2.0
PRINCIPLE
The principle line for bending moment at a section is the graph curves representing the variation of bending moment at a section of a various positions of the load of the span on the beam. The sign convension followed, is shown in the figure 1.
Consider a simply supported beam of span ‘L’ as shown in figure 1 and 2. It is required to draw influence line for bending moment at ‘C’ at a distance ‘a’ from the left support. When the load ‘W’ is toward left of section ‘C’, at a distance ‘x’ from left support ‘A’ 0
( L – a)
……………..[Equati ation 1]
(Considering right side of section c) When the load ‘W’ is towards right of section ‘D’ at a distance ‘x’ from the left support ‘A’ 0
2
2.0
PRINCIPLE
The principle line for bending moment at a section is the graph curves representing the variation of bending moment at a section of a various positions of the load of the span on the beam. The sign convension followed, is shown in the figure 1.
Consider a simply supported beam of span ‘L’ as shown in figure 1 and 2. It is required to draw influence line for bending moment at ‘C’ at a distance ‘a’ from the left support. When the load ‘W’ is toward left of section ‘C’, at a distance ‘x’ from left support ‘A’ 0
( L – a)
……………..[Equati ation 1]
(Considering right side of section c) When the load ‘W’ is towards right of section ‘D’ at a distance ‘x’ from the left support ‘A’ 0
2
The bending moment at ‘D’ (considering left side of section C) M = +Ra • a M=+
3.0
……………..[Equation 2]
OBJECTIVES:
1) To determ determin inee the the shea shearr force force influe influenc ncee line line when when the beam beam is subje subject cted ed to a load load moving from left to right. 2) To plot plot the the shear shear force force influe influenc ncee line line when the beam beam is subje subjecte cted d to a poin pointt load load moving from left to right.
3
4.0
APPARATUS
Sample of beam
The support
Loads
Beam with cut section
Indicator
load hanger
4
5.0
PROCEDURES
The load cell is connected to the digital indicator.
The indicator is switched on. The indicator’s reading must be switch on 10 minutes before taking readings for stability of readings.
The two simple supports is fixed to the aluminium base at a distance equal to the span of the beam to be tested. The support is screwed tightly to the base.
The load hanger is hanged at the 50 mm from the left support.
The indicator reading is noted. If it is not zero, the tare button on the indicator is pressed.
A load is placed on the load hanger
The indicator reading is recorded. This represent the shear force at cut section.
The load is removed from the hanger. The load hanger is moved 200 mm from the left support and step 7-11 is repeated. The distance is increase each time by 50 mm.
Steps 7-11 repeated until the load which is end B.
5
6.0
RESULTS Case 1:
L1
x W1
x R A Beam span Distance of the shear section from the left support W1
R B = 1000mm = 390mm = 2N
Table: case 1. Distance from left support (mm) 100 200 600 800 900
Shear Force at x-x Experimental = R b (N) Theory (N) - 0.1 - 0.2 - 0.1 - 0.4 0.8 0.8 0.4 0.4 0.2 0.2
6
Case 2:
L1
a
x W1
W2
x R A Beam span Distance of the shear section from the left support W1 W2 a = 20mm
R B = 1000mm = 3900mm = 2N = 4N
Table: case 2. Distance from left support (mm) 100 200 600 800 900
Shear Force at x-x Experimental = R b (N) Theory (N) -0.4 -0.68 -0.6 -1.28 2.3 2.32 1.2 1.12 0.5 0.52
7
Case 3:
a
L1
b
x W1
W2
W3
x R A
Beam span Distance of the shear section from the left support W1 W2 W3
R B
= 1000mm = 390mm =2 = 4N = 2N
a = b = 20mm
Table: case 3. Distance from left support (mm) 100 200 600 800 900
Shear Force at x-x Experimental = R b (N) Theory (N) -0.5 -1.16 -1.1 -1.76 3.1 3.04 1.4 1.44 0.7 0.64
8
7.0
CALCULATIONS Case 1 :
P1 100mm
390mm
C
610mm
b/L = 0.61
+ i a/L=-0.39
Load, P1 = 2kN At L1 = 100mm i/100 = -0.39/390 i = -0.1 Vc = - 0.1 x 2N = - 0.2 N At L2 = 200mm i/200 = -0.39/390 i = -0.2 Vc = - 0.2 x 2N = - 0.4 N At L3= 600mm i/400 = 0.61/610 i = 0.4 Vc = 0.4 x 2N = 0.8 N At L1 = 800mm i/200 = 0.61/610 i = 0.2 Vc = 0.2 x 2N = 0.4 N
9
At L1 = 900mm i/100 = 0.61/610 i = 0.1 Vc =0.1 x 2N = 0.2 N Case 2 :
P1 P 2 100mm 20mm
390mm
C
610mm
b/L = 0.61
i
j a/L=-0.39
Load, P1 = 2kN
P2 = 4 kN
At L1 = 100mm j/120 = -0.39/390 j = -0.12 V j = - 0.12 x 4 = - 0.48 N Vc = -0.48 – 0.2 = -0.68 N At L2 = 200mm j/220 = -0.39/390 j = -0.22 V j = - 0.22 x 4 = - 0.88 N Vc = -0.8 – 0.4 = -1.28 N At L3= 600mm j/380 = 0.61/610 j = 0.38 10
V j = 0.38 x 4 = 1.52 N Vc = 0.8 – 1.52 = 2.32 N At L4 = 800mm j/180 = 0.61/610 j = 0.18 V j = 0.18 x 4 = 0.72N Vc = 0.72+0.4 = 1.12N At L5 = 900mm j/80 = 0.61/610 j = 0.08 V j = 0.08 x 4 = 0.32N Vc = 0.32+0.2 = 0.52N Case 3 :
P1 P 2 100mm20mm20mm
390mm
C
610mm b/L = 0.61
i
j
k
a/L=-0.39
11
Load, P1 = 2kN
P2 = 4 kN
P3 = 2 kN
At L1 = 100mm k/140 = -0.39/390 k = -0.14 Vk = - 0.14 x 2 = - 0.28 N Vc = -0.28 – 0.68= -0.96 N At L2 = 200mm k/240 = -0.39/390 k = -0.24 Vk = - 0.24 x 2 = - 0.48 N Vc = -0.48 – 1.28= -1.76 N At L3= 600mm k/360 = 0.61/610 k = 0.36 Vk = 0.36 x 2 = 0.72 N Vc = 0.72+2.32 = 3.04 N At L4 = 800mm k/160 = 0.61/610 k = 0.16 Vk = 0.16 x 2 = 0.32 N Vc = 0.32+1.12 = 1.44 N At L5 = 900mm k/60 = 0.61/610 k = 0.06 Vk = 0.06 x 2 = 0.12 N Vc = 0.12+0.52 = 0.64 N
12
8.0
DISCUSSION AND ANALYSIS
Refer to the graph of Load vs. Shear force.
In this experiment, we used the load 100g, 200g, 600g, 800g, and 900g . From this experiment, the value for the experimental and theoretical are decrease first and then increase after the cut section. The value for the experimental and theoretical value is nearly same. The value is depend on the location of the load. Its shows that the location is one of the causes for the differences between the value. We should know that, influence lines can be used to calculate the shear force at the cut section.
The calculation of percentage of error:
Percentage of error =
Theory – Experiment Theory
X 100%
Table: case 1.
Distance from left support (mm) 100 200 600 800 900
Shear Force at x-x Experimental = R b Theory (N) (N) - 0.1 - 0.2 - 0.1 - 0.4 0.8 0.8 0.4 0.4 0.2 0.2
Percentages of Error, %
Shear Force at x-x Experimental = R b Theory (N) (N) -0.4 -0.68 -0.6 -1.28 2.3 2.32 1.2 1.12 0.5 0.52
Percentages of Error, %
50.0 75.0 0.0 0.0 0.0
Table: case 2. Distance from left support (mm) 100 200 600 800 900
13
41.18 53.13 0.86 7.14 3.85
Table: case 3. Distance from left support (mm) 100 200 600 800 900
Shear Force at x-x Experimental = R b Theory (N) (N) -0.5 -1.16 -1.1 -1.76 3.1 3.04 1.4 1.44 0.7 0.64
Percentages of Error, %
56.89 37.50 1.97 2.78 9.38
From the calculation of error in this experiment, it shows that the error at the distance of 100mm and 200mm is high which is more than 50%. But for the distances of 600mm, 800mm and 900mm, the percentages of error are small. This experimental
results are sometimes different from theoretical results are due to human error and instrument sensitivity as the reading of the instrument keep changing when we conducted the experiment
From the result that we get, there are some errors that make our result not accurate and contribute the error between the experiment and theory. i. Digital indicator is not too accurate Although the value of experiment quite near with the value of theory a there are still have error. The digital indicator is not too accurate. ii.
The digital indicator is too sensitive When we taking the reading, the screen show that the reading not in static. That mean the digital indicator is too sensitive with the wind and the surrounding movement.
iii.
The load hanger is shaking When we taking the reading, we put the load to the hanger. When the load is putting to the hanger, the hanger is shaking and the reading of digital indicator is change. So it effect the reading.
iv.
Parallax error: Reading the ruler scale The ruler scale is in centimeter (cm). So, when the reading process, we can’t get the accurate value, because the scale are not suitable for our eye to read with accurately
v.
The beam is sensitive
14
When we do the experiment, the beam is moving when we try to put the load. When we want to change the holder of hanger to right side, the beam is not in the original position yet. 9.0
CONCLUSION
While doing this experiment, we get the value of the theoretical is exactly the same from the experiment value. Hence, the objective of this experiment is proven. So, we know that our experiment was archived the objective. After the experiment, we have learned how to determine the shear force influence line when the beam is subjected to a load moving from left to right. We also learn how to plot the shear force influence line when the beam is subjected to a point load moving from left to right.
15
LS08 BENDING MOMENT INFLUENCE LINE
1.0
INTRODUCTION
The definition of bending moment which is the internal load generated within a bending element whenever a pure moment is reacted, or a shear load is transferred by beam action from the point of application to distant points of reaction. An influence line for a given function, such as a reaction, axial force, shear force, or bending moment, is a graph that shows the variation of that function at any given point on a structure due to the application of a unit load at any point on the structure. An influence line for a function differs from a shear, axial, or bending moment diagram. Influence lines can be generated by independently applying a unit load at several points on a structure and determining the value of the function due to this load such as shear, axial, and moment at the desired location. Influence lines play an important part in the design of bridges, industrial crane rails, conveyor belts, and other structures where loads move across their span. An influence line represents the variation of the reaction, shear , moment, or deflection at a specific point in a member as a concentrated force moves over the member. Once this line is constructed, one can tell at a glance where the moving load should be placed on a structure so that it creates the greatest influence at the specified point. Furthermore, the magnitude of the associated reaction, shear, moment, or deflection at the point can than be calculated from the ordinates of the influence line diagram.
16
2.0
PRINCIPLES
The principle line for bending moment at a section is the graph curves representing the variation of bending moment at a section of a various positions of the load of the span on the beam. The sign convension followed, is shown in the figure 1.
Consider a simply supported beam of span ‘L’ as shown in figure 1 and 2. It is required to draw influence line for bending moment at ‘C’ at a distance ‘a’ from the left support. When the load ‘W’ is toward left of section ‘C’, at a distance ‘x’ from left support ‘A’ 0
( L – a)
……………..[Equation 1]
17
(Considering right side of section c) When the load ‘W’ is towards right of section ‘D’ at a distance ‘x’ from the left support ‘A’ 0
The bending moment at ‘D’ (considering left side of section C) M = +Ra • a M=+
3.0
……………..[Equation 2]
OBJECTIVE
1) To determine the bending moment influence line when the beam is subjected to a load moving from left to right 2) To determine the value of the function to this load of influence line. 3) To calculate values for each function are then plotted where the load was applied and then connected together to generate the influence line for the function.
18
4.0
APPARATUS
Sample of beam
The support
Loads
Beam with cut section
Indicator
load hanger
19
5.0
PROCEDURE
The load cell is connected to the digital indicator.
The indicator is switched on. The indicator’s reading must be switch on 10 minutes before taking readings for stability of readings.
The two simple supports is fixed to the aluminium base at a distance equal to the span of the beam to be tested. The support is screwed tightly to the base.
The load hanger is hanged at the 50 mm from the left support.
The indicator reading is noted. If it is not zero, the tare button on the indicator is pressed.
A load is placed on the load hanger
The indicator reading is recorded. This represent the shear force at cut section.
The load is removed from the hanger. The load hanger is moved 200 mm from the left support and step 7-11 is repeated. The distance is increase each time by 50 mm.
20
Steps 7-11 repeated until the load which is end B.
6.0
RESULT
6.1
CASE 1
L1
x W1
R A
x
Beam span Distance of the shear section from the left support W1 Distance of load cell from the beam cross section
R B = 1000 mm = 390 mm =2N = 175mm
Table: case 1 Distance from left support (mm) 100 200 600 800 900
6.2
Bending Moment at x-x Experimental = ( F*175) Theory (Nmm) 0.6 x 175 = 105 122 1.4 x 175 = 245 244 1.8 x 175 = 315 312 0.9 x 175 = 157.5 156 0.4 x 175 = 70 78
CASE 2
21
Be a
L1 W1
x
W2
R A
R B x
am span
= 1000 mm
Distance of the shear section from the left support
= 390 mm
W1
=2N
W1 Distance of load cell from the beam cross section
=4N = 175mm
Table: case 2 Distance from left support (mm) 100 200 600 800 950
6.3
Bending Moment at x-x Experimental = ( F*175) Theory (Nmm) 2.5 x 175 = 437.5 414.8 4.6 x 175 = 805 780.8 5.3 x 175 = 927.5 904.8 2.5 x 175 = 437.5 436.8 1.2 x 175 = 210 202.8
CASE 3 a
L1 W1
b
W2
x
W3
R A
x
Beam span Distance of the shear section from the left support W1 W2 W3 Distance of load cell from the beam cross section
22
R B = 1000 mm = 390 mm =2N =4N =2N = 175 mm
Table: case 3. Distance from left support (mm) 100 200 600 800 900
7.0
Bending Moment at x-x Experimental = ( F*175) Theory (Nmm) 3.5 x 175 = 612.5 585.6 6.4 x 175 = 1120 1073.6 7.0 x 175 = 1225 1185.6 3.1 x 175 = 542.5 561.6 1.5 x 175 = 262.5 249.6
CALCULATION For case 1:
P1 100mm
390mm
610mm
C
q
ab/L = 390 x 610 / 1000 = 237.9
Load, P1 = 2kN At L1 = 100mm q/100 = 237.9/390 23
q = 61 Mc = 61 x 2N = 122 Nm At L2 = 200 q/200 = 237.9/390 q = 122 Mc = 122 x 2 = 244 Nm At L3 = 600 q/400 = 237.9/610 q = 156 Mc = 156 x 2 = 312 Nm
At L4 = 800 q/200 = 237.9/610 q= 78 Mc = 78 x 2 =156 Nm At L4 = 800 q/100 = 237.9/610 q= 39 Mc = 39 x 2 = 78 Nm
For case 2:
P1
P2
100mm 20mm
390mm
610mm
q
24
r
Load, P1 = 2kN
P2 = 4kN
At L1 = 100mm r/120 = 237.9/390 r = 73.2 Mr = 73.2 x 4N = 292.8Nm Mc =292.8+122 =414.8 Nm
At L2 = 200 r/220 = 237.9/390 r = 134.2 Mr = 134.2x 4N = 536.8Nm Mc =536.8+122 = 244 = 780.8 Nm
At L3 = 600 r/380 = 237.9/610 r = 148.2 Mr = 148.2 x 4N = 592.8Nm Mc =592.8+312 =904.8 Nm At L4 = 800 r/180 = 237.9/610 r = 70.2 Mr = 70.2 x 4N = 280.8Nm Mc =280.8+156 =436.8 Nm
At L5 = 900 r/80 = 237.9/610 r = 31.2 Mr = 31.2 x 4N = 124.8Nm 25
Mc =124.8+78 =202.8 Nm
For case 3:
P1 P2 P3
100mm 20mm20mm
390mm
610mm
q r
Load, P1 = 2kN
s
P2 = 4kN
P3 = 2 kN
At L1 = 100mm s/140 = 237.9/390 s = 85.4 Ms = 85.4 x 2N = 170.8Nm Mc =170.8+414.8 =585.6 Nm 26
At L2 = 200 s/240 = 237.9/390 s = 146.4 Ms = 146.4 x 2N = 292.8Nm Mc =292.8+780.8 =1073.6 Nm
At L3 = 600 s/360 = 237.9/610 s = 140.4 Ms = 140.4 x 2N = 280.8Nm Mc =280.8+904.8 =1185.6 Nm
At L4 = 800 s/160 = 237.9/610 s = 62.4 Ms = 62.4 x 2N = 124.8Nm Mc =124.8+436.8 =561.6 Nm At L5 = 900 s/60 = 237.9/610 s = 23.4 Ms = 23.4 x 2N = 46.8Nm Mc =46.8+202.8 =249.6 Nm
27
28
8.0
DISCUSSSION AND ANALYSIS
Refer to the graph of Load vs. Bending Moment
P1 P2 P3 100mm20mm20mm
390mm
C
Shear foerce diagram
610mm
+ i1 k 1 j1
Bending moment diagram
+ q1 r1 s1
29
The accuracy: The calculation of percentage of error:
Percentage of error =
Theory – Experiment Theory
X 100%
Table: case 1 Distance from left support (mm) 100 200 600 800 900
Bending Moment at x-x Experimental = Theory (Nmm) ( F*175) 0.6 x 175 = 105 122 1.4 x 175 = 245 244 1.8 x 175 = 315 312 0.9 x 175 = 157.5 156 0.4 x 175 = 70 78
Percentage of Error, %
Bending Moment at x-x Experimental = Theory (Nmm) ( F*175) 2.5 x 175 = 437.5 414.8 4.6 x 175 = 805 780.8 5.3 x 175 = 927.5 904.8 2.5 x 175 = 437.5 436.8 1.2 x 175 = 210 202.8
Percentage of Error, %
Bending Moment at x-x Experimental = Theory (Nmm) ( F*175) 3.5 x 175 = 612.5 585.6 6.4 x 175 = 1120 1073.6 7.0 x 175 = 1225 1185.6 3.1 x 175 = 542.5 561.6 1.5 x 175 = 262.5 249.6
Percentage of Error, %
13.93 0.41 0.96 0.96 10.26
Table: case 2 Distance from left support (mm) 100 200 600 800 950
5.47 3.10 2.51 0.16 3.55
Table: case 3. Distance from left support (mm) 100 200 600 800 900
4.59 4.32 3.32 3.40 5.17
The percentages of the error is small or less than half. It shows here that the accuraccy of the experiment is high. So, we conclude that this experiment is success because the experimental and theoritical value are nearly same.
30
Sometimes, the value of theoritical is too much difference. This is maybe because of the error that occur during the experiment. There are some errors that make result not accurate and contribute the error between the experiment and theory. a. Digital indicator is not too accurate Although the value of experiment quite near with the value of theory a there are still have error. The digital indicator is not too accurate. b. The digital indicator is too sensitive When we taking the reading, the screen show that the reading not in static. That mean the digital indicator is too sensitive with the wind and the surrounding movement. c. The load hanger is shaking When we taking the reading, we put the load to the hanger. When the load is putting to the hanger, the hanger is shaking and the reading of digital indicator is change. So it effect the reading. d. Parallax error: Reading the ruler scale The ruler scale is in centimeter (cm). So, when the reading process, we can’t get the accurate value, because the scale are not suitable for our eye to read with accurately e. The beam is sensitive When we do the experiment, the beam is moving when we try to put the load. When we want to change the holder of hanger to right side, the beam is not in the original position yet.
The ways to overcomes his problem: a. Check the apparatus condition whether it good or not.
b. While putting the load at the hanger, we must put it slowly so that it won’t shake and affect the reading c. When we take the reading, we had to wait until the digital indicator constant and the reading not move anymore. d. We also must wait until the reading shown by the indicator is totally constant before record it.
9.0
CONCLUSION
31
From our discussion, the influence line concept and its application is to specific structural systems reached a certain conclusion at the end of the establishment phrase of structural theory . However , the general influences lines theory actually belongs to the classical phase of structural theory. the result and the theory of left support, middle support and right support. This is may be because of experimental error done during the experiment session. The errors can be the condition during experiment, wind or error in handling the equipments during the experiment.
10.0
REFERENCES
32
1) http://composite.about.com/library/glossary/b/bldef-b604.htm 2) http://en.wikipedia.org/wiki/Influence_line 3) http://www.engr.uky.edu 4) www.bending and shear/htm 5) www.bending/influenceline.com 6) Structural analysis- RC Hibbeler..sixth edition
11.0
APPENDICES
33
