Structures consisting of truss elements are widely used in many engineering fields [1]. These trusses structures are fast to build, lightweight and allow large spans. The design of trusses structures has been the subject of numerous studies [2] [3] [4]. Many methods have been developed for the dimensioning of trusses. For small structures, classical analytical methods such as the node method (Culmann method) and the Ritter method are often used [5] [6]. For large structures, the graphical method was one of the first adopted techniques.

As a result of the increasing performance of computers, finite element software is now widely used for the calculation of trusses. Using numerical simulation tools, some authors have worked on methods for structural optimization of trusses [7] [8] [9]. McKeown [7] presented a displacement optimization approach for the design of linear elastic trusses. The mechanical modeling of truss structures is simple under ordinary conditions to which the classical assumption of linear elasticity applies. However, modern applications of truss structures often require plasticity material analysis, which can accurately predict the structural responses. This observation is justified by structural failures, such as the collapse of the I-35W Bridge over the Mississippi River in Minneapolis [10] [11]. Fan and et al. [12] studied the static and dynamic elasto-plastic stability of cross-linked hulls, while Ferrari et al. [13] numerically studied the non-linear elasto-plastic response of the historical Pardermo truss bridge. All these approaches are strongly non-linear.

In the present work, we propose a numerical simulation approach by FEM for the plane articulated truss. Our modeling will be based on a linearized material law of isotropic plastic strain-hardening (linearization of the elasto-plastic behavioral law: bilinear law) and on a projection method using the modified Newton-Raphson approach.

In this study, we are interested in the elasto-plastic behaviour of articulated truss structures. A simplified linear isotropic strain-hardening model of the Johson Cook type [14] is implemented. Thus, the simplified model is bilinear and characterized by a Young’s modulus E and a tangent modulus E_T, the yield strength σ 0 , yield stress projection σ ′ 0 (see Figure 1). The material constituting our truss is made of a metal alloy, therefore the actual behaviour law is shown in Figure 1(a). The corresponding simplified model is shown in Figure 1(b). In this study, we do not consider the phenomena of damage and failure [15].

This approximation of the real material behaviour law (Figure 1(a)) by a bilinear behaviour law of elasto-plastic type with linear strain hardening (Figure 1(b)) permits to write the following incremental plastic flow law (Equation (1)):

d σ = E T d ε (1)

With d ε = d ε e + d ε p ,

d ε e : elastic deformation increment;

d ε p : plastic deformation increment.

We deduce the evolution law of linear strain-hardening (Equation (2)):

σ = σ s ( σ 0 , E ) + H d ε p (2)

With H = E T 1 − E T E , H is called the strain-hardening modulus, ( σ 0 , E ) : Current

yield strength.

Equation (2) can also be expressed as a σ ′ 0 function as follows (Equation (3)):

σ = σ ′ 0 ( σ 0 , E ) + E T ε (3)

The linearized elasto-plastic evolution allows a simplified writing of the discretized elementary equilibrium, obtained analytically by the FEM an articulated truss (Equation (4)):

( ∫ V ( [ B ] T E T [ B ] d V ) ︸ [ K e ] ) { q e } = [ F ] e − ∫ V [ B ] T σ ′ 0 d V (4)

[ K e ] : local stiffness matrix, [ B ] : strain-displacement relationship matrix, { q e } : Nodal displacement field.

The numerical solving technique used in our elasto-plastic problem is based on the classical incremental, iterative approach and on the minimization of a convergence criterion. The modified Newton Raphson-type projection method, which uses the elastic stiffness matrix at each iteration. Figure 2 presents the algorithm followed in MATLAB program [2].

The projection method on the solution sought is the modified Newton-Raphson method. The linear approximation of the plasticity law allows an easy calculation of the hardening, thus saving the CPU processing time. Figure 3 shows the principle of projection on the boundary of the current elasticity domain for an increase in dε deformation. We have represented the most general case of an element being plasticized. We consider a history of initial stress equal to σ and the state of current strain hardening is denoted σ_h.

The linear approximation of the strain hardening allows to obtain by a simple calculation, the plastic evolution law. From Figure 3(b), are geometrically deduced the relations presented in equations 5 to 8. First, we define a quantity R, as follows (Equation (5)):

R = A B A C = σ + d σ e − σ h d σ e = 1 + σ − σ h d σ e (5)

We deduce from the Figure 3(b) (Equation (6)):

X = R d ε ⇒

| Plasticslope   E T = B σ ′ X → B σ ′ = R E T d ε B C = A C − A B → B C = ( 1 − R ) d σ e (6)

The law of evolution (Equation (7)):

σ ′ = σ + ( 1 − R ) E d ε + R E T d ε ︸ d σ (7)

The increments of elastic and plastic deformations are deduced from the stress increment as follows (Equation (8)):

‖ ε e = 1 E d σ → d ε e = ( 1 − R + R E T E ) d ε d ε p = d ε − d ε e → d ε p = R ( 1 − E T E ) d ε (8)

The numerical implementation is done through the projection algorithm of Figure 4.

The application case structure in this article is a truss consisting of three (03) bars of identical cross-section S (see Figure 5). The inclined bar is of length 2 L and the other of length L. The material behaviour law described is the one presented in Section 2. The numerical values considered are: L = 500 mm, Young’s modulus E= 70.000 MPa, tangent modulus E_T = 510.8 MPa, yield strength σ₀ = 281.559 MPa, bar section S = 50 mm². Table 1 shows the table of connectivities of the 3-bar truss structure.

The resolution by FEM involves either a matrix or a discrete formulation of the problem. In the displacement approach, the main unknowns of the problem are the nodal displacements of the structure. In this article, the stiffness matrix is used to construct the FEM model. The elementary stiffness matrix (for each bar element) is written. In the case of the articulated trusses we consider 1 nodal degree of freedom in the local reference and 2 nodal degrees of freedom in the global reference.

The global stiffness matrix is obtained after assembling the elementary stiffness matrices. Thus, the main unknown, which is the nodal displacement vector is calculated by solving the global matrix equilibrium equation [K]{Q} = {F}. In this equation, [K] is the global stiffness matrix, {F} the nodal force vector and {Q} the nodal displacement vector. When the bar element is in the plastic behaviour phase, the local matrix equilibrium equation is modified as shown in Equation (4). Once the nodal displacement vector has been determined, the internal forces in the bars can be deduced. In the application case of 3-bar truss in this article, the global stiffness matrix is presented in Equation (9). A computational code is developed in MATLAB for the application case.

[ K ] = [ E S 2 L 2 + E S 2 L 2 0 − E S 2 L 2 E S 2 L 2 0 0 − E S 2 L 2 − E S 2 L 2 0 E S 2 L 2 + E S 2 L 2 + E S L E S 2 L 2 − E S 2 L 2 0 − E S L − E S 2 L 2 − E S 2 L 2 − E S 2 L 2 E S 2 L 2 E S 2 L 2 − E S 2 L 2 0 0 0 0 E S 2 L 2 − E S 2 L 2 − E S 2 L 2 E S 2 L 2 0 0 0 0 0 0 0 0 0 0 0 0 0 − E S L 0 0 0 E S L 0 0 − E S 2 L 2 − E S 2 L 2 0 0 0 0 E S 2 L 2 E S 2 L 2 − E S 2 L 2 − E S 2 L 2 0 0 0 0 E S 2 L 2 E S 2 L 2 ] (9)

The results presented in this section are obtained from the finite element computation code developed under MATLAB and, having the architecture of the numerical resolution algorithm presented in Figure 2 and Figure 4. The solving method uses an incremental and iterative method based on minimizing a convergence criterion. Different values of the approximation error of the convergence criterion are used to study its impact on the algorithm used. The MATLAB program is illustrated in the case of the articulated truss shown in Figure 5. A load F = 35 kN is applied to the structure, the load increment dF is chosen to be equal to 100N.

This article presented the development of a FEM calculation code for the simple calculation of an articulated truss structure. A finite element program was developed in the MATLAB environment for a three-bar truss structure. The special feature of the numerical resolution of our computational code is the use of a modified Newton-Raphson projection method, by using a linearized elastoplastic material law. Simulations on a 3-bar articulated truss structure and a comparison with a similar model under ABAQUS have proven the reliability of our approach.

The authors declare no conflicts of interest regarding the publication of this paper.

Rolland, D., Christian, F.B., Bienvenu, K., Abraham, K. and Gerard, M. (2021) A Simple Elasto-Plastic Iterative Method by Fem for the Analysis of Plane Articulated Truss: Case of a 3-Bar Truss. Journal of Materials Science and Chemical Engineering, 9, 12-23. https://doi.org/10.4236/msce.2021.96002