Optimization of space dome trusses considering snap-through buckling

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1 Optimization of space dome trusses considering snap-through buckling Su-Deok SoW, Choon-Wook Park*, Moon-Myung Kang\ Seung-Deong Kim^ ^Department ofarchitectural Engineering, Kyungpook National University, Taegu, Korea ^Department ofarchitectural Engineering, Semyung University, Jaecheon, Korea Abstract In this paper, an analysis program is developed by the finite element method, which considers geometric nonlinearity as well as snap-through buckling in optimization of space trusses. A numerical analysis technique for nonlinearity is used to modify the incremental method and the displacement incremental method. In order to illustrate the optimization of space trusses considering snapthrough buckling, an illustrative example is presented. 1 Introduction Recently, steel space trusses and network domes have become popular in the design of gymnasiums, exhibition halls and other similar buildings which enclose open space truss environments without columns. The authors of this paper focus on the space truss and network dome/ In particular, the single layer dome attracts the attention of structural engineers because the dome is constructed with slender and light-weight materials, and the design of the configuration of these structures is relatively flexible. However, there is the problem of geometric nonlinearity, i.e. snap-through buckling. The purpose of this paper is to develop an analysis program that considers geometric nonlinearity and finds the optimum value considering snap-through

2 140 Computer Aided Optimum Design of Structures buckling of space truss. The optimization of space truss is done by GINO^(General interactive Optimizer) program. The objective function is a volume of the truss and the constraints taken into account are design limits defined by axial force strength, slenderness and deflection. In the optimization procedure, buckling is considered by calculate maximum deflection limit for snap-through buckling. The illustrative example is a star dome truss which is consisted of 13 joints and 24 bars. 2 Stiffness Matrix Considering Nonlinear Theory In this section mechanical fundamentals of this program is described. The stiffness matrix of truss element is leaded by considering the geometrical nonlinear theory. Nonlinear equations are used linearized nonlinear equation retaining only linear term. Here, the error due to eliminating higher order terms can be corrected by caculation of residual force. Local coordinate system of truss element is defined in Figure 1 to derive the stiffness matrix of truss element considering geometrical nonlinear theory. Nodal displacements and force in local coordinate system are as eqn.(l),(2). /-element ~«, / **»"- Local coordinate system : x, y, z Nodal displacement : u(x), v(x), w(x) Figure 1 : Local coordinate system of truss element f ' * \f ' * J x) ) \ J yj } We assume displacements in an element to linear expression about x Representing coefficient a\, a^ with nodal coordinate of eqn(l), we

3 Computer Aided Optimum Design of Structures 141 obtain (4) above expressions can be written in a matrix form as follows : (5) namely, (6) Inverse matrix of [d>] is obtained as 1 0 _.!. Ju Xi X, (7) Coefficients a, 0, 7 using eqn(6) are obtained as follows : a= 0~^d*, $= 0~^dy, r= &~^dz (8) Substituting for eqn(8) in eqn(3) u(x) = aid xi + (bid xi + b^d^x v(x) a^d yi+ (bidyj+ b^d^x (9) w(x) = a^\i 4- ( bidzi 4- byd^x We choose the strain-displacement relation with nonlinear term as (10) Substituting for eqn(9) in eqn(10) and we displacement obtain strain with nodal c,= y,,+ 6z4, (11) Representing above expression into matrix, it is obtained as (12)

4 142 Computer Aided Optimum Design of Structures where, [^ 0 0 bz 0 0 ], B= hi 0 0 bz ^ ^ 0 0 *2 We choose stress-strain relation in incremental region as f,= Ee, (13) Assuming present condition to initial condition and applying the principle of virtual work about increment in present condition, we find (/ + /) (14) Obtaining % from eqn(12) Substituting for eqn(15) in eqn(14) te,= Ai dd^rd^b^b dd (15) where, A : cross-section area, / : length of element. ) = (f^ + fy (16) Substituting for eqn(12) in eqn(13) and again the result in eqn(16) and abbriviating second order term of d, we obtain The residual force can be obtained as r=al- A?o -f (18) Using above expression, incremental equations are obtained as follows: where, /-/ = [Ac4-6J ^

5 3 Nonlinear Solution Algorithm Computer Aided Optimum Design of Structures 1 43 In this section, we considered the solution of nonlinear analysis which was used in developed program. Evaluating the truss load-deformation behavior with geometric nonlinear is required incremental/iterative solution scheme. According to selection of a analysis method, it has merits and demerits respectively about delivering a basic equation, accuracy of analysis, and the time of calculating, etc In this study, in order to solve stress-deformation analysis for load, modified load incremental methods was selected and in order to solve limit point, displacement incremental methods was selected. 3.1 Modified Load Incremental Method The method which is to obtain the solution by linear in each incermental part is the incremental method, and particularly when the load is parameter, it is the load incremental method. Such a load incremental method mixed the Newton-Raphson is a modified load incremental method and a very admirable non-linear analysis method, (see figure 2) calculate procedures are as follows : (1) Calculate D From A ^F- R = [ KE KG*] (point A in figure 2) (2) Calculate R by Using (AB in figure 2) (3) Calculate A D^ from^yl #^ = [ #f + by Using D^and R (4) Calculate D = D + A D (point D in figure 2) (5) Iterate the procedure by next incremental load (incremental path is A >B-»D -» ) Displacement Figure 2 : Modified load incremental method Figure 2 show that A >C is presented load incremental path and B

6 144 Computer Aided Optimum Design of Structures >E is presented the path of Newton-Raphson method. Therefore, the modified load incremental method is as follows : A i(s+l) (20) The modified load incremental method is useful about nonlinear behavior, but if there exist the limit point on equilibrium, such method can't pursuit the limit point of later on path. 3.2 Displacement Incremental Method If there exist limit point in nonlinear equilibrium orbit, analysis method of load incremental type can't trace the path of later limit point. In this case, it is useful to select displacement as incremental parameter. That is called displacement incremental method, (see figure 3) Load AF limit load level Displacement Figure 3 : Displacement incremental method 'A' A{F} = (21) I Aj In incremental parameter, one of (D) select expressed as eqn(21) is (22) Express matrix equation A (23) D

7 Computer Aided Optimum Design of Structures 1 45 If the above equation is calculated, [A,D^...,D»] can be obtained, and external force AF can be obtained from load parameter A. Using this above procedure, calculation is processed by the same procedure as incremental method. 4 Formula of Optimum Problem for Space truss In this study, the objective function is the volume of the space truss and the constraints arc design limits defined by the axial force strength, slenderness and deflection. 4.1 Objective Function The objective function is as eqn(24) l/= & y,= &A,/, (,'=!,2,-,%) (24) 7=1 1=1 Where, Vi is volume of i-th element, At is Cross-section area of i-th element and /,- is length of i-th element. 4.2 Constraints The constraints in this paper are as -follows* : /,(ora--:>0 (,'=!, 2, -,%) (25) a,-,i,:>0 (z'=l,2,-,%) (26) ag-a^o (27) Where, M : axial force of i-th element F ft : allowable tensile stress /, = * i. o fc : allowable compressive stress (when, ^, \, \, 0.277J?\ (when A,> A /,) f c ' A p : slenderness of limit between elastic and plastic buckling

8 146 Computer Aided Optimum Design of Structures Ai : slenderness of i-th element A : deflection of space truss ai : maximum slenderness of element #2 : maximum deflextion of space truss 5 Optimum Process Considering Snap-through Buckling In this study, the element area corresponding to design load are reconstructed by buckling load, and the optimization process can be summarized as follows : (1) Assume initial area (ami). (2) Find snap-through buckling load by using displacement incremental method. (3) Calculate element area(a) corresponding to design load a= # #,,, where a = design load / buckling load. (4) Calculate element stress by using design load. (5) Calculate displacement at control node. (6) Find optimum volume (Vopt) and optimum area (a<>pt). (7) If the optimum value is converged, stop the process Or not, Substitute a,», for a<>pt, and return to (2). 6 Numerical Example : star dome truss The star dome truss has been analyzed by many researchers. The geometry and loading of the truss are shown in figure 4. The truss is consisted of 13 joints and 24 bars and has nodal force P=1000N loaded on top joint vertically. All bars are same material as follows : Elasticity modulus : E = X l(f N/cnf Yielding stress : Fy=23520 N/cm2 43.3cm ' 43.3cm Figure 4 : Geometry and loading for star dome truss

9 6.1 Comparison of Nonlinear Analysis Computer Aided Optimum Design of Structures 147 A difference of 0.52% at the limit point(646.64n) is obtained to compare with the 650N result obtained by Hill and wang(1989).^ Figure 5 shows the load-deflection snap-through response of the truss Z o Z Displacement (cm) Displacement (cm) (a) Node 1 (b) Noad 2 Figure 5 : Load-displacement curves for star dome truss. 6.2 Result of Optimization Table 1 and figure 6 show the result of optimization of the Star Dome Trusses. cm" 350 O-i * -- Spline smoothing \ Optimum value b o. O N Step Number Figure 6 : Optimum volumn No. element «/, d= a d^ dopt a Volume(a) Volume(aopt) * aj(cm) ium Volume Table 1. Result of Star Dome Trusses (unit ; cm\ cnr*) Step 1 Step 2 Step 3 Step ; Maximum deflection of star dome truss

10 148 Computer Aided Optimum Design of Structures The objective value (J/V) is converged at step 3 and the areas (a ) in step4 are equal to the initial area (a,m) 7 Conclusion In this study, we study on the optimization of Space Trusses with snapthrough buckling. The limit point is obtained by using displacement control in numerical analysis procedure, and reliability of developed program is proved through the illustrative example. In the optimization procedure, buckling is considered by the calculated maximum deflection limit for snap-through buckling, and we could know that optimum volume is influenced by the deflection constrain. Considered range of buckling analysis in this paper is only geometric nonlinearity, but it needs to study about material nonlinearity of member and bifurcation buckling. References 1. Kato, S., Takashima, H. and Shibata, R., Effect of Geometrical Initial Imperfections, Relaxation at Connectors and Additional Loads on the Ultimate Strength of A Semi-Rigidly Jointed Single-Layer Reticular Dome, Proceeding of the Third Summer Colloquium on Shell and Spatical Structures, Taegu, Korea, pp , Hill, CD and Wang, S.T., Post-Buckling Analysis of Steel Space Trusses, J. Struct. Engrg. ASCE, vol.115, pp , Liebman, J, Lasdon, Schrage, L and Waren, A.D., Modeling and Optimization with GINO, The Scientific Press, Jagnnethan, D.S., Epstein, HJ. and Christian, P., Nonlinear Analysis of Reticulated Space Trusses, J.Struct. Division ASCE, vol., No.Stl2, pp , Shon, S.D., Mu, Z.G., Kang, S.D. and Kwun, T.G., Optimization of Space Trusses Considering Geometric Nonlinearity, 7%e Third Asian-Pacific Conference on Computational Mechanics, Seoul, Korea, pp , Smith, E.A., Space truss nonlinear analysis, J. Struct. Engrg., ASCE, 110(4), pp , Cassis, J. and Sepulveda, A., Optimum Design of Trusses With Buckling Constraints, Journal of Structural Engineering, Vol. Ill, No.7, pp , 1982.

Improved refined plastic hinge analysis accounting for strain reversal

Engineering Structures 22 (2000) 15 25 www.elsevier.com/locate/engstruct Improved refined plastic hinge analysis accounting for strain reversal Seung-Eock Kim a,*, Moon Kyum Kim b, Wai-Fah Chen c a Department