6. The bending moment (M) along a beam is M = WLx/2 -Wx2/2 kNm where x is the distance of a beam length L from the left hand end. W is the weight per unit length.
(a) Shear force is calculated as the differential of bending moment. Find an expression for shear force and determine the value of shear force at x = L/4 from the left hand end.
If you want to draw it it is a seesaw with no one on it
w pounds per foot of weight down (+)
and a force up of wL = total weight at center :) Note that moment must be zero at both ends and will be maximum at the middle changing signs instantaneously
M = w L (x/2) - w x^2/2
S = w L/2 - 2 w x/2
S = w (L/2 - x)
at L/4
S = w (L/2 - L/4)
= w L/4 (the weight to the left of it :)
the shear, not the moment, changes instantly at the center
thanks alot!
You are welcome.
To find an expression for shear force, we need to take the derivative of the bending moment equation with respect to x.
Given: M = WLx/2 - Wx^2/2 kNm
Taking the derivative of M with respect to x, we have:
dM/dx = (W*L/2) - (W*x/2)
The shear force, V, is equal to the derivative of the bending moment with respect to x. So, V = dM/dx.
Substituting the above expression for dM/dx, we have:
V = (W*L/2) - (W*x/2)
To determine the value of shear force, V, at x = L/4, we substitute x = L/4 into the above expression:
V(L/4) = (W*L/2) - (W*(L/4)/2)
Simplifying this expression further:
V(L/4) = (W*L/2) - (W*L/8)
Combining the terms with a common denominator:
V(L/4) = (4W*L - W*L)/8
Simplifying the numerator:
V(L/4) = (3W*L)/8
Therefore, the expression for shear force is V = (3W*L)/8 kN, and the value of shear force at x = L/4 from the left-hand end is (3W*L)/8 kN.