Material and Geometric Nonlinear Analysis with Calcpad

[Proektsoftbg]

Dear friends,

Here I created a simple example for solution of an asymmetric Biot truss with prestress, geometric and material nonlinearity.

This is a classical example for a linearly unstable structure. It has zero stiffness at the beginning, so it cannot be solved by the methods of the traditional linear structural mechanics. However, after it deflects, it starts to experience a stress stiffening and tensile forces appear in bars due to their elongation. The deflection continues to increase, until their vertical projections balance the external force:

The solution is similar to a weightless rope with a concentrated force. To account for this behavior you have to use 3-rd order geometric nonlinear theory (large displacements). It leads to a system of nonlinear equations, which we solve by using Newton-Raphson's method.

However, the most important question for every engineer is: When will the structure fail? For most structures, we can assume that failure will occur when the steel reaches its yield stress. But this particular kind of structures has a hidden potential to resist after that point. Yielding allows the structure to deflect more and resist larger loads due to the increased vertical projection of internal forces in bars. The actual fail will occur after the steel reaches its ultimate tensile strength.

So, if we approximate the real stress-strain curve for steel with a bilinear material model, we can explore this behavior and predict the actual moment of failure for the structure more accurately. Below is a comparison between linear and nonlinear material models:

Click to run animation:

Until the yielding point, both models behave identically, but after that, they split. Nonlinear model deflects more, but with less internal forces in bars and fails at a higher load. We made a comparison to OpenSeesPy to double check the results.

You can download the complete source code for this sample here:
https://github.com/Proektsoftbg/Calcpad/tree/main/Examples/Mechanics/Structural%20Analysis/Nonlinear