Bending Beam Report

Bending Beam Louisiana State University

Joshua Board

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Table of Contents:

Table of Figures: ........................................................................................................................................... 4

Purpose .......................................................................................................................................................... 5

Introduction ................................................................................................................................................... 5

Apparatus and Test Procedures ................................................................................................................... 11

Summary of Data ........................................................................................................................................ 14

Discussion of Results .................................................................................................................................. 17

Conclusion .................................................................................................................................................. 20

References ................................................................................................................................................... 21

Sample Calculations .................................................................................................................................... 22

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Table of Figures:

Figure 1 - Bending on a Cross Section ........................................................................................................... 5

Figure 2 - Bending Action caused by Transverse Loads ................................................................................ 6

Figure 3 - SM104 Beam Apparatus ............................................................................................................. 11

Figure 4 - Loading Arrangement for Test 1 and Test 2 ............................................................................... 12

Figure 5 - Loading Arrangement for Test 3 ................................................................................................. 13

Figure 6 - Deflection vs. Load Theory .......................................................................................................... 18

Figure 7 - Deflection vs. Load Experimental ............................................................................................... 18

Figure 8 - Deflection vs. Load Steel Theory ................................................................................................. 19

Figure 9 - Deflection vs. Load Steel Experimental ...................................................................................... 19

Table 1 - Test 1 Data ................................................................................................................................... 14

Table 2 - Test 2 Steel 6mm .......................................................................................................................... 15

Table 3 - Test 2 Steel 3mm .......................................................................................................................... 15

Table 4 - Test 2 Brass 6mm ......................................................................................................................... 15

Table 5 - Test 2 Aluminium 6mm ................................................................................................................ 16

Table 6 - Stiffness ........................................................................................................................................ 16

Table 7 - Test 3 Data ................................................................................................................................... 17

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Purpose

The purpose of this laboratory is to become familiar with the SM104 Beam Apparatus by

first verifying its sensitivity and accuracy. The second part of the lab will focus on using the

SM104 Beam Apparatus to determine the deflection of point-loaded simply-supported beams

made of steel, brass or aluminum. Using the deflection measurements, an examination of the

relationship between deflection and material properties will be shown along with a comparison

of the materials based on their strengths and deflections, both theoretical and experimental. The

last objective of this laboratory is to verify the theory of pure bending using the SM104 Beam

Apparatus.

Introduction

Engineers use beams to support loads over a span length. These beams are structural

members that are only loaded non-axially causing them to be subjected to bending. “A piece is

said to be in bending if the forces act on a piece of material in such a way that they tend to

induce compressive stresses over one part of a cross section of the piece and tensile stresses over

the remaining part” (Ref. 1). This definition of bending is illustrated below in Figure 1.

Figure 1 - Bending on a Cross Section

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It can be seen from Figure 1 that the compressive force, C, and the tensile force, T, acting on the

member are equal in magnitude because of equilibrium. Therefore, the compressive force and the

tensile force form a force couple whose moment is equal to either the tensile force multiplied by

the moment arm or the compressive force multiplied by the moment arm. The moment arm is

denoted, e, in Figure 1.

Figure 2 - Bending Action caused by Transverse Loads

Figure 2, shown above, is an illustration of bending action in a beam acted upon by

transverse loads. Bending may be accompanied by direct stress, transverse shear or torsional

shear, however for convenience; bending stresses may be considered separately (Ref. 1). In

order to separate the stresses it is assumed that the loads are applied in the following manner:

loads act in a plane of symmetry, no twisting occurs, deflections are parallel to the plane of the

loads, and no longitudinal forces are induced by the loads or by the supports (Ref. 1).

A beam or part of a beam that is only acted on by the bending stresses is said to be in a

condition of “pure bending.” However for many circumstances bending is accompany by

transverse shear. The term “flexure” is used to refer to bending tests of beams subjected to

transverse loading (Ref. 1). A visual illustration of the transverse shear and bending moment can

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be seen in the shear and bending moment diagrams of the beam. It is important to note that in a

symmetrical 2-point loading scenario, the center portion of the beam will be in a condition of

pure bending as such the bending stresses may be considered separately.

“Deflection” of a beam is the displacement of a point on the neutral surface of a beam

from its original position under the action of applied loads (Ref. 1). Before the proportional limit

of the material, the deflection, Δ, can be calculated using the moment of inertia, modulus of

elasticity along with other section properties that will depend on the given situation imposed on

the beam. The position of the load, the type of load applied on the beam, and the length of beam

are examples of section properties that depend on the situation. The deflection equations for two

common cases are listed below in equations (1) and (2).

Case 1: Center deflection of a simple beam with freely supported ends and concentrated

load, P, at the mid-span (Ref. 1).

Equation (1)

where:

Δ = deflection, (mm)

P = load, (N)

L = length of beam, (mm)

E = modulus of elasticity (N/m2)

I = moment of inertia of section about the neutral axis, (mm4)

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Case 2: Center deflection of a simple beam with concentrated loads, each equal to P, at

third points of span (Ref. 1).

Equation (2)

where:


P = load, (N)

L = length of beam, (mm)

E = modulus of elasticity (N/m2)

I = moment of inertia of section about the neutral axis, (mm4)

Deflection is a measure of overall stiffness of a given beam and can be seen to be a

function of the stiffness of the material and proportions of the piece (Ref. 1). Deflection

measurements give the engineer a way to calculate the modulus of elasticity for a material in

flexure. The stiffness of a given material is calculated using the following equation:

Equation (3)

where:

P = load, (N)


Stiffness (N/m)

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A beam may fail in any of the following ways: A beam may fail by yielding of extreme

fibers, in long span beams compression fivers act like those of a column and fail by buckling, in

webbed members excessive shear stress may occur and stress concentrations may build up in

parts of beam adjacent to bearing blocks (Ref. 1). The scope and applicability of the bending

tests are defined as:

1) Used as a direct means of evaluation behavior under bending loads, particularly for

determining limits of structural stability of beams of various shapes and sizes.

2) Made to determine strength and stiffness in bending.

3) Occasionally made to get stress distribution in a flexural member.

4) May be used to determine resilience and toughness of materials in bending.

5) Uses simple and inexpensive apparatus.

6) Used as control test for brittle materials and not suitable for determining ultimate strength

of ductile materials. (Ref. 1)

Deflection for experiment 1 is calculated with the following:

Equation (4)

where:

R1 = reaction at support 1, (N)

R2 = reaction at support 2, (N)

W1 = load at point 1, (N)


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Percent error for experiment 1 is calculated with the following:

Equation (5)

where:




For experiment 3, y is calculated as follows:

Equation (6)

where:

h1 = reading from gauge 1, (mm)


For experiment 3, h is calculated as follows:

Equation (7)

where:


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Apparatus and Test Procedures

In this laboratory, an SM104 Beam Apparatus was utilized and is shown below in Figure

(3). This apparatus has load cells with a capacity of 0 – 46 N.

Figure 3 - SM104 Beam Apparatus

Procedure for Test 1:

1) Calibrate the dial gauges.

2) Find the mid span of the beam.

3) Set up load cells as shown in Figure 4.

4) Place a beam on apparatus with 1/4 span overhang.

5) Position the two hangers equidistant from the mid-point.

6) Position the dial gauge at the mid-point.

7) Apply the load to the hangers.

8) Record the data.

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Figure 4 - Loading Arrangement for Test 1 and Test 2






5) Place the hanger at the mid-span.

6) Position the dial gauge over the mid-span.

7) Apply the load to the hanger.

8) Record beam deflection.

9) Increase the load and record the dial reading, repeat for 5 different load patterns.

10) Decrease the load and record the dial reading, repeat for 5 different load patterns.

11) Repeat for each beam.





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5) Place one hanger near the left end of the beam and place a second hanger near the right

end of the beam as shown in Figure 5.

6) Position a dial gauge at mid-span and two other dial gauges equidistant on either side as

shown in Figure 5.

7) Zero the gauges.

8) Apply the load to the two hangers.

9) Increase the load and repeat step 8 for at least five different loads.

10) Record the data.

Figure 5 - Loading Arrangement for Test 3

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Summary of Data

The results from Test 1, the verification of the sensitivity and accuracy of the apparatus,

are shown below in Table 1. This table shows the reactions at the supports based on the applied

load. Deflection was calculated using Equation (4) and the percent error was determined with

Equation (5). The average error for Test 1 is also shown in Table 1 and it was determined that the

average error for this Test is 2.5%, therefore the SM104 Beam Apparatus is accurate within

2.5%.

Table 1 - Test 1 Data

R1 (N) W1 (N) W2 (N) R2 (N) R1 + R2 (N) Δ (N) %

4.0 5.0 0.0 1.2 5.2 0.2 4.0

7.0 10.0 0.0 3.3 10.3 0.3 3.0

10.5 15.0 0.0 5.0 15.5 0.5 3.3

14.0 20.0 0.0 6.0 20.0 0.0 0.0

21.0 30.0 0.0 8.0 29.0 -1.0 -3.3

1.5 0.0 5.0 3.8 5.3 0.3 6.0

3.1 0.0 10.0 6.9 10.0 0.0 0.0

5.1 0.0 15.0 10.5 15.6 0.6 4.0

6.2 0.0 20.0 14.7 20.9 0.9 4.5

11.0 0.0 30.0 20.2 31.2 1.2 4.0

5.5 5.0 5.0 5.1 10.6 0.6 6.0

9.8 10.0 10.0 10.2 20.0 0.0 0.0

15.0 15.0 15.0 15.2 30.2 0.2 0.7

21.7 20.0 20.0 19.7 41.4 1.4 3.5

30.5 30.0 30.0 30.8 61.3 1.3 2.2

Average Error 2.5

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The results for Test 2, deflection of a simply supported beam with load, beam thickness and

material, are shown below based on material type in Tables 2 – 5. Each table shows the load

applied, the material properties and both the theoretical and experimental deflections determined

by Test 2. Looking at the tables it is clear that the material with the smallest cross section, 3mm

Steel, experienced the largest deflection which was expected because that specimen has the

smallest moment of inertia.

Table 2 - Test 2 Steel 6mm

Load (N) deflection theory (mm) deflection theory (m) E (N/mm2) Load (N) deflection exp (mm) deflection exp (m)

0 0 0 210000 0 0

5 3.56849 0.00357 Inertia (mm4) 5 5 0.005

10 7.13698 0.00714 342 10 8 0.008

15 10.70547 0.01071 Length (mm) 15 10 0.01

20 14.27397 0.01427 1350 20 15 0.015

25 17.84246 0.01784 25 20 0.02

30 21.41095 0.02141 30 25 0.025

Steel (6mm)

Table 3 - Test 2 Steel 3mm


0 0 0 210000 0 0 0

5 28.54793 0.02855 Inertia (mm4) 5 30 0.03

10 57.09586 0.05710 42.75 10 70 0.07

15 85.64380 0.08564 Length (mm) 15 100 0.1

20 114.19173 0.11419 1350 20 130 0.13

25 142.73966 0.14274 25 170 0.17

30 171.28759 0.17129 30 200 0.2

Steel (3mm)

Table 4 - Test 2 Brass 6mm


0 0 0 105000 0 0 0

5 7.13698 0.00714 Inertia (mm4) 5 8 0.008

10 14.27397 0.01427 342 10 15 0.015

15 21.41095 0.02141 Length (mm) 15 23 0.023

20 28.54793 0.02855 1350 20 31 0.031

25 35.68492 0.03568 25 38 0.038

30 42.82190 0.04282 30 44 0.044

Brass (6mm)

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Table 5 - Test 2 Aluminium 6mm


0 0 0 76000 0 0 0

2 3.94412 0.00394 Inertia (mm4) 2 3 0.003

4 7.88824 0.00789 342 4 7 0.007

6 11.83237 0.01183 Length (mm) 6 11 0.011

8 15.77649 0.01578 1350 8 16 0.016

10 19.72061 0.01972 10 21 0.021

Aluminium (6mm)

Using the applied load and having already solved for the deflection of each material under that

load, the stiffness of the material can be determined using Equation (3). Shown below is Table 6

which had the calculated stiffness for each material.

Table 6 - Stiffness

Material Thickness (mm) Stiffness (N/m) (theory) Stiffness (N/m) (exp)

Steel 6 0.0007 0.0008

Steel 3 0.0057 0.0067

Brass 6 0.0014 0.0015

Aluminium 6 0.002 0.0023

The results from Test 3 are shown in Table 7 below. The table contains the applied load, the

distances to the load, a and b, the dial gauge readings and the modulus of elasticity based on the

data. The average E value is 108544.22 N/mm2 and after accounting for the error found in Test 1

our value is 105830.61 N/mm2.

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Table 7 - Test 3 Data

Load (N) b (mm) a (mm) h1 (mm) h2 (mm) h3 (mm) y (mm) h2 - y E (N/mm2)

5 300 100 2.2 2.4 2.2 2.2 0.2 109649.12

10 300 100 3.2 3.6 3.3 3.25 0.35 125313.28

15 300 100 3.7 4.4 3.8 3.75 0.65 101214.57

20 300 100 7.5 8.3 7.4 7.45 0.85 103199.17

25 300 100 7.8 8.8 7.9 7.85 0.95 115420.13

5 300 150 1.7 2.2 1.8 1.75 0.45 109649.12

10 300 150 3.5 4.4 3.4 3.45 0.95 103878.12

15 300 150 4.6 6 4.6 4.6 1.4 105733.08

20 300 150 6 7.8 6.2 6.1 1.7 116099.07

25 300 150 6.8 9.2 6.8 6.8 2.4 102796.05

5 300 200 1.4 2.2 1.3 1.35 0.85 103199.17

10 300 200 2.8 4.5 2.9 2.85 1.65 106326.42

15 300 200 5 7.6 5.2 5.1 2.5 105263.16

20 300 200 4.7 8 4.8 4.75 3.25 107962.21

25 300 200 4.6 8.5 4.6 4.6 3.9 112460.64

Inertia (mm4) 108544.22

342 105830.61

Test 1

Test 2

Test 3

Average E value

Average E accounting for error

Discussion of Results

Test 1 showed that the SM104 Beam Apparatus was verified to have an average error of

2.5%. This is an acceptable error for this kind of test and we accounted for this error when we

found the average E value in Test 3. Shown below in Figure s 6 - 9 are the graphs that were

obtained from Test 2. Figure 6 shows deflection vs. load based on theoretical data and Figure 7

shows deflection vs. load for the experimental data gathered in the lab. From these charts it is

clear to see that as the modulus of elasticity decreased from material to material, the deflection

increased for the same applied load. This was true for both the theoretical and the experimental

data.

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Figure 6 - Deflection vs. Load Theory

Figure 7 - Deflection vs. Load Experimental

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35

De

fle

ctio

n

Load (N)

Deflection vs. Load Theory

6 mm Steel

3 mm Steel

6 mm Brass

6 mm Aluminum

0

50

100

150

200

250

0 10 20 30 40

De

fle

ctio

n

Load (N)

Deflection vs. Load Experimental

6 mm Steel

3 mm Steel

6 mm Brass

6 mm Aluminium

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Figure 8 and Figure 9 both compare the two types of steels tested. From the figures, you can see

that as the moment of inertia decreases, the deflection of the material increases.

Figure 8 - Deflection vs. Load Steel Theory

Figure 9 - Deflection vs. Load Steel Experimental

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30 35

De

fle

ctio

n

Load (N)

Deflection vs. Load Steel Theory

6 mm Steel

3 mm Steel

0

50

100

150

200

250

0 5 10 15 20 25 30 35

De

fle

ctio

n

Load (N)

Deflection vs. Load Steel Experimental

6 mm Steel

3 mm Steel

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Table 7 displayed previously shows the results from Test 3. From the measured radius of

curvature and the set up shown in Figure 5, the modulus of elasticity was determined. Even

though some of the numbers that were determined were off by some error, the overall average

value of E was determined to be 108,544.22 N/mm2. This number is very close to the expected

value of 105000 N/mm2 and after accounting for the 2.5% error our numbers are very close.

Conclusion

Using the SM104 Beam Apparatus, we were able to determine first that our numbers

were accurate. Secondly, we found from Test 2 that as the modulus of elasticity of each

specimen decreased, the deflection increased which is consistent with Equation (1). We also

determined that the stiffness of a material is a function of the load over the deflection, meaning

that if the load increases but the deflection does not, then the material is considered stiff. Test 3

proved that the theory of pure bending holds true when the material has not exceeded the

proportional limit because as shown in Table 7, the modulus of elasticity is constant over this

region.

Some human error could have occurred during the laboratory, however our results match

up to the reference values which leads me to believe that error could not have played a big factor

in this lab. Overall a greater understanding of deflection and how different materials react to the

same applied load was achieved.

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References

1. Jacobs, C., CE 3410 Notes – “Bending summary”, received in class on March 16, 2009.

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Sample Calculations

Steel 6mm

Test 1:

Deflection, Δ

Δ = (4.0 + 1.2) – (5 + 0) = 0.2

% error

= 4.0%

Test 2:

Deflection theory (mm)

= 3.57 mm

Stiffness (N/m)

= 0.008 N/m

Test 3:

E = = = 7.97 x 1011

N/m2

Documents

Bending Beam Report