The internal bending moment varies linearly over the length
(L) of the element. Since the moment (m) is equal to a
constant (the modulus of elasticity of the material, E and the
moment of inertia of the member, I) times the second derivative
of the vertical displacement, which is a cubic polynomial, the
internal bending moment must be a linear polynomial. The
previously mentioned relationship is: .gif){kind=small}
and if there is no
applied force acting on an element, then the moment will be
constant (pure bending) and mI = mJ, since
no moment can be applied within the element, only at the nodes.
This constant value will be equal to the constant term of the
polynomial. If there is a force and a moment, then the moment
will vary linearly with the distance x along the element
as in the figure below.
