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Converted document EXAMPLES: BOUNDARY CONDITIONS FOR BEAMS IN BENDING

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 images/frame.png
Figure 1 Frame of reference
Pinned-pinned (simply supported) beam
Figure 2↓ shows a beam with pinned boundary conditions at both ends.
figure images/pinnedpinned.png
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 images/Clampedpinned.png
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
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