1.3.1 Algorithm
So far we have derived everything about a nonlinear truss element in addition to Updated Lagrange formulation. The algorithm used is based on Generalized Displacement Control Method by Yang (1990). It is very powerful to solve bifurcation buckling problems (see Reference [4]).
1.3.2 Analysis Example
2-truss member example is chosen as a benchmark problem. It was widely used in the studies of Yang (see references) because it’s the simplest but challenging problem due to capturing the bifurcation points. The geometric and material properties are as follows:
E = 1884.694
A = 1
h = 25.847
α = 63.4°
In the first case only the vertical force is assigned so that \(\left[\begin{array}{cc} P_u & P_v \\ \end{array}\right] = \left[\begin{array}{cc} 0 & – 1 \\ \end{array}\right]\). As can be seen from the animation below (vertical deflection vs. vertical load), there’re 2 limit points where stiffness of the system is equal to zero.
However, if 5% of the vertical load is applied as horizontal load \(\left[\begin{array}{cc} P_u & P_v \\ \end{array}\right] = \left[\begin{array}{cc} 0.05 & – 1 \\ \end{array}\right]\), the system behavior is changed by bifurcation points so that we observe 4 limit points. The vertical deflection vs. vertical load animation is given:
Also, horizontal deflection vs. vertical load graph is obtained:
References:
[1] Torkamani, M. A. M., & Shieh, J.-H. (2011). Higher-order stiffness matrices in nonlinear finite element analysis of plane truss structures. Engineering Structures, 33(12), 3516-3526.
[2] Yang, Y.-B., & Leu, L.-J. (1991). Constitutive laws and force recovery procedures in nonlinear analysis of trusses. Computer Methods in Applied Mechanics and Engineering, 92(1), 121-131.
[3] Yang, Y. B., Leu, L. J., & Yang, J. P. (2007). Key Considerations in Tracing the Postbuckling Response of Structures with Multi Winding Loops. Mechanics of Advanced Materials and Structures, 14(3), 175–189.
[4] Yang, Y.B. & Shieh, Ming-Shan. (1990). Solution method for nonlinear problems with multiple critical points. Aiaa Journal – AIAA J. 28. 2110-2116. 10.2514/3.10529.