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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
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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
Wong Siew Hung AF040176
b
d
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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
Experiment 2 : Span = 300 mm
No. Mass* (N) Deflection
(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
Experiment 3 : Span = 200 mm
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
No. Mass* (N) Deflection
(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.
No. Mass* (N) Deflection
(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
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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
For Experiment 2 : Span = 300 mm
When, N = 0.981 N
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
= – 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
For Experiment 3 : Span = 200 mm
When, N = 0.981 N
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
= – 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
For Experiment 1 : Span = 400 mm
When, N = 0.981 N
% Difference = {– 0.15 – (– 9.66 x 10-5 )}÷ (– 9.66 x 10-5) x 100
= 1.55 x 105 %
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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 %
For Experiment 2 : Span = 300 mm
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 %
When, N = 2.943 N% Difference = {– 0.21 – (– 1.22 x 10-4)}÷ (– 1.22 x 10-4) x 100
= 1.72 x 105 %
For Experiment 3 : Span = 200 mm
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 %
When, N = 2.943 N% Difference = {– 0.07 – (– 3.62 x 10-5)}÷ (– 3.62 x 10-5) x 100
= 1.93 x 105 %
8.0 DISCUSSION
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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.
For Experiment 2 (span 300mm), when the load of 100g or 0.981 N was
place at the mid span, test indicator give us the reading of deflection with -0.07.
When the load is increased to 1.962 N and 2.943 N respectively, the deflection
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.
For Experiment 3 (span 200mm), when the load of 100g or 0.981 N was
place at the mid span, test indicator give us the reading of deflection with -0.03.
When the load is increased to 1.962 N and 2.943 N respectively, the deflection
recorded by test indicator are -0.05 and -0.07. The value of deflection for this
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
= – 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
Wong Siew Hung AF040176
BFC 2091 Structure Lab – Span Deflection (Double Integration Method)
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.
Wong Siew Hung AF040176