Cemb111 Civil Engineering Materials Laboratory Exp1

1

CEMB111 CIVIL ENGINEERING MATERIALS

LABORATORY EXPERIMENT (NO. 1)

(DEFLECTION OF A SIMPLE SUPPORT BEAM)

SECTION : 03

GROUP NO. : 03

GROUP MEMBERS

1. TSHIAMO KETLHOILWE CE084412 2. SAIF ELESLAM ISMAIL ABDELGADIR CE084409 3. MOHAMED LEMINE OULD MOHAMED CE080687 4. AHMED MOHAMMED ABDULLAH MOGALLI CE084850 5. TANG WENG KEAT CE084411

DATE OF LABORATORY SESSION: 20-7-2010

DATE OF REPORT SUBMISSION: 27-7-2010

LAB INSTRUCTOR: PUAN SITI ALIYYAH Bt. MASJUKI

LAB REPORT MARKING Scale

Poor Acceptable Excellent A. Appearance, formatting

and grammar/spelling 1 2 3 4 5

B. Introduction and objective 1 2 3 4 5 C. Procedure 1 2 3 4 5 D. Results: data, figures,

graphs tables, etc 1 2 3 4 5

E. Discussion 1 2 3 4 5 F. Conclusions 1 2 3 4 5

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TABLE OF CONTENT

EXPERIMENT PART 1 INDEX CONTENT PAGE NUMBER

1 Summary 3 2 Objective 3 3 Apparatus 4 4 Procedure 4 5 Results and Analysis 5 6 Discussion 7 7 Conclusion 7





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Deflection of simple support beam part 1

Summary For this experiment, we can summarize that the purpose is to carry out a laboratory

investigation to identify the relationship between deflection and the applied load at the centre of the beam, thus using the deflection data to obtain the Modulus of Elasticity or Young’s Modulus.

By applying the formulae provided we can calculate the theoretical value of Modulus of Elasticity that also requires the experimental deflection data. By referring to the experimental deflection data, the accuracy is low due to several sources of error such as in the gauge meter and other systematic and human errors, i.e. parallax error, shaking and vibrating of the table, windy environment, etc. Hence, the calculated Modulus of Elasticity is not very accurate.

By using the deflection data to plot the graph, we can calculate the Modulus of Elasticity, through the slope of the graph obtained, assuming a linear relationship between load and deflection. The graphing and linear equations of the curve are generated by computerization for better accuracy rather than manual sketching and drawing.

Finally we achieved the objective of the experiment that is to establish the relationship between deflection and applied load and determine the elastic modulus of the beam specimen from the deflection data.

Objective To establish the relationship between deflection and applied load and determine the elastic modulus of the beam specimen from the deflection data.

Theory

From Theory, The mid-span deflection of a beam loaded with a load W at mid-span is given by :

L/

W

L

4

𝛿 =WL3

48EI

The theoretical mid-span deflection, 𝛿 = bt3

12

Rewriting, 𝐸 = L3

48I⦁Wδ

or, 𝐸 = L3

48I × Slope of the load de�lection curve

Apparatus 1. A support frame 2. A pair of knife- edge support 3. A load hanger 4. A dial gauge with 0.01 accuracy to measuring deflection 5. Beam specimens with constant width and depth. 6. A micrometer to measure the depth and width of the beam specimen 7. A meter ruler to measure the span of the beam 8. A set of weights

Procedure 1. Firstly we bolted the knife-edge support to the support frame using the plate and bolt

supplied with the apparatus. The distance between the two supports was be equal to the span of the beam to be tested.

2. Secondly we measured the width and depth of specimen and noted the readings 3. The we placed the beam specimen on the supports 4. And fixed the load hanger at the mid-span of the beam 5. We positioned the dial gauge at the mid-span at the beam to measure the resulting

deflection. 6. And zero the dial gauge reading. 7. We placed a suitable load on the load hanger 8. Then made a note of the resulting dial gauge reading 9. Increased the load on the load hanger 10. Then we repeated the steps 8 and 9 for a few more loads increments 11. We also repeated the two above steps for two other beams. The span of the beam

should be similar to the first beam.

5

Results and Analysis Span of tested beam, L = 600mm

Width of beam specimen, b = 25mm

Depth of beam specimen, d = 5mm

Moment of inertia of beam specimen, (bd3/12) = 260.42mm4

Dial gauge reading, 1 div = 0.01mm

Table 1

Applied Load

Experimental Deflection

Theoretical Deflection

Test 1 Test 2 Average

N div mm div mm mm mm

5 55 0.55 55 0.55 0.550 0.547

10 110 1.11 109 1.09 1.100 1.093

15 164 1.64 163 1.63 1.635 1.640

20 219 2.19 218 2.18 2.185 2.186

25 274 2.74 273 2.73 2.735 2.733

30 330 3.30 327 3.27 3.285 3.279

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1. Using the tabulated data in Table 1, the graph of load versus experimental deflection is plotted and the best fit curve through the plotted points is drawn.

2. As the experimental deflection increases, so will the load increase steadily. The load is proportional to experimental deflection.

3. Modulus of Elasticity

𝐸 = 𝐿3

48 𝐼× 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑐𝑢𝑟𝑣𝑒

= 6003

48 (260.42) × 9.1478

= 158071.961 N/mm2

4. Calculation of theoretical deflection in Table 1

5N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (5)(600)3

48(158071.961)(260.42) = 0.547mm

10N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (10)(600)3

48(158071.961)(260.42) = 1.093mm

15N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (15)(600)3

48(158071.961)(260.42) = 1.640mm

20N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (20)(600)3

48(158071.961)(260.42) = 2.186mm

y = 9.1478x - 0.018

0

5

10

15

20

25

30

35

0.000 0.500 1.000 1.500 2.000 2.500 3.000 3.500

Load vs. Experimental Deflection

Plotted Point

Best Fit Curve

Experimental Deflection (mm)

Load (N)

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25 N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (25)(600)3

48(158071.961)(260.42) = 2.733mm

30N: 𝛿 =𝑊𝐿3

48 𝐸𝐼 = (30)(600)3

48(158071.961)(260.42) = 3.279mm

Discussion 1. The relationship between the applied load and the resulting displacement is that the

greater the load applied, the greater the resulting displacement or deflection, whenever we increase the load, the deflection increases proportionally.

2. There is room for errors and mistakes throughout this whole experiment. - The main error is that the dial gauge was way too sensitive to any movement. The

needle fluctuated even a small movement on the table it is lying on. The unstable swinging of the load on the hanger also caused the reading to be inaccurate.

- Zero error occurred easily, the initial reading of the dial gauge was not always zero.

- Parallax error happened when the reading was taken. - The wrong readings when shifting the beam to its desired location on the support

frame. One could even be so careless that the wrong midpoint of the beam is

wrongly calculated.

Conclusion From this experiment, we are able to reach its objective, which is to establish the relationship

between deflection and applied load and determine the elastic modulus of the beam specimen

from the deflection data. We can observe from the graph experimental deflection versus load

that as the experimental deflection increases, so will the load increase steadily. The load is

proportional to experimental deflection. The modulus of elasticity calculated for this

experiment is 158071.961 N/mm2. The results achieved through this experiment may not be

entirely accurate due to the various factors of inaccuracy, i.e. human or systematic error. It is

indeed essential to learn the relationship between deflection and span of the beam specimen

to know how much a beam can take when constructing a building.

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Summary For this experiment, we can summarize that the purpose is to carry out a laboratory

investigation to identify the relationship between deflection and the span of the beam specimen using different loads, thus using the deflection data to decide the elastic modulus.

By applying the formulae provided we can calculate the theoretical value of Modulus of Elasticity that also requires the experimental deflection data. By referring to the experimental deflection data, the accuracy is low due to several sources of error such as in the gauge meter and other systematic and human errors, i.e. parallax error, shaking and vibrating of the table, windy environment, etc. Hence, the calculated Modulus of Elasticity is not very accurate.

By using the deflection data to plot the graphs, we can calculate the Modulus of Elasticity. The slope of the graph log (δ/W) vs. log L represents the power of the span and the vertical intercept represents the value for log C.

Finally we achieve our purpose of this experiment and get to know that the span of the beam specimen and deflection can be used to determine the elastic modulus of a beam specimen

Objective To find the relationship between deflection and span of the beam specimen

Theory

From Theory, The mid-span deflection of a beam is given by the equation,

𝛿 =WL3

48EI

L

L/2

W

9

In order to study the affect of span upon deflection δ, the power 3 for the span is replaced by n.

Thus, the deflection equation can be written as

δ/W = 148EI

× L𝑛

δ/W = C×L𝑛

Where the constant C = 148EI

The deflection equation can be written in the log from as below

Log (δ/W)= n log L + log C

Based on linear equation,

𝑦 = 𝑚𝑥 + 𝑐

Thus,

Log(δ/W) = n log L + log C

y represents Log (δ/W)

m represents n

x represents log L

c represents log C

This represents the equation of a straight line. The slope of the graph represents the power of the span and the vertical intercept represents the constant.

Apparatus 1. A support frame 2. A pair of knife- edge support 3. A load hanger 4. A dial gauge with 0.01 accuracy to measuring deflection 5. Beam specimens with constant width and depth. 6. A micrometer to measure the depth and width of the beam specimen 7. A meter ruler to measure the span of the beam 8. A set of weights

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Results and Analysis Beam specimen dimension:

Width, b = 25mm

Depth, d = 5mm

Moment of Inertia, I = 260.42mm4

Dial gauge reading 1 div = 0.01mm

Table 1:

Applied load

Experimental Mid-Span Deflection

Span L1, 450mm Span L2, 700mm Span L3, 800mm

N div mm div mm div mm

5 23 0.23 88 0.88 125 1.25

6 26 0.26 106 1.06 152 1.52

7 32 0.32 122 1.22 178 1.78

8 37 0.37 141 1.41 206 2.06

9 42 0.42 160 1.60 229 2.29

10 46 0.46 176 1.76 256 2.56

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Table 2:

Span, L Log (L) Slope, 𝛿 /W Log (𝛿 /W)

450 2.653 0.048 -1.319

700 2.845 0.1774 -0.751

800 2.903 0.2611 -0.583

1. a. Graph 1 (450mm span)

y = 0.048x - 0.0167

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted Points

Best Fit Curve

Load (N)

Deflection (mm)

12

b. Graph 2 (700mm span)

c. Graph 3 (800mm span)

2. Graph 1 (450mm span): Slope = 0.048 mm/N Graph 2 (700mm span): Slope = 0.1774 mm/N Graph 3 (800mm span): Slope = 0.2611mm/N • All slopes are obtained by computerization for greater accuracy.

y = 0.1774x - 0.009

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted Points

Best Fit Curve

Load (N)

Deflection (mm)

y = 0.2611x - 0.0486

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted Points

Best Fit Curve

Load (N)

Deflection (mm)

13

3. Information from step 2 has been filled in Table 2

4. Using the data from Table 2, the graph of log (𝛿 /W) versus log L is plotted.

5. From the graph of log (δ/W) vs. log L, the power for the span obtained is 2.9476.

(Slope is generated by computerization for greater accuracy)

6. The intercept = -9.1386 = log C C = 10−9.1386 = 7.26775× 10−10 E = 1

48 𝐶𝐼 = 1

48×7.26775×10−10×260.42 = 110073.924 N/mm2 ≈ 0.11 MN/mm2

Discussion 1. We can observe from the graph experimental deflection versus load for each span, as

the span increases so will the deflection increase more rapidly. In other words, the longer the span, the bigger slope of the graph. The span is directly proportional to experimental deflection.

2. From theory, we can calculate the exact elastic modulus, however, as we conduct the experiment, the experimental value will be slightly different than the actual value, possibly caused by several unavoidable errors during the experiment. There is room for errors and mistakes throughout this whole experiment. - The main error is that the dial gauge was way too sensitive to any movement. The



- Parallax error happened when the reading was taken.

y = 2.9476x - 9.1386

-1.400

-1.200

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

2.600 2.650 2.700 2.750 2.800 2.850 2.900 2.950

log (δ/W) vs. log L

Plotted Points

Linear

log L

log (δ/W)

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- The wrong readings when shifting the beam to its desired location on the support frame. One could even be so careless that the wrong midpoint of the beam is wrongly calculated.

We can increase the accuracy by

- Use digital dial gauge instead of analog - Calibrate the gauge each time the load is applied. - Make sure that position of eyes of observer is parallel to the scale. - Check for at least twice to make sure that the midpoint and the span length of the

beam are located correctly.

Conclusion Through this experiment, we have learnt the relationship between deflection and span of the beam specimen. From the graph experimental deflection versus load for each span, as the span increases so will the deflection increase more rapidly. In other words, the longer the span is, the bigger slope the graph has. The span is directly proportional to experimental deflection. The modulus of elasticity calculated for this experiment is 110073.924 N/mm2 ≈ 0.11 MN/mm2. There is a massive room for errors and mistakes that cause inaccuracy due to human and systematic error.

15


Summary For this experiment, the purpose is to perform a series of investigation to identify the

relationship between deflection and depth, and hence determine the Modulus of Elasticity for each different thickness.

By applying the formulae provided we can calculate the theoretical value of Modulus of Elasticity that also requires the experimental deflection data. The deflection data is obtained from tests for different depth or thickness of the beam. By referring to the experimental deflection data, the accuracy is low due to several sources of error such as in the gauge meter and other systematic and human errors, i.e. parallax error, shaking and vibrating of the table, windy environment, etc. Hence, the calculated Modulus of Elasticity is not very accurate.

By using the deflection data to plot the graphs, we can calculate the Modulus of Elasticity, through the slope of the graph obtained, assuming a linear relationship between beam span length and deflection

Finally we achieve the purpose of this experiment and also get to know that the depth of beams and deflection can be used to determine the elastic modulus of a beam specimen.

Objective To establish the relationship between deflection and depth and hence determine the elastic modulus for the beam specimen.

Theory

From Theory, The mid-span deflection is given by the equation:

L

W

L/

16

𝛿 =WL3

48EI

The section modulus 𝐼 = 𝑏𝑡3

12

The deflection can be written as follows:

𝛿 = 𝑊𝐿3

48 𝐸𝐼 ×

12𝑏𝑡3

In order to study the affect of thickness, t, upon deflection, 𝛿, the power 3 for the thickness is replaced by n.

𝛿 = 𝑊𝐿3

48 𝐸𝐼 ×

12𝑏𝑡𝑛

Or,

𝛿 𝑊⁄ = 𝑊𝐿3

4 𝑏𝐸 × 𝑡(−𝑛)

Or 𝛿 𝑊⁄ = 𝐶 × 𝑡(−𝑛)

Where C is a constant, the above equation can be rewritten in log form as well which is as follows:

log(𝛿 𝑊) = log(𝐶) − 𝑛 log(𝑡)⁄

This represents the equation of a straight line of:

y = mx+c

Apparatus 1. A support frame 2. Sa pair of support stand 3. A pair of knife- edge support 4. A load hanger 5. A dial gauge with 0.01 accuracy to measuring deflection 6. 3 beam specimens having similar width but of different depth. Each beam must have

constant depth and width throughout its depth 7. A micrometer to measure the depth and width of the beam specimen 8. A meter ruler to measure the span of the beam 9. A set of weights

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Results and Analysis Span of beam, L = 600mm

Width of beam, b = 26mm

Dial gauge reading 1div = 0.01mm

Table 1:

Load Experimental Mid-span Deflection

Thickness, t1 Thickness, t2 Thickness, t3

N div mm div mm div mm

5 111 1.11 52 0.52 21 0.21

6 149 1.49 63 0.63 25 0.25

7 189 1.89 74 0.74 29 0.29

8 235 2.35 85 0.85 34 0.34

9 268 2.68 97 0.97 39 0.39

10 313 3.13 108 1.08 42 0.42

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Table 2:

Thickness of beam, t log (t) Slope, δ/W log (δ/W)

3 0.477 0.4037 -0.394

5 0.699 0.1123 -0.950

7 0.845 0.0434 -1.363

1. For each thickness, the graph of deflection against load is plotted.

a. Graph 1 (thickness t1)

y = 0.4037x - 0.9195

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted points

Best Fit Curve

Load (N)

Deflection (mm)

19

b. Graph 2 (thickness t2)

c. Graph 3 (thickness t3)

2. Graph 1 (thickness t1): Slope = 0.4037 N/mm2 Graph 2 (thickness t2): Slope = 0.1123 N/mm2 Graph 3 (thickness t3): Slope = 0.0434 N/mm2

*all slopes are obtained by computerization for greater accuracy

3. Information from step 2 has been filled in Table 2.

y = 0.1123x - 0.0438

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted Points

Best Fit Curve

Load (N)

Deflection (mm)

y = 0.0434x - 0.009

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 1 2 3 4 5 6 7 8 9 10 11

Deflection vs. Load

Plotted Points

Best Fit Curve

Load (N)

Deflection (mm)

20

4. Using the data from Table 2, the graph of log (δ/W) versus log t is plotted.

5. The power for the thickness = -0.3809 (Slopes generated by computer for greater accuracy)

6. Elastic modulus for each thickness:

𝐸 = 𝐿3

48 𝐼 × 𝑆𝑙𝑜𝑝𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 , 𝐼 = 𝑏𝑑

3

12 = 𝑏𝑡

3

12

t1, 3mm: 𝐼 = (26)(3)3

12= 58.5𝑚𝑚4

𝐸 = 6003

(48)(58.5)× 0.4037 = 31053.8462𝑚𝑚−1

t2, 5mm: 𝐼 = (26)(5)3

12= 270.833𝑚𝑚4

𝐸 = 6003

(48)(270.833)× 0.1123 = 1865.91𝑚𝑚−1

t3, 7mm: 𝐼 = (26)(7)3

12= 743.1667𝑚𝑚4

𝐸 = 6003

(48)(743.1667)× 0.0434 = 262.7943𝑚𝑚−1

7. Sample calculation of percentage error: Load 5N on t1 (3mm) beam:

Theoretical deflection, 𝛿 = 𝑊𝐿3

48 𝐸𝐼× 12

𝑏𝑡3= (5)(600)3

48×31053.8462×58.5× 12

26×33= 0.21𝑚𝑚

Experimental deflection, δ = 1.11mm Percentage error = 0.21−1.11

0.21× 100% = 428.57%

y = -0.3809x + 0.33

-1.600

-1.400

-1.200

-1.000

-0.800

-0.600

-0.400

-0.200

0.000

0.000 0.200 0.400 0.600 0.800 1.000

log (δ/W) vs. log t

Plotted Points

Linear

log (δ/W)

log t

21

Average percentage error = 428.57+225+403

= 231.19% Table 3: 5N:

Beam thickness

Theoretical deflection (mm)

Experimental deflection (mm)

Percentage Error (%)

Average percentage error (%)

t1, 3mm 0.21 1.11 428.57 231.19 t2, 5mm 0.16 0.52 225

t3, 7mm 0.15 0.21 40

6N: Beam

thickness Theoretical

deflection (mm) Experimental

deflection (mm) Percentage Error (%)


t1, 3mm 0.25 1.49 496 247.53 t2, 5mm 0.20 0.63 215

t3, 7mm 0.19 0.25 31.58

7N: Beam





t1, 3mm 0.30 1.89 530 261.19 t2, 5mm 0.23 0.74 221.74

t3, 7mm 0.22 0.29 31.82 8N:

Beam thickness





t1, 3mm 0.34 2.35 591.18 284.7 t2, 5mm 0.26 0.85 226.92

t3, 7mm 0.25 0.34 36 9N:

Beam thickness





t1, 3mm 0.38 2.68 605.26 289.29 t2, 5mm 0.30 0.97 223.33

t3, 7mm 0.28 0.39 39.29

22

10N: Beam





t1, 3mm 0.42 3.13 645.24 302.66 t2, 5mm 0.33 1.08 227.27

t3, 7mm 0.31 0.42 35.48 Overall percentage error = 231.19+247.53+261.19+284.7+289.29+302.66

6= 269.43%

Discussion 1. The relationship between the thickness and deflection of a beam is that the thicker the

beam is, the less the deflection is. Deflection is inversely proportional to thickness. 2. From theory, we can calculate the exact elastic modulus, however, as we conduct the

experiment, the experimental value will be slightly different than the actual value, possibly caused by several unavoidable errors during the experiment. There is room for errors and mistakes throughout this whole experiment. - The main error is that the dial gauge was way too sensitive to any movement. The



- Parallax error happened when the reading was taken. - The wrong readings when shifting the beam to its desired location on the support

frame. One could even be so careless that the wrong midpoint of the beam is wrongly calculated.

We can increase the accuracy by

- Use digital dial gauge instead of analog - Calibrate the gauge each time the load is applied. - Make sure that position of eyes of observer is parallel to the scale. - Check for at least twice to make sure that the midpoint and the span length of the

beam are located correctly.

Conclusion Deflection is inversely proportional to thickness. When the thickness of the beam increased, the deflection of mid span decreased. Thus, this can take more force. This can be investigated by drawing a graph of log (δ/W) vs. log t. The modulus of elasticity for each thickness has also been found; for 3mm, E = 31053.8462𝑚𝑚−1; for 5mm, E = 1865.91𝑚𝑚−1; for 7mm, E = 262.7943𝑚𝑚−1. The results achieved through this experiment may not be entirely accurate due to the various factors of inaccuracy, i.e. human or systematic error. The overall percentage error for this experiment is 269.43%.

Cemb111 Civil Engineering Materials Laboratory Exp1

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Transcript of Cemb111 Civil Engineering Materials Laboratory Exp1