Introduction

We shall consider here only beams which have an axis of symmetry in the loading plane (usually vertical). The dimension in the loading plane will be referred to as the depth. Additionally, for best results, the length should be greater then ten times the depth of the beam.

In the study of beam deflections the following conditions are generally assumed:

1. linear elastic theory,
2. homogeneous isotropic material,
3. small deformations,
4. pure bending.

While the condition of pure bending (bending moment without accompanying shear) is rarely if ever met in practice, pure bending is assumed in developing the theory. In practice the shear contribution is generally small enough that it does not seriously limit the use of deflection equations derived from pure bending theory. For beams with a length greater than ten times the depth the shear contribution to deflection would typically be no more than two to three percent of the total deflection.

The assumption of small deformations requires clarification since small is obviously relative. In the derivation of the differential equation relating deflection y to bending moment M (and thus load) we utilize the expression for curvature. In any good calculus book you will find the expression for curvature (the reciprocal of the radius of curvature):

Small deflections are defined as deflections for which the slope of the curve dy/dx < 0.1. Under this restriction the denominator of the curvature expression above is approximately one and the curvature is given by

The error in this approximation is quite small, being only about 1.5% in the worst case, where dy/dx = 0.1.

From the study of flexure stresses in beams we found the curvature was related to bending moment by the relationship

where E is the modulus of elasticity and I is the moment of inertia with respect to the neutral axis of bending. Thus the differential equation for beam deflection is

For relatively simple beam loading situations it is easy to write the bending moment M as a function of x. If EI is constant, we simply integrate twice to obtain the deflection y. The integration constants are evaluated utilizing the boundary conditions. In most beam problems EI is constant.

 

Experiment

The object of this experiment is to compare experimental beam deflections for two simple beam loading situations with the defections predicted by theory.

Load two beams as follows:

1. Simply support a beam and load it at its center (Figure 1). Measure and record the deflections at two locations as assigned by your instructor. It is suggested that measurements be made at the quarter point and at a location near the center of the beam. Use about ten equal loading increments. Be sure to record all pertinent dimensions including the measurement locations.

2. Cantilever a beam and load it near the end (Figure 2). Measure and record deflection at two locations as specified by your instructor. It is suggested that measurements be made at the mid point and near the loading point for each of ten load increments.

Loading increments and elastic constants will be supplied by your laboratory instructor. Be sure to record beam dimensions, loading point, support length and elastic constants on your data sheet. This information should be shown in conjunction with a sketch of beam and loading geometry.

 

Report

1. Plot a graph of deflection vs load P for each beam (use a separate graph for each beam)(Figure 3).
2. Obtain the slopes of the curves (deflection per unit load) by the method of least squares.
   
3. Obtain the theoretical deflections per unit load from the equations.
4. Compare the theoretical and experimental deflections per unit load and determine the percentage by which the experimental values deviate from the theoretical values.

 

Option

Utilize the data from one of the tests (instructor's option) to determine the modulus of elasticity for the member. You should calculate a value of E for each set of deflection data and then compare the two results.

It is suggested that a wooden beam be used in order to show that a) wood acts as a linear elastic material (but note that it is not isotropic) and b) to show that E of wood is about an order of magnitude smaller than that of aluminum and steel.

 

Summary of Results Format

SIMPLY SUPPORTED BEAM

	Experiment	Theory	% Deviation from Theory
dy(a)/dP
dy(b)/dP

CANTILEVER BEAM

	Experiment	Theory	% Deviation from Theory
dy(a)/dP
dy(b)/dP

OPTION
(replaces one of the above sets of values)

	Experimental	Ref. Value (if known)
E from dy(a)/dP data
E from dy(b)/dP data

 

References

Mechanics of Materials by Higdon, et al., 4th Ed., Wiley, 1985, Chapter 7.
Mechanics of Materials by Hibbeler, Macmillan, 1991, pp 557-562.
Mechanics of Materials by Hibbeler, Macmilian, 2^nd Ed, 1994, pp 581-589.