3.0 0. 0 4.295 2.685 0.0 0. 002 4.510 2. 820 1.6 0.004 4.730 2.955 3.2 0. 006 4.930 3. 080 4.9 0. 008 5. 120 3.200 6.5 0. 010 5. 270 3. 300 •8.1 0. 02 6.010 3.760 16 . 0 c T h e o r e t i c a l R e s u l t s . Hinges Load L f d b f/L d/b O ' e r O ' e r * k L 2 case in. in. i n . in. [kips] s EI 3 2 144.0 56.0 1.5 0.5 J 0. 39 3.0 0.03 6. 600 4. 120 24.0 0.04 7. 100 4.440 32.0 0. 05 7.550 4.720 40.4 0. 06 7. 980 4.990 49.0 0. 08 8.700 5-440 65.0 0. 10 9. 340 5. 820 81.0 0. 12 9. 910 6.190 97.0 0.14 10.420 6.510 113.0 0.16 10.900 6.810 129. 0 0.18 11.340 7. 080 145.5 0. 20 11.790 7.370 162.1 0. 22 12.160 7.600 ' 178.0 0. 24 12.520 7. 830 194. 2 0. 26 12.830 8.020 209.0 2 1 144.0 56.0 2.0 0.4 0.39 5.0 0. 0 .664 0.415 0. 0 0.05 1 . 620 1.015 60. 8 3 1 144.0 56.0 2.0 0.4 0.39 5.0 0. 0 .559 0. 349 0.0 0.05 1.620 1.015 60.8 2 2 144.0 56.0 2.0 0.4 0.39 5.0 0.0 2.182 1.368 0.0 3 2 144.0 56.0 2.0 0.4 0.39 5.0 0.0 2. 090 1.306 0.0 0.05 4. 130 2.585 60.8 T h e o r e t i c a l R e s u l t s . Hinges Load case L in. f in. d i n . b in. f/L d/b [kip] 0" cr [kips] Ccr <> — 1 0 k L 2 s E I 2 1 144.0 56.0 2.0 0.5 , 0. 39 4.0 0. 0 . 995 0. 622 0. 0 0. 01 1. 292 0. 808 6.5 0.015 1.419 0. 886 9.7 0. 020 1 • 530 0. 956 13.0 0. 025 1. 630 1.018 16. 1 0 . 0 3 1.719 1.072 19.0 0.04 1.872 1. 170 26. 0 0. 05 2. 000 1.251 32.0 3 1 144.0 56.0 2.0 0.5 0.39 4.0 0. 0 0. 01 0.015 0. 02 0. 025 0. 030 0. 040 0. 05 0. 06 0.08 0. 10 0. 12 0.14 0. 16 0.18 0. 20 0. 22 0. 24 0. 26 .838 1.192 1.345 1.475 1.595 1 .692 1.865 2. 000 2.120 2. 315 2. 480 2. 620 2.760 2. 880 2. 995 3. 100 3.210 3. 300 3.390 0. 525 0.745 0. 841 0. 927 0. 996 1.058 1. 165 1. 251 1. 325 1.446 1 .550 1.640 1.725 1.800 1.870 1.938 2.015 2. 060 2. 120 0. 0 6.5 9.7 13.0 16.1 19.0 26. 0 32.0 39.0 52.0 65.0 • 80.0 90.0 103.0 116.0 129. 0 142.0 155.0 168.0 Theoretical Results. Hinges Load case L i n . f i n . d i n . b i n . f/L d/b k [kip] Ocr [kips] o c r 10 -3 k L' s E I 144.0 56.0 2.0 0.5 0. 39 4.0 0 . 28 0 . 30 0 . 32 0 . 34 0 . 36 0 . 38 3.460 3.550 3.630 3.700 3.780 3.850 2.160 2. 220 2 . 270 2 . 3 1 0 2. 360 2 . 4 1 0 1 8 1 . 0 1 9 3 . 0 2 0 6 . 0 2 2 0 . 0 232.0 2 4 5 . 0 144.0 56.0 2.0 0.5 0.39 4.0 0 . 0 0 . 0 1 0 . 02 0 . 03 0 . 04 0 . 0 5 3. 2 9 0 3. 8 1 0 4. 235 4 . 5 8 0 4. 8 9 0 5 . 1 7 2 2. 050 2. 380 2. 640 2.870 3. 060 3. 240 0 . 0 6.5 1 3 . 0 1 9 . 0 26. 0 3 2 . 0 144.0 56.0 2.0 0.5 0.39 4.0 0 . 0 0 . 0 1 0 . 02 0.03 0 . 04 0 . 05 0.06 0 . 08 0 . 10 0 . 12 0 . 1 4 0 . 1 6 0 . 18 0 . 20 3.150 3.750 4. 200 4.570 4. 890 5.172 5.440 5.900 6. 310 6. 660 7. 000 7. 320 7. 590 7.850 1.970 2. 340 2. 620 2. 860 3. 060 3. 240 3.400 3.690-3-940 4.160 4. 370 4. 560 4.740 4. 900 0 . 0 6.5 1 3 . 0 1 9 . 0 26.0 3 2 . 0 3 9 . 0 52.0 65.0 8 0 . 0 9 0 . 0 103.0 1 1 6 . 0 1 2 9 . 0 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f / L d/b [ k i p ] 0" c r [ k i p s ] O c r « E 10 k L 2 S EI 3 2 144.0 56.0 2.0 0.5 , 0.39 4.0 0. 22 0. 24 0. 30 0. 36 0.40 8. 100 8. 330 8. 980 9. 540 9. 870 5. 060 5. 200 5-620 5.960 6.170 142. 0 155.0 193.0 232.0 258.0 2 1 144.0 56.0 3.5 0.35 0.39 10.4 0.0 0.01 0. 02 0.03 0. 04 0. 06 0. 08 0. 10 0.12 . 178 .254 . 308 . 344 . 374 .401 • 465. . 502 .535 0.135 0. 158 0. 193 0. 215 0.234 0. 251 0. 291 0. 314 0.334 0.0 10. 3 20. 6 30. 9 41.2 62.0 83-0 103.0 124.0 3 1 144.0 56.0 3.5 0.35 0.39 10.0 0. 0 0. 01 0.02 0. 03 0. 04 0. 06 0.08 0. 10 0.14 0.16 0.18 0. 20 .148 . .241 . 302 .344 .376 .426 .466 .536 .566 .595 . 622 .647 0.111 0.151 0.189 0. 215 0.235 0.266 0. 291 0.335 0.354 0.372 0. 389 0. 405 0 10. 3 20. 6 . 30. 9 41.2 62. 0 83.0 103.0 144.0 165.0 186.0 206. 0 Theoretical Results. Hinges Load L f d b f/L d/b O c r O c r - k L 2 case i n . i n . i n . i n . [kip] [kips] ~ 1 0 s EI 3 1 144.0 56.0 3.5 0.35 , 0.39 10.0 0. 22 .672 0.421 227.0 0. 24 .694 0.434 248.0 0. 26 .714 0.446 268.0 0. 30 .754 0.472 309. 0 0. 40 .836 0. 522 413.0 0. 50 .887 0.554 516.0 0. 60 .925 0. 578 620. 0 0. 70 . 940 0. 586 723.0 2 2 144.0 56.0 3.5 0.35 0.39 10.0 0.0 . 926 . 579 0. 0 0. 01 1.142 .714 10. 0 0. 02 1. 296 . 808 20. 0 0.03 1.421 .889 31.0 ,0. 04 1. 530 .956 41.0 0. 06 1.712 1.070 62. 0 3 2 144.0 56.0 3.5 0.35 0.39 10.0 0. 0 0.885 . .553 0. 0 0. 02 1.298 .811 20. 0 0.03 1.423 . 890 31.0 . 0. 04 1.534 .958 41.0 0.06 1.713 1.071 62. 0 0. 08 1.860 1.161 83.0 0. 10 1.995 1.245 103. 0 0.12 2. 125 1.328 124.0 0. 14 2. 220 1. 390 144.0 0. 16 2.315 1.447 165.0 0.18 2.410 1. 505 186.0 0. 20 2.490 1.555 206. 0 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f / L d/b k g [ k i p ] a c r [ k i p s ] a c r . E ~ 1 0 k L 2 s EI 3 2 144.0 56.0 3.5 0.35 < 0.39 10.0 0. 22 2. 560 1 . 600 227. 0 0.24 2. 640 1.650 248.0 0. 26 2.700 1.688 268.0 0. 28 2.765 1.728 289.0 0. 30 2. 830 1.768 309. 0 0.40 3.090 1.934 412.0 0. 60 3.470 2.170 618.0 0. 70 3.610 2. 260 722. 0 2 1 144.0 56.0 3.5 0.875 0.39 3.0 0.0 0. 001 0.005 0.010 0.025 0.03 0. 04 0.05 0. 06 0. 08 0. 10 0. 12 0. 14 0. 16 0.18 0. 20 0. 22 0. 26 0. 30 0. 34 1.475 1.482 1.499 1.523 1.591 1.616 1.660 1.700 1.720 • 1.826 1. 905 1.972 2.045 2.105 2. 170 2. 225 2. 285 2.385 2.480 2. 560 0. 922 0. 926 0.937 0.954 0. 006 1.011 1.038 1. 062 1.075 1.042 1. 190 1.234 1.279 1.316 1.357 1.391 1.429 1.491 1.550 1.600 0.0 0. 1 0. 3 0.7 1.7 2.0 2.7 3.3 4.0 5.3 6.7 ' 7.9 9-3 10.7 12.0 13.3 14.6 .17.3 20. 0 22.6 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f/L d/b [ k i p ] o'er [ k i p s ] k L 2 s EI 3 1 144.0 5 6.0 3 . 5 0 . 8 7 5 . 0 . 3 9 3.0 0 . 0 1 . 2 1 6 0 . 761 0 . 0 0 . 0 2 1 . 3 3 5 0 . 8 3 4 1 . 3 0 . 0 3 1. 392 0 . 8 7 0 2 . 0 0 . 0 4 1 . 4 4 8 0 . 9 0 5 2 . 7 0 . 0 5 1 . 5 0 3 0 . 942 3 . 3 0 . 0 6 1 . 5 5 8 0 . 9 7 4 4 . 0 0 . 0 8 1 . 6 5 8 1 . 0 3 7 5 . 3 0 . 1 0 1 . 7 5 8 1 . 1 0 4 6 . 7 0 . 1 2 1 . 8 5 0 1.157 7 . 9 0 . 14 1 . 9 3 2 1 . 2 0 8 9 . 3 0 . 16 2 . 0 1 8 1 . 2 6 0 10 .7 0 . 18 2 . 0 9 5 1 . 3 0 8 1 2 . 0 0 . 2 0 2.165 1.351 13 .3 0 . 2 2 2 . 2 3 5 1 . 3 9 5 14 .6 0 . 24 2 . 2 9 5 1 . 4 3 2 16 .0 0 . 2 6 2 . 3 5 5 1 . 4 7 0 17 .3 0 . 2 8 2.410 1 . 5 0 5 18 .7 0 . 3 0 2 . 4 6 5 1 . 5 4 0 2 0 . 0 2 2 144.0 5 6.0 3 . 5 0 . 8 7 5 0 . 3 9 3.0 0 . 0 0 . 0 1 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6 0 . 0 8 0 . 1 0 5 . 4 6 0 5 . 5 5 0 5 . 6 6 0 5 . 7 6 0 5 . 8 4 0 5 . 9 9 0 6 . 0 0 0 6 . 1 4 0 6 . 3 3 0 3.410 3 . 4 8 0 3 . 5 4 0 3 . 6 0 0 3 . 6 5 5 3 . 7 4 5 3 . 7 6 0 3 . 8 4 0 3 . 9 5 5 0 . 0 0 . 7 1 . 3 2 . 0 2 . 7 3 . 3 4 . 0 5 . 3 6 . 7 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f / L d/b k s a c r a c r ^ k L 2 s [ k i p ] [ k i p s ] EI 2 2 144.0 56.0 3.5 0.875. 0.39 3.0 0. 12 0.14 6 . 4 7 0 6. 6 0 0 4. 0 4 0 4.135 7.9 9.3 3 2 144.0 56.0 3.5 0 . 8 7 5 0 . 3 9 3.0 0.0 0 . 0 0 5 0 . 0 1 0. 0 2 0. 03 0 . 0 4 0. 10 0 . 1 2 0.14 0.16 0. 20 0. 22 0. 24 0. 26 0. 28 0. 30 5. 223 5. 2 8 0 5. 3 7 0 5 . 4 7 0 5 . 5 7 0 5.710 6. 235 6. 4 0 0 6 . 5 4 0 6 . 6 8 0 6. 9 6 0 7 . 0 7 0 7. 2 0 0 7. 320 7 . 4 3 0 . 7 . 5 5 0 3. 2 7 0 3. 310 3. 3 6 0 3.420 3 . 4 8 5 3 . 5 7 0 3. 9 0 0 4. 000 4 . 0 8 0 4.175 4. 3 5 0 4. 420 4. 5 0 0 4. 5 7 0 4 . 6 5 0 4 . 7 3 0 0.0 0. 3 0. 7 1. 3 2.0 2.7 6.7 7.9 9.3 10.7 13.3 14. 6 16.0 17.3 18.7 20.0 2 1 144.0 56.0 5.0 0.5 0.39 10.0 0.0 0 . 0 1 0. 0 2 0 . 0 3 0. 04 0 . 0 5 0. 08 0. 10 0.431 0 . 4 7 8 0 . 5 2 4 0 . 5 6 4 0. 601 0 . 6 3 4 0 . 7 1 6 0 . 7 5 7 0. 269 0.299 0. 328 0 . 3 5 3 0. 376 0. 396 0 . 4 4 7 0 . 4 7 4 0. 0 2.5 5.0 7.4 9.9 12.4 19.8 24.8 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f/L d/b [ k i p ] O c r [ k i p s ] O c r i k L 2 s E I 2 2 1 4 4 . 0 56.0 5.0 0.5 0. 39 1 0 . 0 0 .08 1 . 6 8 8 1 . 0 5 6 1 9 . 8 0 . 10 1.770 1 . 1 0 8 2 4 . 8-0 . 1 5 1 . 9 5 0 1. 221 3 7 . 2 0 . 20 2 . 1 0 5 1. 316 4 9 . 6 0 . 25 2. 240 1 . 4 0 0 62 .0 0 . 30 2. 370 1 .481 7 4 . 5 0 .40 2. 590 1 . 5 5 5 9 9 . 3 3 2 1 4 4 . 0 56.0 5.0 0.5 0 . 3 9 10.0 0.0 0 . 01 0. 02 0 . 03 0 . 04 0 . 05 0 . 0 8 0 . 10 0 . 12 0 . 15 0 . 20 0 .25 0. 30 0 . 4 0 0 . 50 0 . 80 1 . 0 0 1 . 20 1 . 4 0 1 . 6 0 1 . 1 7 7 1.270 1 . 3 5 0 1 .421 1 . 4 8 5 1.540 1 . 6 9 0 1.772 1 . 8 4 8 1 . 9 5 3 2 . 1 1 0 • 2 . 250 2-375 2 .590 2 .780 3. 270 3.530 3 . 7 6 5 3 . 9 7 0 4 . 1 5 0 0 . 7 3 7 0 . 7 9 5 0 . 8 4 4 0 . 889 0 . 930 0 . 963 1.056 1 . 0 7 8 1 . 1 5 3 1 .221 1. 317 1.406 1 . 4 8 3 1 . 6 1 8 1 . 7 3 8 2 . 0 4 5 2 . 207 2 . 3 5 5 2 . 4 8 0 2 . 5 9 0 0 . 0 2 . 5 5 . 0 7 . 4 9 . 9 1 2 . 4 1 9 . 8 2 4 . 8 2 9 . 8 3 7 . 2 4 9 . 6 62 .0 7 4 . 5 9 9 . 3 1 2 4 . 0 1 9 8 . 2 2 4 8 . 0 2 9 8 . 0 3 4 7 . 0 397.0 T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f/L d/b k s acr a cr ^ E - 1 0 k L 2 s [ k i p ] [ k i p s ] EI 2 1 144.0 56.0 5 . 0 0 . 5 , 0.39 1 0 . 0 0 . 20 0 . 4 0 0 . 60 0 . 8 0 0 . 9 3 6 1.176 1.372 1.547 0 . 5 8 5 0 . 7 3 6 0 . 8 5 8 0 . 9 6 6 4 9 . 6 9 9 . 3 148.8 198. 2 3 1 144.0 56.0 5.0 0 .5 0 . 39 1 0 . 0 0 . 0 0 . 01 0 . 02 0.03 0 . 04 0 . 0 5 0 . 08 0 . 10 0 . 20 0 . 4 0 0 . 60 0 . 80 1 . 00 1 . 20 1 . 4 0 1 .60 0 . 346 0 . 4 0 6 0 . 464 0 . 516 0 . 5 6 4 0 . 606 0 . 7 0 9 0 . 760 0 . 9 3 4 1.175 1.373 1.547 1.701 . 1 .842 1.956 2 . 0 6 5 0.217 0 . 254 0 . 290 0 . 322 0 . 352 0 . 3 7 9 • 0 . 4 4 2 0 . 4 7 5 0. 584 0 . 7 3 4 0 . 862 0 . 9 6 8 1 .062 1.152 1.223 1 .290 0 . 0 2 . 5 5 . 0 7 . 4 9 . 9 1 2 . 4 19.8 24.8 4 9 . 6 9 9 . 3 148.8 198. 2 2 4 8 . 0 2 9 8 . 0 3 4 7 . 0 . 3 9 7 . 0 2 2 144.0 56.0 5.0 0 .5 0.39 10.0 0 . 0 0 . 01 0 . 02 0 . 0 3 0 . 04 0 . 05 1.229 1. 305 1.374 1 .436 1.491 1.542 0 . 7 6 8 0 . 816 0 . 8 5 9 0 . 8 9 6 0 . 9 3 3 0 . 964 0 . 0 2 .5 5 . 0 7 . 4 9.9 12 .4 Theoretical Results Hinges Load case L i n . f i n . d i n . b i n . f/L d/b k [kip] 0" c r [kips] a c r -r - 1 0 k L 2 s E I 3 2 144.0 56.0 5.0 0.5 , 0.39 10. 0 1.8 4.320 2.695 446.0 2.0 4.470 2.795 496. 0 2.2 4.610 2.885 545.0 2.4 4.740 2. 960 595.0 2.6 4.850 3.030 645.0 2.8 4. 960 3. 100 694.0 3.0 5.050 3.160 745.0 3.2 5.150 3. 220 794.0 3.4 5.230 3. 270 843.0 3.6 5. 320 3. 330 894.0 3.8 5.340 3. 340 943.0 c T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f / L d/b k g [ k i p ] a c r [ k i p s ] a c r 1 0 " J E k L 2 s EI 2 5 1 4 4 . 0 56.0 1 . 5 2.0 3.5 0.375 0.284 0. 50 0.39 4.0 7.0 7.0 0 . 0 2 . 4 9 2 1.050 1.764 1 . 5 5 5 0 . 656 1.103 0 . 0 2 4 1 4 4 . 0 56.0 1.5 2.0 2.0 3.5 0.375 0. 284 0. 500 0. 50 0.39 4.0 7.0 4.0 7.0 0 . 0 3.353 1.546 3 . 9 1 1 1.764 2 . 1 6 6 0 . 966 2.445 1.104 0 . 0 2 3 1 4 4 . 0 56.0 1.5 2.0 2.0 3.5 0.375 0. 284 0.5 0. 50 0.39 4.0 7.0 4.0 7.0 0 . 0 3.016 1.266 3 . 9 1 7 2.097 1.883 • 0 . 793 2.445 1 . 3 1 0 0 . 0 2 6(.68)» 6C.63) 6(.43) 6(.32) 6(.27) 6(.87) 6(1.05 1 4 4 . 0 56.0 1.5 1.5 1.5 3.5 3.5 3-5 3.5 0.150 0 . 1 5 0.15 0.875 0.875 0.875 0.875 0.39 0. 39 0.39 0.39 0. 39 0.39 0.39 10.0 10.0 10.0 4.0 4.0 4.0 4.0 0 . 0 0 . 2 1 3 0 . 1 5 9 . 0 . 1 3 6 1.822 1 . 7 1 8 4.735 8.097 0 . 1 3 3 0 . 099 0 . 085 1 . 140 1.072 2 . 960 5.060 0 . 0 *The number i n the b r a c k e t i n T h e o r e t i c a l R e s u l t s . Hinges Load case L i n . f i n . d i n . b i n . f/L d/b k [ k i p ] acr [ k i p s ] acr ^ — 1 0 k L 2 s EI 2 1 144.0 24.0 1.5 1.5 2.0 2.0 3.5 3.5 0. 213 0.375 ' 0.284 0. 500 0. 500 0.875 o. 17 7.0 4.0 7.0 4.0 7.0 4.0 0. 0 0. 181 0. 554 0. 241 0.748 0. 430 1. 323 0.113 0. 346 0.151 0.468 0. 269 0. 826 0. 0 2 2 144.0 24.0 1.5 1.5 2.0 2.0 3.5 3.5 0. 213 0. 375 0. 284 0. 500 0. 500 0.975 o. 17 7.0 4.0 7.0 4.0 7.0 4.0 0.0 1.077 3. 310 1. 352 4.193 2. 066 6.357 0. 666 2. 070 0. 846 2.615 1.290 3.980 0. 0 2 6(.67) 6(.85) 6(.66) 6(.85 6(1.00) 6(1.05) 144.0 24.0 1.5 3.5 3.5 1.5 • 1.5 0.213 0.875 0.875 0. 213 0. 213 o. 17 7.0 4.0 7.0 7.0 0. 0 0. 480 0.769 3.372 . 5.554 1.247 1.502 0. 300 0.480 2.110 3.480 0.781 0. 938 0.0 c 70. 4.0-10~3 3.0 1.0 0"cr E 2 Hinged Arches Loadcase 1(^=0.0) f/L = .39 d/b =3.0 L/d =96 d/b = 4.0 L/d = 72 d/b =10.0 L/d =29 d/b = 10.0 2 L/d =41 ^ 0 50 100 150 P i g . ( 5 . 2 ) Graph of ° c r/E VS. -g—- two-hinged arches, Loadcase 1, f/L = 0 .>9 71 4.0-10" 3.0 2.0 1.0 ^ c r E 3 Hinged Arches Loadcase 1 (^ =0:0) f/L = .39 d/b =3.0 L/d =96 d/b = 4.0 L/d = 72 d/b =10.0 L/d =29 d/b =10.0 2 L/d = 41 k s L 0 50 Pig-(5-3) Graph of .cr^/E VS. k L c  s EI 100 for three-hinged arches, 150 Loadcase 1, f/L = O.39 8.0-10 2 Hinged Arches ksL2 I I I E I | 0 50 100 150 k L Fig. (5.4) Graph of a c r / E VS. for two-hinged arches, Loadcase 2, f/L = 0.39 8.0 -10"3 3 Hinged Arches Loadcase 1(^=.96) ksL2 I | E I 0 50 100 150 Fig. (5-5) Graph of a^yE vs., -|-- for three-hinged arches, Loadcase 2, f/L = 0.39 74. 75-a _b a T f/L L/d c 0 0.133 0.682 0.39 96.0 1.282 0.099 0.528 0.39 96.0 O.958 O.085 0.427 0.39 96.0 O.819 0.078 0.360 0.39 96.O 0.750 0.697 0.308 0.39 96.0 O.706 1.855 0.637 0.39 41.3 1.220 1.431 0.478 0.39 41.3 0.936 1 .040 0.320 0.39 41.3 0.750 0.345 0.0 0.39 96.O 0.532 0.136 0.0 0.39 41.3 0.553 0.178 0.0 0.39 41.3 0.556 0.922 0.0 0.39 164.8 0.582 1.741 0.976 0.39 96.O 2.600 0.587 0.940 0.39 41.3 2.420 2.950 0 .,865 0.39 41.3 1 .940 3.650 . 1.080 0.39 72.0 3.630 0.468 1.050 0.39 72.0 3.370 0.151 0.0 0.17 72.0 0.543 2.075 0.960 0.17 96.0 3.200 1.292 0.900 0.17 41.3 2.600 0.300 0.673 0.17 96.O 1 .432 2.105 0.660 0.17 41.3 1.390 0.480 0.850 0.17 96.O 2.290 0.778 1 .000 0.17 96.O 3.710 0.940 1.050 0.17 96.0 4.490 1.338 1.095 0.17 96.O 6.36O TABEL IV c o values 76. 77-CHAPTER VI ' NUMERICAL- DESIGN EXAMPLE 6.1 INTRODUCTION I t i s the object of t h i s chapter to consider the design of a p r a c t i c a l arch. The purpose of the example i s to show that design f o r l a t e r a l buck-l i n g of an arch according to the presented theory must be considered, as w e l l as design f o r in-plane buckling and maximum shear' and normal s t r e s s . The design applies to roof systems using glulam two- and three-hinged arches of narrow rectangular cross-sections. The arches w i l l have t h e i r top edges continously r e s t r a i n e d against l a t e r a l displacement, and the bottom edge .unsupported. T o r s i o n a l .rotation .and l a t e r a l t r a n s l a t i o n of the ends of the arch must be prevented. . In the f o l l o w i n g sample design c a l c u l a t i o n the arch i s designed f o r : 1. In-plane buckling 2. L a t e r a l buckling without spring e f f e c t 3>. L a t e r a l buckling with spring e f f e c t The spring e f f e c t i s a t o r s i o n a l r e s t r a i n t of the top edge from the pur-l i n g . 6.2 A Sample Design C a l c u l a t i o n To i l l u s t r a t e the importance of considering l a t e r a l buckling i n the design of a t y p i c a l p r a c t i c a l arch, a sample design i s given. The design c a l c u l a -t i o n i s s a t i s f a c t o r y according to e x i s t i n g code C.S.A. 086, but the s t r u c -78. ture i s not safe for lateral buckling without spring effect. However i t i s shown that taking the purlin spring effect into account the structure i s safe for later a l buckling. . Consider an arch with following dimensions: Span L = 100 f t Rise f = 25 f t f/L = 0.25 Depth d = 28 inch With b = 4 inch d/b = 7 Modulus of E l a s t i c i t y E = 1600 ksi Allowable axial stress P = 1600 psi EL Allowable bending stress P^ = 2400 psi Purlins 2x8" at 16 inch center to center span B = 16 f t Loads DL = 20.0 psf LL = 26.0 psf The maximum moment i n the arch due to DL and LL on one side i s then, from a linear analysis: M = 906 kip inch max and the thrust: . P = 26.4 k , The design, considering in-plane buckling using existing codes, w i l l then be: f b - £29§ .- 1 -7*5 ksi 100*12 _ PI R 2x28 " 0 K a 79-a / T \2 d ^ L _ \ .21.5" 2dy F,_ = 2400 ksi b The stresses should satisfy the following equation: a b Therefore the design i s satisfactory according to the existing code. The design using the preceding theory gives: a. Without purlin effect °* = 1.545 ksi • & .•• Assuming the same factor of safety for l a t e r a l buckling as for in-plane buckling the allowable bending stress F^ corresponding to a c r i t i c a l . stress °" =1.545 k s i and a slenderness ratio of-: cr 0* - 1 12-1.545 ~ y cr w i l l then be: pb=4£=1:f^= - 5 1 5 k s i The stresses should satisfy the equation: a b Therefore the design i s not safe considering late r a l buckling. b. Including purlin effect. The moment of inertia for the purlins i s distributed as a uniform spring-effect over the length of the arch. 8 0 . T 1 b d 1 1 -5x7-53 7 0 n . . 4 , . . 1 = 12 16- = 12 T o - = 5 - 2 9 i n c h / i n c h The spring constant i s (see fig ( 6 . 0 l ) ) ' „ EI 4x1600x5.29 . . Q . . k = 4 — = — — - = 108 kips s B .16x12 Taking approximately 26 o/o of this purlin effect into account"will in-crease the c r i t i c a l stress to CT = 6960 k s i . cr * Hence the structure i s safe B B B Figure(6.01) Determination of springconstant i n a real structure. Method of connecting purlins to the arch r i b It i s seen from the numerical example that a l i t t l e purlin effect gives a considerable increase of the c r i t i c a l stress of the arch. To obtain a spring effect at the top edge the connection between the purlin and the arch must be a moment connection. In a practical structure the ..roof deck can be mounted to the arch by various methods. In fig(6.02) a detail i s shown of such a moment connection, where the purlins are resting on the top of the arch and mounted with a steel plate nailed to the purlin and the arch. This connection i s only able to produce small spring effects. 81. Fig.(6.03) shows a moment connection, where the p u r l i n s are down beside the arch with a plywood roof deck going over the top. This connection i s able to give a considerable spring e f f e c t . The creep over the width of the arch and the length of the p u r l i n s w i l l give some slack. Furthermore the inaccuracy i n the structure w i l l give a gap between the p u r l i n s and the arch which might cause buckling before the moment connection i s e f -f i c i e n t . Steel plate connectors nailed to the purlin and archrib. Purlin Cross-section of the arch. P i g (6,02) Moment connection with the p u r l i n on the top of the a r c h r i b . .. ' Purlin Cross-section of the arch. P i g (6.03) Moment connection with the p u r l i n down beside the a r c h r i b . 83. CHAPTER VTI  CONCLUSIONS In t h i s Thesis i t has been shown by checking against model t e s t s , that '. the s t i f f n e s s matrix method using the modified matrix of Charlwood gave s a t i s f a c t o r y p r e d i c t i o n s of the c r i t i c a l buckling loads f o r two- and three-hinged arches with and without p u r l i n e f f e c t . The d i f f e r e n c e between experimental and t h e o r e t i c a l r e s u l t s was w i t h i n the l i m i t of e r r o r which might be expected of experiments with timber- -s t r u c t u r e s . The design of a p r a c t i c a l arch according to C.S.A. 086 w i l l not always give a safe structure because l a t e r a l buckling i s not considered t h e r e i n . However, i t i s shown that i n many cases the p u r l i n e f f e c t can increase the c r i t i c a l s t r e s s to provide a safe s t r u c t u r e . • In order to include the p u r l i n e f f e c t i n the design process i t w i l l be necessary to determine the v a l i d i t y of the p r a c t i c a l connections used be-tween the p u r l i n s and the arch r i b . I t w i l l be a u s e f u l extension of t h i s work to evaluate such connections. When these p r a c t i c a l p u r l i n to arch connections have been evaluated the designer w i l l then be i n a p o s i t i o n to take advantage of the economy o f f e r e d by a deep and narrow s e c t i o n . 84. LIST OF REFERENCES 1. CANADIAN STANDARDS ASSOCIATION, 086-1959-Code of Recommended Practice f o r Engineering Design i n Timber. C. S.A., Ottawa, 1959. 2. CHARLW00D, R.G., L a t e r a l S t a b i l i t y of Glulam Arches, M.A.Sc.Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1968. 3 . TIMOSHENKO, S.P.> and J.M.GERE, Theory of E l a s t i c S t a b i l i t y , 2nd ed., McGraw-Hill, New York, 1961. 4 . GERE, J.M., and W. WEAVER, Analysis of Framed Structures., D. Van Nostrand, Princeton, N,J., 1965« 5. CRANDALL, S.H., Engineering Analysis, McGraw-Hill, New York, 1956. 6. WILKINSON, The Algebraic Eigenvalue Problem,Oxford. 7. A.S.C.E. Report, Wind Forces on Structures, Paper No.3269, A.S.C.E., New York, 1965. 

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