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Is bending stress dependent on non-localized thickness?
Last Post 27 Aug 2014 01:49 PM by Jeff Ehler. 1 Replies.
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Mike B
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05 Aug 2014 01:01 PM
    I had a disagreement with another engineer if bending stress in a certain location is affected by varying the thickness that is far from where the bending stress is to be calculated.

    Consider a simply supported beam with a concentrated load in the center. As far as I know, the shear and bending moment diagrams have nothing to do with the thickness of the beam. The center of the beam always has a bending moment of WL/4 regardless if the beam is thin in some parts and thick in other parts. Therefore if you have a known cross section in the center of say 1"X1", the bending stress could easily be calculated my finding the moment of inertia, and using the formula stress = My/I. That stress in that center cross section never changes even if the thickness in a different cross section changes. If the thickness in a different cross section was changed, the deflection in the center would change, but not the bending stress.

    The other engineer I was discussing this was saying that if the thickness somewhere between the supports and the center of the beam were made thinner, the center would deflect more. This I agree with. And he says that because it deflects more there is a greater curvature on the top surface of the beam and therefore the bending stress is greater at the center. This seems to contradict the equations described in the previous paragraph.

    Can someone tell me who is correct?
    Jeff Ehler
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    27 Aug 2014 01:49 PM
    The moment diagram won't change. Remember that deflection is not directly related to the moment diagram but to the M/EI diagram. Therefore if you change the I of the section somewhere you will change the deflection. But the moment doesn't change at all across the span.

    This is based on simple spans. If you have a redundant system (say multiple spans) then the change in section somewhere does affect the load distribution across the system and the moment will change.

    Your friend is correct that the simple span beam will deflect more in terms of translational downward movement but the curvature along the beam at the midpoint (i.e. the moment) is the same.

    Where the curvature WOULD change would be in those areas where you changed the cross section. But even with a change in curvature (i.e. change in deformation) you would not have more moment there since the I is now smaller.
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