# Statics Question (Using Modulus of Rigidity)

by papasmurf
Tags: modulus, rigidity, statics
 P: 22 1. The problem statement, all variables and given/known data Find the displacement (mm) in the horizontal direction of point A due to the force, P. P=100kN w1=19mm w2=15mm 2. Relevant equations $\tau$ = G * $\gamma$ $\tau$ = Shear stress = P / A $\gamma$ = Shear strain = (pi / 2) - $\alpha$ 3. The attempt at a solution I haven't attempted to work out a solution here yet, but I do have a question regarding the separate G values that are given. Can I just look at the top layer, the layer where P is acting, and use that G value to determine $\delta$? Or do I need to do something with the other G value as well? If I were to try something, I would find tau by doing 100[kN] / (100[mm] * 2[mm]). So tau would be equal to 1[kN]/2[mm2] = 0.5[GPa]. Next I would find gamma by dividing tau by G (100[GPa]) giving me $\gamma$ = .005rad. I can use trig to define gamme as $\gamma$=sin-1($\delta$/40). Setting this equal to .005 I would get $\delta$= .20[mm]. Even if I do have to do something with both of the G values, I feel like my method is correct. Any help is appreciated, thanks in advance. Attached Thumbnails
 P: 22 I'm getting closer to the correct answer. First I set V/A, where V is the internal shear force and A is the area of the cross section where the shear force is acting, equal to G*$\gamma$, where G is the modulus of rigidity and gamma is the shear strain. I rewrote gamma as pi/2 - θ, where θ=cos-1($\delta$/h), h is the height of the "layer", and put it all together so that my equation looks like this: V/A = G * ( pi/2 - cos-1($\delta$/h) ) Solving for $\delta$ I come up with $\delta$ = h * cos( (pi/2) - V/AG) I used this formula for each "layer" and added up all of the deltas. However after plugging my numbers in and making sure of correct units, I still am off by fractions of a millimeter.