Mechanics of Materials - Beam Deflection Test

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Mechanics of Materials Laboratory Beam Deflection Test David Clark Group C: David Clark Jacob Parton Zachary Tyler

Transcript of Mechanics of Materials - Beam Deflection Test

Page 1: Mechanics of Materials - Beam Deflection Test

Mechanics of Materials Laboratory

Beam Deflection Test

David Clark

Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10/20/2006

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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 10%, for locations close to supports.

As much as 30% error was experienced when analyzing positions closer to the center of

the beam.

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Table of Contents

1. Introduction & Background.............................................................................3

1.1. General Background................................................................................3

1.2. Determination of Curvature.....................................................................3

1.3. Central Loading.......................................................................................3

1.4. Overhanging Loads..................................................................................5

2. Equipment and Procedure................................................................................6

2.1. Equipment................................................................................................6

2.2. Experiment Setup.....................................................................................6

2.3. Central Loading.......................................................................................7

2.4. Overhanging Loads..................................................................................7

3. Data, Analysis & Calculations.........................................................................8

3.1. Central Loading.......................................................................................8

3.2. Overhanging Loads................................................................................10

4. Results............................................................................................................12

5. Conclusions....................................................................................................13

6. References......................................................................................................13

7. Raw Notes......................................................................................................14

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1. Introduction & Background

1.1. General 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.

1.2. Determination of Curvature

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

1.3. Central Loading

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

as shown below.

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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

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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

1.4. 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.

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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

2. Equipment and Procedure

2.1. Equipment

1. Frame with Movable Knife Edge Supports

2. Metal beam: In this experiment, 2024-T6 aluminum was tested. The beam

should be fairly rectangular, thin, and long. Specific dimensions are

dependant to the size of the test frame and available weights.

3. Calipers, Dial Gages, and a Tape Measure: Calipers should be used to

measure the width and thickness of the beam. Dial gages will be used to

measure deflection along the length of the beam. The tape measure is used

to measure the length of the test region.

4. Hangers and Weights:

2.2. Experiment Setup

Set the knife supports at determined positions along the frame and mount the

beam to be tested. The material, width, thickness, and length between supports should be

measured and recorded for later use.

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2.3. Central Loading

Place dial gages along lengths of the test area (the area between the knife

supports) and set the gages to read zero with no load applied. Adding the hook and

hanger to the center of the beam, record the new readings for the gages. Add new loads

onto the hanger, recording the new deflections for each gage after every loading.

Figure 5

2.4. Overhanging Loads

Dial gages were placed along lengths of the test area and set to read zero with no

applied load. Adding a hook and hanger on each ends extending outside the knife

supports, record the new readings on each of the gages. In discrete intervals, add weights

to both ends of the beam with the hooks applied previously. Record the new deflections

read by the dial gages after each new loading.

Figure 6

Gage 1 Gage 2 Gage 3 Gage 4

Load

Gage 1 Gage 2 Gage 3

Load

Load

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3. Data, Analysis & Calculations

3.1. 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

Table 3 returns the results from six different load configurations.

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Table 3

Deflection Resulting on a Centrally Loaded Beam

-0.160

-0.140

-0.120

-0.100

-0.080

-0.060

-0.040

-0.020

0.000

0.000 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.000

Position (inches)

De

fle

cti

on

(in

ch

es

)

Step 0

Step 1

Step 2

Step 3

Step 4

Step 5

Step 0 Theoretical

Step 1 Theoretical

Step 2 Theoretical

Step 3 Theoretical

Step 4 Theoretical

Step 5 Theoretical

Figure 7

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3.2. Overhanging Loads

Table 4

Table 5

Table 6

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Deflection Resulting from Overhanging Loads

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

2 4 6 8 10 12 14 16 18

Position (inches)

De

fle

cti

on

(in

ch

es

)

Step 0

Step 1

Step 2

Step 3

Step 4

Step 5

Step 0 Theoretical

Step 1 Theoretical

Step 2 Theoretical

Step 3 Theoretical

Step 4 Theoretical

Step 5 Theoretical

Figure 8

4. 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. Though the difference in inches was small, the

percent error could be as high as 30%.

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.

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Figure 9

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 10

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.

5. Conclusions

When an 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.

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6. References

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

Mechanics of Materials Laboratory Manual. June 2000.

7. Raw Notes

Figure 11

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Figure 12

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