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.
Find the position and value of the maximum bending moment.
first go here
http://www.jiskha.com/display.cgi?id=1480944543
Calculus applications - Damon, Monday, December 5, 2016 at 9:24am
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
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 :)
Now you want the max moment and where
the moment will be max where its derivative, the shear, is zero
S = w (L/2 - x) = 0
so the moment is max where x = L/2, the middle of course
How much is it?
M = w (L^2/4 -L^2/8)
= wL^2/8 which is wl (L/8) which is the weight * L/8
To find the expression for shear force, we need to differentiate the bending moment equation with respect to x.
M = (WLx/2) - (Wx^2/2)
Differentiating M with respect to x, we get:
dM/dx = W(L/2) - Wx
The shear force (V) is equal to the differential of the bending moment, which means V = dM/dx.
Therefore, the expression for shear force is:
V = W(L/2) - Wx
Now, to determine the value of shear force at x = L/4 from the left hand end, we substitute x = L/4 into the expression for V:
V = W(L/2) - W(L/4)
V = (WL/2) - (WL/4)
V = WL/4
So, the shear force at x = L/4 is WL/4.
To find the position and value of the maximum bending moment, we can take the derivative of the bending moment equation with respect to x, set it to zero, and solve for x.
Taking the derivative of M with respect to x:
dM/dx = W(L/2) - Wx = 0
Setting this equation to zero, we have:
W(L/2) - Wx = 0
Simplifying the equation, we get:
L/2 - x = 0
x = L/2
Therefore, the maximum bending moment occurs at x = L/2. Now, to find the value of the maximum bending moment, substitute x = L/2 into the bending moment equation:
M = (W(L/2)*(L/2))/2 - W(L/2)^2/2
M = (WL^2/4)/2 - W(L^2/4)/2
M = WL^2/8 - WL^2/8
M = 0
So, the value of the maximum bending moment is 0.