MOMENT INFLUENCE LINE 1.0 OBJECTIVE
Part 1 : To plot monet influence line. Part 2 : To apply the use of moment influence on a simply support beam.
2.0 LEARNING OUTCOME
1. Application the engineering knowledge in practical application. 2. To enhance technical competency co mpetency in structural civil engineering through laboratory application. 3. Communicate effectively in group. 4. To identify problem, solving finding out appropriate solution through laboratory application.
3.0 INTRODUCTION
Influence line are important in the design of o f structures that resist large live loads. We learned that shear and moment diagrams are importantin determining the maximum internal force in a structure. An influence line represents the variation of the reaction, shear, moment, or deflection at a specific point in a member as a concenstrated force moves over the member. Moving loads on beams are common features of design. Many road bridges are constructed from beam and as such have to be designed to carry a knife edge load or a string of wheel loads, or a uniformly distributed load, or perhaps the worst combination of all three. To find the critical critical moment at a section, influence lines is used.
4.0 THEORY
Definition :Influence line is is defined as a line line representing the changes in either moment, moment, shear force, reaction or displacement at a section of a beam when a unit load moves on the beam. Part 1 :This experiment examines how momentvaries at a cut section section as a unit load moves from from one end to another ( see diagram diagram 1). From the diagram, moment influence line equation can be written. For 0 ≤ x ≤a shear line is given by : Sy = -x/ -x/ L………(1) For a ≤ x ≤b shear line is given by : Sy = 1-x/ 1-x/ L……(2) For a unit load between0 between0 ≤ x ≤a,
M X = (L-x)a L
…
1(a- x)………….(1) 1(a- x)………….(1)
For unit load between a ≤ ≤ b MX = xb/L- (x-a)……….……(2 (x-a)……….……(2
1 ( Unit Load )
x
‘Cut’
MX
x
RA = (1- X/L)
RB=(X/L)
MX
a
b
LK
Figure 1 5.0 Part 2 : If the beam are loaded as shown below, the shear force at the “ cut” can be calculated using the influence influence line. (See diagram diagram 2). Moment at ‘cut ’section = F 1Y1 + F2Y2 + F3Y3 …(3)
(Y1, Y2 and Y3 are ordinates derived from the influence line in term of X1, X2, X3, a,b and L)
6.0 APPARATUS
1. Bending moment in a beam apparatus
2. Load and hanger
7.0 PROCUDER
Part 1
1. Check the Digital Force Meter reads Zero with n o load. 2. Place hanger with any mass between 150g to 300g at the first grooved hanger support at the left support and record the Digital Digital Force reading in Table 1. 3. Repeat the procedure to the next n ext grooved hanger until to the last grooved hanger at the right hand support. 4. Complete the calculation in Table 1.
Part 2
1. Place three load hangers with any load 50g to 400g on it and place it at any position between the supports. Record the positions and the Digital Force Display reading in Table 2. 2. Repeat the procedure with three other locations. lo cations. 3. Complete the calculation in Table .
8.0 RESULT Part 1
Location of load
Digital Force
Moment force
Experimental
Theory influence
from left hand
Display
at cut section
Influence line value
line value (Nm)
support (m)
Reading (N)
(N)
(Nm)
0.04
0.3
0.0375
0.025
0.013
0.06
0.4
0.05
0.034
0.019
0.08
0.5
0.0625
0.042
0.025
0.10
0.6
0.075
0.051
0.032
0.12
0.7
0.0875
0.059
0.038
0.14
0.9
0.1125
0.076
0.045
0.16
1.0
0.125
0.085
0.051
0.18
1.1
0.1375
0.093
0.057
0.20
1.2
0.15
0.102
0.064
0.22
1.3
0.1625
0.11
0.07
0.24
1.4
0.175
0.119
0.076
0.26
1.6
0.2
0.136
0.083
0.34
-0.6
-0.075
-0.051
0.068
0.36
-0.5
-0.0625
-0.042
0.054
0.38
-0.4
-0.05
-0.034
0.041
0.40
-0.2
-0.025
-0.017
0.027
Notes : 1. Moment at cut section = Digital force x 0.125 2. Experimental Influence line values =
() ()
3. Calculate the theoretical value using the equation 1 for load position 40 to 260mm and equation 2 for load position 320mm to 380mm
Part 2 :
Location
1 2 3 4
Position of hanger from left hand support (m)
100g
200g
300g
0.04 0.08 0.36 0.26
0.1 0.16 0.34 0.4
0.2 0.26 0.08 0.6
Shear force Digital Reading (N)
Experimental Shear Nm
Theoretical Shear Nm
2.3 2.7 2.1 1.6
0.288 0.334 0.263 0.200
0.261 0.366 0.260 0.190
Notes : 1. Experiment Moment = Digital force Reading × 0.125 2. Theoretical Moment is calculated Using equation (3)
Calculation
Part 1 Moment at cut section = 0.2×0.125
=0.025 Experimental influence line values =
=
() ()
0.0375 1.962
=0.025 m
Theoretical influence lines value
Equation 1 for load position 40 to 260 mm
Mx = (0.44 – (0.44 – 0.04) 0.04) (0.3) – (0.3) – 1(0.3 – 1(0.3 – 0.04) 0.04) 0.44 = 0.013 Nm
Equation 2 for load position 320mm to 400
When x = 0.34 m Mx = (0.34) (0.14) – (0.14) – (0.34 – (0.34 – 0.3) 0.3) 0.44 =0.068 Nm
Part 2 F1 = =
100g 100 x 9.81 1000
= F2 = =
0.981N
200g 200 x 9.81 1000
= F3 = =
1.962N
300g 300 x 9.81 1000
=
2.943N
*For location 1
Experimental moment at cut section (Nm) = Digital force reading × 0.125 = 2.3 × 0.125 = 0.288 Nm
Moment at cut : ∑Mx = 0 Mx = 1(0.3)-
x
(0.3) – (0.3) – 1 1 (0.3-x)
0.44 = 0.3 - 0.3x – 0.3 0.3 + x 0.44 Mx = 0.318
When x = 0.3 Mx = 0.318x =0.318(0.3) =0.095 Nm
use interpolation to get y1, y2, y3 y1, 0.095 = y1 0.3
0.04
0.3y = 0.0038 y1 = 0.013 m
y2,
0.095 = 0.3
y2 0.1
y2 = 0.032 m
y3,
0.095 = 0.3
y3 0.2
y3 = 0.063 m
Theoritical moment at cut section (Nm) =
F1y1 + F2y2 + F3y3
=
0.981 (0.013) + 1.962 (0.032) + 2.943 (0.063)
=
0.261 Nm
*for location 2
Experimental moment (Nm) = 0.363 Nm When y1 = 0.025 m , y2 = 0.051 m , y3 = 0.082m Theoritical moment (Nm) = 0.366 Nm
*for location 3
Experimental moment (Nm) = 0.263 Nm When y1 = 0.054m , y2 = 0.068m , y3 = 0.025m Theoritical moment (Nm) = 0.260 Nm
*For location 4,
Experimental moment (Nm) = 0.4125 Nm When y1 = 0.082m , y2 = 0.027m , y3 = 0.019m Theoritical moment (Nm) = 0.190 Nm
9.0 DISCUSSION
1. Derive equation 1 and 2
ΣFx = 0
ΣFy = R A + R B – 1 1 =0 R A + R B = 1 R A( L ) – ) – 1( 1( L – L – x x ) = 0 R AL = 1(L- x)
R B
R A = 1( L – L – x x ) L =1- x L = 1 – 1 – (1 – (1 – x) x) = x L L
Equation 1 ; 0 ≤ x ≤ a -Mx + R A(a) – (a) – 1(a 1(a - x) = 0 Mx = (1 – (1 – x/L)a – x/L)a – 1(a 1(a - x) = (L – (L – x)a – x)a – 1(a 1(a - x) L
Equation 2 ; a≤ x ≤ b Mx – Mx – R R B(b) + 1(x - a) = 0 Mx = R B (b) – (b) – 1(x 1(x - a) = x/L (b) – (b) – 1(x 1(x -a) = xb/L – xb/L – 1(x 1(x -a)
2. On the same graph paper, plot the theoretical and experimental values against distance from left hand support.
GRAPH EXPERIMENT VALUE (Nm) VERSUS THEORETICAL VALUE (Nm) VERSUS DISTANCE 0.25 0.2 0.15 ) m 0.1 N ( e u l a 0.05 v
theoretical value
0 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.34 0.36 0.38 0.4 -0.05 -0.1
experimental value
distance (m)
From the graph, a peak shaped graph can be obtained. The peak is the weakest point of the beam where there is a hinge in the beam. As load is being moved on the beam, the influence line which was constructed can be used to obtain the value of the moment. As load is moved across near to it, the moment will increase. So does the other way round when load is moving further than the hinge, the value of moment will decrease as the load is moving towards the support at the end. As the load is moving along towards the hinge from both side of support, it will come to a peak where the value of moment is the same. 3. Comment on the experimental results and compare it to the theoretical results.
The experimental results that we obtained are quite accurate and compare to the theoretical results, the experimental results are only slightly different with theoretical results. When we were conducted the experiment, we tried to minimize the error by ensuring the Digital Force Meter reads zero with no load before we place the hangers.
PART 2
1.
Calculate the percentage difference between experimental and theoretical results in table 2. Comment on why the results differ.
Experimental Moment (Nm)
Theoretical moment (Nm)
Percentage Different (%)
0.263
0.261
0.77
0.363
0.366
0.82
0.263
0.26
1.15
0.2
0.19
5.26
The experimental results are slightly 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.
10 CONCLUSION
As a conclusion, both objectives were achieved. Moment influence line could be plot and the influence influence line can be use to determine determine the moment. We were able to identify the reaction and behaviour of a beam in terms of its moment reaction value. This method is useful to check every cross section for a particular beam.
