Span Deflection

BFC 2091 Structure Lab – Span Deflection (Double Integration Method)

TITLE : SPAN DEFLECTION (DOUBLE INTEGRATION METHOD)________________________________________________________________________

1.0 OBJECTIVE

To determine the relationship between span and deflection.

2.0 INTRODUCTION

A beam must posses sufficient stiffness so that excessive deflections do

not have an adverse effect on adjacent structural members. In many cases,

maximum allowable deflections are specified by Codes of Practice in terms of the

dimensions of the beam, particularly the span. The actual deflections of a beam

must be limited to the elastic range of the beam, otherwise permanent distortion

results. Thus in determining the deflections of beam under load, elastic theory is

used.

3.0 THEORY

Beam with point load at mid span

Wong Siew Hung AF040176

C

X

P X

L/2 - x

A B

L/2 L/2

x


When x = 0 ;

When x = L/2 ; y = 0 ;

When x = 0 ; (mind span ; c)

x = L/2 (at support)

where E can be obtained from the backboard

4.0 APPARATUS

4.1 Specimen beam (your group may choose one of the following material :

Aluminiun, Brass or Steel)

4.2 Digital Dial Test Indicator

4.3 Hanger and Masses

5.0 PROCEDURE


b

d


5.1 Positioned the moveable knife-edge supports so that they are 400 mm

apart.

5.2 Place the chosen beam on the support.

5.3 Place the hanger and the digital dial test indicator at mid span. Zeroed the

digital reading.

5.4 Apply an incremental load and record the deflection for each increment in

the table below.

5.5 Repeat the above using span of 300 mm and 200 mm.

6.0 RESULT

Experiment 1 : Span = 400 mm

No. Mass* (N) Deflection

(Experimental)

Theoretical

Def. (ymax)

% Difference

1 0.981 - 0.15 - 9.66 x 10-5 1.55 x 105

2 1.962 - 0.28 - 1.93 x 10-4 1.45 x 105

3 2.943 - 0.42 - 2.90 x 10-4 1.45 x 105



(Experimental)

Theoretical

Def. (ymax)

% Difference

1 0.981 - 0.07 - 4.07 x 10-5 1.72 x 105

2 1.962 - 0.14 - 8.15 x 10-5 1.72 x 105

3 2.943 - 0.21 - 1.22 x 10-4 1.72 x 105





(Experimental)

Theoretical

Def. (ymax)

% Difference

1 0.981 - 0.03 - 1.21 x 10-5 2.48 x 105

2 1.962 - 0.05 - 2.42 x 10-5 2.01 x 105

3 2.943 - 0.07 - 3.62 x 10-5 1.93 x 105

Used any mass between 10 to 500 g

For Extra Calculation/Experiment with 400 mm span and x=L/3 (400 (from

experiment 1, no. 3), the hanger and the digital dial test indicator is place at the

L/3 (400mm / 3) of the span.


(Experimental)

Theoretical

Def. (ymax)

% Difference

3 2.943 - 0.41 - 2.90 x 10-4 1.45 x 105

Given,

Esteel = 207 GNm-2

= 207 x 109 Nm-2

Width and Thick of the Span ;

Reading Width / b (m) Thick / d (m)

1 19.19 x 10-3 3.54 x 10-3

2 19.13 x 10-3 3.48 x 10-3

3 19.00 x 10-3 3.33 x 10-3

Average 19.11 x 10-3 3.45 x 10-3

7.0 DATA ANALYSIS / CALCULATION



Given, Esteel = 207 x 109 Nm-2

Width, b = 19.11 x 10-3 mThick, d = 3.45 x 10-3 m

From Equation,

= 6.54 x 10-11 m4

For Experiment 1 : Span = 400 mm

When, N = 0.981 N

= – 9.66 x 10-5 m

When, N = 1.962 N

= – 1.93 x 10-4 m

When, N = 2.943 N

= – 2.90 x 10-4 m


When, N = 0.981 N



= – 4.07 x 10-5 m

When, N = 1.962 N

= – 8.15 x 10-5 m

When, N = 2.943 N

= – 1.22 x 10-4 m


When, N = 0.981 N



= – 1.21 x 10-5 m

When, N = 1.962 N

= – 2.42 x 10-5 m

When, N = 2.943 N

= – 3.62 x 10-5 m

Percentage of Differences Between the Experimental Deflection and Theoretical

Deflection


When, N = 0.981 N

% Difference = {– 0.15 – (– 9.66 x 10-5 )}÷ (– 9.66 x 10-5) x 100

= 1.55 x 105 %



When, N = 1.962 N

% Difference = {– 0.28 – (– 1.93 x 10-4)}÷ (– 1.93 x 10-4) x 100

= 1.45 x 105 %

When, N = 2.943 N% Difference = {– 0.42 – (– 2.90 x 10-4)}÷ (– 2.90 x 10-4) x 100

= 1.45 x 105 %


When, N = 0.981 N

% Difference = {– 0.07 – (– 4.07 x 10-5)}÷ (– 4.07 x 10-5) x 100

= 1.72 x 105 %

When, N = 1.962 N

% Difference = {– 0.14 – (– 8.15 x 10-5)}÷ (– 8.15 x 10-5) x 100

= 1.72 x 105 %


= 1.72 x 105 %


When, N = 0.981 N

% Difference = {– 0.03 – (– 1.21 x 10-5)}÷ (– 1.21 x 10-5) x 100

= 2.48 x 105 %

When, N = 1.962 N

% Difference = {– 0.05 – (– 2.42 x 10-5)}÷ (– 2.42 x 10-5) x 100

= 2.01 x 105 %


= 1.93 x 105 %

8.0 DISCUSSION



Comment on the different between the theoretical and experimental results.

Referring to the results from the calculation, we can conclude that, the

different between the theoretical and experimental results are very big for both

Experiment 1, 2, and 3. Thus, the percentage (%) of the difference between the

theoretical and experimental results are extremely big and high. From the

experiment done, we can notice that, the span with longer length will give us the

bigger value of deflection when the load is place at the mid span for both

theoretical and experimental results. While for the span with shorter length, the

deflection is slightly small compare to the longer span.

For Experiment 1 (span 400mm), when the load of 100g or 0.981 N was

place at the mid span, test indicator give us the reading of deflection with -0.15.

When the load is increased to 1.962 N and 2.943 N respectively, the deflection

recorded by test indicator are -0.28 and -0.42. The values of the deflection for

both theoretical and experimental results increase proportionally to the load when

the load of 100g, 200g and 300g is place on the mid span.




recorded by test indicator are -0.14 and -0.21. But, the value of deflection for this

experiment is smaller than the experiment 1. This is because the length of the

span used, 300mm, is shorter than experiment 1. The values of the deflection for

both theoretical and experimental results increase proportionally to the load when

the load of 100g, 200g and 300g is place on the mid span.




recorded by test indicator are -0.05 and -0.07. The value of deflection for this



experiment is smaller than the experiment 1 and experiment 2. This is because the

length of the span used, 200mm, is shorter than the span used for experiment 1

and experiment 2. The values of the deflection for both theoretical and

experimental results increase proportionally to the load when the load of 100g,

200g and 300g is place on the mid span.

From the results we get from this experiment, though the different between the

theoretical and experimental results are very big, but the deflection in the span

increase when the load is increase. Besides that, the value of deflection also

increase when the length of span used is longer. Thus, we conclude that, the

deflection of span is proportional to the load we place on it and the length of the

span we used.

9.0 EXTRA QUESTIONS

9.1 Calculate the deflection when x = L/3 (experiment 1, no. 3). Check the

result by placing the digital dial at this position.

Calculation :

When x = L/3, this mean that x = 133.33mm (400/3), the value for

Deflection (Experimental) we get is – 0.41 and the Theoretical Deflection

we get from the calculation is – 2.90 x 10-4 m. The percentage (%) of the

difference between the theoretical and experimental results for this extra

experiment is 1.45 x 105 %.

When, N = 2.943 N



= – 2.90 x 10-4 m

When, N = 2.943 N

% Difference = {– 0.41 – (– 2.90 x 10-4)}÷ (– 2.90 x 10-4) x 100

= 1.45 x 105 %

9.2 Calculate Vmak in experiment 2, no.2.

Given, Esteel = 207 x 109 Nm-2

Width, b = 19.11 x 10-3 m

Thick, d = 3.45 x 10-3 m

From Equation,

= 6.54 x 10-11 m4

From Equation,

= 2.45 x 10-4 m

10.0 CONCLUSION

From this experiment, our group managed to determine the relationship

between span and deflection. In determining the deflections of the beams under

load, elastic theory is used. From the experiment and the results we get from this

experiment, we notice that, the span with longer length will give us the bigger

value of deflection when the load is place at the mid span for both theoretical and



experimental results. While for the span with shorter length, the deflection is

slightly smaller compare to the longer span though the load used is same with the

longer one. Though the different between the theoretical and experimental results

are very big, but the deflection in the span also increase when the load is increase.

Thus, we conclude that, the deflection of span is proportional to the length of the

span and the load we place on it.

11.0 REFERENCES

Yusof Ahamad (2001). “Mekanik Bahan Dan Struktur.” Malaysia: Universiti

Teknologi Malaysia Skudai Johor Darul Ta’zim.

R. C. Hibbeler (2000). “Mechanic Of Materials.” 4th. ed. England: Prentice Hall

International, Inc.


Span Deflection

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