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# How to calculate stress in beam with varied section?

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## How to calculate stress in beam with varied section?

(OP)
Hello guys,

Recently I've tried to calculate a beam with varied section. In FEA it's really simple, but by hand it isn't.
I've tried to use the Theory like below:

#### Quote:

σ=(M*y)/I

This calculation works very well with linear beam, but when it's varied it doesn't work.
Somebody could help me explaining how could I calculate this beam by hand?
In FEA, I just fix one side and applied a load in other side.

Follow attached a picture with FEA calculation in a beam with varied section:

Thanks in advance.
Henry.

### RE: How to calculate stress in beam with varied section?

This isn't really a "by hand" calculation. Not that it can't be done....

Why don't you just find your critical cross-section and work from there?

### RE: How to calculate stress in beam with varied section?

This depends on the materials, whether they are linear or not, whether the cross section is symmetric etc.

Jason McKee
proud R&D Manager of
Cross Section Analysis & Design
Software for the structural design of cross sections
Moment Curvature Analysis
Interaction Diagrams
Reifnorcement Design etc.

### RE: How to calculate stress in beam with varied section?

Henry,
looking at the proportions of the FEA model you showed I suspect the theory doesn't match for few possible reasons:

1, Not a slender beam. Formula σ=(M*y)/I is valid only for fairly slender beams
2, The section variation is too quick along the axis. Meaning that the transmission phase from one section to another affects the outcome more than the section "straight" section.
3, Related no 1. Section is too narrow. I understand the bending is around Y axis.

Can't tell from the picture but I take it the section profile is a rectangle (no rounds or anything that would make it a non-flat beam)
It is also difficult to say whether your verification model in ANSYS was corresponding to your hand calculation.

If you indeed need to analyze section like the one shown AND not too very frequently. I recommend sticking with FEA to save yourself a hassle. If it would be still a frequent matter, then APDL scripts might help with automation. Otherwise you need to deal with some of the problems when doing hand calculation.

I hope this helps.

### RE: How to calculate stress in beam with varied section?

(OP)
Thanks for all the input.

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