g.alshamsi
Civil/Environmental
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!
(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!