The general method of obtaining internal forces at certain cross-section of a system under a given loading
(and support) condition is by applying the concepts of equilibrium . To illustrate, let us consider
the beam-column AB in Figure below for which we have to find the internal forces at section a - a . As we have
learned earlier, equilibrium conditions are best studied through free body diagrams. We can find the
reactions at supports A and B using a free body diagram of the whole beam-column AB . We
solve the three equations for static equilibrium for this free body:
If a system is in static equilibrium condition, then every segment of it is also in equilibrium. So, we can
consider the equilibrium for each of AC or CB independently. Let us consider the equilibrium of part AC , and
draw its free body diagram . In addition to externally applied forces and the support reaction
and , this free body is acted upon by forces P , V and M on the surface a - a . These are nothing but the
internal forces (axial force, shear force and bending moment, respectively) acting at the cross-section a - a
of AB . Note that these forces are drawn in their respective positive directions in order to avoid sign
confusion. Solving the three static equilibrium equations for AC we find these internal forces:
Thus we obtain the internal forces at section a - a . These could also be obtained by considering the
equilibrium of the part at the other side of section a - a , that is of part CB . Figure shows the free
body diagram of CB . Again, the internal forces are drawn in their positive directions on surface a - a , which
is a negative x-surface for this free body. Solving the three equations we find the values for these internal
forces:
Note :- That these values match exactly with the values obtained previously by considering the equilibrium of
segment AC . This is true for any system because there is always a unique set of internal forces on an
internal surface for a given loading condition.
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