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# Relationship of stresses vs. loads in linear analysis

## Relationship of stresses vs. loads in linear analysis

(OP)
Let's assume we have the most general structural FEM model, which is being simulated with a classical linear FEA solution (no source of non-linearity is being considered). We know that if we run with load 1.0*P, 2.0*P, or even 3.0*P, we can predict deflections (displacements) will be the corresponding 1.0*u, 2.0*u, and 3.0*u, since there exists a linear relationship by means of the stiffness matrix.

But what conclusion can we make about the stresses? Will stresses scale linearly in the same fashion if we duplicate or triplicate the applied load? What about the stress components? Can we still assume the stress tensor will scale in the same factor?

What about von Mises stresses (by definition its formula does not seem to be linear)?

### RE: Relationship of stresses vs. loads in linear analysis

The same applies to stresses. To see this use a simple example (like a cantilever beam) with different load cases.

### RE: Relationship of stresses vs. loads in linear analysis

as above, these basic questions you should answer for yourself ... running simple tests if you can't convince yourself from the theory.

Linear means linear, for 2*P everything in the FEA "should" be 2*(1P) ... displacements, stresses, principal stresses, von Mises, etc. Run 1P and @P loads, then factor 1P results (most FEA allows you to do this) and see the difference (2*1P-2P) (most FEA allows you to do this too)

### RE: Relationship of stresses vs. loads in linear analysis

Hence the elegance and utility of solving for unit applied loads, then superposing/scaling the results in your favourite software (invariably excel) to achieve a particular loadcase. Very handy if you're dealing with hundreds of cases.

### RE: Relationship of stresses vs. loads in linear analysis

Principal of superposition is used extensively to combine the different load analysis results for "linear analysis" to form different load combinations as Ng2020 stated. So yes your results will be in the same multiples if the analysis type is "linear"

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