Non linear finite element theory

[chandran]

I wanted to get some thought in non linear fea.

Let me say there is a spring of initial stiffness k.

Let me say in a non linear analysis the load is divided into 20 steps of 1N each(Total load is 20N).
1.1st iteration.1N is applied and the displacement is arrived
2.2nd iteration 2N is applied. The stiffness matrix(tangent stiffness) is derived from iteration 1. The displacement(u) is calculated. F=ku is the calculated value of resisting force of the spring.

Now i want to understand how any fea software is able to calculate the actual resisting force the spring produces.Because after this step only the software is able to calculate the residual force(residual force=Calculated resisting force-actual resisting force).

can anyone explain this for a rod
(Take the example of a axially loaded rod.

The non linearity arrises from material non linearity).

 

[PerennialII]

Hi Chandran,

A decent example is to consider the behavior of the modified Newton-Raphson algorithm in a 1D case when material nonlinearities appear. In the modified routine the tangent stiffness, the stiffness matrix more usually, is not updated or is updated only at times so it applies well to your question.

Consider first that you derive a solution for an initial loadstate, call that for example state "1" (this could be an initial linear step in a nonlinear analysis).

We now seek a solution in state "2", which can contain a nonlinear response. We can write the force in state "2" as

$$P_{2}=P_{1}+k_{1}\Delta u_{1}$$

where $k_{1}$ is the stiffness in state "1" and $\Delta u_{1}$ is an increment of displacement. Now in nonlinear FE analysis, we can solve the $\Delta u_{1}$ since we know the applied loads and the stiffness in state "1". From this solution we obtain a new displacement estimate for state "2", which is

$$u^{i+1}=u_{1}+\Delta u_{1}$$

Now we compute a new residual force between the state corresponding to $u^{i+1}$ and "2", and redo the iteration. When the force residual converges towards 0, $u^{i+1}$ converges to $u_{2}$. Notice that I used "i" in there rather than state "2", being at the crux of it all, since several iterative steps, "i"s, may be required before reaching state "2".

So the keys are understanding the force residual driving the solution and updates to the displacement done on the basis of it, the stiffness acting as a 'projection' direction in the nonlinear P-u curve (you can think of it totally analogous to classic Newton's method).

 

[ravenprp]

Nonlinear finite element theory is a powerful tool used in structural engineering and other fields to analyze complex systems that exhibit non-linear behavior. Unlike linear finite element analysis, which assumes that the relationship between the applied loads and the resulting displacement is linear, nonlinear finite element theory takes into account the nonlinear behavior of materials and structures.

In your example, the spring has an initial stiffness of k, which means that for small displacements, the relationship between the applied load and the resulting displacement is linear. However, as the load increases, the spring will start to exhibit nonlinear behavior, such as yielding or buckling. In order to accurately model this behavior, the load is divided into smaller steps and the stiffness matrix is updated at each step based on the current displacement.

The software is able to calculate the actual resisting force produced by the spring by solving the nonlinear equations using numerical methods, such as the Newton-Raphson method. This method iteratively updates the solution until it converges to a solution that satisfies the equilibrium and compatibility equations. The residual force is then calculated by comparing the calculated resisting force with the actual resisting force, allowing for a measure of the accuracy of the solution.

For a rod under axial loading, the nonlinearity arises from material nonlinearity, such as plastic deformation or strain hardening. Similar to the spring example, the software will divide the load into smaller steps and update the stiffness matrix at each step based on the current displacement. The actual resisting force is then calculated using the stress-strain relationship of the material, taking into account any nonlinearity.

In conclusion, nonlinear finite element theory allows for a more accurate and realistic analysis of structures and materials that exhibit nonlinear behavior. By dividing the load into smaller steps and updating the stiffness matrix at each step, the software is able to calculate the actual resisting force and provide valuable insights into the behavior of the system.