Boundary Conditions

It is a general mathematical principle that the number of boundary conditions necessary to determine a solution to a differential equation matches the order of the differential equation. The static beam equation is fourth-order (it has a fourth derivative), so each mechanism for supporting the beam should give rise to four boundary conditions.

Cantilevered Beams

Figure 5: A cantilevered beam.

For a cantilevered beam, the boundary conditions are as follows: If a concentrated force is applied to the free end of the beam (for example, a weight of mass m is hung on the free end), then this induces a shear on the end of the beam. Consequently, the the fourth boundary condition is no longer valid, and is typically replaced by the condition where g is the acceleration due to gravity (approximately 9.8 m/s^2). We note that we could actually use this boundary condition all the time, since if m=0 , it reduces to the previous case.

Simply-Supported Beams

Figure 6: A simply-supported beam.

A simply-supported beam (or a simple beam , for short), has the following boundary conditions:

Question 7

A simply-supported beam of length L is deflected by a uniform load of intensity q . We assume that we know E, I, L and q . Let's use this fact to solve for the deflection of the beam under the load.

Other Beam Supports

There are many other mechanisms for supporting beams. For example, both ends of the beam may be clamped to a wall. Or one end may be bolted and the other end is free to rotate. Or the beam may be clamped at one end but "overhang" a support placed at some point along its length.

Figure 7: Other mechanisms for supporting beams.

Question 8

Each support mechanisms has an associated set of boundary conditions. In order to gain some intuition for boundary conditions, sketch idealized beams whose support mechanism gives rise to the following boundary conditions. The beams should be shown in a "deflected" position, as shown in the figures on this page. In all cases, the beam is supported only at the ends.
Next: Exploring Static Deformations of Beams
Return to: Modeling Deflections in Beams
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The Geometry Center Calculus Development Team

Copyright © 1996 by The Geometry Center. Last modified: Fri Apr 12 15:38:15 1996