Computer Aided Optimum Design in Engineering IX 165 Compression modelling of pin-jointed trusses S. D. Waller & K. A. Seffen Department of Engineering, University of Cambridge, U.K. Abstract A method is presented for including imperfections within the analysis of pin- jointed trusses, without requiring several beam elements to model each strut. A non-linear material model is used to represent the loss of stiffness due to elastic buckling, before the critical Euler load is reached. The model is used to assess the validity of assuming a perfect structure, which is common for size, shape and topology optimisation of trusses. Applying the model to investigate the stiffness of tree-like structures for roofing supports is presented, along with the future exten- sion of the model to truss-columns. Keywords: non-linear buckling, truss optimisation, critical slenderness, bowing deformation. 1 Introduction Pin-jointed trusses are used for applications such as transmission towers, space and roofing structures. Triangulated truss structures are inherently efficient and lightweight, because the members are used in tension and compression only, modes in which they are more effective than when subject to bending. Finite ele- ment analysis of such trusses is comparatively simple, because each node only requires one degree of freedom, to account for axial force in the local co-ordinate system. The treatment of compression is also greatly simplified because each mem- ber is pin-ended and undergoes Euler buckling. When optimising the size, shape and/or topology of truss structures, it is neces- sary to evaluate the structural performance of many different configurations, with minimal computational cost. It is common to find the minimum weight structure that satisfies various constraints, such as stress, displacement and buckling. Each finite element analysis is usually linear, and assumes perfectly straight members. WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 166 Computer Aided Optimum Design in Engineering IX Many studies limit compressive stresses to avoid yielding or simple Euler buck- ling (e.g. Xu et al [1]); some studies include a safety factor in the buckling stress (e.g. Smith [2]), while others also include an arbitrary maximum slenderness con- straint (e.g. Achtziger [3]). Some studies use a more complex approach, which lim- its compressive stresses according to codes of practice (e.g. Guerlement et al [4]). In this paper, an appropriate maximum slenderness constraint is derived math- ematically, such that the perfect structure assumption in not violated. A method is also presented for simplified stiffness analysis of struts that are too slender to be considered perfect. A non-linear material model is used to represent the loss of stiffness caused by elastic buckling of the imperfect struts. This material model can be incorporated into any standard finite element package (if non-linear materials are permitted), to include the loss of stiffness due to lateral bowing of imperfect slender members. 2 Single element deformation An imperfect strut is considered for analysis, with length L,areaA, second moment of area I, radius of gyration r, and slenderness ratio, λ = L/r.Thestrut is axially compressed by a force P , which produces an average stress, S = P/A: σ is reserved to represent the stress at a single point within the bar. The axial short- ening is δ, which corresponds to an axial strain, = δ/L. The initial and deflected shapes are assumed to be y0 = e0 sin(πx/L) and y = e sin(πx/L) respectively, where e0 is the central imperfection and e is the central deflection. The ends are pinned at x =0and x = L. The eccentricity ratio is ξ = e0/L. The maximum value for the eccentricity ratio ξ is limited to 0.001 by the materials delivery stan- dards of most industrialised nations [5]; so this value is used for all of the examples in this paper. The critical Euler load PE, and stress SE,are π2EI π2E P S E = L2 ; E = λ2 and a dimensionless loading ratio can be defined as α = P/PE = S/SE.A complete description of the assumptions required is given in [6]. The material is linear elastic, with Young’s modulus E, and yield stress Y . Member deflections are small, so the square of the slope is negligible compared to unity. Compressive loads and deflections are assumed to be positive. The yield stress of the material is not exceeded during the deformation. The cross-section is taken to be a uniform hollow tube (due to its widespread practical application), with a radius-to-thickness ratio, µ = R/t: µ is assumed to be sufficiently large so that the average radius is identical to the outer-fibre radius. Calculation of the load shortening characteristic is summarised here, and explained fully in [6] (which also provides a comprehensive review of associ- ated literature). The central lateral deflection can be obtained as a function of the applied load using engineers beam theory. The axial shortening can then be found as a function of the deflected shape, using a path-length equation. Thus, the total WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Computer Aided Optimum Design in Engineering IX 167 axial shortening can be found as a function of the applied load by combining the axial shortening due to lateral bowing with that due to squashing, which gives PL ξ2λ2(2 − α) δ = +1 (1) AE 4(1 − α)2 3 Limit of validity for analysing a perfect structure The validity of analysing a perfect structure can be assessed by considering Sbow, which is defined as the average stress that causes significant axial shortening due to bowing for an imperfect member. The parameter β is used to characterise the percentage of axial shortening due to squashing that is considered significant. The value of Sbow can be found from Eqn (1), by setting the bowing component equal to β ξ2λ2 (2 − α) 2 = β 4(1− α) The resulting quadratic equation is solved for α, then converted to Sbow, such that π2E ξ2λ2 ξλ ξ2λ2 S = 1 − − + β (2) bow λ2Y 8β 2β 16 Sbow can be considered as the stress that causes the strut to have significantly lost its stiffness, due to lateral bowing of the imperfect member. The maximum strength of an imperfect strut is the Perry-Robertson stress SPR, which can be found from [7], as a function of the yield stress Y , and critical Euler stress SE,as √ √ 2 Y +(1+ 2ξλ)SE Y +(1+ 2ξλ)SE S = − − YSE (3) PR 2 2 If Sbow is greater than SPR, then the axial shortening of each member is PL/AE, all the way up to the maximum strength: thus, a linear finite element analysis that assumes perfectly straight members and limits member stresses to SPR is valid. Furthermore, the Perry-Robertson stress SPR is compared to that required to cause significant axial shortening due to bowing Sbow, along with yielding and Euler buckling in Fig. 1, with the stiffness to yield ratio, E/Y = 600, and the percent- age of axial shortening due to squashing considered significant, β =5%. λbow is defined as the crossover slenderness ratio such that significant axial shortening due to bowing occurs before the maximum strength according to Perry-Robertson; for this case λbow =85. λbow is plotted as a function E/Y and β in Fig. 2. If a structure has any strut with slenderness greater than λbow, then it is essential to model that strut using a non-linear analysis that includes the loss of stiffness due to lateral bowing. A method is now presented to achieve this, without using several finite elements to model each strut. WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 168 Computer Aided Optimum Design in Engineering IX 1 0.8 0.6 S/Y 0.4 0.2 0 0 20 40 60 80 100 120 140 160 180 200 220 λ Figure 1: Ratio of average stress S, compared to yield stress Y , for various failure modes. The solid line represents Euler buckling and yielding. The dotted line represents the Perry-Robertson failure from Eqn (3). The dashed line represents the average stress required to cause significant axial shorten- ing due to bowing for an imperfect member, from Eqn (2). E/Y = 600, ξ =0.001, and the percentage of axial shortening due to squashing con- sidered significant is β =5%. 120 110 100 90 bow λ 80 70 60 200 300 400 500 600 700 800 900 1000 E/Y Figure 2: Slenderness ratio λbow, such that significant axial shortening due to bow- ing occurs yielding at the outer-fibre. For this example β =1%(solid line), β =5%(dotted line), β = 10% (dashed line), and ξ =0.001. 4 Finite element implementation for pin-jointed structures For a strut in tension, the response is linear elastic S − Y ≤ S ≤ = E for 0 WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) Computer Aided Optimum Design in Engineering IX 169 For a strut in compression, Eqn (1) can be rearranged to give S ξ2λ2 (2 − α) = 2 +1 for 0 ≤ S ≤ SPR (4) E 4(1− α) It is assumed that the strut fails at the Perry-Robertson stress SPR, because the elastic theory of buckling is not valid after yield has been reached in the outer fibre. In this case structural modification, or a more complex analysis including post-buckling is required.