Mechanics of Materials Laboratory

Beam Deflection Test

Date Performed: 2/28/11

Date Due: 3/14/11

Richard Dyar

Group B:

Yazmin Ince

Richard Dyar

Abstract

If a beam is supported at two points, and a load is applied anywhere on the

beam, the resulting deformation can be mathematically estimated. Due to improper

experimental setup, the actual results experienced varied substantially when compared

against the theoretical values. The following procedure explains how the theoretical and

actual values were determined, as well as suggestions for improving upon the

experiment. The percent error remained relatively small, around 11%, for locations close

to supports. Error was experienced when analyzing positions closer to the beam, with

the exception of odd values on gage 4.

1

Background

If a beam is supported at two points, and a load is applied anywhere on the

beam, deformation will occur. When these loads are applied either longitudinally

outside or inside of the supports, this elastic bending can be mathematically predicted

based on material properties and geometry.

Curvature at any point on the beam is calculated from the moment of loading

(M), the stiffness of the material (E), and the first moment of inertia (I.) The following

expression defines the curvature in these parameters as 1/ρ, where ρ is the radius of

curvature.

Equation 1

Equation 1 does not account for shearing stresses.

Curvature can also be found using calculus. Defining y as the deflection and x as

the position along the longitudinal axis, the expression becomes

Equation 2

Central Loading

Central loading on a beam can be thought of as a simple beam with two supports

as shown below.

2

Figure 1

Applying equilibrium to the free body equivalent of Figure 1, several expressions

can be derived to mathematically explain central loading.

Equation 3, 4, and 5

Figure 2 and 3 act as free body diagrams for the section between AB and BC

respectively.

Figure 2

Figure 3

Solving the reactions between AB and BC, equation 1 can be expressed as

3

Equation 6, 7

Integrating twice, Equation 6 becomes

Equation 8, 9

To determine the constants, conditions at certain positions on the beam can be

applied. Knowing the deflection at each of the supports, as well as the slope at the top

of the curve is zero, the constants can be derived to

Equation 10, 11, 12, and 13

Combining Equations 8 and 9 with 10 through 13, the expressions for deflection

can be expressed as

Equation 14, 15

Overhanging Loads

Overhanging loading on a beam is similar to that of central loading. In

overhanging loading, a simple beam is supported with two supports and two loads as

shown below.

4

Figure 4

Using similar methods used previously for central loading, the equation for

determination of deflection as a function of position, load, length, stiffness, and

geometry can be derived as

Equation 16

Procedure

See lab manual section 11

Central Loading

Gage 1 Gage 2 Gage 3 Gage 4

Load

5

Overhanging Loads

Data & Calculations

Central Loading

Table 1 and 2 catalog the dimensions of the beam, as well as the position of the

gages as measured from one of the two fixed supports.

Table 1

Table 2

Gage 1 Gage 2 Gage 3

Load

Load

6

Table 3

Overhanging Loads

Table 4

Table 5

7

Table 6

Results

The theoretical results were not as expected or experienced. There was

significant error between the actual results and theoretical value, especially as the

distance studied approached the midpoint of the beam.

The main source of error within this experiment occurs due to the improper

testing procedure. As seen in Figure 9, the theory used within this exercise is based

upon a beam with one fixed support allowing one degree of freedom, a second support

allowing two degrees of freedom, and a central load.

8

Figure 5

This produces dramatically different results when compared against the actual

setup. When using two knife supports, the setup contains two supports allowing two

degrees of freedom and a central load. This is pictured in Figure 10.

Figure 6

Since both ends are under-constrained, the analysis for the experiment with the above

theory is not accurate.

Another cause of error in the theoretical is the effect of gravity on the beam.

With no applied load, the equations above would return a zero result. This is inaccurate

for beams that are not specifically supported such that gravitational factors are

overcome. Also the measurements and position of gages for overhanging loading may

be incorrect, and the experiment most likely needs to be re-run.

Conclusions

When a load is applied to a beam, either centrally over at another point, the

deflection can be mathematically estimated. Due to the error that occurred in this

exercise, it is clear that margins in safety factors, as well as thorough testing, is needed

when utilizing beam design. It is also important to ensure the scope of the testing

closely models real-world practicality.

9

References

Gilbert, J. A and C. L. Carmen. "Chapter 11 – Beam Deflection Test." MAE/CE 370 –

Mechanics of Materials Laboratory Manual. June 2000.

10