can someone please help me on this question ASAP? I understand the logic on this

Question

can someone please help me on this question ASAP? I understand the logic on this but I don't understand how to prove it for part a? And I don't understand how to do part b.

A proposed solution (based on dimension analysis and using separation of variables) for a wedge loaded by a load "P" at an angle beta from the x-axis as shown is given by: (M = 0) sigma_r = -2P/r (cos beta cos theta/alpha + sin alpha + sin beta sin theta/alpha - sin alpha), sigma_theta = 0, tau_r theta = 0 Prove that the b.c. are satisfied on the faces theta = plusminus alpha/2. Show that equilibrium in the x-direction (only) is satisfied by this solution by considering a circular domain of radius "R". (The end result should NOT contain the value of R, right?)

Explanation / Answer

The dimensional analysis works on the constant of the wedge formula (2P/r) if P/r has units of stress. This happens if P has units of force/length, ie force per unit length, divided by length gives force/area which is stress.

now to prove that the b.c are satisfied when theta = +- alpha/2

The resolved force P must match that given by the expression:

Pcos(beta) and Psin(beta) must equal the integrals

INT{(expression for stress)*r*costheta} over -apha/2 +alpha/2

and INT{ (expression for stress)*r*sin( theta ) over the same range of angles

After simplifying the trig expressions and knowing the integral of cos^2 and sin2theta, the force is found to agree.

(b) in x direction put the angles theta =0

( is it for a half plane, then the result is still dependent on r inversely. take alpha pi beta =0 and you get this solution which is the same as that obtained by complex variable integration)

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