Beam Support

In this module, we will consider two different methods for supporting a beam. The first method is called a cantilever , which is obtained by firmly clamping or bolting the beam at one of its ends, and allowing the beam to hang freely on the other end. The second method is called a simply-supported beam. In this case, the beam is placed upon two supporting structures and the beam is pinned so that it can flex, but not translate.

In the model of static beams we use in this lab, the deflection of a beam is describe by a deflection function w(x) . The value of w (x ) is the amount of vertical displacement at the position on the beam x units from the left end.

The way the beam is supported translates into conditions on the function w (x ) and its derivatives. These conditions are collectively referred to as boundary conditions .

Boundary Conditions and Beam Supports

In the exposition that follows, we will let L be the total length of the beam and we let x measure the position along the beam (with x=0 being the left-hand endpoint of the beam.)

The values of w (x ) and its derivative all have physical interpretations relating to the way a beam is supported. Here is a "dictionary" to help translate between boundary conditions and supports:

Cantilevered Beams

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 Consequently, the the fourth boundary condition is no longer valid, and is typically replaced by the condition where g is the acceleration due to gravity (approximately 9.8 m/s^2).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

For a simply-supported beam, we use the following boundary conditions:

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.

Each of these support mechanisms have associated boundary conditions, but we will not pursue these mechanisms here.

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The Geometry Center Calculus Development Team

Copyright   1996 by The Geometry Center. Last modified: Fri Apr 12 15:54:29 1996