Compression Modelling of Pin-Jointed Trusses

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 element 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 member 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.

Many studies limit compressive stresses to avoid yielding or simple Euler buckling (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 constraint (e.g. Achtziger [3]). Some studies use a more complex approach, which limits 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 shortening 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 percentage 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.

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 shortening 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 considered signiﬁcant is β =5%.

120

110

100

90 bow λ

60 200 300 400 500 600 700 800 900 1000 E/Y

Figure 2: Slenderness ratio λbow, such that signiﬁcant axial shortening due to bowing 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

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. SPR can be found from Eqn (3). Eqn (4) can be implemented as a non-linear material model for a two-noded truss element, to capture the loss of stiffness caused by elastic buckling of an imperfect strut: this is designated as the “truss-material method”. For a general case, the material model must be specified for each element, because it is a function of the element length. An example material model is shown in Fig. 3, which illustrates a substantial non-linear compressive region prior to yielding.

−50

−100

[MPa] −150 S −200

−250

−300

−350

−1.5 −1 −0.5 0 0.5 1 [10−3]

Figure 3: Material model used in truss-material method, from Eqn (4) (solid line). The squares show the limit of the model’s validity from Eqn (3). The dashed line is the critical Euler stress SE. Compressive loads and deﬂec- tions are positive. The strut is a hollow tube, with R/t =10, λ = 150, E = 210GPa, Y = 350MPa, and eccentricity ratio ξ =0.001.

The stress results from any solution using the truss-material method refer to the average stress S. For tensile elements, this is identical to the maximum stress σmax. For compression elements, σmax can be found from [7] √ 2ξλSE σmax = S +1 (5) SE − S

The method can be extended to trusses of multiple members very simply; each strut is modelled by one truss element, with a solid cross-section that has the same area

WIT Transactions on The Built Environment, Vol 80, © 2005 WIT Press www.witpress.com, ISSN 1743-3509 (on-line) 170 Computer Aided Optimum Design in Engineering IX as the tube. The actual section shape and initial eccentricity are implicity speciﬁed within the material model from Eqn (4). The nodal geometry does not need to be updated after each increment. The truss-material method can be implemented as a material model in any standard ﬁnite element program. It is suitable for including imperfections within concept design, because of the reduced computation required for one linear truss element compared to several quadratic beam elements for each strut. The truss-material method is invalid after yield has been reached; however for concept design, the objective is to keep all stresses below yield so the model is within its range of applicability. Further research discussing the effect of including imperfections within size, shape and topology optimisation is recommended. The application of the truss-material method for some case studies is now presented.

5 Stiffness of branched tree structures

Branched tree structures can be used to provide support for large span roofing structures such as Stuttgart airport, which is shown in Fig 4(a). In the first instance, a two dimensional branched structure that supports a line-load is considered for analysis, Fig 4(b). Branched structures can be useful for roofs because they provide frequent support, in a structurally efficient manner, and without excessively clut- tering the interior space; their similarity to natural tree structures also makes them aesthetically pleasing [8]. However, the slender nature of such structures means they are prone to stiffness loss due to imperfect members, so it is crucial to investigate deflections. The vertical deflection of the roof can be calculated using a work based implementation of the strain-stress characteristic from Eqn (4) instead of finite element analysis; provided the structure is statically determinate. All of the members are in compression, so for each strut the work done Wstrut, to achieve a stress S, can be calculated by integrating Eqn (4). S S2V ξ2λ2 W V Sd strut = = 2 +1 (6) 0 2E 2(1 − α) where V is the volume of the strut. The total work done Wtot, is found by summing Wstrut for all struts. For any statically determinate structure, the bar stresses can be found as a function of the applied load; Eqn (6) can then be used in incremental form to find the displacement of the load point directly. The load-displacement response calculated using this method is identical to that found using finite element analysis with several beam elements modelling each strut, see Fig 5.

6 Stiffness of truss-columns

The efﬁciency of a compression strut can be improved through increased hierarchy, with structural detail at different orders of magnitude. For a constant volume of

(a) (b)

Figure 4: (a) is a photo of the branched roofing structure of Stuttgart airport c Architektur-Dokumentation, München. (b) is a two-dimensional ana- lytical model for a branched roofing structure. The horizontal joint posi- tions are calculated by equilibrium to achieve equal vertical reaction forces at the upper level. All struts are hollow tubes, with R/t =10, E = 210GPa, and ξ=0.001. The total height is 20m, and the upper support points are spaced 2m apart. material, with constrained minimum material thickness, a truss-column composed of tubes has a higher failure load than a single tube [9]. The maximum strength of truss-columns is influenced by the imperfection sen- sitivity from failure mode interaction at different orders of magnitude, hence it must be calculated using the load-displacement response. Elyada and Babcock [10], and Miller and Hedgepeth [11] both analysed a three- legged truss-column with local and global imperfections, see Fig. 6. A form of Eqn (1) is used to find the local load-shortening response, which is then inserted into curvature differential equations, and the truss-columns mid-span lateral deflection can be found as a function of the applied load. Future research is recommended to calculate the corresponding axial shortening, such that a truss-column could be modelled by a single truss element, with a modified material characteristic. This would enable simplified analysis of structures created from truss-columns, enabling an investigation for optimal sizing.

7Conclusion

The truss-material method has been presented for modelling compression of imperfect members within pin-jointed truss structures. A non-linear material

5 x 10 6

3 BBB

0 0 0.002 0.004 0.006 0.008 0.01 0.012 AAA

Figure 5: Results from problem deﬁned in Fig. 4. The squares represents the ABAQUS [12] ﬁnite element solution using 25 quadratic beam elements to model each imperfect strut. The solid line represents the incremental solution of the work based truss-material method from Eqn (6).

Figure 6: Local and global imperfection modes for a three legged truss-column. The column’s lateral deﬂection is found as a function of the applied load in [10] and [11].

model for truss elements is used to represent the loss of stiffness due to elastic buckling, and this can be implemented in any finite element program. For any truss, it is essential to include the stiffness loss due to imperfect members if the slenderness ratio λ is large enough so that significant axial shortening due to bowing occurs, before yielding at the outer-fibre. Future research is required to assess the impact of including imperfections within size, shape and topology optimisation. The truss-material method has been applied to branched roofing structures, and gives identical results to finite element analysis with several beam elements modelling each strut. The method has potential application for a simplified analysis of truss-columns.

Acknowledgements

Financial support for S.D.W. was provided by the Schiff Foundation. Thanks to our C.U.E.D. research colleagues, particularly Dr A. Britto, for their discussion and interest.

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