Shear
Force and Bending Moment Diagrams for a SimplySupported Beam
Under A Uniform Load
After the
support reactions are calculated, the shear force and bending
moment diagrams can be drawn.
Shear
force is the force in the beam acting perpendicular
to its longitudinal (x) axis. For design purposes, the beam's
ability to resist shear force is more important than its ability
to resist an axial force. Axial
force is the force in the beam acting parallel
to the longitudinal axis.
The following
is a drawing of a simplysupported beam of length L under a
uniform load, q:
This beam
has the following support reactions:
where R_{l}
and R_{r} are the reactions at the left and right ends
of the beam, respectively.
The shear
forces at the ends of the beam are equal to the vertical forces
of the support reactions. The shear force F(x) at any other
point x on the beam can be found by using the following equation.
where x is
the distance from the left end of the beam.
Shear force
diagrams are simply plots of the shear force (on the yaxis)
versus the position of various points along the beam (on the
xaxis). Thus, the following is the generalized shear force
diagram for the beam shown above.
The bending
moment at any point along the beam is equal
to the area under the shear force diagram up to that point.
(Note: For a simplysupported beam, the bending moment at the
ends will always be equal to zero.)
To calculate
the bending moment the beam must be broken up into two sections:
(a) 
one
from x = 0 to x = L/2 and 
(b) 
the
other from x = L/2 to x = L. 
The bending
moment M(x) at any point x along the beam can be found by using
the following equations:
Bending moment
diagrams are simply plots of the bending moment (on the yaxis)
versus the position of various points along the beam (on the
xaxis). Thus, the following is the generalized bending moment
diagram for the beam shown above.
