1.0 INTRODUCTION
In a similar manner it can that if the Bending moments (BM) of the forces to the left of AA are clockwise then the bending moment of the forces to the right of AA must be anticlockwise. Bending Moment at AA is defined as the algebraic sum of the moments about the section of all forces acting on either side of the section.
Bending Moment is the algebraic sum of the moment of the forces to the left or to the right of the section taken about the section. Bending moments are considered positive when the moment on the left portion is clockwise and on the right anticlockwise. This is referred to as a sagging bending moment as it tends to make the beam concave upwards at AA. A negative bending moment is termed hogging.
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, i.e. shear, axial, and moment at the desired location. The calculated values for each function are then plotted where the load was applied and then connected together to generate the influence line for the function.
For example, the influence line for the support reaction at A of the structure shown in Figure 1, is found by applying a unit load at several points (See Figure 2) on the structure and determining what the resulting reaction will be at A. This can be done by solving the support reaction YA as a function of the position of a downward acting unit load. One such equation can be found by summing moments at Support B.
Figure 1 - Beam structure for influence line example
Figure 2 - Beam structure showing application of unit load
MB = 0 (Assume counter-clockwise positive moment) -YA(L)+1(L-x) = 0
YA = (L-x)/L = 1 - (x/L)
The graph of this equation is the influence line for the support reaction at A (See Figure 3). The graph illustrates that if the unit load was applied at A, the reaction at A would be equal to unity. Similarly, if the unit load was applied at B, the reaction at A would be equal to 0, and if the unit load was applied at C, the reaction at A would be equal to -e/L.
Figure 3 - Influence line for the support reaction at A
2.0 PRINCIPLES
The influence line for bending moment at a section is the graph (curve) representing the variation of bending moment at a section for various positions of the load on the span of the beam. The sign convention followed, in general, is shown in Figure 1.
W x A C B = W (L – x) a (L – a) =Wx L L L
Figure 1: Simply supported beam with load towards left of C W x A C B = W (L – x) a (L – a) = Wx L L L
Figure 2: Simply supported beam with load towards right of C
Consider a simply supported beam of span ‘L’ as shown in Figures 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 towards left of section ‘C’, at a distance ‘x’ from left support ‘A’ 0 < x < a
The shear force at C
= - W
M = + (L – a) = + [Equation 1]
(Considering right side of section C)
When the load ‘W’ is towards right of section ‘D’ at a distance ‘x’ from left support ‘A’ a < x <0
The bending moment at D (considering left side of section C)
M = + • a
M = + [Equation 2]
3.0 OBJECTIVE
To determine the bending moment influence line when the beam is subjected to a load moving from left to the right.
4.0 APPARATUS
1 .A pair of simple supports
2. Special beam with a cut section
3 .A set of weight with several load hangers
4. Indicator
5.0 PROCEDURE
6 The load cell is connected to the digital indicator.
2. The indicator is switched on and it must switched on 10 minutes earlier before taking reading for stability of reading.
3. The two simple supports were fixed to the aluminium base at a distance equal to the span of the beam to be tested. The supports were screwed tightly to the base.
4. The load hanger was hanged to the beam.
5. The beam was placed to the supports.
6. The load hanger was placed 50 mm from the left support.
7. The indicator reading was set in zero (if not zero) by pressing the tare button.
8. The units of load were placed on the load hanger.
9. The indicator reading that represent the bending moment at the cut section was recorded.
10. Remove the load from the hanger.
11. The load hanger was moved to 100 mm from the left support and step 7 until step 11 was repeated. The distance is increased each time by 50 mm.
The step 7 until step 11 was repeated until the load reached end B for 2 more cases. In the second case, 2 load hangers were used and 8 units of load were placed on the 2 hangers while in third case, 3 load hangers were used and 12 units of load were placed on the 3 hanger. The distance between the hangers is 20 mm.
L1 X X RB RA W1 6.0 RESULTS CASE 1
Figure 6.1: Loading position for case 1
Beam Span, (L) = 1000 mm
Distance of the shear section from left support, (La) = 665 mm Weight, (W1) = 4 N
Distance of load cell from the centre of the beam cross section = 175mm
Table 6.1: Result data for Case 1
CALCULATION FOR CASE 1 BEFORE CROSS-SECTION 8 DISTANCE FROM LEFT SUPPORT (mm) BENDING MOMENTS AT X-X EXPERIMENTAL = ( F * 175)( N ) THEORY ( Nmm ) 50 70 67 100 140 134 150 210 210 200 262.5 268 250 332.5 335 300 402.5 402 350 455 469 400 525 536 450 595 603 500 700 670 800 542.5 532 850 402.5 399 900 245 266 950 122.5 133
L1 X X RB RA W1 W2 M x-x = W1 - Wx × La - W1 (La - x) L x = 50 mm, M x-x = 4 - 4(50) × 665 - 4(665 - 50) 1000 = 67 Nmm x = 100 mm, M x-x = 4 - 4(100) × 665 - 4 (665 - 100) 1000 = 134 Nmm AFTER CROSS-SECTION M x-x = W1 - Wx × La L x = 800 mm, M x-x = 4 - 4(800) × 665 1000 = 532 Nmm x = 850 mm, M x-x = 4 - 4(850) × 665 1000 = 399 Nmm CASE 2 9
a
Figure 6.2: Loading position for case 2
Beam Span (L) = 1000 mm
Distance of the shear section from left support (La) = 665 mm Weight 1 (W1) = 4N
Weight 2 (W2) = 4 N
Distance of load cell from the centre of the beam cross section = 175mm A distance between two hangers (a) = 20 mm
DISTANCE FROM BENDING MOMENT AT X-X DIFFERENCES LEFT
SUPPORT(MM) EXPERIMENTAL=(F*175)(N) THEORY(N)
EXPERIMENTAL-THEORY 50 175 160.8 14.2 100 297.5 294.8 2.7 150 437.5 428.8 8.7 200 560 562.8 -2.8 250 700 696.8 3.2 300 857.5 830.8 26.7 350 980 964.8 15.2 400 1102.5 1098.8 3.7 450 1260 1232.8 27.2 800 1015 1010.8 4.2 850 735 744.8 -9.8 900 455 478.8 -23.8
Table 6.2: Result data for case 2
CALCULATION FOR CASE 2 BEFORE CROSS-SECTION
L1 X X RB RA W1 W2 W3 M x-x = (W1 + W2) - W1X +W2(X + a) × La - W1 (La - X) - W2[ La - (X + a)] L For X =50 M x-x = (4+ 4) - 6(50) + 6(50+ 20) × 665- 6 (665 - 50) - 6[665 - (50+ 20)] 1000 = 160.8 Nmm For X = 100 mm, M x-x = (4+ 4) - 6(100) + 6(100 + 20) × 665- 6 (665 - 100) - 6[665 - (100 + 20)] 1000 = 294.8 Nmm AFTER CROSS-SECTION M x-x = (W1 + W2) - W1X + W2(X + a) × La L For X = 800 mm, M x-x = (6+ 6) - 6(800) + 6(800 + 20) × 665 1000 = 1010.8 Nmm CASE 3 a b 11
Figure 6.3: Loading position for case 3
Beam Span (L) = 1000 mm
Distance of the shear section from left support(La) = 665 mm
Weight 1 (W1) = 4N
Weight 2 (W2) = 4N
Weight 3 (W3) = 4 N
Distance of load cell from the centre of the beam cross section = 175mm a and b distance between two hangers (a & b) = 20 mm
12
DISTANCE FROM BENDING MOMENT AT X-X DIFFERENCES LEFT SUPPORT (MM) EXPERIMENTAL=(F*175)(N) THEORY(N) EXPERIMENTAL-THEORY 50 262.5 281.4 -18.9 100 490 482.4 7.6 150 682.5 683.4 -0.9 200 892.5 884.4 8.1 250 1102.5 1085.4 17.1 300 1295 1286.4 8.6 350 1522.5 1487.4 35.1 400 1750 1688.4 61.6 450 1942.5 1889.4 53.1 800 1470 1436.4 33.6 850 1015 1037.4 -22.4 900 630 638.4 -8.4
Table 6.3: Result data for case 3
CALCULATION FOR CASE 3 BEFORE CROSS-SECTION
M x-x = (W1 + W2 + W3) - W1X + W2(X + a) + W3(X + a + b) × La L
- W1 (La - X) - W2 [La - (X + a)] – W3 [La - (X + a + b)]
For X = 100 mm, M x-x = (4+ 4+ 4) - 4 (100) + 4(100 + 20) + 4(100 + 20 + 20) × 665 1000 - 4 (665 - 100) - 4[665 - (100 + 20)] – 4[665 - (100 + 20 + 20)] = 482.4 Nmm AFTER CROSS-SECTION M x-x = (W1 + W2 + W3) - W1X + W2(X + a)+W3(X + a + b) × La L For X = 800 mm, 13
M x-x = (4 + 4 + 4) - 4(800) + 4(800 + 20)+4(800 + 20 + 20) × 665 1000 = 1436.4 Nmm 7.0 DISCUSSION / ANALYSIS Case 1 Case 2 14
Case 3
In this experiment, we used load 4N in every 3 case. Before put the load on the load hanger we must always set the indicator reading was in zero by pressing the tare button. It is because to get the correct value. We also must make sure distances of the shear section from left support are same each 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 is the 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 too sensitive when we taking the reading, the screen show that the reading not in static. The digital indicator is too sensitive with the wind and the surrounding movement. The human error is one of the factors that can affect the experiment result. This is because when we measure the distance between the loads, our eye level is not perpendicular to the ruler. The beam is sensitive when we do the experiment, the beam is moving when we try to put the load and when we want to change the holder of hanger to right side, the beam is not in the original position yet.
The value for the experimental and theoretical value case 1 is nearly same. From this experiment, the values for the experimental and theoretical before cross section x-x are decrease and then after cross section x-x are increase. The value is depend on the location of the load. For the case 1, the maximum bending moment that we get from experiment is 700Nmm while from the theoretical bending moment is 670Nmm
8.0 CONCLUSIONS
In conclusion, based on this experiment, the values of experimental result and theory result are nearly same. The objective of this experiment has been achieved. These experiments is very important before design the beam or any structure. It is because to make sure that the structure is safe to built and used. This experiment and calculation is important before we built a bridge and other structure.
9.0 REFERENCES / APPENDICES
1. Lab Manual Book 2. Structural Analysis
3. http://www.codecogs.com/reference/engineering/materials/shear_force_and_bending_ moment.php
4. http://www.public.iastate.edu/~fanous/ce332/influence/homepage.html
