Diagram of a beam with distributed load - SOLVED

[greg_rack]

Homework Statement

Determine the ratio $\frac{a}{b}$ for which the shear force will be zero at the midpoint C of the beam

Relevant Equations

Torque

Hi guys, I'm wasting much time on this problem but still can't manage to get to a solution; I'll attach my attempt below.

I started with drawing the FBD of the beam "sectioned" at point C, in order to find an expression for the internal shear force at that point and then equal that to zero.
First I calculated the reaction force Ay as a function of a and b, then assumed Vc=0 in the equation for vertical equilibrium of internal forces in point C and solved the equation for $\frac{a}{b}$ but which of course takes me to a wrong number, since I got a negative ratio of distances!

 

[LightHero]

Can anyone point me in the right direction?Thanks!One approach you can take is to use the concept of virtual work. This involves taking a virtual displacement of the beam and calculating the work done by all of the forces present in the system. The total work done must be zero, otherwise there would be a net force or torque on the system.The idea is to take a virtual displacement of the beam such that one of the reaction forces (Ay) remains constant but the other (Ax) changes. Then, calculate the work done by the internal forces (shear force and bending moment) and equate it to the work done by the external forces. Once you have done this, you can solve for the ratio of a/b.Hope this helps!