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Load sharing between stringers in gravel decked log bridges Lansdowne, Matthew Wayne 2006

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L O A D SHARING B E T W E E N STRINGERS IN G R A V E L D E C K E D L O G BRIDGES by M A T T H E W W A Y N E L A N S D O W N E B. Sc, The University of British Columbia, 2003 A THESIS SUMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Forestry) THE UNIVERSITY OF BRITISH C O L U M B I A May 2006 © Matthew Wayne Lansdowne, 2006 A b s t r a c t In British Columbia bridge designers have shifted to using steel or concrete for more permanent structures in forest roads. For temporary structures gravel decked log stringer bridges can still be a cost effective alternative. However, as companies move into smaller second growth timber, load sharing between the stringers becomes an important consideration. There are many ways that load sharing between log stringers is achieved. Examples of methods used to distribute the live loads between the log stringers are gravel surfaces, cable lashing wrapped around the stringers, and cross members including cedar cross puncheon and needle beams. In this paper a Finite Element Model (FEM) of a gravel decked log stringer bridge was developed. This model represented the stringers as beam elements and the cable lashing as elements that only transfer vertical loads. The load spread effect of the gravel surface was accounted for by using equations that predict the stress at the base of the gravel deck due to live and dead loads. For simple configurations where hand calculations were possible the results from the F E M were compared to the hand calculations. The comparison of the F E M to the hand calculations indicates the F E M is calculating values correctly. The F E M was then calibrated based on data available from the Forest Engineering Research Institute of Canada for an in service bridge. The F E M model was calibrated by varying the stiffness of the lashing element stiffness, a parameter affecting the load sharing between the stringers. . When varying the stiffness of this parameter it was found through comparison of the F E M to the in-situ data that setting the stiffness of the lashing elements to very low values underestimated the load sharing between the stringers, and setting the stiffness to i i high values overestimated the load sharing. An error minimization analysis was run to determine the optimum lashing stiffness to calibrate the model to the in-situ data.. As the calibrated F E M deflections approximated the deflected pattern of the in-service stringers the forces, moments, and reactions of the F E M should approximate the internal conditions experienced by the stringers of the in-service bridge. When considering the calibrated F E M results for the in-situ data it was found that full load sharing between all stringers of the superstructure was not occurring. This indicates it is inappropriate i f design assumes full load sharing between all stringers when designing gravel decked log stringer bridges with lashing. Additionally, it was found when lashing the bridge at thirds, the sections of the stringers in the span between the lashing points were acting independently. This action produced elevated stress levels in the stringers due to bending. The elevated stress levels between the lashing points could be of particular concern to designers using smaller second growth stringers. i i i T a b l e o f C o n t e n t s Abstract i i Table of Contents iv List of Tables v List of Figures vi Chapter 1: Introduction 1 1.1 Objectives .• 6 Chapter 2: Stringer and lashing elements 7 2.1 Background 7 2.2 Stringer elements 8 2.3 Lashing elements 10 Chapter 3: Loading Program 13 3.1 Stress at the gravel base 13 3.2 Area partitioned to stringer nodes 17 3.3 Vertical loads due to live and gravel dead loads 20 3.4 Forces and moments from the self weight of stringers 20 3.5 Total node loads 23 Chapter 4: Gravel decked log stringer bridge F E M construction 24 4.1 Background..... 24 4.2 Parameter Input 25 4.3 Stringer and cable lashing elements constructed 26 4.4 Boundary conditions set 27 4.5 Global stiffness matrix construction and reordering 27 4.6 Nodal Loads Input from Loading Algorithm 28 4.7 Displacements Solved from Simultaneous Linear Equations 28 4.8 Results from F E M 28 Chapter 5: Testing of the F E M 30 5.1 Comparison of F E M to hand calculations 30 5.3 Comparison of F E M displacements to measured deflections for in-service gravel decked log stringer bridge 39 5.3.1 Description of deflection test of gravel decked log stringer bridge 39 5.3.2. F E M of Bear Lake Bridge 45 5.4 Load Sharing Between the Stringers of Bear Lake Bridge 58 Chapter 6: Conclusions 67 Literature Cited 71 iv L i s t o f T a b l e s Table 3.1. Coefficients for a Gaussian curve of the unit load data including depth and position 16 Table 4.1. User Input Geometric Properties 26 Table 4.2. User Input Material Properties 26 Table 4.3. User Input Load Requirements 26 Table 4.4. Boundary conditions used within the gravel decked log stringer bridge F E M 27 Table 5.1, Unknown nodes in Problem_5.1 31 Table 5.2, F E M inputs for Problem_5.1 31 Table 5.3. Comparison of calculated versus F E M predicted deflections 35 Table 5.4. Comparison of calculated versus F E M predicted deflections for the outside stringers 38 Table 5.5. IBD of stringers in Bear Lake Bridge 40 Table 5.6. Distance in the x-direction of instrumentation locations for the stringers of Bear Lake Bridge 40 Table 5.7. Coordinates of the gravel truck tires producing the greatest bending moment 41 Table 5.8. Measured and total deflections of each stringer at lashing 1 42 Table 5.9. Measured and total deflections of each stringer at mid-span 42 Table 5.10. Measured and total deflections of each stringer at lashing 2 42 Table 5.11. Required Bridge Inputs for Bear Lake Bridge F E M Model 45 Table 5.12. Required Input for Stringers for Bear Lake Bridge F E M Model 46 Table 5.13. Proportion of live load supported by each stringer 60 v L i s t o f F i g u r e s Figure 1.1. Side view of a mid-span lashed gravel decked log stringer bridge 2 Figure 1.2. Cross-sectional view of a gravel decked log stringer bridge 2 Figure 2.1 Degrees of freedom in element coordinates of stringer elements 8 Figure 2.5. Degrees of freedom in element coordinates for interaction element...:. 11 Figure 3.1. Finite element model of the gravel surfacing 15 Figure 3.2. Example of rectangular areas attributed to stringer nodes 18 Figure 3.3. Fixed end stringer elements 21 Figure 3.4. Reaction forces and moments for a fix end beam under a uniform distributed load 22 Figure 4.1. Gravel decked log stringer bridge F E M program framework 25 Figure 5.1. Diagram of Problem_5.1 30 Figure 5.2. Free body diagram of middle stringer in Problem_5.1 32 Figure 5.3. Free body diagram of lateral stringer in Problem_5.1 32 Figure 5.4. Internal Forces and Moments on Middle Stringer Over Interval 0 < x < 2 33 Figure 5.5. Internal Forces and Moments on Middle Stringer Over Interval 2 < x < 3 34 Figure 5.6. Internal Forces and Moments on Outside Stringers Over Interval 0 < x <2 '. 36 Figure 5.7. Internal Forces and Moments on Outside Stringers Over Interval 2 < x <3 37 Figure 5.4. Cross-sectional view of measured stringer deflections at lashing 1 43 Figure 5.9. Cross-sectional view of measured stringer deflections at mid-span 44 Figure 5.10. Cross-sectional view of measured stringer deflections at lashing 2.... 45 vi Figure 5.11. Plan View of Gravel Truck Wheel Locations to Stringer Neutral Axis. 47 Figure 5.12. Comparison of F E M and measured stringer deflections at lashing 1, no interaction element 48 Figure 5.13. Comparison of F E M and measured stringer deflections at mid-span, no interaction element 49 Figure 5.14. Comparison of F E M and measured stringer deflections at lashing 2, no interaction element 50 Figure 5.15. Comparison of F E M and measured stringer deflections at lashing 1, high interaction element stiffness 51 Figure 5.16. Comparison of F E M and measured stringer deflections at mid-span, high interaction element stiffness 52 Figure 5.17. Comparison of F E M and measured stringer deflections at lashing 2, high interaction element stiffness 53 Figure 5.18. Sum of squares of the difference between the measured and F E M predicted displacements 54 Figure 5.19. F E M versus measured stringer deflections at lashing 1 using minimization of error the interaction element stiffness 55 Figure 5.20. F E M versus measured stringer deflections at mid-span using minimization of error the interaction element stiffness 56 Figure 5.21. F E M versus measured stringer deflections at lashing 2 using minimization of error the interaction element stiffness 57 Figure 5.22. The F E M predicted reaction forces at support x = 0 for all stringers of Bear Lake Bridge 59 Figure 5.23. The F E M predicted reaction forces at support x = Span for all stringers of Bear Lake Bridge 60 Figure 5.24. Diagram of Bear Lake Bridge loading problem assuming full load sharing through the lashing and gravel 61 Figure 5.25. Free body diagram of Bear Lake Bridge loading problem 61 Figure 5.26. Support reactions assuming full load sharing compared to the F E M predicted support reactions for all stringers at end x = 0 in Bear Lake Bridge 63 vn Figure 5.27. Support reactions assuming full load sharing compared to the F E M predicted support reactions for all stringers at end x = Span in Bear Lake Bridge.. 64 Figure 5.28. Longitudinal distribution of the normal stress on a transverse cross-section for each stringer within the Bear Lake Bridge 66 Chapter 1 Introduction Gravel decked log stringer bridges are structures used for creek crossings in British Columbia. The construction of gravel decked log stringer bridges involves placing several whole logs, termed stringers, side-by-side over the span of the creek. Nagy et al (1980) note wire rope or cable is used to lash all the stringers together to spread or share the load between the stringers directly loaded by vehicle traffic, and the unloaded stringers. The wrapping technique and location of the cable lashing is selected by the bridge designer, but typically the lashing location is either mid-span or at thirds of the span. Cross members including needle beams or cross puncheon can be used to further provide load sharing between the stringers. A geosynthetic material is placed over the stringers, and the gravel material is then laid over the geosynthetic to produce the running surface. The geosynthetic allows drainage of water and retention of the gravel material. Guards are placed on top of the outside stringers to retain the gravel running surface, and to form a railing preventing vehicles from driving off the bridge surface. A side-view and cross-sectional view of a mid-span lashed gravel decked log stringer bridge without needle beams or cross-puncheon is shown in Figures 1.1 and 1.2. 1 Figure 1.2. Cross-sectional view of a gravel decked log stringer bridge. Bradley and Pronker (1994) note that large diameter old growth logs are decreasing in availability; therefore, companies are moving away from using log stringer bridges in favour of steel or concrete structures. For temporary crossings gravel decked log stringer bridges are still an economical alternative as higher initial capital investments are required for steel or concrete alternatives. Pronker and Bradley (1994) note for a 6.1m span the initial cost in 1994 value dollars for a gravel decked log stringer bridge is $7,600. A n equivalent steel bridge costs $17,300, though there is a residual value of $7,200 for the steel girders, neglecting transportation and dismantling costs. Thus, there is a cost savings of $2,500 for a 6.1m crossing when using a gravel decked log stringer bridge. Forest engineers may find it desirable to use gravel decked log stringer bridges for temporary crossings given the cost efficiencies. Current design practice assumes a group of adjacent stringers are the primary load bearing members of the superstructure. The grouping of stringers is assumed to share the load evenly by acting as a unit under loading. The groupings can consist of a group of stringers under a wheel path, or sometimes design considers the grouping to consist of all stringers comprising the superstructure. When considering all stringers of the superstructure to fully share the load, load sharing between the stringers is defined as the point when the support reactions at a given stringer end are equal. The normal stress on a transverse cross-section developed in a stringer from the grouping due to vehicle loading is calculated using a combined section modulus for the group. The normal stress on a transverse cross-section developed in a stringer from the grouping due to vehicle loading is calculated using a combined section modulus for the group. The combined section 3 modulus can be based on either the average mid-span diameter of the stringer groupings as discussed by Nagy et al. (1980), or by using the actual geometric properties of each stringer. For log stringer bridges employing needle beams or cross puncheon, the assumption of full load sharing between the stringers within the groupings may be realistic due to the stiff nature of these cross members. For gravel decked log stringer bridges without cross-puncheon or needle beams, load sharing is a function of only the gravel surfacing and cable lashing. Before full load sharing can be assumed, the ability of the gravel and cable lashing to share the load between the stringers should be investigated. Analysis of gravel decked log stringer bridges is complicated due to the nature of the structure. Stringers vary in geometric and material properties. Stringers are prone to rot over time, reducing the effective cross-sectional area and related stiffness of the stringer (Moody et al., 1979). The gravel acts to spread the live load over the stringers through the load distribution angle of the gravel (Jewell, 1996), but Nagy et al. (1980) note these . effects are difficult to determine. The cable lashing transfers vertical force between stringers directly loaded through vehicle trafficking and adjacent stringers, but the actual magnitude of force transferred is unknown. These factors must be accounted for in the analysis of a gravel decked log stringer bridge. Finite element modeling (FEM) sections a structure into discrete elements connected at common nodes, producing a set of finite elements. Logan (1993) notes the finite element method allows analysts to consider structures composed of members with differing 4 properties. Previous F E M have been successfully implemented by authors such as Lyons and Bennett (2004) to model log stringers under transverse loading conditions. Commercially available F E M programs vary in cost and nodes available to the user. This software is not readily available to practitioners interested in using F E M for analysis of gravel decked log stringer bridge structures. To circumvent this issue, a gravel decked log stringer bridge F E M was created in Visual Basic for Applications for execution within Microsoft Excel, a common Microsoft Office program. This removed the necessity of special software, and expanded the availability of F E M for analysis of gravel decked log stringer bridges to interested parties. To create a gravel decked log stringer bridge F E M , two sub programs are required. One sub program models the stringer and cable elements using material and geometric inputs. The other sub program determines the total loads acting on the gravel decked log stringer bridge, accounting for the load spread effects of the gravel surfacing. The two sub programs are combined to create the gravel decked log stringer bridge F E M . . Once constructed, the F E M requires testing to ensure the program operates properly. Through comparison to hand calculations the gravel decked log stringer bridge F E M can be assessed as working correctly. The F E M can then be calibrated based on available data for full scale deflection testing of an in-service gravel decked log stringer bridge done by the Forest Engineering Research Institute of Canada. The F E M can be calibrated by varying the lashing element stiffness thereby affecting the ability of load transfer 5 between stringers within the F E M . When the F E M approximates the deflected pattern of the in-service stringers, the forces, moments, and reaction forces of the F E M should approximate the internal conditions experienced by the in-service stringers. The F E M results for the calibrated model can then be examined to determine the ability of the lashing and gravel to share the load between the stringers for the in-service bridge. 1.1 Objectives The objectives of this study include the following. 1. Develop a F E M of a gravel decked log stringer bridge. 2. Test the gravel decked log stringer F E M against hand calculations and calibration of the model to available FERIC data for an in-service bridge. 3. Use the calibrated F E M results to consider the ability of cable lashing and gravel surfacing to share the load between stringers of the in-service bridge. 6 C h a p t e r 2 S t r i n g e r a n d l a s h i n g e l e m e n t s 2.1 Background The stringer and cable lashing are modeled following the direct stiffness method as discussed by Weaver and Gere (1980). Using this method, the stringers and cable lashing are discretized into a finite set of elements. The element forces are related to the displacements through the element stiffness matrix (2.1) f = sa v (2.i) Here f are the loads applied to the element, S is the element stiffness matrix, and d are the displacements of the element. Due to the differing nature and behaviour of the stringers and lashing, these elements are modeled separately. The stringers are modeled as beam elements, and as such they are based on elementary beam theory. Beam elements have successfully been used to model log stringers under transverse loading conditions by Lyons and Bennett (2003). The following assumptions were used to model the stringer elements following work by Bennett et al. (2004) on full scale bending strength and stiffness testing of log stringers: • Log stringers are considered an isotropic homogenous material. • Log stringers are assumed to be of a span to depth ratio greater than 25, thus shear deformations are neglected. 7 In design of log stringer bridges, the stringers are considered to work within their linear elastic range, as such, the stringers are considered linearly elastic. The log stringers are assumed to have a linear taper along the length of the stringer. The log stringer elements are considered of constant circular cross-section. The lashing transfers vertical load in response to differential displacement between adjacent stringers. The lashing is not capable of transmitting a moment or torque. 2.2 Stringer elements The log stringers are modeled as 3-dimensional beam elements capable of resisting translations and rotations about the x, y, and z axis. For a 3-dimensional beam element there are 6 possible degrees of freedom at each end, for 12 degrees of freedom per stringer element. The possible degrees of freedom in element coordinates for the stringer elements are shown in Figure 2.1. AZ2 Figure 2.1 Degrees of freedom in element coordinates of stringer elements *adopted from ANSYS® 8.0 (2003) 8 The element stiffness matrix S of the stringer elements as given by Weaver and Gere (1980) is as follows s = EA 0 0 EA 0 0 0 0 0 0 0 0 L 1 2 £ / „ 6EI^ L 0 Z 0 0 0 Z 0 Z 0 0 0 6EIy Z I? \2EIV 6EIV L2 I? L2 0 0 Y 0 Y 0 0 0 0 GI„ 0 I3 GIV L2 L3 I2 0 0 0 6EIV X L 0 AEIV 0 0 0 0 6Ely X L .0 2EIy 0 0 0 Y L2 0 Y L 0 0 0 L2 0 L 0 6EI„ AEI^ 6EIZ L2 2EIr7 0 Z L2 0 0 0 Z L 0 0 0 0 Z L EA EA 0 0 0 0 0 0 0 0 0 0 L 1 2 £ / „ 6EI„ L YIEI^ 0 Z L3 0 0 0 6EIV Z L2 0 Z L3 0 l2EIy 0 0 6EIy Z L2 0 0 Y L3 0 GI„ Y L2 0 0 0 L3 0 GI„ L2 0 0 0 0 6EIV X L 0 2EIV 0 0 0 0 6Ely X L 0 4EIy 0 0 0 Y L2 0 Y L 0 0 0 6EI„ L2 0 L 0 4EI„ 0 Z L2 0 0 0 Z L 0 Z L2 0 . 0 0 Z L (Weaver and Gere, 1980) Here E is the modulus of elasticity of the stringer, A is the stringer element's cross-sectional area (considered constant circular), L is the stringer element's length, Tzis the stringer element's moment of inertia about the Z axis, Iyis the stringer element's moment 9 of inertia about the Y axis, G is the modulus of rigidity of the stringer, and Ix is the stringer element's polar moment of inertia. The forces and moments f capable of acting on a stringer element are Fyi r Z l Mx\ Mn Mz\ FX2 Fy2 FZ2 MX2 (2.3) The displacements d possible for the stringer elements are Ay, A Z 1 &x\ dY\ Ay2 A z 2 &X2 6Y2 _&Z2 _ (2.4) 2.3 Lashing elements The lashing transfers only vertical loads in response to differential displacement between adjacent stringers. To model this, the lashing is modeled as a 1-dimensional element capable of resisting only differential vertical translation between the element ends. To 10 model an element in this manner, an element that accounts for only deformations due to shear was created. This element is termed an interaction element, as the element transfers vertical force or "interacts" adjacent stringers in response to differential displacement. The interaction elements have one degree of freedom at each end, for two degrees of freedom per interaction element. The possible degrees of freedom for the interaction element are shown in Figure 2.5. &Z2 Z Figure 2.5. Degrees of freedom in element coordinates for interaction element. The interaction element stiffness matrix is GA GA cL cL GA GA cL cL Here c is the shear deflection constant for a circular cross section. The forces and moments f capable of acting on an interaction element are FZ2 (2.5) (2.6) The displacements d possible for the interaction elements are 11 A 2 , (2.7) To facilitate combination of the stringer and interaction elements, both element stiffness matrices require the same dimensions. The interaction element stiffness matrixes were augmented with zeros producing a 12x12 matrix that allowed addition in a 3-dimensional model. This modified cable element stiffness matrix S is as follows S = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 GA cL 0 0 0 0 0 GA cL 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 GA cL 0 0 0 0 0 GA cL 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Modeling the lashing in this method allows only vertical force to be transferred between adjacent stringers. The stiffness coefficient of the interaction element can be modified to increase or decrease the vertical force transferred between stringers for a given differential displacement. 12 Chapter 3 Loading Program 3.1 Stress at the gravel base The gravel surfacing is considered a continuous, homogenous, isotropic, linear-elastic material. In this study, the surfacing material is assumed to be composed of grains that are much smaller than the surfacing depth; therefore, the assumption of continuous homogenous material is adequate. Coduto (1999) notes most materials nearly meet isotropic criteria, and can be modeled as linear-elastic if the material undergoes small strains, so large plastic deformations such as rutting are not considered in this study. Compaction of the gravel surfacing is necessary for the gravel to be limited to small strains. During construction processes and through in service wheel trafficking the gravel is well compacted; therefore, the assumption of small strains is valid. Nonlinear behaviours on the gravel surfacing including creep are not considered. At the base of the gravel surfacing, the vertical stress distribution is due to the live load and gravel self-weight. The shape of the vertical stress distribution at the base of the gravel surfacing resulting from the live loads is a function of the depth of the gravel. To model the gravel surfacing requires modeling a large 3-dimensional volume. This significantly increases the size of the structure stiffness matrix within the gravel decked log stringer bridge F E M , increasing the virtual memory and processing time necessary for computational purposes. Reducing the size of the F E M program by using an equation that determines the stress at the base of the gravel surfacing from the live loading and gravel self-weight is advantageous. 13 Work by Lyons and Lansdowne (2004) provided evidence that i f finite boundary conditions exist, the stress at depth within a gravel surfacing may be underestimated using equations based on semi-infinite boundary conditions. As the stringers themselves act as a finite boundary to the stringers, classical soil mechanics equations for predicting stress at depth are not appropriate for calculating the stress at the base of a gravel surfacing in a gravel decked log stringer bridge. Lyons.and Lansdowne (2004) constructed a F E M model of a gravel volume using the commercially available finite element software ANSYS® version 8.0. The gravel volume was modeled as an isotropic, linear elastic, homogeneous material using eight node brick elements (SOLID 45). The gravel volume model was 5m by 5m in dimensions to eliminate the effects of the lateral boundary conditions on the vertical stress distribution at the base of the gravel. The tire contact patch was given dimensions of 0.245m wide and 0.32m long following work done by Bradley (1993) for a tire load of 23,642N and a tire pressure of 414kPa. As the number of nodes available was limited to 100,000 the volume around the tire contact patch was meshed finely, and the rest of the volume received a courser mesh. The F E M model of the gravel surfacing is shown in Figure 3.1. 14 Figure 3.1. Finite element model of the gravel surfacing A unit load was applied as a uniform stress over the tire contact patch for various finite depths of gravel. The stress field at the base of each gravel depth was examined and used to construct unit load stress curves. As the size of the tire contact patch is dependant on the magnitude of the applied load and tire inflation pressure (Jewel, 1996), the effect of varying the tire contact patch size on the stress field at the base of the gravel was examined. The stress fields for a unit load for an extreme range of tire patch sizes showed that the magnitude of the maximum stress directly under the applied load differed, but the stress distribution at the base of the gravel did not. Therefore, multiplying the unit load stress distribution by the wheel load to obtain the wheel load stress distribution at the base of the gravel is appropriate. For the different gravel depths the unit load stress curves plot the stress across the base of the gravel as a function of the 15 lateral distance from the applied load. For all gravel depths modeled the stress curves were related by a power function that fit a Guassian profile of the form <j = a{depthh)ey^{-c(depthd)x2) (3.1) Here <x is the vertical stress at the gravel base due to a I N load, depth is the depth of the gravel surfacing, x is the lateral distance from the point of interest to the applied load, and a, b, c, and d are coefficients given in Table 3.1 determined through a nonlinear least-squares fitting routine. Table 3.1. Coefficients for a Gaussian curve of the unit load data including depth and position Coefficient a , Coefficient b Coefficient c Coefficient d 0.7839 -1.8002 2.4684 -1.7731 Using (3.1), the stress at any point along the gravel base from an applied wheel load can be calculated as o~L = a(depthh)exp(-c(depthd)x2)* WheelLoad (3.2) Here oL is the vertical stress from the live load at the gravel base, and WheelLoad is the applied live load. Using (3.2), the stress from the wheel loads at the gravel base over any stringer node can be calculated based on the stringer node's lateral distance to the applied wheel loads. The stress at the base of the gravel surfacing due to the gravel self-weight is calculated from the specific weight and depth of the gravel surfacing assuming a constant surfacing thickness over the span of the gravel decked log stringer bridge (3.3) 16 crD=/depth (3.3) Here aD is the vertical stress at the base of the gravel due to the gravel self-weight, and / is the specific weight of the gravel. The total vertical stress at the base of the gravel surfacing is the combination of the stress from the live loads, and the stress from the gravel self-weight (3.4) crT=cTL+crD (3.4) Here o~r is the total vertical stress at a point along the base of the gravel surfacing. The sub program calculates crT for points along the gravel base coinciding with the stringer nodes defined during discretization of the stringers. 3.2 Area partitioned to stringer nodes The base of the gravel surfacing is partitioned into discrete rectangular areas based on the stringer nodes defined during construction of the stringer elements. These rectangular "nodal areas" are shown in Figure 3.2. 17 ^- 24 Area Boundary Stringers Figure 3.2. Example of the rectangular areas attributed to stringer nodes. There are three possible width sizes for a nodal area based on the stringer location. The first possibility is stringer 1, the outside stringer located at x = 0. The width associated with the nodal area for all nodes on stringer 1 is calculated as „ , . ,, Y(i + NodesPerStringer)-Y(i) _ _ . Width, = —* — ' - ^ + RadiusStringer, (3.5) For 1 < i < NodesPerStringer Here i is an integer ranging from 1 to the number of nodes per stringer, NodesPerStringer is the number of nodes along each stringer, and Radiust is the radius of stringer 1 at the location of the node This routine uses the y-coord of the node on stringer 1 to find the y-coord of the node located adjacent on stringer 2. The average width between the two 18 nodes is then calculated. This average width is added to the radius of the stringer at the node location calculated as a linear taper. The outside stringers are considered to extend to the end of the running surface. The second possibility is a stringer contained between the first and end stringer. The width associated with the nodal area for all nodes on contained stringers is calculated as W i d t h _ Y{i + NoderPerStringer)- YJi) + Y(i)-Y(i - NodesPerStringer) For NodesePerStringer < i < FirstNodeEndStringer Here FirstNodeEndStringer is the node number of the node at x = 0 on the first stringer. This routine uses the y-coord of a node to determine the y-coords of the nodes on both adjacent stringers. The width is calculated as the addition of the average of both distances. The third possibility is the end outside stringer. The width associated with the nodal area for all nodes on the outside stringer is calculated as w m = Y(i)-Y(i-NodesPerStringeri) + ^ . ^ . ^ ( 3 7 ) For FirstNodeEndStringer < i < NJ Here NJ is the total number of nodes. This routine uses the y-coord of a node on the end outside stringer to determine the y-coord of the node on the stringer adjacent. The width is taken as the average distance between these nodes plus the radius of the outside stringer at the node location, calculated as a linear taper. 19 The length of the rectangular area for every node is calculated as , X(i +1) - X(i) Xii) - Xii -1) Length, = —* ^ ^ + w v £ (3.8) For 1 < z < NJ If a negative number is generated, it is given a value of 0. Here i is a integer ranging from 1 to the number of stringer nodes, and NJ is the number of stringer nodes. Using the determined length and width values, the area attributed to each node is calculated as Area, = Width,* Length, (3.9) For \ <i<NJ 3.3 Vertical loads due to live and gravel dead loads The total vertical force acting on the gravel decked log stringer bridge is calculated by Fz - <TTi * Area, (3.10) F o r l < i < 7 v 7 Here Fzi is the vertical force applied at the stringer node resulting from the stress due to the live loads and gravel self-weight. 3.4 Forces and moments from the self weight of stringers The stringer self-weight is a distributed load per meter over each stringer element that requires transformation into forces and moments applied to the stringer nodes as 20 discussed by Weaver and Gere (1980). The stringer elements are of constant circular cross-section; so the self-weight is a uniform distributed load over each element. The magnitude of the uniform distributed load for each element is (3.11) w = ps*g*A (3.11) Here w(x) is the uniform distributed load over the element, ps is the density for the species of stringer (kg/m3), g is the acceleration due to gravity (9.81 m/s2), and A is the cross-sectional area of the stringer element. Equivalent forces and moments at the stringer nodes from the uniform distributed load over the element are determined by considering the element to be a fixed end beam as shown in Figure 3.3. \ \ \ \ w V 4 ± / V Figure 3.3. Fixed end stringer elements. 21 The reaction forces and moments to the uniform distributed load are shown in Figure 3.4. Ma Ml) jfc ± ± i_ k-Figure 3.4. Reaction forces and moments for a fix end beam under a uniform distributed load. From static equilibrium, the equivalent forces and moments due to the uniform distributed load as given by Weaver and Gere (1980) are coL2 12 coL (3.12) (3.13) These forces and moments are equal and opposite the uniform distributed load. The equivalent loads due to the stringer self weight are co{x)L2 M„ = —Mk = 12 Pa=Ph = co(x)L (3.14) (3.15) 22 3 .5 Total node loads The total loads applied to the stringer nodes is the combination of live loads, the dead load due to the gravel self-weight, and dead load due to the stringer self-weight. These loads are organized according to the node to which they are applied, and form the load vector for the gravel decked log stringer bridge F E M . 23 C h a p t e r 4 G r a v e l d e c k e d l o g s t r i n g e r b r i d g e F E M c o n s t r u c t i o n 4.1 Background The gravel decked log stringer bridge FEM is based on matrix analysis of framed structures as described by Weaver and Gere (1980). This method relates the total loads acting on the gravel decked log stringer bridge to the possible displacements of the gravel decked log stringer bridge through the structure stiffness matrix (4.1) fT=SGdT (4.1) Here fx is the structure load vector, Sa is the structure stiffness matrix, and dx is the structure displacement vector. The framework of the gravel decked log stringer bridge FEM program is shown in Figure 4.1. 24 Parameter input Sub program Stringers and Interaction Element constructs elements Boundary Conditions Set Stiffness matrices combined and reordered Sub program Loading calculates nodal loads Displacements solved from simultaneous linear equations Forces and moments for elements and reactions calculated and output Figure 4.1. Gravel decked log stringer bridge F E M program framework. 4.2 Parameter Input Various inputs are required for the gravel decked log stringer bridge F E M construction. These inputs include the geometry of the model, loading details, and the material properties of the gravel and stringers. The geometric, material, and load inputs required for the model are listed in Tables 4.1, 4.2, and 4.3. 25 Table 4.1. User input geometric properties. Stringer Lengths Number of Elements Per Stringer Diameters of the stringers Depth of gravel surfacing Table 4.2. User input material properties. Modulus of elasticity Modulus of rigidity Density of the stringers Density of the gravel Table 4.3. User input load requirements Wheel Locations Wheel Loads 4.3 Stringer and cable lashing elements constructed The parameters input into the gravel decked log stringer F E M are input into sub program Stringer and Interaction Elements to discretize the stringers, define the node and element locations in space, and construct the stringer and interaction element stiffness matrices. 26 4.4 Boundary conditions set The nodes located at the stringer ends are. given boundary conditions. The boundary conditions are shown in Table 4.3. Table 4.4. Boundary conditions used within the gravel decked log stringer bridge F E M . Node Located x = 0 Node Located x = Span Rotations Restrained X -Translations Restrained X, Y, Z Y, Z These conditions model the stringer as a simply, supported beam as suggested by Bennett et al. (2004) to approximate actual conditions experienced by in use stringers within a gravel decked log stringer bridge. 4.5 Global stiffness matrix construction and reordering The stringer and interaction element stiffness matrices are combined to produce the global stiffness matrix for the gravel decked log stringer bridge. Once combined, the global stiffness matrix is reordered, systematically moving the degrees of freedom to the top of the matrix, and the restrained degrees of freedom to the bottom of the structure matrix producing a structure matrix as shown in Figure 4.2. 7a" Sap "V Spa SPP_ Figure 4.2. Reordered Structure Stiffness Matrix Here a represent free node displacements, and p represent restrained node displacements. 27 4.6 Nodal Loads Input from Loading Algorithm The total load vector is calculated in sub program Loading. The total load vector is organized to apply the correct loads to the corresponding node. 4.7 Displacements Solved from Simultaneous Linear Equations The ordered structure force matrix and structure stiffness matrix form a system of simultaneous linear equations. This system of linear equations relating the nodal loads to the corresponding degrees of freedom though the global stiffness matrix is solved using a modified factorization method as discussed by Weaver and Gere (1980). 4.8 Calculating Stress The calculated displacements and element stiffness matrices are used to calculate element forces and moments as discussed by Weaver and Gere (1980). The element forces and moments are f M = SMd (4.2) Here fM is the element end actions. The reaction forces and moments are f R = - f N + V d (4.3) 28 Here f R is the support reaction, f N is the applied nodal load at the support restrained node, and SRF is a pattern within the structure stiffness matrix relating the effects of free node displacements on the restrained support nodes. As the stringers are considered circular in cross-section, the distance from top outer distal fibre from the neutral axis is equal to the distance from the bottom outer fibre to the neutral axis. The magnitude of maximum normal stress due to bending on a transverse cross-section on the tension side is equal and opposite the magnitude of the maximum normal stress on a transverse cross-section on the compression side. As the stringer elements are based on elementary beam theory, the normal stress on a transverse cross-section is a linear function of the distance from the neutral axis. The gravel decked log stringer bridge is considered to be under no axial loading. The maximum normal stress on a transverse cross-section of a stringer element is Mivr, (4-4) for 1 < i < StringerElements Here i is an integer, StringerElements is the number of stringer elements, cr, is the normal stress on a transverse cross-section from bending on the stringer element, M,y is the stringer element moment about the y-axis, r, is the stringer element radius, and 7,z is the stringer element moment of inertia about the z-axis. The gravel decked log stringer bridge outputs these calculated values to an organized spreadsheet. 29 C h a p t e r 5 T e s t i n g a n d C a l i b r a t i o n o f t h e F E M 5.1 Comparison of FEM to hand calculations To determine i f the gravel decked log stringer F E M was functioning properly, the F E M was compared to hand calculations for a simple structure. The problem considered in this section will be called Problem_5.1, Problem_5.1 includes three stringers connected at thirds with rigid links and subject to a point load of 30kN applied at mid-span of the center stringer (Figure 5.1). To model this problem with the F E M the stringers were constructed of beam elements and links connecting the stringers were constructed with the interaction elements set to be very stiff. Figure 5.1. Diagram of Problem_5.1. 30 . In this section the nodal displacements calculated by the F E M will be compared to hand calculations using beam theory. In order to perform the hand calculations it is necessary to determine the loads applied to each stringer. The 30 kN point load applied at mid-span on the center stringer is known; however, the following loads are not known a priory (Table 5.1). Table 5.1, Unknown loads in Problem_5.1 Stringer Reaction at x = 0m Lashing load x=lm Point load x=2m Lashing load x=3m Reaction at x = 4m Stringer 1 R l l L l l N A L12 R12 Stringer 2 R21 L21 Known L22 R22 Stringer 3 R31 L31 N A L32 R32 The unknown loads for Problem_5.1 were found by running the F E M with the inputs listed in Table 5.2. The results from the F E M give the loads applied to the nodes at each of the points listed in Table 5.1; therefore, these loads were used in the hand calculations. The free body diagrams for each of the stringers used in the hand calculations are presented in Figures 5.2 and 5.3, Note only one of the lateral stringers is required since the loading is symmetric. Table 5.2, F E M inputs for Problem_5.1 Str inger D iamete r D iamete r M O E Str inger T o p (m) Bot tom (m) ( G P a ) Densi ty(kg/m3) 1 0.5 0.5 11.75 0 2 0.5 0.5 11.75 0 3 0.5 0.5 11.75 0 31 1 1.500N 1 1.500N 1 1 " f ' 1 t 3,500N 30,000N 3,500N 2 m — l m — — l m — 2 m , Figure 5.2. Free body diagram of middle stringer in Problem_5.1. Y X " f i : i t 5,749.99N 5.749.99N 5.749.99N 5,749.99N 2m 2m *"« 2m Figure 5.3. Free body diagram of lateral stringer in Problem 5.1. For the stringers, the transverse deflection at any position x is related to the bending moment by M A"= (5.1) MOExIz Here A is the transverse deflection at position x along the stringer, prime (') is the derivative with respect to x, M i s the internal bending moment for the stringer, MOE is the modulus of elasticity, and Iz is the moment of inertia about the z axis. The point loads applied to the stringers will cause discontinuities in the moment functions for the stringers; therefore, it is necessary to develop displacement equations for the continuous sections and then link these though their boundary conditions. 32 For the middle stringer, on the continuous interval from 0 < x < 2, the point of the first lashing location, the internal forces and moments for the middle stringer is shown in Figure 5.4. Z 3,500N Figure 5.4. Internal Forces and Moments on Middle Stringer Over Interval 0 < x < 2. From static equilibrium conditions I X =0 M / = 3,500x (5.2) Substituting (5.2) into (5.1) and integrating with respect to x, then EIA\ = l,750x2 +cx (5.3) Integrating again with respect to x, then l,750x3 EIAX - -J—^ + cxx + c2 (5.4) Here cj and C2 are constants of integration. At x = 0, A, =0, therefore C2 = 0. On the interval from 2 < x < 3 for the middle stringer, the internal forces and moments are shown in Figure 5.5. 33 2311 V 11 ,500N mi 3,500M Figure 5.5. Internal Forces and Moments on Middle Stringer Over Interval 2 < x < 3. M2 =15,000x-23,000 Substituting (5.5) into (5.1) and integrating with respect to x, then (5.5) EIA\ = 7,500x2 - 23,000x + c 3 (5.6) Integrating again with respect to x, then EIA2 = 7 ' 5 Q Q x — l l , 5 0 0 x 2 +c 3x + c 4 (5.7) Here cj, and C4 are constants of integration. A t x = 3, A ' 2 = 0. From (5.6) 0 = 7,500(3)2 - 23,000(3)+ c 3 c 3 = 1,500 At x = 2, A \ = A ' 2 . From (5.3) and (5.6) 1,750(2)2 +c, =7,500(2)2 - 23,000(2) +1,500 34 c, =-21,500 At x = 2, A, = A 2 . From (5.4) and (5.7) 1,750(2)3 •21,500(2) = 7,500(2)3 •11,500(2)2 +1,500(2)+ c 2 c2 =-15,333.33 With the constants of integration solved, the deflection equation for the middle stringer becomes EIA, = 1 , 7 5 ° X - 21,500x (0 < x < 2) EIA, 7,500x3 •ll,500x 2 +l,500x-15,333.33 (2<x<3) (5.8) (5.9) Deflection results for (5.8) and (5.9) are compared to the F E M predictions for various positions along the middle stringer neutral axis in Table 5.3. Table 5.3. Comparison of calculated versus F E M predicted deflections. Position x Along Stringer (m) F E M Predicted Deflections (m) Equation (5.8) Deflections (m) Equation (5.9) Deflections (m) 0 0 0 -0.5 -0.000296336 -0.000296186 -1 -0.000580529 -0.000580236 -1.5 -0.000840437 -0.000840013 -2 -0.001063917 - -0.001063381 2.5 -0.001232178 - -0.001231557 3 -0.001299829 - -0.001299174 The hand-calculated displacements for the loaded stringer match the displacements calculated by the F E M for the loading shown in Figure 5.2 to 7 decimal places. This represents an error of 0.05% of the calculated deflection. Differences between the hand calculated and F E M predicted deflections can partly be attributed to rounding error as the 35 Gravel Decked Log Stringer Bridge program is capable of retaining values in excess of 32 decimal places. For the unloaded stringers, the deflections at any position x along the unloaded stringer can be solved following the same manner as the loaded stringer. For the problem from Figure 5.3, solving for the internal forces and moments on the interval 0 < x < 2 gives Figure 5.6. Internal Forces and Moments on Outside Stringers Over Interval 0 < x < 2. x V 5,749.81N 5 X = o M 5 = 5,749.81x (5.10) Substituting (5.10) into (5.1) and integrating with respect to x, then EIM3 = 5,749.81x2 2 (5.11) Integrating again with respect to x, then EIA3 = 5749.8 lx (5.12) 6 + c5x + c6 Here cs and are constants of integration. At x = 0, A 3 =0, therefore = 0 36 On the interval from 2 < x < 3 for the outside stringers, the internal forces and moments are shown in Figure 5.7. ,n mm 4v X 5.749.81N 2m 5,749.81N V M4 Figure 5.7. Internal Forces and Moments on Outside Stringers Over Interval 2 < x < 3. ]LX=o M 4 = 11,500 Substituting (5.13) into (5.1) and integrating with respect to x, then £/A' 4 = 1 l,500x + c 7 Integrating again with respect to x, then £7A 4 = 5,575x2-+ c 7 x + c 8 Here c-j and cs are constants of integration. A t x = 3, A ' 2 =0. From (5.14) 0 = 11,500(3)+ c 7 cT= -34,500 (5.13) (5.14) (5.15) A t x = 2, A ' 3 = A ' 4 . From (5.11) and (5.13) 37 5,749.81(2)' + c5 =11,500(2)-34,500 c5 = -23,000 At x = 2, A 3 = A 4 . From (5.4) and (5.7) 5 749 81C2"!3 ' y - K J — 23,000x = 5,575(2)2 - 34,500(2) + c 8 6 c 8 = 7,666.65 With the constants of integration solved, the deflection equation for the outside stringers becomes £/A 3 = 5,74981x3 23,000x (0 < x < 2) (5.16) EIA, =5,575x2 -34,500x +7,666.65 (2 < x < 3) (5.17) Deflection results for (5.16) and (5.17) are compared to the F E M predictions in Table 5.4 for various positions x along the outside stringers neutral axis. Table 5.4. Comparison of calculated versus F E M predicted deflections for the outside stringers. Position x Along Stringer (m) F E M Predicted Deflections (m) Equation (5.8) Deflections (m) Equation (5.9) Deflections (m) Difference (m) 0 0 0 - 0 0.5 -0.00031585 -0.000315691 - -1.59195E-07 1 -0.000611752 -0.000611444 - -3.08305E-07 1.5 -0.000867757 -0.00086732 - -4.37243E-07 2 -0.001063917 - -0.001063381 -5.35831 E-07 2.5 -0.001183608 - -0.001183012 -5.96193E-07 3 -0.001223505 - -0.001222888 -6.16341 E-07 The hand-calculated displacements for the loaded stringer match the displacements calculated by the F E M for the loading shown in Figure 5.2 to 7 decimal places. This represents an error of 0.05% of the calculated deflection. Differences between the hand 38 calculated and F E M predicted deflections can partly be attributed to rounding error as the Gravel Decked Log Stringer Bridge program is capable of retaining values in excess of 32 decimal places. The similarity of the hand-calculations to the F E M predicted translations for the load condition provides evidence that the F E M is calculating values correctly. 5.3 Calibration of FEM to measured deflections for in-service gravel decked log stringer bridge 5.3.1 Description of deflection test of gravel decked log stringer bridge In May 2005, the Forest Engineering Institute of Canada (FERIC) in conjunction with International Forest Products Limited (Interfor), instrumented and monitored stringer displacements for a gravel decked log stringer bridge (Bear Lake Bridge) composed of 9 green Douglas-fir {Pseudotsuga mensziesii) stringers lashed at thirds, under transverse loading by a 3 axle gravel truck. For the bridge prior to loading, the following data was measured: 1. Outside bark diameter (OBD) of the stringers at support x = 0. 2. OBD of the stringers at mid-span. 3. OBD of the stringers at support x = Span. 5. Gap between stringers. 6. Location of cable lashing along stringers. 7. Span of the superstructure. 7. Width of gravel surfacing. 39 8. Depth of gravel surfacing From the measured outside bark diameters, the inside bark diameter (IBD) was estimated. The IBD for each stringer at different positions along the stringer neutral axis and the gap'between the stringers is given in Table 5.5. Table 5.5. IBD of stringers in Bear Lake Bridge Stringer IBD Bottom (m) IBD Middle (m) IBD Top (m) Gap (m) 1 0.57 0.618 0.75 0 2 0.777 0.643 0.622 0.03 3 0.69 0.7205 0.8 0 4 0.77 0.715 0.664 0.02 5 0.6055 0.64 0.763 0.09 6 0.7 0.637 0.572 0.05 7 0.6445 0.72 0.798 0.06 8 0.79 0.732 0.6975 0.03 9 0.569 0.617 0.73 0 The location of the first lashing, the mid-span, and the location of the second lashing for each stringer of the superstructure were instrumented with rulers of 0.001 meter increments. The location along the x-axis of the stringers where the rulers were located is given in Table 5.6. A high precision level equipped with a micrometer was used to note the ruler scale values that were level with the instrument ( A i ) prior to loading the bridge with the gravel truck. Table 5.6. Distance in the x-direction of instrumentation locations for the stringers of Bear Lake Bridge. Lashing 1 (m) Mid-Span (m) Lashing 2 (m) 2.53 5.0 5.64 40 The gravel truck's axles were positioned at the location producing the greatest bending moment in the bridge. This location was calculated and marked symmetrically about center line on the bridge running surface prior to loading. The coordinates of the gravel truck tires considered to produce the greatest bending moment are shown in Table 5.7. Table 5.7. Coordinates of the gravel truck tires producing the greatest bending moment Axle Tire X(m) Y(m) Wheel Load (Newtons) Steer 1 0.85 2.495 18973.43 Steer 2 0.85 4.505 18884.25 Drive 1 3 4.9 2.265 36163.23 Drive 1 4 4.9 2.665 36163.23 Drive 1 5 4.9 4.335 35271.41 Drive 1 6 4.9 4.735 35271.41 Drive 2 7 6.32 2.265 38794.09 Drive 2 8 6.32 2.665 38794.09 Drive 2 9 6.32 4.335 38860.98 Drive 2 10 6.32 4.735 38860.98 Post loading, the high precision level was used to determine the ruler scale that were level with the instrument (A2).. The difference between Ai and A2 represents the stringer vertical deflection due only to the gravel truck loading (AT) . Here j is the incremental deflection of the stringer from the gravel truck measured at the point of ruler attachment. The measured deflections and incremental deflections at the ruler positions for each stringer are given in Tables 5.8 to 5.10. 41 Table 5.8. Measured and total deflections of each stringer at lashing 1 Stringer A i (m) A 2 (m) A T (m) 1 0.51765 0.51876 -0.00111 2 0.55910 0.56153 -0.00243 3 0.63991 0.64637 -0.00646 4 0.59285 0.60148 -0.00863 5 0.62387 0.63240 -0.00853 6 0.69262 0.70136 -0.00874 7 0.59146 0.59863 -0.00717 8 0.57437 0.57705 -0.00268 9 0.60232 0.60366 -0.00134 Table 5.9. Measured and total deflections of each stringer at mid-span Stringer A i ( m ) A 2 (m) A T ( m ) 1 0.52060 0.52190 -0.005219 2 0.58182 0.58475 -0.0058475 3 0.69122 0.69888 -0.0069888 4 0.62015 0.63087 -0.0063087 5 0.64530 0.65496 -0.0065496 6 0.70557 0.71595 -0.0071595 7 0.57950 0.58792 -0.0058792 8 0.60323 0.60591 - -0.0060591 9 0.62390 0.62513 -0.0062513 Table 5.10. Measured and total deflections of each stringer at lashing 2 Stringer Ai (m) A 2 (m) A T ( m ) 1 52.265 52.4 -0.00144 2 57.365 57.64 -0.00279 3 69.471 70.182 -0.00697 4 60.95 61.98 -0.00996 5 66.508 67.378 -0.00854 6 72.065 73.008 -0.00918 7 57.956 58.72 -0.00747 8 62.56 62.805 -0.00245 9 66.347 66.45 -0.001105 Based on the measured deflections at the ruler points, the deflections of the stringers in cross-sectional view for lashing location 1, mid-span, and lashing location 2 are given in Figure 5.8 through Figure 5.10. Lashing 1 Stringer Number 0 -0.002 -0.004 -0.006 Is -0.008 c" -0.01 o o a> S -0.012 Q -0.014 -0.016 -0.018 -0.02 Figure 5.8. Cross-sectional view of measured stringer deflections at lashing 1. 43 Mid-span Stringer Number Figure 5.9. Cross-sectional view of measured stringer deflections at mid-span. Lashing 2 Stringer Number Figure 5.10. Cross-sectional view of measured stringer deflections at lashing 2. 5.3.2. Calibration of FEM to Bear Lake Bridge The inputs required for modeling Bear Lake Bridge using the gravel decked log stringer bridge F E M are listed in Tables 5.7, 5.11, and 5.12. Table 5.11. Required Bridge Inputs for Bear Lake Bridge F E M Model. Span Length (m) 10.0 Span Width (m) 7.0 Fill Depth (m) 0.28 45 Table 5.12. Required Input for Stringers for Bear Lake Bridge F E M Model. Stringer X=0 Y Coord X=Span Y Coord Diameter Diameter Modulus Neutral Axis (m) Neutral Axis (m) Y = 0(m) Y = Span (m) Elasticity (Pa) 1 0.285 . 0.39 0.57 0.75 1.175E+10 2 0.996 1.131 0.777 0.622 1.175E+10 3 1.747. 1.867 0.69 0.8 1.175E+10 4 2.517 2.644 0.77 0.664 1.175E+10 5 3.30725 3.4675 0.6055 0.763 1.175E+10 6 4.0175 4.1975 0.7 0.572 1.175E+10 7 4.7585 4.955 0.6445 0.798 1.175E+10 8 5.5245 5.749 0.79 0.6975 1.175E+10 9 6.219 6.4815 0.569 0.73 1.175E+10 The stringer's ends were located using the known stringer diameter and measured gaps between the stringers. Based on the known end positions of the stringers, the neutral axes were located assuming a straight neutral axis between the bottom and top input locations. The stringer diameter was considered constant linear taper between the bottom and top of the stringer. Assuming straight stringer neutral axis, a plan view of the F E M modeled Bear Lake bridge can be seen in Figure 5.11. 46 f < X / V A v ' V * v J f V V V V \ A | 1 1 - W v * ? 1 ^ kvvvvvl Stringer 9 Stringer 8 Stringer 7 Stringer 6 Stringer 5 Stringer 4 Stringer 3 Stringer 2 Stringer 1 X Figure 5.11. Plan View of Gravel Truck Wheel Locations to Stringer Neutral Axis. No testing was conducted to determine the actual M O E of the stringers in Bear Lake Bridge. Based on work done by Bennett et al. (2004) on determining the stiffness of green second growth Douglas-fir log stringers, each stringer was given an M O E value of 11.75GPa. To calibrate the F E M , the stiffness of the interaction element was varied as this parameter effects the sharing of the load between the stringers. The stiffness of the interaction element was set to zero for the initial modeling. A cross-sectional view of the deflections predicted by the F E M to the measured data at lashing location 1, mid-span, and lashing location 2 are shown in Figure 5.12 through Figure 5.14. 47 Stringer Number • Measured Deflection at Lashing 1 • FEM Predicted Deflection at Lashing 1 -0.025 Figure 5.12. Comparison of F E M and measured stringer deflections at lashing 1, no interaction element. 48 Stringer Number -0.005 -0.02 -0.025 2 9 4 5 6 7 8 j • • 1 • • • • • m a • Measured Deflection at Mid-span ra FEM Predicted Deflection at Mid-span m Figure 5.13. Comparison of F E M and measured stringer deflections at mid-span, no interaction element. 49 Stringer Number •Measured Deflection at Lashing 2 a FEM Predicted Deflection at Lashing 2 -0.025 J ' Figure 5.14. Comparison of F E M and measured stringer deflections at lashing 2, no interaction element. The F E M does not correctly predict the measured deflections with the interaction element stiffness set to zero. The stringers under the wheel loads deflect, but little load is transferred to the unloaded stringers in the F E M through the gravel. The stiffness of the interaction elements were set to a high value and the model run again producing the predicted deflections shown in Figure 5.15 through 5.17. 50 Stringer Number • Measured Deflection at Lashing 1 • FEM Predicted Deflection at Lashing 1 Figure 5.15. Comparison of F E M and measured stringer deflections at lashing 1, high interaction element stiffness. 51 Stringer Number C -0.008 % B *= -0.01 2 3 4 5 6 7 8 < | i • * * • • a m " ' • • • • Measured Deflection at Mid-span a FEM Predicted Deflection at Mid-span • • Figure 5.16. Comparison of F E M and measured stringer deflections at mid-span, high interaction element stiffness. 52 Stringer Number -0.002 -0.004 e -0.008 S -0.01 £ -0.012 -0.014 -0.016 -0.018 -0.02 I Measured Deflection at Lashing 2 a FEM Predicted Deflection at Lashing 2 Figure 5.17. Comparison of F E M and measured stringer deflections at lashing 2, high interaction element stiffness. The high interaction element stiffness value forces all stringers to approximately deflect the same amount at the interaction element attachment points, which does not accurately predict the measured stringer deflections. To determine the most appropriate stiffness of the interaction element for matching the measured deflections, a minimization of error analysis was performed Let the measured displacements be di and the predicted displacements be d{, then the total squared error given as a percent of measured displacements is 1=1 d :100 (5.17) 53 The total squared error as given by (5.17) is plotted as a function of interaction element stiffness in Figure 5.18. • 0.0074 -, . 0.0073 0.0072 0.0071 £ 0.007 U J 0.0069 0.0068 0.0067 0.0066 •) 1 . , , , , 1 3500000 4000000 4500000 5000000 5500000 6000000 6500000 7000000 Interaction Element Stiffness (N/m) ' Figure 5.18. Sum of squared of difference between the measured and F E M predicted displacements. As can be seen in Figure 5.18 the interaction element stiffness that minimizes the total squared error is 4,850,000 N/m. Running the F E M with the interaction element stiffness set to 4,850,000 N/m produces results the deflected pattern shown in Figure 5.19 through 5.21. 54 Stringer Number • Measured Deflection at Lashing 1 B FEM Predicted Deflection at Lashing 1 Figure 5.19. F E M versus measured stringer deflections at lashing 1 using minimization of error the interaction element stiffness. 55 Stringer Number -0.002 -0.004 -0.006 ' e -coos E if -0.01 £ -0.012 a -0.014 • • m m B • ta a • • Measured Deflection at Mid-span B FEM Predicted Deflection at Mid-span • • -0.016 -0.018 -0.02 Figure 5.20. F E M versus measured stringer deflections at mid-span using minimization of error the interaction element stiffness. 56 Stringer Number -0.002 -0.004 -0.006 12 -0.008 aj a> E. "= -001 2 -0.012 -0.014 -0.018 -0.02 • Measured Deflection at Lashing 2 • FEM Predicted Deflection at Lashing 2 Figure 5.21. F E M versus measured stringer deflections at lashing 2 using minimization of error the interaction element stiffness. Given the uncertainty associated with the stringer's MOE, the F E M predicted deflections approximate the behaviour of the stringers in Bear Lake bridge. Other discrepancies that can be attributed to differences between the measured and F E M predicted deflections include uncertainties in the exact location of the stringers neutral axis in relation to the wheel loads. The F E M considers the stringer neutral axis straight between the location of the top and bottom nodes. Stringers are not necessarily this shape, and can lead to offsets of the live loads relative to the stringer neutral axis. 57 5.4 Load Sharing Between the Stringers of Bear Lake Bridge As the gravel decked log stringer F E M approximates the measured displacements of Bear Lake Bridge, the element forces and moments, and reaction forces can be examined based on the deflected stringers pattern predicted by the F E M . As the predicted deflections approximate the measured deflections, the trends between the internal forces, moments, and support reactions for the stringers in Bear Lake Bridge should be similar to those predicted by the F E M . Thus, the load sharing between the stringers modeled by the gravel decked log stringer F E M of Bear Lake Bridge corresponds to the load sharing experienced by the stringers of Bear Lake Bridge. To determine the amount of load sharing between stringers, the vertical support reactions were examined for each stringer at both support ends. This data is shown in Figures 5.22 and 5.23. 58 50000 45000 40000 35000 30000 _ S 25000 20000 15000 10000 5000 Stringer Figure 5.22. The F E M predicted reaction forces at support x = 0 for all stringers of Bear Lake Bridge. 59 50000 45000 35000 30000 g 25000 10000 5000 5 Stringer Figure 5.23. The F E M predicted reaction forces at support x = Span for all stringers of Bear Lake Bridge. Based on the F E M results in Figures 5.22 and 5.23, the proportion of the live load supported by each stringer is given in Table 5.13. Table 5.13. Proportion of the live load supported by each stringer. Stringer Number Proportion of Live Load Supported % Stringer 1 2 Stringer 2 4 Stringer 3 12 Stringer 4 28 Stringer 5 13 Stringer 6 13 Stringer 7 28 Stringer 8 10 Stringer 9 3 60 When considering full load sharing between all stringers of the superstructure, full load sharing is defined as where the support reactions at any given end for a stringer group supporting the live load are equal. To determine the support reaction at every stringer end, the Bear Lake Loading problem was considered as shown in Figure 5.24. 37.857.68N 142.869.3N 155.310.1N D 0.85nr < 4.05m • *—^.42m—* 10m 3.68m-Figure 5.24. Diagram of Bear Lake Bridge loading problem assuming full load sharing through the lashing and gravel. From Figure 5.24, the free body diagram of Bear Lake Bridge is shown in Figure 5.25. 37.857.68N 142.869.3N 155.310.1N X P A B PB Figure 5.25. Free body diagram of Bear Lake Bridge loading problem. 61 From static equilibrium conditions, the sum of vertical forces is _ ^ F Z = PA +PB -37,857.68-142,869.3-155,310.1 = 0 i^+Pg =336,037.1 (5.18) (5.19) Here PA is the reaction force at stringer end A (x = 0), and PB is the reaction force at stringer end B (x = Span). Summing the moments about point A gives ^_MA = -(0.85) * (37857,68) - (4.9) * (142,869.3) - (6.32) * (155,310.1) + (10) * (PB) = 0 (5.20) PB = 171,379.8437V From (5.20) P< = 164,657.2577V These values represent the total vertical support reaction force for all the stringers at the respective stringer end. For full load sharing, the reaction force for each stringer can be calculated by P P _ X_A J p — B r ZPerStringerA M , c . T n d l m r ZPerSlringerB NumberStringersInGroup NumberStringersInGroup (5.21) Here FzperStnngerA is the average support reaction force at support A , NumberStringerlnGroup is the number of stringers per group, FzperStnngerA is the average support reaction force at support B. 62 Based on (5.21), considering all stringers to fully share the supported load, the vertical support reaction for each stringer at end A is _ 164,657.257 ZPerSfringerA F z r e r » A = 18,295.257V Similarily for end B F. ZPerStringerB = 19,042.27V Full load sharing support reactions were compared to the support reactions predicted by the F E M for the stringers of Bear Lake Bridge as shown in Figures 5.26 and 5.27. 40000 35000 30000 z g 25000 o u. 20000 15000 B FEM Predicted Support Reaction Force ^ Reaction Forces, Full Load Sharing 4 5 6 Stringers Figure 5.26. Support reactions assuming full load sharing compared to the F E M predicted support reactions for all stringers at end x = 0 in Bear Lake Bridge. 63 50000 40000 30000 g 25000 5000 4 5 6 Stringer EU FEM Predicted Support Reaction Force 1 Reaction Forces, Full Load Sharing Figure 5.27. Support reactions assuming full load sharing compared to the F E M predicted support reactions for all stringers at end x = Span in Bear Lake Bridge. The stringers directly loaded under the wheel path of the gravel truck can be ascertained in Figures 5.12 to 5.14; and visually through Figure 5.11. In Figures 5.12 to 5.14 the directly loaded stringers deflect only through the transfer of load through the gravel surfacing as no interaction elements are present. This sharing of the load between the stringers is a function only of the load spread capability of the gravel. Examining these figures, the directly loaded stringers are 4, 6, 7, and to a small degree 3. The amount of load transferred through the gravel to stringer 3 is not enough to consider this stringer directly loaded, so it is considered proximal to the live loads. It can be seen in Figures 5.22 and 5.23 that the support reactions predicted by the F E M for the directly loaded stringers are significantly higher than the support reactions for the unloaded stringers. 64 The support reactions for the. loaded stringers are also significantly higher than the anticipated support reactions assuming full load sharing. Stringer 4 is approximately 250% of the full load sharing support reaction value, stringer 7 is approximately 206%, and stringer 6 is approximately 115%. In comparison, stringer 1 is 6% of stringer 4's reaction value, and 15% of the equal load sharing value. As the reaction forces are significantly higher for the directly loaded stringers, the maximum normal stress on a transverse cross-section is greater in these stringers. Examining the longitudinal stress distribution for the stringers, it can be seen the normal stress on a transverse cross-section is significantly higher in the directly loaded stringers as shown in Figure 5.28. 65 6000000 -1000000 -1 : 1 Distance along Stringer (m) Figure 5.28. Longitudinal distribution of the normal stress on a transverse cross-section for each stringer within the Bear Lake Bridge. It can be seen in Figure 5.28 stringers 4, 6, and 7 are subjected to significantly higher stress. Stringer 1 is subjected to a peak stress of approximately 10% Stringer 4's peak stress. Examination of Figure 5.28 shows that the stringer stress at lashing locations (2.54, 5.64) act somewhat dependently as loaded stringers stress slope decreases while the unloaded stringer stress slopes increase. The support reactions shown in Figures 5.26 and 5.27; and the longitudinal distribution of the normal stress on a transverse cross-section for each stringer in Figure 5.28 show that the assumption of full load sharing between all the stringers comprising the superstructure is not representative of in service conditions for Bear Lake Bridge. 66 Chapter 6 Conclusions The calibrated gravel decked log stringer F E M approximated the measured displacements of Bear Lake Bridge. Using the gravel decked log stringer bridge F E M , the element forces and moments, and reaction forces were examined based on the deflected stringers pattern predicted by the F E M . As the calibrated deflections approximated the measured deflections, the trends between the internal forces, moments, and support reactions for the stringers in Bear Lake Bridge should be similar to those predicted by the F E M . Based on the predictions of the calibrated gravel decked F E M model assuming full load sharing between all stringers of the superstructure is not representative of the load sharing between the stringers of Bear Lake Bridge. The calibrated F E M deflections for Bear Lake Bridge were similar to results measured from full scale testing. Differences between the F E M and measured results can be attributed to unknown exact locations of the live loads relative to the stringer's neutral axis, unknown M O E for the stringers, and a constant stiffness value used for all the cable interaction element constants within the superstructure. Even with these uncertainties, varying the load sharing parameter the stiffness value of the interaction element modeled the deflected pattern of the stringers well. This allowed the support reaction forces and the normal stress on a transverse cross-section due to bending to be examined for each stringer. The output trends of the F E M approximate the actual trends experienced by in -service stringers as the deflected shapes were similar. This provided insight into the load 67 sharing ability of the lashing and gravel. The directly loaded stringers share a substantial load with the immediately adjacent stringers. These stringers form the group of stringers primarily supporting the live load, with little load transferred to stringers outside this group. Design considering the outside stringers to add strength to the structure is not justified. Considering the wheels directly under the wheel path is more conservative and based on the measured data provided by FERIC, and the predictions of the Gravel Decked Log Stringer F E M , a more realistic approach. Further, examination of distribution of the normal stress on a transverse cross section showed that at points of lashing, the stresses acted somewhat dependently as load was shared from the loaded stringers to the unloaded stringers. However, as the distance from the lashing increased, the stress increased in the loaded stringers independently, leading to elevated stress levels within the loaded stringers, a concern to gravel decked log stringer bridge designers. Use of the gravel decked log stringer bridge program for analysis of in-service log stringer bridges is limited due to the uncertainties associated with these structures. The lashing tension varies due to construction practices, aging and slacking of the lashing. This influences the ability of the lashing to share the live load, as greater differential displacement maybe required before the transfer of the vertical load is possible. Given the capacity of the gravel and lashing to share the live load evidenced in Bear Lake Bridge, and the known uncertainties associated with the lashing within any in service structure, care should be taken in the evaluation of any structure using only the F E M . However, using the F E M to analyse load sharing behaviour of the stringers provided evidence that the practice of using a combined section modulus for all stringers of the 68 superstructure for design and analysis of gravel decked log stringer bridges is not justified. ^ Within the gravel decked log stringer bridge F E M program, the deflections are sensitive to the location of the live loads relative to the neutral axis of the stringers. Work by Lyons and Lansdowne (2005) showed that for shallow gravel depths (40cm) the stress field had a lateral distribution distance of =< lm. Within this range, the small changes in lateral position have corresponding influences on the applied stress to the stringers. Shifting the location of the live loads across the running surface of the bridge would affect the deflection pattern predicted by the F E M . Differences of 10cm greatly influences the stress determined over the nodes along the neutral axis as the algorithm that determines stress across the gravel base is sensitive to changes in the lateral distance between the nodes and live loads. As the gravel decked log stringer bridge F E M uses straight beam elements to represent the stringers, using exact locations of the wheel loads with respect to the stringer's neutral axis maybe useful to future practitioners using the F E M for simulating log stringer bridges as logs used for stringers are not essentially straight. Future work for modeling gravel decked log stringer bridges will require clarification of some of the mentioned uncertainties in modeling the Bear Lake Bridge. The M O E of the stringers can be determined through non-destructive testing. Known M O E values for the stringer removes variation due to this parameter, and the deflection of the stringers is then a function of only the sharing of the load. The precise location of the neutral axis relative 69 to the live loads could be recorded to remove uncertainty involved with the stress subjected to a stringer node. The stiffness of the interaction elements could be varied individually until the F E M deflections coincide with the measured displacements. Tests such as these could provide evidence of the variability of the lashings ability to transfer load between all stringers within a gravel decked log stringer bridge structure, and possibly these tests could be repeated in time to determine service life effects on the load transfer through gravel decked log stringer bridges. Further work can be done examining the ability of adding more lashing to further spread the load between stringers. As it has been shown that the lashing does account for the sharing of the live load between stringers, it is of interest to determine i f different lashing locations, or using more lashing, can increase the load shared between the stringers. Also of interest is the distance between lashing locations to determine how this effects the independent behaviour of the loaded stringers to determine an optimum lashing pattern. 70 L i t e r a t u r e C i t e d Bradley, H . Allan., 1993, "Testing a Central Tire Inflation System In Western Canadian Log-Hauling Conditions", Forest Engineering Research Institute of Canada, FERIC, Technical Note TN-197, Vancouver. Bradley, H . Allan. Pronker, Vera., 1994, "Standard Design for Using Railcar Subframes as Superstructures for Temporary Bridges on Forest Roads in British Columbia", Forest Engineering Research Institute of Canada, FERIC Special Report No. SR - 98, Vancouver. Bennett, D. M . Modesto, R. Ewart, J. Jokai, R. Parker, S.P. Clark, M.L . , 2004, "Bending Strength and Stiffness of Douglas-fir and Western Hemlock Log Bridge Stringers", Forest Engineering Research Institute of Canada (FERIC), Vancouver. Coduto, P. D. 1999, "Geotechnical Engineering: Principles and Practices," Prentice Hall Inc., New Jersey (759p). Hibbler, R. C , 1997. "Mechanics of Materials", Prentice Hall, Inc., New Jersey. Jewell, R.A., 1996, "Soil Reinforcement with Geotextiles", Construction Industry Research and Information Association, CIRIA Special Publication 123, London (332p). Logan, Daryl L. , 1993, " A First Course in the Finite Element Method", PWS Publishing y Company, Boston. Lyons, C. K. Lansdowne, M . , 2005, "Vertical Stress in the Gravel Decking of Log Bridges," Western Journal of Applied Forestry, Accepted for Publication August 2005. Moody, C. R. Tuomi, L. R. Eslyn, E. W. Muchmore, W. F., 1979, "Strength of Log Bridge Stringers After Several Year's Use in Southeast Alaska", United States Department of Agriculture, Forest Service, Forest Products Laboratory Research Paper FPL 346. 71 

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