EXAMPLES: BOUNDARY CONDITIONS FOR BEAMS IN BENDING

In this section, several examples are shown, illustrating the boundary conditions for beams in bending. In all of the following examples l is the length of the beam and x = 0 is the left end of the beam. Figure 1↓ shows the frame of reference for all the examples in this section, where w is the displacement about z axis.

Figure 1 Frame of reference

Pinned-pinned (simply supported) beam

Figure 2↓ shows a beam with pinned boundary conditions at both ends.

Figure 2 Pinned-pinned beam

(1) x = 0 and x = l: w = 0 M = 0

with the Euler-Bernoulli assumption:

(2) x = 0 and x = l: w = 0 M = EI(∂²w)/(∂x²) = 0

Clamped-pinned beam

Figure 3↓ shows a beam with clamped boundary condition at the root and pinned at the tip.

Figure 3 Clamped-pinned beam

(3) x = 0 : w = 0 θ = 0 x = l : w = 0 M = 0

with the Euler-Bernoulli assumption:

(4) x = 0 : w = 0 θ = (∂w)/(∂x) = 0 x = l : w = 0 M = EI(∂²w)/(∂x²) = 0