Optimal Mass Design of 25 Bars Truss with Loading Conditions on Five Node Elements

The optimal design of a twenty-five bar space truss commonly involves multiple loading conditions acting on 4 node elements in the linear elastic model. In this paper, we describe the behavior of the truss system with our experimental loading conditions on five node elements subject to minimum displacement and stresses that are used to formulate the constrained nonlinear optimization problem. Numerical computations are developed with the objective of mass minimization and the best structural design is selected by applying the interior point method with the guidance of Matlab Optimization Toolbox. Our numerical results show the optimal values of cross-sectional areas, material densities, and internal forces which satisfy the minimum weight design. These results provide the appropriate mass to the experimental data and allow substantial changes in size, shape, and topology. KeywordsLinear elastic model, Variational problem, Structural optimization, Interior point method, Trusses


Introduction
The twenty-five bars space truss problem is typically characterized by their large numbers of design variables and constraints within allowable limits. It is therefore difficult but not impossible to describe this truss system with all stresses and displacements for minimum weight, deflection and for the maximum fundamental natural frequency of vibration. This structure has been optimized by many workers including Rajeev and Krishnamoorthy (1992), Chow (1995a, 1995b), Adeli and Park (1996), Erbatur et al. (2000), Park and Sung (2002). In these studies, the loading conditions were acting on four node elements using the size and shape symmetry in certain study case. Other studies were made with the same loading conditions including Venkayya (1971), Schmit and Farshi (1974), Schmidt and Miura (1976), Adeli and Kamal (1986), Saka (1990), Lamberti (2008), Farshi and Ziazi (2010). To make the structure economical and save materials, the truss profiles were selected for optimal mass design using STADD.Pro in Bora (2016) and also in Jha and Paliwal (2017).
In this paper, we examine the case where the loading conditions are acting on five nodes elements in a linear elastic model in order to select the economical profile of the 25 bars space truss. For this, we find the displacement field from the variational problem and look for the design requirements and structural criteria that leads to the objective of mass minimization. Numerical The structural design of a twenty-five bars truss commonly involves a mathematical model in term of partial differential equations, a set of objective functions and a set of admissible constraints or stresses. In general, one denotes by the Hilbert spaceℒ 2 (Ω), the set of measurable and square integrable functions in Ω and ℋ 1 (Ω) = {ω ∈ ℒ 2 (Ω) | ∀i ∈ {1,2, … , N}, (2) u = u ad , on ∂Ω u 1 = ∂{{1}, {2}} ( (4) F = σ. n = 0, on ∂Ω F 0 The solution space to model (1-5) is ℋ s (Ω) = {u ∈ ℋ 1 (Ω) | u = 0 on ∂Ω u 0 and u = u ad on ∂Ω u 1 } and the variational problem can be stated as: for all v ∈ ℋ s (Ω), with i ∈ {1, 2, 3, 5, 6}.
The Dirichlet conditions (2) and (3) are satisfied in the solution space ℋ s (Ω) and the Neumann conditions (4) and (5) are imposed in the variational formulation. We assume that the unique solution of the variational problem (6) is also the unique solution of the boundary value problem (1-5). Its stability depends continuously on the surface force g i ∈ ℒ 2 (Ω) and the volume force f ∈ ℒ 2 (Ω) . This solution is obtained by minimizing the energy functional J(v) = where the bilinear form ∫ A∇u. ∇vdx Ω ∈ ℋ s (Ω) can admit a quadratic form denoted by q(v).
The problem (7) provides the minimum displacement u ∈ ℋ s (Ω) which is used to attain the objective of the mass minimization.

Stress and Displacement Constraints of Our Approach
One denotes by U ad ⊂ ℋ s (Ω) and σ ad ⊂ ℋ s (Ω), the set of admissible displacements and stresses in tension or compression. Also, we assume that the coordinates at each node of the twenty-five bars truss are given in Table 1. As it is shown in Table 2, one denotes by (l i ) 1≤i≤25 > 0, the ith length or line joining two node elements and by (S i,j ) 1≤i,j≤25 ∈ Ω, the members of cross-sections. The formula used to compute the lengths are given by: l i = √∆x i 2 + ∆y i 2 + ∆z i 2 .

External Stability Conditions of the Truss System
For stability, we denoted by F = {F i,j ∈ ℋ 1 (Ω) | 1 ≤ i ≤ 6 and 1 ≤ j ≤ 10 } the set of external forces Fi,j which satisfies the equilibrium state of 25 bars space truss. The projections are represented in Ω on the node elements with respect to x, y and z axis and the equations describing this equilibrium are given as follows: Projections on the node 1: Projections on the node 2: Projections on the node 3: Projections on the node 5: Projections on the node 6:

Formulation of the Structural Optimization Problem
Size and shape optimization of truss structure is widely used despite large computation costs for massive structures. This optimization is often applied to obtain the optimum values of the cross section area Si,j and lengths li which lead to minimum weight design. Our approach imposes the loading conditions and movements of nodes which include the calculations of cross sections and internal forces hi,j with the objective of mass minimization.

Linear Design Constraints
The limitations of the truss system are behavior and side constraints in matrix form A eq x = b eq and A ineq x ≤ b ineq .

Equality Constraints
The first design requirement for A eq ∈ IR 19×75 , b eq ∈ IR 19×1 and x ∈ IR 75×1 are represented by the equality constrains expressed by: An additional equality constraint on limited material density is given by:

Nonlinear Objective Function
One represents the structural mass of the truss system by the nonlinear objective function of several variables stated as: +3(x 51 x 1 + x 52 x 10 + x 53 x 11 + x 54 x 12 + x 55 x 13 ) where Li, ρi, Si and n=25 are respectively the length, material density, the cross-section area of the i th element and the number of elements in the structure.

Optimization Statement
The structural optimization problem consists of finding the material density, stresses and cross section area which minimize the objective function m(x) under equality (41-60) and inequality (61-72) constraints. This can be stated as: The interior point method is applied through the solver of type fmincom to solve the nonlinear minimization problem (74-75) using Matlab Optimization Toolbox (Coleman et al., 1999).

Applications of the Structural Analysis 4.1 Additional Data
The loading cases imposed for this structural analysis is given by the next Table 8. We define CSA, VIF, SN, and MD respectively as the cross-section area, values of internal forces, stress nature and material density.

Result of the Loading Case 1
Through the loading case 1, the minimal value of the structural mass m1 = 25679.755 kg is found at the fifth iteration and the decisions parameters that provide this mass are presented in Table 9.
In Table 9, we find that the material density is the same at each cross section and it values 7.701 kg/m 3 . The internal force vanishes where the cross-section areas x10, x14 and x18 are not equal to zero. This result shows that the internal forces are inactive at the cross section areas x10, x14 and x18. We observe that there are 13 elements in compression and only 9 in tension in the results of the loading case 1.

Result of the Loading Case 2
The minimal value of the structural mass m2 = 1963.908 kg is found at the second iteration and the decisions parameters that provide this mass are presented in the Table 10. Through the loading case 2, we have also the material density equal to 7.701 kg/m 3 . The same observation was done in loading case 1 is made with loading case 2, namely, the internal forces are inactive at the cross section areas x9, x14 and x18. These results show that the structural mass corresponding to the worst loading is case 1 with m1 = 25679.755 kg compared to case 2 with lesser effects on the structure.