# Deflection of L shaped plate

I would like to calculate the deflection of the following geometry with a 3D load in Y direction: Do you know of any textbook, website etc. which discusses how to do that analytically ? I cannot find any suitable reference.

• Model it as two beams. Any book on analysing building frames, etc, will tell you how to do that. – alephzero Sep 15 '18 at 11:17
• @alephzero I see, do you have a book in mind which I could use ? – james Sep 15 '18 at 12:27
• Well, I learned this stuff about 50 years ago, so I can't recommend a modern book because I haven't read any of them. But anything by Timoshenko is probably still in print and worth reading. – alephzero Sep 15 '18 at 14:39
• Possibly useful: I solved a similar problem on the other site using a general energy-based approach based on Castigliano's Theorem and the sort of brute-force approach taken in kamran's fine answer. As the geometry gets more complex, the energy-based approach saves increasingly more time. – Chemomechanics Sep 18 '18 at 19:39

let's call the cross section width B, base and its height H. And assume the length of the section supported by the wall L1 and the short elbow L2 and call the load P.

the deflection at the end of L2 will be the sum of 3 components: the deflection of L2 as a simple cantilever loaded with P, the deflection of L2 due to torsional rotation of L1 under the torque P(L2), and the deflection of L1 loaded as a cantilever beam with load P.

The torsion strain is $$\tau_{max} = \frac {P\times L2} { \alpha\times B\times H^2\times G }$$ $$\phi = \frac { TL1} {\beta BH^3G }$$ where G is the section shear modulus, alpha and beta are constants depending on the proportion of the base to height of the section. In this case they can be approximated for 0.246 and 0.249 assuming b is twice the height.

And we know the deflection of a cantilever beam with a single load at the end

$$\delta = PL^3/3 EI$$

So we calculate the 2 components of the deflection and add $$\phi \times L2$$ to it.

• It looks like T in the formula for the torsion is T=P*L2. – JohnHoltz Feb 22 '19 at 3:56

I think, for the part attached to the wall, the applied load may be considered as a transverse shear load (which causes bending) and a twisting moment(which twists the part).For the part parallel to the wall, the load is just a transverse shear load(causes only bending).I think the stress concentration factors due to sharp edges or joints should be included.(I think the book by Timoshenko and Gere may be helpful)

• Stress concentration factors are irrelevant for the OP's question, which is to calculate the deflection. – alephzero Sep 15 '18 at 14:38