Lateral Buckling Analysis of a Steel Pony Truss

Lateral Buckling Analysis of a Steel Pony Trussby Derek Matthies A study submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Civil Engineering (Structural Engineering) Committee Members: Fouad Fanous - Major Professor Robert Abendroth - Committee Member Vernon Schaefer - Committee Member Iowa State University of Science and Technology Ames, IA 2012 ii Contents List of Symbols and Abbreviations................................................................................................ iv List of Figures ................................................................................................................................. v 1. 2. Introduction and Objective ...................................................................................................... 1 1.1 Introduction ...................................................................................................................... 1 1.2 Objectives ......................................................................................................................... 2 Background ............................................................................................................................. 3 2.1 Buckling Behavior............................................................................................................ 3 2.2 Euler Buckling.................................................................................................................. 4 2.3 Buckling of Bars on Elastic Supports .............................................................................. 6 2.4 Buckling of Un-braced Top Chord Truss Members......................................................... 7 2.4.1 Analysis according to Engesser ................................................................................ 7 2.4.2 Buckling Load using the Energy Method ................................................................. 9 2.4.3 Buckling Solution with Variable Axial Load ......................................................... 10 2.4.4 Buckling of a Pony Truss Top Chord with Elastic Ends ........................................ 12 2.4.5 Analysis of a Pony Truss Top Chord According to Holt ....................................... 14 2.4.6 Buckling Load with Initial Out-of-plane Deformations ......................................... 16 2.5 3. 4. Pony Truss Design according to AASHTO Specifications ............................................ 17 Finite Element Analysis ........................................................................................................ 18 3.1 Finite Element Model of the Compression Chord ......................................................... 18 3.2 Analysis of Top Chords as a Bar on Elastic Supports.................................................... 19 3.3 Finite Element Model of the Pony Truss ....................................................................... 20 Discussion and Results of the Analysis of a Pony Truss Top Chord .................................... 23 4.1 Effective Buckling Length Factor – Lateral Support Stiffness Relationships ............... 23 4.2 Example Calculations for the Buckling Load of a Pony Truss ...................................... 24 4.2.1 Calculations following Engesser’s Procedure ........................................................ 25 4.2.2 Calculations following Bleich’s Procedure ............................................................ 25 4.2.3 Calculations following Timoshenko’s Procedure ................................................... 26 4.2.4 Calculations following Lutz and Fisher’s Procedure .............................................. 26 4.2.5 Calculations following Holt’s Procedure ................................................................ 26 4.2.6 Calculations using the Energy Method ................................................................... 27 4.3 Analysis of the Pony Truss using Finite Element .......................................................... 28 iii 5. 4.3.1 Two Dimensional Analysis .................................................................................... 28 4.3.2 Three Dimensional Analysis .................................................................................. 30 4.4 Effects of Compression Chord Moment of Inertia on the Stiffness of the Elastic Supports ......................................................................................................................... 32 4.5 Analysis with Modified Elastic Stiffness ....................................................................... 35 Summary, Conclusions, and Recommendations ................................................................... 38 5.1 Summary ........................................................................................................................ 38 5.2 Conclusions .................................................................................................................... 39 5.3 Recommendations .......................................................................................................... 39 Appendix A ................................................................................................................................... 40 Appendix B ................................................................................................................................... 43 Appendix C ................................................................................................................................... 45 Appendix D ................................................................................................................................... 47 References ..................................................................................................................................... 50 iv List of Symbols and Abbreviations A b C Ce CE C0 c d E Et h I Ib Ic Id K k ks l L Ld Le M m n P Pcr Pd Q q qo r U v V x y α ∆ δ πp θ ϕ ψ area of compression chord length of floor beams spring stiffness of interior supports spring stiffness of end supports Engesser’s spring stiffness required spring stiffness with rigid end supports ratio of C to C0 length of diagonal end members modulus of elasticity tangent modulus of elasticity height of truss moment of inertia moment of inertia of floor beam moment of inertia of vertical web member moment of inertia of diagonal web member column stiffness: K2 = P/EI effective length coefficient joint spring stiffness distance between panels total length of the truss length of diagonal web member effective length bending moment number of buckling modes number of bays compression load critical buckling load axial load on diagonal end members virtual load distributed compressive force maximum compressive load with a varying load distribution radius of gyration: r2 = I/A internal energy factor of safety external work distance from end support displacement of chord at point x elastic foundation constant maximum displacement of compression chord relative displacement of vertical web members potential energy angle of rotation at joints stiffness of compression chord Schweda’s elastic end support coefficient v List of Figures Figure 1.1 Pony Truss Bridge………...…...….……….……………………………...……….1 Figure 2.1 Equilibrium Path for Initially Straight Column…..…......………..……………….4 Figure 2.2 Equilibrium Path for Slightly Crooked Column…….…......……..……………….4 Figure 2.3 Euler Buckling………………………………...………..……...………………….5 Figure 2.4 Buckling Modes for a Bar with Pin Ends.....………………..…………………….6 Figure 2.5 Elastically Supported Bar......………….………………………………………….7 Figure 2.6 Column on elastic supports……………………………………………...………...9 Figure 2.7 Varying Axial Load Distribution………...…….………………………………...10 Figure 2.8 Compression Chord with Elastic Ends…..…………………………………........12 Figure 3.1 Finite Element Idealization of the Top Chord as a Bar on Elastic Supports…….19 Figure 3.2 Finite Element Idealization of the Pony Truss…………………………………...21 Figure 3.3 Stress-Strain Curve…..………………………………………………………......22 Figure 4.1 Compression Chord Design Curve……..……………………………………......23 Figure 4.2 Energy Method Design Curve ……....………………………………………......24 Figure 4.3 2-D Compression Chord Elements.....…….…………………………………......29 Figure 4.4 3-D Compression Chord Elements………………..…………………………......30 Figure 4.5 Nonlinear Load vs. Displacement Curve.……….. ………………….………......30 Figure 4.6 Pony Truss Top Chord Analysis……..………………………………………......31 Figure 4.7 Rigid Frame Boundary Conditions.…..……………...………………………......33 Figure 4.8 Rigid Frame Displacements…..……...………………………………………......33 Figure 4.9 Load Application of Pony Truss Frames…..……...…………………………......34 Figure 4.10 Pony Truss Lateral Displacement at Frame.…………………………………......35 Figure 4.11 Compression Chord with New Stiffness………………………..……………......37 Figure 4.12 Compression Chord with Elastic Ends………………………………………......37 1 1. while no longer used in constructing new bridges. 1. 1. unbraced compression flange of steel girders. may find applications in similar situations such as a walkway for a conveyer system between grain elevators. 1. In the following chapter.1c. The calculation of the critical load for a pony truss top chord using published relations has been examined and compared to the results obtained using an analytical method. 1983). The pony truss. The compression chord of the pony truss structure. The buckled shape of the bar will fall somewhere between the extreme limits of a half-wave length of unity and the number of .1 Introduction Lateral stability of steel members under compression has been of interest to researchers for years. which together with the floor beams form rigid frames as show in Fig. where vertical clearance prohibits lateral bracing. and the top chord of a pony truss for which vertical clearance requirements prohibit direct lateral bracing. 1. The structural behavior of the previously listed members has been studied by several researchers.1a Pony Truss Elevation View A Fig. Introduction and Objective 1. is elastically supported in the horizontal plane by the truss vertical and diagonal web members. Among these members: columns under axial compression load. the behavior of an axially loaded bar and the top chord of a pony truss are briefly summarized.1c Section A-A A Fig. n Panels CL Fig.1b Pony Truss Plan View To analyze the compression chord of a pony truss. This member with intermediate elastic restraints will buckle in half-waves depending on the stiffness of the elastic restraints. the chord can be treated as a bar on elastic supports (Ballio. Conduct a literature search to review available information that is related to the stability of the top chords in truss structures.2 spans between the end restraints. From this assumption. In other words. k. 1988) was one of the first researchers to investigate the problem and develop an approximate formula to determine the required stiffness for the elastic restraints that corresponds to a specified effective wave length. This theory used the frame consisting of the floor beam and vertical and diagonal members at each panel point location to provide stiffness for the compression chord. Verify the results of analyzing a top chord of a pony truss using the approaches given in published literature and the results obtained using the finite element method. k. one may argue that investigating the behavior of the bridge as a three dimensional system may result in a higher stiffness coefficient of these lateral supports. Bleich (1952) and Holt (1952). . Recommend the most applicable published analysis technique for determining the critical load of an unbraced top chord of a truss system. 1. The method on how to determine the effective length has long been the focus of compression chord buckling. However. From the buckled shape. 3. Engesser (sited in Galambos. To the writer’s knowledge. 2. Engesser’s approach for determining the stiffness of the elastic restraints and its effects on the compression chord was based off the assumption that the connection between the web members and the floor beam is rigid. such as Timoshenko (1936). others. The failure of several pony truss bridges at the end of the nineteenth century prompted the research of compression chord buckling. provided methods of solving for the effective buckling length factor. all of the research for determining the critical buckling load on a compression chord with elastic supports is based on Engesser’s assumption. the effective length of the compression chord can be used to determine the critical load. These objectives were accomplished by performing the following tasks: 1. the theory in question is if the idealized structure is a conservative approach of the actual frame stiffness.2 Objectives The objective of the work presented herein was to verify the results of the published solutions for determining the effective length factor using the finite element method. shows the loaddisplacement relation (referred to equilibrium path) for an axially loaded column. Background 2.e.1. Euler Load. and hence the load displacement relation will not follow that shown in the Fig. buckling presents a failure mode due to high compressive stresses which causes the member to no longer be in equilibrium. is a disproportionate increase in displacement resulting from a small increase in load. from Brush (1975). deflection. In linear mechanics of deformable bodies. which in turn cause the member to collapse.1 but rather a different load displacement will be obtained. but as the critical load is reached. The elastic buckling of an axially loaded column in compression occurs when a certain critical load is reached causing the member to suddenly bow out. a secondary path is formed representing the bent equilibrium configurations.e. Although not a limit state.2 for the straight and crooked columns. 2. The essence of buckling..1 and 2. The load at which collapse occurs is referred to as the buckling load and is thus a design criterion for compression members. 2. For example. which may result in yielding or rupture. The buckling strength of compression members has long been studied to relate the empirical methods of analysis to the actual results. 2. Each point of this path represents an equilibrium configuration of the structure.3 2. The load displacement relationship of an imperfect axially loaded column is shown in Fig. Points along the primary (vertical) equilibrium path represent the configuration of a compressed. Of course. Fig. displacements are proportional to the applied loads.2. slenderness or clearance). perfectly straight column.1 Buckling Behavior The failure of an axially loaded bar in compression is defined by limit states which are an identifying condition of design criteria. vibration. When comparing Fig 2. Usually buckling occurs before the column reaches the full material strength. the figures show that the equilibrium paths generally converge as the lateral . 2. the equilibrium path will follow the secondary path shown in Fig. i. as the applied load reaches a critical value. The deviation of the member axis will result in additional bending that gives rise to large deformations. However. however. The critical load is defined as the minimum load for which the structure remains in equilibrium before instability is reached and failure occurs.1. no real column can be perfectly straight. or serviceability limits states (i. Limit states for a structural member include strength limit states. Analyses for both columns. P = . straight and slightly crooked.4 displacements increase. lead to large lateral displacements at the critical load.
  C1 = .
  P Secondary path Primary path -Δ Δ at x = L/2 +Δ Fig. His work was based on a straight. however.2 Equilibrium Paths for Slightly Crooked Column 2. The discrepancies.1 Equilibrium Paths for Initially Straight Column Δ at x = L/2 Fig. M. According to Salmon (2009). Figure 2.2 Euler Buckling Buckling of axially loaded bars in compression was investigated by Leonhard Euler in 1744. 2. the bar remained straight and underwent only axial compression. Et. at a distance. the following summarizes the derivation of the Euler buckling load. Euler’s formula was finally validated in 1889 when Considère and Engesser independently published works showing that one must use the tangent elastic modulus. to account for the fibers beyond the proportional limit. In his work. 2. For the reader’s interest. were due to the fact that the elastic limit was exceeded before the elastic buckling was attained. can be related to the curvature as follows:
   =  =    (2. Euler stated that if the applied load. P.1) . prismatic. Euler’s formula was not widely accepted initially since the test results on columns did not agree with his theory.3 illustrates the deflected shape on an axially loaded bar. was less than the critical value. concentrically-loaded column with pin-ended connections. x. By that definition all fibers would remain elastic until buckling occurred. The bending moment. 2.5 P x y L/2 Δ L L/2 P Fig.4 can then be written as and 0 =  sin  (2. 2. the boundary conditions can be set as y = 0 at x = 0 and y = 0 at x = L. K.3) where.3 can be calculated utilizing the support conditions at both ends of the bar. For a pinned-end column. The constants A and B in Eq. is equal to /
.4) 0 = sin    = .3 Euler Buckling (Thandavamoorthy 2005) The solution for the linear differential equation above can be written as  =  sin  +  cos  (2. These conditions will result in: B=0 Equation 2. 6) By substituting K =  /
 into equation 2.5 and solving for P. (2. ! "  #$ % .5)  = (2. Euler buckling equation yields the critical buckling load. Pcr 4Pcr m=1 9Pcr m=2 m=3 Fig. .5a. 2. As the elastic stiffness varies between the two extreme limits. 2.6 where m = 0. then bar will behave similar to a bar not supported by restraints and deflect in one-half wave as shown by Fig. 2. the bar will buckle somewhere between one half wave and the number of spans between the rigidly supported ends such as Fig. if the elastic supports are very flexible.3 Buckling of Bars on Elastic Supports A bar supported by rigid supports at the ends with equally spaced elastic restraints between the ends can have several modes of buckling depending on the stiffness of the supports. the bar will buckle in half-waves of a length equal to the distance between supports as shown by Fig.4 Buckling Modes for a Bar with Pin Ends 2. The bar will then behave similar to a bar on rigid supports. Therefore. 2…… is referred to as the number of buckling modes. 2. the stiffness of the elastic supports that are provided by the vertical and diagonal members of a pony truss is vital in controlling the buckling load and the buckling length of the top chord.4. If the stiffness of the elastic support is sufficiently large.5b. However.5c. The deformed shapes for the first three buckling modes are shown in Fig. 2. 1. can be idealized for buckling analysis as a continuous beam that is braced by elastic springs. he used his work to explain the failures of pony truss bridges and provide a rational method of design for similar structures.1 Analysis according to Engesser The analysis proposed by Engesser in the late 1800’s can be applied with some reasonable accuracy to analyze a bar that is pinned at its ends and is supported on equally spaced intermediate elastic springs provided that the half-wavelength of the buckled shape is at least 1. Therefore.9 (2007) addresses these issues and gives recommendations on the design of the vertical web as well as the connection to the floor beam. one must realize that Engesser’s solution can only be used as a preliminary design tool and more comprehensive analysis is needed.2. such as a pony truss. unless one considers the effect of imperfections of the compression chord. which correspond to the stiffness of the transverse frames at each panel point. the compression chord from the unbraced top chord in a steel truss. However. Engesser developed a simple formula to calculate the required stiffness. In the following years. design of the transverse frames formed by the web members and floor beams will have a direct effect of the critical buckling load of the chord members.8 times the spring spacing (See Galambos 1988). the calculated critical load is an upper limit. 1952) 2.4 Buckling of Un-braced Top Chord Truss Members As mentioned above. However. Creq.7 (a) P P P P (c) P P (b) Figure 2. Engesser examined the top-chord buckling problem of pony trusses and summarized his findings in a paper that was published in 1884. 2.5 Elastically Supported Bar (Bleich.14. The following sections summarize some of the published work that is related to the analysis of the unbraced top chord of steel trusses. of the .4. AASHTO section 6. of the provided lateral restraint. Several researchers suggest that one needs to assume a factor of safety. In addition. 2. the required spring constant is & '( = -.7) If Creq is met at each frame location. By combining Euler’s buckling equation and Eq. the load Pcr. The top chord. In his work. EtI. can be taken as vP. is straight and of uniform cross section. The equally spaced elastic supports have the same stiffness and can be replaced by a continuous elastic medium. 2. Its ends are taken as pin-connected and rigidly supported. C.8 elastic support to reach the desired critical load that is based on a specific buckling length. Engesser suggested that one needs to assume an effective length factor. where P is the calculated top chord member load.*+ . Introduce a factor of safety not less than 2. . Carry out a structural analysis to calculate the maximum load in the top chord members. k. Engesser also provided the following assumptions: 1. of two when calculating the design load. of 1. v.8) The use of Engesser’s original approach in design is summarized as follows: 1.3. l. P = v*load from step one above. Engesser’s solution for the required stiffness of a pony-truss transverse frame which is derived in Appendix C is & '( = )*+  . ") (2. 3. Use an admissible structural analysis technique to calculate the elastic constant.0 and calculate the design load. Pcr. the flexural rigidity EI should be modified using the tangent modulus.7. will achieve the specified design load. In other words. The axial compressive force is constant through the chord length. -#$ (2. once the calculations show that the stress induced in the member exceeds the limit specified in the design specifications. including the end posts. 2. 3. the chord with the length between panels. 4. is the number of bays. the deflected shape can be defined by the equation  = / sin !" % (2. (a) P A B P L = nl (b) P P Figure 2.9 4. and the number of modes can be related to n-1. 5. One approach that can be used to calculate the provided lateral supports stiffness. k. Cl/P. Using the energy method. is detailed in Appendix A. found in step five. to calculate the effective length factor. 6. l. n.6 Column on Elastic Supports To solve for the external work and internal energy of the member. Using the sin curve. C.2 from Holt (1956) or Fig. multiplied by each bay length. 2. Use the above calculated ratio and the number of panels. The total length.2 Buckling Load using the Energy Method Similar to the approach for a simply supported column. the boundary conditions are y = 0 at x = 0 and y = 0 at x = L. using Table 2.6. n. can be defined by an equation which represents the buckled shape. the first and second differential equations for the line can be solved as ′ = 1!" % cos !" % (2.9) where m is the mode number. 4. Apply k. a column simply supported at both ends with equally spaced interior elastic supports. as shown in Fig. 2.1 citing other authors. to the equations found in Chapter E of the AISC (2011) manual to determine the nominal compression capacity of the compression chord. L. Utilize the information calculated above to estimate the ratio.10) . 2. Since the energy method uses an assumed buckled shape for the chord. the energy method provides a satisfactory approximation.10 ′′ = 1! "  % sin !" % (2. U to find the critical buckling load. For the case presented in Fig. the potential energy of the system can be set equal to the external work.11) With P as the axial load.6. V. the solution is obtained by some degree of approximation. A full derivation of a bar on two elastic springs using the energy method is presented in Appendix B. If the assumed shape is properly chosen to satisfy the boundary conditions. plus the internal energy. the potential energy can be represented as . 2.7 Varying Axial Load Distribution (Timoshenko.4. In addition. 1936) . the critical buckling can be found as a function of the mode number. He assumed the compression load varies parabolically along the length of the chord with the load equal to zero at the ends then reaching a maximum value at the center (see Fig. By solving for P. L P P x Figure 2. 2.12 represents the energy of the elastic supports as a function of the spring constant C.2 = 3 + 4 = 5) 68 ′ 7 + 68 ′′ + & ∑  % #$ % 9 (2.12) The final term in equation 2. Timoshenko’s solution assumed that the ends were pin connections.7).3 Buckling Solution with Variable Axial Load Timoshenko (1936) extended the work of Engessers’ to include the effects of a varying the axial load along the top chord of a truss structure. the compressive forces that are transmitted to the chord by the diagonals are proportional to the distance from the middle of the bridge span as C = C8 D1 − .G (2. Eq. 2. 2. ∑?@A ?@9 > / + B.15) where x is the distance from the left support in the figure and qo is the maximum force of the axial load represented by C8 = H ) (2.11 Similar to Engesser. l. = < 4 = " = #$ . Timoshenko also stated that if the bridge is uniformly loaded. α.8.19 can be developed to relate the critical buckling load to the effective length factor by combining Eqs. is related to the lateral support spring constant.16 and 2. since the elastic supports are treated as a continuous elastic medium. can then be calculated using the information given above as follows  3 = − . )* " % 9 = N H D. C. and the distance between each lateral support.I 68 JK − L D G 7 .14 and 2.19) . V.18) Finally. one can then obtain the following: D (M .18. (2.17) Substituting the information given in Eqs. ( (2. - ∑?@A ?@9 / (2. 2. . - G  = "  #$ % (2.16) The external work. <.14) In his solution. Timoshenko assumed an equivalent elastic foundation. as . 2.17 into the total potential energy relationship.13) The strain for the chord energy can be represented as -. G  (2. 1 Analysis According to Bleich (1952) Bleich obtained his solution by using finite difference as an exact approach to quantify the buckling load of the chord.4. As n increases.8 Compression Chord with Elastic Ends (Bleich. simplifying the stiffness to ϕ=. which varies per the axial load. Et. Euler’s buckling equation is substituted into equation 2.22) For a chord with n spans in equation 2.4.20. Schweda extended Bleich’s results to include chords with elastic ends. 95 W ST XY Z[\ ]U / = ^5 R_` ^ />7 a = ^ (2. Later.21. increases to a limiting value. Bleich also assumed the chord had a constant moment of inertia and constant axial compressive force over the entire length. there are n -1 different half-wave buckling configurations. 1952) The theoretical exact solution proposed by Bleich for a chord supported on rigid ends was where vP = Pc and &= ST O) D95 PQR U G1 5 V .21) To eliminate the tangent modulus.12 2. His solution was based on the ends being pin connections and equally spaced intermediate supports of equal rigidity. C∞.4 Buckling of a Pony Truss Top Chord with Elastic Ends 2. Bleich (1952) showed that Cn can be replaced by C∞ for any . " (2. the spring constant required for an infinite number of spans. vPd Ce vPd -n C C vP l C -r Ce vPd +r d +n 2nl d Figure 2.20) ^J95PQR^L ^5R_`^ The stiffness of the chord is represented by ϕ which is equal to ϕ = K c# *$ ) d (2.4. 86 0.822 0.94 1.361 0.692 0.54 0.264 0. in order to continue on Bleich’s exact buckling theory.5 0. 1952) 1/k Φ 1/k Φ 1/k Φ 1/k Φ 0.102 0.74 0.1 Values of Φ in Eq.32 0. Although the previous assumption of designing the bridge for the center span only would yield conservative results. The spring constant of the end supports is denoted by Ce and the intermediate supports by C.036 0.70 0.652 0.375 0.652 1.36 0.388 0.1 from Bleich (1952) and is valid for the elastic and plastic range of buckling.530 0.544 0. the assumption of rigid ends can result in unsafe buckling loads when using Bleich’s theory.261 0.218 0.84 0. equation 2.219 0.46 0. Schweda provides results to determine the required stiffness for a chord supported elastically on the ends. which is rarely the case in practice.447 0.126 0. So.76 0. Table 2. the center bay of the chord should be designed with the appropriate k value for the maximum load and then used for the remaining bays.478 0.335 0.82 0.614 0.976 0.13 span where n is greater than six which corresponds to an error less than 1%.56 0.239 0.80 0.734 0.66 0.444 0.42 0. 2.92 1.58 0.62 0.38 0.309 0.68 0.97 1.48 0. Φ (2.00 2.72 0. for trusses with more than six spans.870 0.921 0.510 0.142 0.90 1.160 0.52 0.99 1.285 0.34 0.44 0.138 0.777 0. e^ f R_` ^g = )* .179 0.88 1.95 1.64 0.177 0.40 0.78 0.000 Since Bleich’s theory assumes a constant axial force.91 1.316 0. Thus.417 0.198 0. Schweda assumed the load was a constant axial force throughout the length of the .93 1.96 1.111 0.23 (Bleich.3 0.578 0.60 0.23) where Φ is given in Table 2.98 1.16 simplifies to & '( = )* ^J95PQR^L . C0. As shown by Bleich (1952). His research not only tested the primary constraints mentioned above but also the effects of secondary factors. the value of C = cC0 where c > 1.1. Error introduced by considering the chord and end post to be a single straight member. Thus. His conclusion stated that the load capacity of a pony truss bridge would be satisfactorily predicted by previous buckling analyses mentioned.24) where CE is Engesser’s equation (Eq. Schweda calculated ψ as a function of the number of bays with respect to c and k. the appropriate .14 chord with equally spaced elastic supports similar to Bleich’s theory.8 shows the compression chord with the diagonals extended a length. These values are listed in Appendix D. 1956): 1.24 is equal to C and the equation becomes i & = J9 5 kL O) (2. 2. Effect of non-parallel-chord trusses.25) 2. If all of the crossframes are identical. Figure 2. 4. The following secondary factors were considered in his research (Holt. if the truss has at least ten panels. 5.5 Analysis of a Pony Truss Top Chord According to Holt Holt’s research in the 1950s tested pony truss bridges in an attempt to compare the actual buckling load of the compression chord with the design equations. then the effective length factor depends only on the stiffness of the transverse frames.8).4. 3. Lateral support given to the chord by the diagonals. Torsional stiffness of the chord and web members. Thus. d. The required spring constant with the axial load in the diagonal is &' = O)i  + j&# (2. Effect of web-member axial stresses on the restraint provided by them. Ce in equation 2. The results of Holt’s analysis proved that the error in determining the critically buckling load by neglecting the above factors was relatively small. 2. The spring constant must be larger the spring constant of the chord with rigid end. subjected to the compressive force vPd and pinned at points –n and +n. 383 0.035 1.643 2. a transverse force of 0.794 1.940 0.800 0.900 0.15 effective length factor is a function of Cl/Pc which is shown in the results section.200 1.045 1.112 0.000 2.068 0.031 0.754 3.460 2.254 2.980 0.595 2.600 0. A summary of Holt’s results can be seen in Table 2.960 0.950 0.088 0.183 2.101 1.00 0.007 0.2 (Galambos.3% of its axial load at its upper end.850 0.107 0.985 1.053 0.750 0.550 0.055 0.282 2.940 0. 1/k 1.593 2.787 3.480 2.714 0.094 2.434 0.263 2.665 2.097 0.750 1.100 0.350 0.785 2.139 0.114 0.013 1.961 2.362 1.448 2.708 0.555 0.158 0.070 0.450 0.700 0.087 1.010 0 0 .668 0.338 1.627 0.273 1.988 0.135 0.981 1.442 1.029 0.595 1. Based on the results of his research.133 0.025 0.600 0.750 1.878 0.660 3.694 1. in addition to its axial load.501 1.944 2.739 1.249 0.252 2.616 3.284 2.774 2.808 0.003 0 0.681 1.465 1.754 2.211 1.955 1.111 0.454 0.323 0.187 0 0.250 0.085 4 3.629 1.121 1.829 0.500 0.121 0.639 1.045 0 0.860 1.951 2.454 16 3.428 0.889 1.236 1.147 1.809 2.170 0.500 0. 1988) 1/k for Various Values of Cl/Pc and n n 6 8 10 12 3.359 1.121 0 Table 2.479 2.750 1.768 0.047 0.517 1.280 0.806 2.847 0.300 0.968 1.262 1. Holt noted that Bleich’s analysis showed adequate results for the entire range of effective length values where Timoshenko’s results show adequate results for k > 2.344 1. Holt (1957) also recommended the following on the design of the end posts The end post should be designed as a cantilever to carry.017 0 0.650 0.303 2.352 0.2.686 3.259 0.180 0.292 0.232 0.456 1.352 2.714 3.187 0.313 1.920 0.624 0.886 0.103 0.709 2.456 2.771 2.537 0.203 14 3.400 0.146 2.542 2.150 0.530 0.293 0.200 0. 5 + 1.26) where Le = kl and π2/4 ≅ 2.27 provides a minimum value of stiffness for the compression chord to reach the required critical load however. needed to fully brace the compression member over the length.5. . C. there are currently no design procedures available to account for initial imperfections.) . Creq. For trusses with a small l relative to Le equation 2. is equal to ideal stiffness l)*+ .4. -)*+ . The stiffness equation by Engesser was & '( = 2.3 going as low as 1.26 provides an accurate bracing stiffness for the solution. Lutz and Fisher used Engesser’s formula for a perfectly straight compression chord and developed a factor of safety to account for the out-of-plane stiffness. as l increases relative to Le.5 D% G s %*+ .16 2.5 %*+ ) . The fully braced case where k = 1 corresponds to Winter’s stiffness of -)*+ . Thus. is usually twice the . Such lateral deflections would reduce the maximum load capacity of the chord. A vehicle load on the floor beams would cause a displacement of the chord at the location of the load creating initial lateral displacements in the chord. Winter (1960) proposed the ideal stiffness. & = r2. then equation 2. p p (2. Lutz and Fisher proposed the following empirical equation to determine the required stiffness . Design recommendations by AASHTO only make a note of the design vertical truss members and the connection to the floor beam. . Their work was similar to the stiffness criteria George Winter proposed in 1960.6 Buckling Load with Initial Out-of-plane Deformations Initial out-of-plane deformations of the compression chord can reduce the critical buckling load determined by the previously mentioned methods. Equation 2. p (2.0. l. However. where the stiffness required. Lutz and Fisher (1985) addressed this issue in their publication to the Structural Stability Research Council in 1985. The chord could also have initial crookedness and unintentional eccentricities due to manufacturing.26 will result in unsafe errors.27) Then they extended the applicability of k factors to less than 1. There are two primary causes of out-ofplane deformations that need to be taken into consideration. 5 Pony Truss Design according to AASHTO Specifications AASHTO Specifications (2007) for the LRFD design of half-through trusses recommends design loads for both the top chord of the truss and the web verticals.14. By applying the appropriate vehicle or other live load cases to the truss. AASHTO states in section 6. .2. provide the elastic lateral supports at the panel points.9 that The top chord shall be considered as a column with elastic lateral supports at the panel points.17 2. the floor beam can be designed. The floor beams. The vertical truss members and the floor beams and their connections in half-through truss spans shall be proportioned to resist a lateral force of not less than 300 pounds per linear foot applied at the top-chord panel points of each truss considered as a permanent load for Strength 1 Load Combination and factored accordingly. in addition to the vertical truss members designed with the 300 plf applied load. The Young’s modulus for the members was assumed to be 29. In the following paragraphs. which were W27x84 sections. The combin14 element is an element with no mass and was used as the spring in all of the analysis. Finite Element Analysis The following investigation focused on validating the published results for calculating the effective buckling length factor.2) in his book. The investigation by Galambos was carried out assuming a factor of safety of 2. beam4. y and z axes. the translations in the z direction must be restrained to simulate the actual buckling properties of the compression chord since the chord has considerably more stiffness against buckling in the z direction. ANSYS is a commercial engineering software that is capable of analyzing different engineering properties on the structure very quickly with a host of different elements available. y and z axes and rotations about the x.1 general purpose finite element program. k.1 Finite Element Model of the Compression Chord The buckling analysis that is presented in this paper was conducted using the ANSYS 12. Stress stiffening and large deflection capabilities are also included. Using ANSYS. which provided the analysis for the basic compression chord case. The beam4 element is similar to the beam3 element except the beam4 element has six degrees of freedom: translations in the x. A beam3 element was used in the compressions chord model and is a uniaxial element with tension. 3. This element is a 2-D element. to show the adequacy of model. This combination element has longitudinal or torsional capabilities in 1-D. which was used in this analysis. When using the beam4 element. 3. The example used for the analysis was studied by Galambos 1988 (see Fig. These results were compared to the results for a 3-D element. The top chord of the truss consisted of a 10”x10”x5/8” box section that was designed for a maximum compressive force of 360 kips.000 ksi. that can be determined by the methods mentioned in the previous chapter. the element type and why it was used will be explained in more detail. compression. and bending capabilities. the nonlinear material properties of the compression chord can also be investigated. The vertical and web members were composed of a W10x33 sections in addition to the floor beams for the rigid frame.18 3. 2-D or 3-D applications. is a . Each node has three degrees of freedom: translations in the x and y axes and rotation about the z axis. The longitudinal spring-damper option. y and z directions. and height..2 Analysis of Top Chords as a Bar on Elastic Supports Figure 3.1 shows the boundary conditions of the 2-D model. which is supported in the x and y directions at the base and only in the x direction at the top to simulate the pin and roller connections. and bending capabilities. This element has three degrees of freedom at each node: translations in the x and y directions and rotation in the z direction.1 (SAP Inc. The beam23 element is a uniaxial element with tension-compression and bending capabilities which also has plastic.19 uniaxial tension-compression element with up to three degrees of freedom at each node: translations in the nodal x. 3. the beam23 element was used. A more in-depth description of each element is available from ANSYS 12.1 Finite Element Idealization of the Top Chord as a Bar on Elastic Supports . creep. A beam3 element was used in the compressions chord model and is a uniaxial element with tension. When analyzing the nonlinear properties of the compression chord. moment of inertia. and swelling capabilities. compression. 3. 2009). Fig. The element is defined by the area. as shown in Fig.3. In the work presented herein. The model was composed of beam4 (3-D beam elements) and 3-D link8 elements.4Fy. By rearranging Eq. the user needs to provide the stress-strain relationship of the material.2. 3-D beam elements were used to model the top and bottom chords of the truss structure. the tangent modulus of elasticity can be calculated as 7344 ksi for the chord. The diagonal members were modeled using these 3-D truss elements. the portion as the material reaches yield was defined using very small slope. The moment of inertia for the 10x10 box section was calculated to be 418. the analysis was carried out considering the effects of the nonlinear material behavior only.20 3. 3. The tangent modulus was found above using Engesser’s equation and varies as the critical load changes.3. . and the yield value for this model was 36 ksi. 3. This was done in order to compare the loads with the analysis mentioned above in chapter 2 since those authors neglected the effects of the diagonal web members. For this purpose. The ANSYS program allows the user to carry out a nonlinear buckling analysis. Link8 is a 3-D truss element with three translation degrees of freedom at each node.3 Finite Element Model of the Pony Truss The pony truss described above was also modeled.7. The proportional limit was specified as per the AISC manual as 0. The truss was restrained in the x. This tangent modulus is used in the nonlinear model. y and z directions on one end and only the y and z directions on the other end for pin and roller connections as seen in the Fig 3. This was necessary to avoid overshooting and any problems that may cause non-convergence to the solution. In the nonlinear model of the compression chord. 2. 3. Notice that in Fig.3 in4.2. the material nonlinearity was modeled using a multilinear isotropic hardening option (MISO) for the material with the stressstrain profile as shown in Fig. the program calculates a new element stiffness matrix based on the element strains in the stress-strain profile provided.2 Finite Element Idealization of the Pony Truss The nonlinear solution of the compression chord in ANSYS used the Newton-Raphson option to converge on the displacements of the solution.1%. An initial load larger than the predicted buckling load was applied to the chord and ANSYS then uses load steps to continuously apply the load in small increments to iterate the solution. For each iteration.21 Fig. . it was assumed that a converged solution was reached when the difference in displacements between load steps was equal to or less than 0. For the nonlinear model in this research. The results of analyzing the truss described above using the different available analyses techniques are summarized in the following chapter. 3. 004 0.01 Strain (in/in) Fig.007 0.002 0.006 0.3 Stress-Strain Curve .009 0. 3.003 0.22 Stress-Strain Curve 40 35 Stress (ksi) 30 25 Et = 7344 ksi 20 15 10 5 E = 29000 ksi 0 0 0.008 0.001 0.005 0. 500 1.200 1. Pcr. the equations needed to be rearranged for the calculation of the effective length factor. 4.600 Bleich Timoshenko 0.000 0.000 0.000 Cl/P Fig. Effective Length Factors 1.200 0. Pe.500 .23 4.000 1.400 Lutz-Fisher 0.800 1/k Engesser 0. This factor is required to check the capacity of the top chord member following the recommendation used in AISC compression calculations.1 and 4.500 2. The relationships of the suggested procedures that can be used in calculating the effective length factor are shown in the Figures 4.000 0.2 below. of the member. 4.500 4.1 Compression Chord Design Curves 4. Discussion and Results of the Analysis of a Pony Truss Top Chord 4. k.2 depicts the energy relation between the stiffness of the lateral elastic support and Euler’s load. and Fig.1 Effective Buckling Length Factor – Lateral Support Stiffness Relationships For the comparison of the equations in chapter two with the FEM. Figure 4. to the critical compressive load.1 relates the inverse of the effective length factor to the stiffness of the lateral support.500 3.000 2.000 3. From the specified truss geometry. C. a structural analysis of the truss system illustrated that the top chord member being investigated was subjected to a compression force of 360 kips. 4.24 Critical Buckling Load 120 100 Pcr/Pe 80 60 40 20 0 0 500 1000 1500 2000 2500 3000 (CL/Pe) Fig. the provided lateral spring stiffness. The dimensions of the truss structure were previously listed in chapter three of this document. was used in the analysis.2 Energy Method Design Curve 4.2 Example Calculations for the Buckling Load of a Pony Truss Given below are the calculations that were used to develop the buckling load of the truss structure that was examined by Galambos (1988). In this example. was calculated using Eq. A1. as suggested in the literature. A factor of safety of 2.1 as C = u w z f s xyZ yz v r = {888 XM xM f ‚ x∗X~M ∗€M 9 8 | = 6.75 kip/in . 5 ksi Hence. Pn. one can calculate the value of 1/k according to the different analysis techniques that were previously summarized. l∗9Š8 -.2.8l F‘ = ’ u “”  D G • = 51. using the ASD and LRFD approaches.1 Calculations following Engesser’s Procedure 1/k = 0.658 NŠ] 98{.30 ˆ Ž = 9.5 ∗ 25.2 ksi = D0.78 (From Fig.1 = FPŽ = –0.28 ˆ Ž = = 50.9L ˜™ ] ˜‘ š F› = 109. Fe.1) k = 1.1. G 36 = 31.3 9.8 ksi = D0.0 = Ÿ   ¡¢£¤ This procedure is then repeated using the other methods mentioned in chapter two.25 Then.768 (From Fig. one can calculate the critical buckling stress. 4.658 NŠ] 99N. the ratio of Cl/P was determined as ‰ = ‡ˆ = 1. respectively as F‘ = ’ u “”  D G • = FPŽ = –0. 4. 4. the critical buckling load is: P` = FPŽ Až = 31.1) k = 1.658 ’ {888 JŒ9.2.N8∗9Š8 -.NL ˜™ ] ˜‘ š F› = 113.4 ksi .‹Œ∗9Š8 ‹ 8 Using the calculated ratio Cl/P in conjunction with Fig 4.l G 36 = 31. and the nominal buckling load. 4.50 Š.658 ’ {888 JŒ8.8l Following the design steps that are given in the AISC (2011) manual.2 Calculations following Bleich’s Procedure 1/k = 0. 2.NL ˜™ ] ˜‘ š F› = 104.1) k = 1.8 ∗ 25.5 = FPŽ = –0.{L ˜™ ] ˜‘ š F› = 67.0 = ŸŸª ¡¢£¤ 4.NN∗9Š8 -.3 Calculations following Timoshenko’s Procedure 1/k = 0.2.8 ksi P` = FPŽ Až = 30.33 ˆ Ž = 9.658 ’ {888 JŠ-.719 (From Fig.5 Calculations following Holt’s Procedure 1/k = 0.658 NŠ] Š‹.8l F‘ = ’ u “”  D G • = 54.2 ksi P` = FPŽ Až = 31. G 36 = 30.‹ G 36 = 31. 4.66 ˆ Ž = 9.4 Calculations following Lutz and Fisher’s Procedure 1/k = 0.4 ∗ 25.658 NŠ] 98-.8 ksi P` = FPŽ Až = 28.2.2) k = 1.3 = FPŽ = –0.N{∗9Š8 -.9 ksi = D0.604 (From Fig.ŒL ˜™ ] ˜‘ š F› = 96.658 ’ {888 JŒ .{ G 36 = 28.8l F‘ = ’ u “”  D G • = 64.39 ˆ Ž = 9. 4.658 NŠ] {Š.0 = Ÿ § ¡¢£¤ 4.2 ∗ 25.0 = Ÿ¨©¡¢£¤ 4.2 ksi = D0.ŠŠ∗9Š8 -.8l F‘ = ’ u “”  D G • = 52.26 P` = FPŽ Až = 31.8 ∗ 25.7 ksi = D0.0 = ŸŸ« ¡¢£¤ .9 = FPŽ = –0.1) k = 1.658 ’ {888 JŒ-.75 (From Table 2. 6 Calculations using the Energy Method P‘ = J­L = ‡­ ‰® ‰Z• ‰® = ’ u¬ ’ ∗‹N--∗-9l. 4.28 1.30 1. the results should be compared to the design load of P=360*2.2) PPŽ = P‘ R = 11.1. . it was later addressed that the compression chord in Galambos example was designed for the maximum panel load using AASHTO’s 1983 formula which uses a factor of safety of 2.12 (Ziemian.14 2.00 2.16 2.S 2.27 4.2. 2010).2 = 11.67 (From Fig.12 = 763 kips.84 ∗ 61.67 = Ÿ°ª ¡¢£¤ A summary of these results is shown in Table 4. it may be noted that using the energy method underestimates the critical buckling load since it is based on an assumed deformed buckling shape.‹Œ∗9Š88 99. However.1 Critical Load Results Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method k 1.39 1. Table 4.03 The results above show the methods of analysis in chapter two reasonably predict the critical load of the compression chord for the pony truss example.33 - Pn 788 785 721 770 779 730 F. The factor of safety listed in this table was calculated as the ratio between the estimated critical buckling load and the applied member load of 360 kips.18 2.66 1.84 kips = 61. As noted in Galambos (1988) the factor of safety for the compression chord on elastic supports was 2.l- = 912. However.N J9Š88L Š.0 when determining the Cl/P ratio. In essence.19 2. Et = 7344 ksi. Using a 2-D beam3 element with the tangent modulus. 4. . the validity of these tests was checked with both a 2-D and 3-D element model.28 4.3. the truss was analyzed in ANSYS to solve for the critical buckling load. the critical buckling load was calculated to be 719. 4.5 the buckling load in the nonlinear model was equal to 720 kips. All of the FEM solutions prove the reasonability of the calculations determined in the previous section with C = 6.3.3 Analysis of the Pony Truss using Finite Element 4.1 Two Dimensional Analysis To verify the critical load calculations of the compression chord. As mentioned in chapter three. This buckling load was found by plotting the Load vs. there are two changes in slope which account for the change in modulus at the proportion limit and the yielding limit. Displacement in the vertical direction and then noting the load at which large displacements occur with only a small increase in load.213 kips as shown in Fig. Using Fig. The compression chord buckling load was also determined using a nonlinear approach.4) validating the model. The next step of the analysis was to check the critical load on the compression chord when the entire truss was modeled.213 kips (Fig. As seen in the graph. The 3-D element analysis of the compression yielded the exact same buckling load of 719.75 and Et = 7344 since the three compression chords analysis all had approximately the same buckling load of 720 kips. 4. 4.3 2-D Compression Chord Elements Fig.4 3-D Compression Chord Elements .29 Fig. 4. 30 Fig. Displacement Curve 4. the truss was restrained in a manner to create two symmetrical sides of the truss. increased dramatically to a load of 1121 kips on the compression chord which can be seen in Fig. It is the opinion of this author that the increase in critical buckling load is due to the effect of inertia from the compression chord. In order to only focus on the lateral displacements of the compression chord. By applying these boundary conditions. the resultant load could be compared to the compression chord models. when analyzing the compression . 4.5 Nonlinear Load vs. The spring stiffness. when modeling the entire pony truss. the truss was restrained against translation in the z direction and rotations in the x and y directions at the center of the floor beams. 4. To accomplish this symmetrical model.6.2 Three Dimensional Analysis The pony truss model was analyzed similar to the compression chords in that a compression load was applied to the top chord on each side of the truss. The critical buckling load.3. 31 chord by previous work. 4. Fig.6 Pony Truss – Top Chord Analysis . was composed only of the frame stiffness as calculated in Appendix A. The goal in the preceding paragraphs is to analyze this theory using finite element models. However. the lateral stiffness of the chord itself provides an addition stiffness which explains the significant increase in load when modeling the entire truss. 3.7. From this table. 4. and the inertia of the compression chord was varied from 425 in4 to 0. This table shows that the moment of inertia for the compression chord should be accounted for when determining the lateral stiffness at the frames locations and that the current determination used for the frame stiffness is an underestimation. the compression chord seems to have an effect on the overall lateral stiffness against buckling. a value of C = 1/0. This discrepancy was examined using the finite element method.8) which can be used to find the stiffness of the frame by taking the load divided by the displacement.576 k/in is determined. Using the displacement of the frame to calculate the stiffness. To prove this point. the new stiffness should be used.148014” (Fig. The load was applied at each frame location individually to get the stiffness at each restraint point. The frame was analyzed as a simply supported structure with a load of 1 kip placed on each vertical web as seen in Fig.9. In order to determine if the inertia of the compression chord has an effect on the stiffness of the compression chord.3 in4. which can be seen in Fig. 4. a transverse load was applied to the center panel. The results of this analysis were then used to calculate the stiffness of the lateral supports.4 Effects of Compression Chord Moment of Inertia on the Stiffness of the Elastic Supports When calculating the elastic stiffness restraining the compression chord from buckling in Appendix A. several models of the truss structure considering different moments of inertia of the top chord were analyzed by applying two lateral loads at the panel points as seen in Fig 4. In order to compare the procedures from the published works.2 were determined using a compression chord tangent modulus of 7344 ksi and the moment of inertia of 418. The same procedure was then repeated except with entire pony truss modeled as a simply supported structure. A summary of these results can be found in Table 4. The results in Table 4. similar to before. This is the same stiffness that the equation from Appendix A yielded.2 which references the panel locations in Fig 4. 4. .32 4.9.001 in4. the stiffness is a product of the rigid frame only and ignores the contributions of the stiffness from the top chord. the frame itself was analyzed as a control procedure before examining the entire truss. To analyze this theory.9.148014 = 6. The results of this process can be seen in Table 4. The load causes a displacement of 0. 8 Rigid Frame Displacements .33 Fig.7 Rigid Frame Boundary Conditions Fig. 4. 4. ∆ (in) 0.07249 0.07226 0.839 13.07226 0.795 13.07226 0.9 Load Application of Pony Truss Frames Table 4.07227 0.07249 0.837 13.34 ` 1 2 3 4 5 6 7 8 9 10 11 Fig.10612 Stiffness. C (k/in) 9.07227 0.07518 0.837 13.301 13.10612 0.795 13.301 9.839 13.839 13.2 Resultant Stiffness at Each Bay Panel Location 1 2 3 4 5 6 7 8 9 10 11 Displacement.426 13. 4.423 .07518 0. 069192 0.100502 0.10 Pony Truss Lateral Displacement at Frame Location Table 4.3 Effects of Top Chord Inertia on the Lateral Stiffness Moment of Inertia (in4) 425 325 225 125 25 0.052882 0. C (k/in) 13.4 summaries the critical buckling loads of these calculations. 4.98 11.058840 0. by completing the same procedure mentioned in section 4. Engesser’s approach was not valid for new analysis since his theory is based off a minimum k value of 1.75 (using Eq.89 12.45 8.141938 Stiffness.048816 0.35 Fig. all of the methods underestimate the critical load of the . C = 13.04 6. A1.3 Displacement.08 7.87 10. With the new stiffness value.001 418. ∆ (in) 0.3.5 Analysis with Modified Elastic Stiffness The calculation for each of the methods in chapter 2 was redone using the new stiffness value.1) 4.8 found in the previous section.2. Table 4. This buckled shape essentially shows the “exact” buckling load and shape of the compression chord treated as a single member.8.30 2.01 - Pn NA 827 807 819 828 1032 F.24 2. . The buckling load was found to be 979.S NA 2.08 1. Although this load is closer to the pony truss analysis of 1121 kips.2. Table 4.87 Using ANSYS. the compression chord modeled with the new stiffness provided a large increase in the compression capacity.12. 4.02 1. The next figure.30 2. C Method Engesser Bleich Timoshenko Lutz & Fisher Holt Energy Method K NA 1. shows the buckling shape with the elastic supports having the new stiffness values determined in Table 4.4 Critical Load Results with New Stiffness. Fig.36 compression chord when analyzing the entire truss since the formulas do not account for the inertia stiffness of the compression chord and the increase in stiffness with respect to k is not a linear response.28 2. Figure 4.03 kips. the end supports in this model were rigid which does not accurately account for the elasticity at these supports.11 shows the compression chord modeled with rigid supports and the interior restraints having a stiffness of 13.16 1. 4.37 Fig.11 Compression Chord Analysis with New Stiffness Fig. 4.12 Compression Chord Analysis with Elastic Ends . A summary of these results can be seen in Table 5. Table 5.01 828 Energy Method 730 1032 FEM Chord . This discrepancy was examined using the finite element method. the formulas in Appendix A ignore the contributions of the moment of inertia of the top chord.39 770 1. Summary.1 Summary Previously published work closely relates the critical buckling load of the compression chord to the actual load when the chord is taken as a single member with the stiffness provided according to Appendix A. a new stiffness was calculated and used to calculate the new buckling load.16 807 Lutz & Fisher 1.38 5.8 Method k Pn k Pn Engesser 1. and Recommendations 5.1: Buckling Load Analysis Results C = 6. From the FEM. This analysis shows that the omission of the top chord inertia when calculating the frame stiffness will cause an underestimation of the critical buckling load.1.02 827 Timoshenko 1.33 779 1. which is compared to full pony truss model load of P = 1121 kips.28 788 NA NA Bleich 1.Rigid Ends 719 979 FEM Chord – Elastic Ends 644 886 FEM Pony Truss Pcr = 1121 kips .30 785 1.08 819 Holt 1. However.75 C = 13. It is evident that the capacity of the compression chord can be affected by including the inertia effects of the chord. Conclusions.66 721 1. k.39 5.2 Conclusions The following are the conclusions that can be attained from the study presented herein: • Current design of a compression chord for a pony truss bridge would be best accomplished by using the effective length factor. Due to the complexity of a full pony truss and the necessity of empirical confirmation of any design model. 5. provided by Holt to determine the critical load from Chapter E in the AISC manual. it is important to be able to calculate this stiffness factor. • The finite element method will provide satisfactory results when using the appropriate member rigidity. testing of physical models is required before determining any definitive conclusions.e.3 Recommendations Past testing on pony truss bridges is limited to Holt’s work (1957) which focused on the effective length factor on the compression chord and not the stiffness supplied by the frames. However. Empirical testing of a pony truss model could reveal a better understanding of the actual stiffness supplied by the compression chord inertia. • Test a model to verify the effect of inertia on the compression chord stiffness. using the correct tangent modulus of elasticity. However. its application depends on the number of the provided elastic supports. • Determine a method to verify the rigidity of the frame connection. • The energy method provides a close answer for calculating the critical buckling load for the truss top chords with rigid supports. This means that one must consider the type of connection between the floor beams and the vertical as well as with diagonal members when calculating the lateral spring stiffness.. i. such an effective length factor is dependent on the stiffness of the elastic lateral supports. Hence. . • All available procedures require the knowledge of the provided elastic stiffness. The following is the derivation of the spring constants of the lateral supports if: 1. The connection between the vertical and the floor beam is assumed rigid. C ± ∗ ² = 6 ³! #$ Moment due to virtual loads. For this purpose. Q. This can be accomplished by calculating the force.40 Appendix A Calculations of the stiffness of the lateral support to a pony truss A h B Ic Ib C D b The calculation of the stiffness for the lateral supports to the top chord of an unbraced truss can be calculated using the energy method. (Q was assumed = 1) 7 = #$ D &ℎ ∗ ℎ ∗ N ℎG + #$ J&ℎ ∗ a ∗ ℎL * 9 h 9 W . C C Q Q Ch Ch h Ch h Moment due to applied real load. C. the virtual work method was utilized. that is required to induce a unit displacement in the lateral direction at the panel point. is a virtual load that is applied in the horizontal direction at the points where the displacement is to be calculated. a term of Ld3/3Id is added in the denominator where the additional stiffness is an addition to the vertical web. The moment. one needs to substitute a value of 2 for the displacement. In these in calculations.2) µ W f º · ¶W µ x x¶* ¸x¶i – š ¹i 3. at the vertical-floor beam connection is:  = »¼ ½ = & ∗ ℎ  ½ = & ∗ ℎ/»¼ Following the analysis that was summarized in the section above. & = µx # (A1. a virtual load.41 where. the relative displacement between points A and D can be calculated as: .1) 2. in the equation above. to calculate the elastic constants. ∗<Hx N#$* & = + # <H V #$W µx µ W s r f x¶* ¶W (A1. Therefore. To account for truss diagonals. This yields to: 1 ∗ 2 = Or. Q. Q. M. of unity was assumed. ∆. The effect if joints C & D are not rigid 1 1 A D θ ks B ks C Where. is the relative displacement between points A and B. Notice that. ks is the joint rotational stiffness. ∆. C. of the lateral spring. ¿ ‚+ Á # ‚ . would underestimate the capacity of the chord. However. Bleich determined that disregarding the torsion on the compression chord would lead to an unsafe design for members with only one axis of symmetry. which have a high torsional rigidity. and hence the bending of the chord will be accompanied by twisting.¿ <H . This problem was studied in detail by Bleich (1952). in most cases. then it is a completely rigid connection Notice that the above equations did not take into account the actual shape of the cross section of the top chord members. thin-walled sections having only one axis of symmetry. the above relations for calculating the spring constant must be modified to account for such effects. the cross section of the chord consists of open.42 ¾ = 2 = 1 = 1 = 1 = & = <H # <H # <H # ∗<H # | ‚ + H N$* | ‚ + | H N$* H N$* À| H + N$* <H V #$W V $W + V #$W +2∗½∗ℎ +| ‚∗ℎ ‚+| $W <H V <H .3) if ks = ∞.¿ # µ W  H r f f s x¶* ¶W ÿ (A1. . Therefore. box sections. the problem of bucking of the chord will be a caused by flexural and torsion. Thus. Conversely. External Work V = 5‰ V = 5‰È Æ π Ç V = 5‰ 68 y′ dx ­ |68 cos ­ ­ Δ2 ÆπÇ ­ | + dx‚ È Æ π sin -­ Æπ ­ ÆπÇ ­ ­ ‚ 8 y = ∆ = a sin Æπ N . Initial conditions EI P L/3 P L/3 L/3 2.43 Appendix B Energy Method to determine the buckling load for a bar with 2 springs 1. Displaced shape x Δ1 y = a sin y′ = y′′ = ÈÆπ ­ ÆπÇ ­ cos ÈÆ π ­ ÆπÇ ­ sin ÆπÇ ­ Where: y9 = ∆9 = a sin Æπ N and 3. 0 0.1) .44 V = 5‰È Æ π -­  4.0 = ‰Æ π ­  5‰ÈÆ π ­  + Ca |sin = C |sin Æπ N + sin P = Æ ’ |sin ‡­ Æπ N Æπ N + sin Æπ N ‚+ + sin Æπ N ‚+ ÈÆ= π= u¬ ­x Æ= π= u¬ ­x Æπ N ‚ + Æ π u¬ ­ (B1. Combine external work and internal energy and solve for Pcr π= V+U π= ∆π ∆È 5‰È Æ π -­  + ‡È |sin Æπ + sin N Æπ N ‚+ È Æ= π= u¬ -­x = 0. Internal Energy U = C∆9 + C∆ + 68 ′′ dx 9 9 ­ u¬ U = C∆9 + C∆ + 68 9 9 9 Æπ Ca |sin U= U= N ‡È + sin |sin ­ È Æ= π= u¬ Æπ N Æπ N ­= ‚+ + sin sin ÆπÇ ­ dx È Æ= π= u¬ Ç | − ­= Æπ N Æπ sin -­ ÆπÇ ­ ­ ‚ 8 È Æ= π= u¬ ‚+ -­x 5. Initial conditions EI P L/3 P L/3 L/3 2. External Work V = 5‰ V = 5‰È Æ π Ç V = 5‰ 68 y′ dx ­ |68 cos ­ ­ V = ÆπÇ ­ | + È Æ π sin -­ Æπ 5‰È Æ π -­ dx‚  ­ ÆπÇ ­ ­ ‚ 8 Δ2 .45 Appendix C Engesser’s Method to determine the buckling load for a bar with 2 springs 1. Displaced shape with continuous elastic medium x Δ1 y = a sin ÈÆπ y′ = y= ­ ÆπÇ ­ cos ÈÆ π ­ ÆπÇ ­ ÆπÇ sin ­ 3. Combine external work and internal energy and solve for P π= V+U π= ∆π ∆È 5‰È Æ π  -­ + έÈ - + È Æ= π= u¬ -­x = 0. < = )*+  . Solve for required stiffness coefficient P = ’ ud ¬ ‡Ï + ’ . -#d .46 4. Ï where: α = C/L and v = L/m (wavelength) Note that Pcr will be at a minimum when 0 = − ’ ud ¬ ‡Ï + ’ . Ïx v = πc = C '( #d $.1) 6. back into the equation above: (C1.2) . v. Internal Energy U = 68 y dx + 68 ­ Î U= u¬ ÎÈ Ç Æπ | − -­ sin έÈ U= - ­ È Æ= π= ­= ÆπÇ ­ ‚ + ­ sin ÆπÇ ­ dx È Æ= π= u¬ Ç 8 ­= | − where: α = C/(L/3) Æπ -­ sin ÆπÇ ­ ­ ‚ 8 È Æ= π= u¬ + -­x 5.0 5‰Æ π 0. )* O =0 Substituting wavelength.0 =  ­ P = + έ + Æ= π= u¬ ­x Æ ’ u¬ έ + ­ Æ ’ (C1. J2> ÒÓ¼H B?±PQR Õ? 1L ÜÝÙ ± ÜÝÙ J2> − 1L cosh .47 Appendix D Schweda’s extension of Bleich’s analysis j= -.1) . = Jß + 1L + Û + Jß − 1L + Û 2 cos Ù = Jß + 1L + Û + Jß − 1L + Û (D1.  ß = |1 + cos . G‚ 9 " " " " á = D1 − Ü l. + Ü lã Ýâ> . " " " " 2 cosh .  G ÜÝ . − Ü l.ÚÞG − 1L sin Ù ± sinh . D. sin ÙJ2> − 1LÚ + 2Û×ÜÝℎ .  D1 "  − 9 Ö×sinh . − Ýâ> .J2> − Where:  = 2Jß − 2L + Ü Û = √á − ß " -. 16 1.96 0.07 2.17 0.77 1.92 1.1 2.2 2.35 1.0 2.37 1.79 2.39 1.32 1.3 1.81 0.4 1.25 1.82 0.97 c k 2n=8 1.18 1.20 1.08 2.54 1.32 0.06 1.9 0.3 2.94 1.04 1.66 1.90 .00 1.00 1.02 1.90 0.12 2.40 1.90 0.85 0.0 2.03 1.29 1.24 1.55 1.10 1.87 0.80 0.30 1.87 0.35 1.25 1.6 1.71 0.25 1.83 0.86 0.69 1.65 2.41 1.73 1.44 1.5 1.27 1.26 1.3 0.47 1.78 0.95 0.97 1.94 1.5 1.36 1.89 2.4 2.75 0.25 1.54 1.8 1.9 2.08 1.94 1.2 1.12 1.04 1.5 1.8 1.6 1.06 0.96 2.17 1.96 1.6 1.93 1.85 0.01 2.99 1.94 1.5 1.85 0.04 1.7 1.91 0.62 0.90 0.1 2.28 1.85 2.00 1.84 0.94 1.12 1.64 1.8 1.03 1.71 1.2 1.85 0.2 1.93 1.74 0.19 1.17 1.48 Tables of Factor ψ: c k 2n=6 1.42 1.12 1.13 2.06 1.04 1.55 1.32 1.20 0.08 1.97 1.10 1.96 1.4 0.46 1.11 1.41 2.98 2.54 1.48 1.4 0.97 2.11 1.06 1.2 2.76 1.94 0.7 1.43 2.39 1.87 0.93 2.89 1.9 2.69 2.3 2.6 1.58 0.85 0.38 1.4 2.92 0.83 3.60 1.90 0.48 1.4 1.3 0.17 1.07 0.99 1.7 1.05 1.27 1.65 0.26 1.7 1.09 1.5 1.27 1.94 0.2 0.06 1.36 0.3 1.5 1.66 1.8 1.01 1.9 0. 17 1.49 1.60 1.4 0.34 1.24 1.87 0.7 1.7 1.34 1.39 0.01 1.4 0.92 0.21 0.96 2.93 0.9 2.82 0.32 1.87 0.19 0.25 1.63 1.80 1.60 1.92 0.95 1.6 1.81 0.36 0.94 1.00 1.94 0.05 1.06 1.78 0.06 1.66 2.9 0.1 2.28 2.3 0.82 0.83 0.23 1.0 2.99 1.2 2.63 0.06 0.11 1.74 2.20 1.97 1.4 1.6 1.07 1.14 2.26 1.10 1.77 0.60 .06 1.5 1.48 1.93 1.65 1.63 1.3 2.8 1.64 1.5 1.94 0.2 1.91 0.87 0.83 0.93 1.46 1.61 1.49 c k 2n=10 1.00 1.2 0.86 0.8 1.8 1.86 0.35 1.17 1.03 1.2 1.4 2.85 2.03 1.5 1.4 1.3 2.05 1.57 1.0 2.74 1.75 0.69 0.00 1.2 0.3 1.45 1.92 0.30 1.84 0.27 1.5 1.62 0.12 1.88 0.89 0.92 1.26 1.13 1.54 1.7 1.98 1.00 1.91 0.6 1.5 1.84 0.00 1.99 1.3 0.65 1.99 1.06 1.42 2.08 0.80 0.93 1.9 0.12 2.21 1.18 1.12 1.2 2.74 0.7 1.82 0.40 1.89 1.97 0.43 1.37 1.05 1.14 1.5 1.83 0.88 0.36 1.84 0.78 1.98 1.09 1.41 1.93 1.34 1.22 1.17 2.8 1.70 0.12 1.96 0.87 c k 2n=12 1.87 0.72 1.55 1.73 1.4 2.9 2.95 1.1 2.19 1.14 1.6 1.12 2.22 1.66 1.83 0.99 2.3 1. J. 1983. The Analysis and Design of Single Span Pony Truss Bridges. 2009. and Malhas. American Institute of Steel Construction.. B. Steel Structures Design and Behavior 5 Ed. Holt. E. AASHTO. SAP Inc. Stability of Bridge Chords without Lateral Bracing. . E. J. New York. Pearson Prentice Hall. D. 3. New Jersey. F. Buckling of a Pony Truss Bridge. Column Res. E. McGraw-Hill Book Company. No. 1957. 2005. G. Galambos. Guide to Stability Design Criteria for Metal Structures 4th Ed. Lutz. Salmon. 4. Chapman and Hall. Inc. T. Buckling of Bars. and Fisher. Vol. A Unified Approach for Stability Bracing Requirements. Holt. Theory and Design of Steel Structures. T. F. 1951. 2009. AISC Eng. and Shells. L. McGraw-Hill Book Company. AASHTO LRFD Bridge Design Specifications. 1988.S. & Almroth. New York. Column Res. Johnson. No.O. F. McGraw-Hill Book Company. Washington. 1936.1. 4. Buckling of a Continuous Beam-Column on Elastic Supports. 1975.O. and Mazzolani. S. 1956. Stability of Bridge Chords without Lateral Bracing. Brush. Tests on Pony Truss Models and Recommendations for Design. No. Bleich.C. Council Rep. 1952. Holt. Timoshenko.50 References American Institute of Steel Construction (AISC). American Association of State Highway and Transportation Official (AASHTO).M. New York. 2007.. J. New York. No. Oxford University Press. D. Analysis of Structures: Strength & Behavior. Column Res. 1985. Buckling Strength of Metal Structures. Theory of Elastic Stability. New York. Ballio. Inc. E. 1. Column Res. Steel Construction Manual. Holt. 2011. Stability of Bridge Chords without Lateral Bracing. John Wiley & Sons. C. 1952. ANSYS User's Manual for Release 12. No. 2. Plates. New York Thandavamoorthy. Council Rep. Inc. Inc. Council Rep. Stability of Bridge Chords without Lateral Bracing. Council Rep. 22. 2010. 807-845.D. 125. pp. Lateral Bracing of Columns and Beams. R. John Wiley & Sons. New York. Guide to Stability Design Criteria for Metal Structures 6th Ed. ASCE. . Trans. Vol.51 Winter. G. 1960. Ziemian. 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