Steel Plate Shear Walls (SPSW) angle of inclination

[g.alshamsi]

Hey all, have some trouble deriving the angle of inclination for a SPSW bounded by rigid columns/frames. I have made the same post in several math forums without any help so hopefully I'll get a few responses here. The angle of inclination is derived by applying the principle of least internal work. The work expression is:

(V^2*h)/2E [(1/(w*L*sin[sup]2[/sup]⁡α*cos[sup]2[/sup]⁡α))+(1/(2A[sub]c[/sub]*tan[sup]2[/sup]⁡α))+((h*tan[sup]2[/sup]⁡α)/(L*A[sub]b[/sub]))]....eq1.

The angle is then obtain by finding the first derivative w.r.t α and setting the expression to 0. The other variables (V, E, h...) are material properties/applied force and can be considered as constants. The final form of the equation is:

tan[sup]4[/sup]α = [1+(L*w)/2A[sub]c[/sub]]/[1+(h*w)/A[sub]b[/sub]].....eq2.

Basically I'm having trouble going from equation1 to equation2. I have attached my solution in which all terms were converted to sines and cosines. The final step of my equation has a similar form to eq2 however there are terms that I can't cancel out.

Any help is greatly appreciated

[URL unfurl="true"]https://res.cloudinary.com/engineering-com/image/upload/v1675367331/tips/Alpha_-_Wagner_ys11kj.pdf[/url],,

Thanks!

 

[centondollar]

There exist many tools for symbolic differentiation (mathematica, wolframalpha, mathcad, a graphical calculator etc.). Try one of those and plug in some numerical values. The "equation 2" (the result) probably involves some simplifications based on trigonometric relations, which may be difficult to spot, but if the results are the same, there shouldn't be any issue.

 

[g.alshamsi]

Thanks for the input!