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First of all, this is not a homework question. I simply have a hard time finding relevant sources or help. Therefore, I am asking this question here. Hopefully, someone can help me.

I would like to calculate the analytic solution of the deflection curve of the L-Shaped cantilever beam as shown in the following image:

enter image description here

It is difficult, because the force acts out of the plane, and will therefore induce an out of plane bending of the cantilever. The cantilever will exhibit torsinal and bending stress.

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    $\begingroup$ Anything analytic will be an approximation. What do you actually need to know? (how accurate?) $\endgroup$ – agentp Mar 21 '18 at 14:18
  • $\begingroup$ @agentp Your are correct. I would like to have an analytical model, because in fact, I would like to have a rough estimation of the deflection curve before doing FEM analysis. $\endgroup$ – james Mar 21 '18 at 14:29
  • $\begingroup$ You need to treat as two parts. I expect you can find formulas for the twist of a beam with a torque on the end. $\endgroup$ – agentp Mar 21 '18 at 14:38
  • $\begingroup$ @agentp this is a good idea, however to find the torque at the end of the first beam, I would need to calculate the deflection of the second beam, which in turn is attached to the first beam, so I cannot assume a fixed constraint at the interface $\endgroup$ – james Mar 21 '18 at 14:41
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    $\begingroup$ The way this has been done for centuries is to divide the beam into pieces that are of a size such that the important forces are constant or vary linearly, and the deflection angles are reasonably small. Then you hand crank the problem matching boundary conditions piece-wise. This is what those rooms full of "computers", mostly girls, were doing. $\endgroup$ – Phil Sweet Mar 21 '18 at 15:28
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For the L2 part the deflection is $$PL^3/3EI $$ considering it a simple cantilever beam with L= L2-W1

The L1 with the L = L1+ W2 has two deflections. first acting as a cantilever under load P. Second twisting under the torque P(L2-W1).

This is simplifying the stresses in the small corner rectangular W1.W2.

From Roark Formulas hanbook, twist per inch length = T/(KG) $$K = ab^3*[16/3-3.36*(b/a)*(1-((b/a)^4)/12)]$$

a = long side, W ; b = short side, T.

T =Torque in-lb

L= Length in inches

G = Modulus of rigidity

This is twist per inch length, you multiply by L.

You now add the deflection of the L1 under cantilever action to the axis of L1.

You can modify theese results by considerin the small rectangula W1.W2 strains too.

This is kind of a rough first estimate. Just to give you preliminary numbers.

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