Deflection of the Beam, What Is The Best Method To Consider?

Considering deflection of the beam, first thing to consider is if the beam is statically indeterminate. If it isn’t, the beam is statically determined, then you can simply integrate to receive the deflection, like so:

The Formula for Deflection of the Beam

Shear:

V(x)=∫−q(x)dxV(x)=∫−q(x)dx

Moment:

M(x)=∫V(x)dxM(x)=∫V(x)dx

Curvature:

κ=M(x)EIκ=M(x)EI

Before you continue, some conditions should be met, for example: M(0)=0M(0)=0 and M(l)=0M(l)=0 (l is here the full length of the beam)

Angular rotation:

ϕ(x)=∫M(x)EIdxϕ(x)=∫M(x)EIdx

Deflection:

ω(x)=∫−ϕ(x)dxω(x)=∫−ϕ(x)dx

If your beam is statically indeterminate, then I suggest trying gap comparison (I don’t know if this translates well), making use of angular rotation.

Here is a link to some common formulas to use:

https://www.wagemaker.nl/wp-content/uploads/WAGEM0032_Bureaugadget-230220102.pdf

Lets say you have a beam with three supports and a simple static load, the fields of the beam want to bend downward, whereas the in the middle support the beam wants to bend upward.

The first conditions we can give is that the moment in A and C are both zero MA=MC=0MA=MC=0 and the angular rotation on the left is equal to the one one on the right ϕBA=ϕBCϕBA=ϕBC:

Deflection of the Beam
Deflection of the Beam

We can write the following equations:

ϕBA=qd1l324EI−MBl33EIϕBA=qd1l324EI−MBl33EI

ϕBC=−qd2l324EI+MBl33EIϕBC=−qd2l324EI+MBl33EI

Your shear and moment graph will look something like this (note that ever zero point on our shear graph corresponds to the highest moment in the field of the beam):

Considering deflection of the beam, first thing to consider is if the beam is statically indeterminate. If it isn't, the beam is statically determined, then you can simply integrate to receive the deflection, like so:

The same formulas given in the link can be used to find the deflection (you can use those formulas for static determined beams as well).

Another method to find the deflection, and that is one most software such as SCIA or REFM uses, is the Finite Element Methode (or FEM in short), but going into further detail will take some time. I think this answer will suffice for now.

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