The vertical shear at C in the figure shown in previous section (also shown to the right) is taken as
VC = (ΣFv )L = R1 − wx
where R1 = R2 = wL/2
The moment at C is
If we differentiate M with respect to x:
|dM/dx = V|
Thus, the rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point.
Differentiate V with respect to x gives
dv/dx = 0-w
Thus, the rate of change of the shearing force with respect to x is equal to the load or the slope of the shear diagram at a given point equals the load at that point.
Properties of Shear and Moment Diagrams
The following are some important properties of shear and moment diagrams:
The customary sign conventions for shearing force and bending moment are represented by the figures below. A force that tends to bend the beam downward is said to produce a positive bending moment. A force that tends to shear the left portion of the beam upward with respect to the right portion is said to produce a positive shearing force.
An easier way of determining the sign of the bending moment at any section is that upward forces always cause positive bending moments regardless of whether they act to the left or to the right of the exploratory section.
Without writing shear and moment equations, draw the shear and moment diagrams for the beams specified in the following problems.
Give numerical values at all change of loading positions and at all points of zero shear. (Note to instructor: Problems 403 to 420 may also be assigned for solution by semi-graphical method describes in this article.)