Question

Write equations for v(x) and m(x) for the entire legth of the beam. in some caces there will be different sets of equations for segments of the beam

Answer 1

Before proceeding further, let us begin by understanding what a beam is. A beam is a structural member whose length along one direction, called the longitudinal axis, is larger than its dimensions on the plane perpendicular to it and is subjected to only transverse loads (i.e., the loads acting perpendicular to the longitudinal axis). A typical beam with rectangular cross section is shown in figure 8.1. Thus, this beam occupies the points in the Euclidean point space defined by B = {(x,y,z)|- l ≤ x ≤ l,-c ≤ y ≤ c,-b ≤ z ≤ b} where l, c and b are constants, such that l > c ≥ b with l∕c typically in the range 10 to 20.
Next let us understand which moment is a bending moment and how it is related to the components of the stress. The component of the moment parallel to the longitudinal axis of the beam is called as torsional moment and the remaining two component of the moments are called bending moments. Thus, for the beam with the axis oriented as shown in figure 8.1, the Mx component of the moment is called the torsional moment and the remaining two components, My and Mz the bending moments. To relate these moments to the components of the stress tensor, first we find the traction that is acting on a plane defined by x = xo, a constant. This traction is computed for a general state of stress as

( ) ( )
σxx σxy σxz { 1}
t(ex) = ( σxy σyy σyz) ( 0) = σxxex + σxyey + σxzez.
σxz σyz σzz 0
(8.1)
Now we are interested in the net force acting at this section, i.e.,

∫ ∫
F = P e +V e +V e = t (x ,y, z)dydz = (σ e +σ e +σ e )dydz,
x y y z z a (ex) o a xx x xy y xz z
(8.2)
where P is called the axial force, V y and V z the shear force. Equating the components in equation (8.2) we obtain