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

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............................................................4 1.1. General Background..............................................................4 1.2. Determination of Curvature....................................................4 1.3. Central Loading.....................................................................4 1.4. Overhanging Loads................................................................6 2. Equipment and Procedure............................................................7 2.1. Equipment..............................................................................7 2.2. Experiment Setup...................................................................7 2.3. Central Loading......................................................................8 2.4. Overhanging Loads................................................................8 3. Data, Analysis & Calculations.......................................................9 3.1. Central Loading......................................................................9 3.2. Overhanging Loads..............................................................11 4. Results........................................................................................12 5. Conclusions.................................................................................13 6. References..................................................................................14 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. 1 M = ρ E⋅I 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 d2y dx 2

1 = 3 ρ  2 2  dy   1 +      dx   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. + → Fx = 0 = Rax PL P ∑ M A = 0 = − 2 + RC ⋅ L ⇒ RC = 2 P + ↑ Fy = 0 = Ray − P + Rc ⇒ Ray = 2 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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d2y P x EI 2 = 2 dx Px PL d2y EI 2 =− + 2 2 dx

0≤ x≤

L 2

L ≤x≤L 2

Equation 6, 7 Integrating twice, Equation 6 becomes P x3 + C1 x + C2 12 P x3 P L x 2 EI y=− + + C3 x + C4 12 4 EI y=

0≤ x≤

L 2

L ≤x≤L 2

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 C1 = −

P L2 16

C2 = 0 C3 = −

3 P L2 16

C4 =

P L3 48

Equation 10, 11, 12, and 13 Combining Equations 8 and 9 with 10 through 13, the expressions for deflection can be expressed as P x 3 P L2 x − 12 16 3 2 Px PLx 3 P L2 x P L3 EI y=− + − + 12 4 16 48 EI y=

0≤ x≤

L 2

L ≤x≤L 2

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 EI y=

2 P x3 ( a − b ) − P a x + P L ( 2a + b ) x 0 ≤ x ≤ L 6L 2 6

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. Load

Gage 1

Gage 2

Gage 3

Gage 4

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. Load

Load Gage 1

Gage 2

Gage 3

Figure 6

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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. Beam Dimensions (inches) Test Length 20.000 Width 1.060 Thickness 0.140

Table 1 Position of Gages (inches) x1 2.150 x2 6.500 x3 13.500 x4 17.500

Table 2 Table 3 returns the results from six different load configurations.

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Strep

Type

0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5

Actual Theoretical Error Actual Theoretical Error Actual Theoretical Error Actual Theoretical Error Actual Theoretical Error Actual Theoretical Error

Deflection Data for Central Loading Load (lb) Gage 1 (in) Gage 2 (in) Gage 3 (in) 2.500 6.500 13.500 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.00% 0.00% 0.00% 0.110 -0.003 -0.008 -0.008 0.110 -0.003 -0.006 -0.006 10.92% 31.31% 31.31% 0.610 -0.017 -0.041 -0.041 0.610 -0.015 -0.034 -0.034 13.35% 21.36% 21.36% 1.110 -0.030 -0.076 -0.075 1.110 -0.027 -0.061 -0.061 9.92% 23.62% 22.00% 1.610 -0.045 -0.110 -0.110 1.610 -0.040 -0.089 -0.089 13.68% 23.36% 23.36% 2.110 -0.059 -0.145 -0.145 2.110 -0.052 -0.117 -0.117 13.73% 24.08% 24.08%

Gage 4 (in) 17.500 0.000 0.000 0.00% -0.004 -0.003 49.79% -0.019 -0.015 28.30% -0.034 -0.027 26.17% -0.050 -0.039 27.92% -0.066 -0.051 28.84%

Table 3 Deflection Resulting on a Centrally Loaded Beam 0.000

-0.020 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

Deflection (inches)

-0.040

-0.060

-0.080

-0.100

-0.120

-0.140

-0.160 0.000

2.000

4.000

6.000

8.000

10.000

12.000

Position (inches)

Figure 7

10

14.000

16.000

18.000

20.000

3.2.Overhanging Loads Beam Dimensions Test Length Width Thickness Distance from left support to edge Distance from right support to edge

20.000 1.060 0.140 13.000 13.000

Table 4 x1 x2 x3

Position of gages 2.5 10 17.5

Table 5 Strep 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5

Deflection Data for Overhanging Loads Type Load (lb) Gage 1 (in) Gage 2 (in) 2.500 10.000 Actual 0.000 0.000 0.000 Theoretical 0.000 0.000 0.000 Error 0.00% 0.00% Actual 0.110 0.014 0.035 Theoretical 0.110 0.012 0.028 Error 12.82% 23.40% Actual 0.330 0.043 0.107 Theoretical 0.330 0.037 0.085 Error 15.51% 25.75% Actual 0.550 0.070 0.172 Theoretical 0.550 0.062 0.142 Error 12.82% 21.28% Actual 0.770 0.101 0.252 Theoretical 0.770 0.087 0.199 Error 16.27% 26.92% Actual 0.990 0.131 0.320 Theoretical 0.990 0.112 0.255 Error 17.30% 25.36%

Table 6

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Gage 3 (in) 17.500 0.000 0.000 0.00% 0.015 0.012 20.88% 0.048 0.037 28.94% 0.076 0.062 22.49% 0.112 0.087 28.94% 0.142 0.112 27.15%

Deflection Resulting from Overhanging Loads

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

0.350

0.300

Deflection (inches)

0.250

0.200

0.150

0.100

0.050

0.000 2

4

6

8

10

12

14

16

18

Position (inches)

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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