1.0 OBJECTIVE

1.1 Part 1 : To plot moment influence line

1.2 Part 2 : To apply the use of a moment influence on a simply supported

beam

2.0 LEARNING OUTCOMES

2.1 Application of engineering knowledge in practical application

2.2 To enhance the technical competency in civil engineering through

laboratory application.

2.3 Communicate effectively in group.

2.4 To identify problem, solving and finding out appropriate solution through

laboratory application.

3.0 INTRODUCTION

Moving loads on beam 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 moment in section, influence line is

used.

4.0 THEORY

Definition : Influence line is define as a line representing the changes in either

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 moment varies at a cut section as a unit

load moves from one end another ( see diagram 1 ). From the diagram,

moment influence equation can be written.

For a unit load between 0 < x < a ,

Mx = ( L – x ) a - 1 (a – x )……….(1)

L

For unit load between a < x < b ,

Mx = xb / L – ( x – a )…………..(2)

‘ cut ‘

1 ( unit load ) Mx

x

Mx

RA = (1-x/L) RB = x/L

a b

L

Figure 1

Part 2 : If the beam is loaded as shown below, the moment at the ‘cut’ can be

calculated using the influence line. ( See diagram 2 ).

Moment at the ‘cut’ section = F1y1 + F2y2 + F3y3 ……….(3)

( y1, y2, and y3 are coordinates derived from the influence line in terms of x1, x2, x3, a,

b and L )

a+b = L

x1

x2

x3

y1 y2 y3

Moment influence line for cut section

Figure 2

5.0 APPARATUS

Figure 3

6.0 PROCEDURES

Part 1 :

1. The digital forces meter reads zero with no load.

2. Hanger with any mass between 150 – 300 g was place at the first

grooved hanger support at the left support and the digital forces reading

were recorded in Table 1.

3. The procedure repeated to the next grooved hanger until to the last

groove hanger at the right hand support.

4. Calculation in Table 1 completed.

Part 2 :

1. Three load hangers with any load between 50 – 400 g was placed on it

and placed it at any position between the supports. The position and the

digital forces display reading recorded in Table 2.

2. The procedure repeated with three other location.

3. The calculation in Table 2 completed.

7.0 RESULT

Location of load

from left hand

support (m)

Digital Force

Display

Reading (N)

Moment at

cut section

(N)

Experimental

influence line

value (N)

Theoretical

Influence lines

value (Nm)

0.04 0.2 0.025 0.013 0.013

0.06 0.3 0.038 0.019 0.019

0.08 0.4 0.05 0.025 0.026

0.10 0.5 0.063 0.032 0.032

0.12 0.6 0.075 0.038 0.038

0.14 0.7 0.088 0.045 0.045

0.16 0.8 0.10 0.051 0.051

0.18 0.9 0.113 0.058 0.057

0.20 1.0 0.125 0.064 0.064

0.22 1.1 0.138 0.07 0.07

0.24 1.2 0.15 0.077 0.076

0.26 1.3 0.163 0.083 0.083

0.30 1.5 0.188 0.10 0.096

0.32 1.3 0.163 0.083 0.082

0.34 1.1 0.138 0.07 0.07

0.36 0.8 0.10 0.051 0.055

0.40 0.4 0.05 0.025 0.027

Table 1

Notes :

1. Moment at cut section = Digital force reading x 0.125

2. Experimental Influence line values = Moment (Nm)

Load (N)

3. Calculate the theoretical value using the equation 1 for load position 40 – 260

mm and equation 2 for load position 320mm and 400mm.

Part 2,

Location Position of hanger from left hand support (m)

Digital force reading (N)

Experimental Moment (Nm)

Theoretical moment (Nm)100

gram200 gram

300 gram

1 40 100 200 2.1 0.263 0.2612 80 160 260 2.9 0.363 0.3663 360 340 80 2.1 0.263 0.2604 260 400 60 1.6 0.200 0.190

Table 2

Notes :

1. Experimental moment = Digital force reading x 0.125

2. Theoretical moment is calculated using equation (3)

8.0 CALCULATION

EXAMPLE CALCULATION

PART 1

Moment at cut section= 0.2 x 0.125= 0.025 N

Experimental Influence line values = Moment (Nm)

Load (N)

= 0.025

1.962

= 0.013 m

Theoretical Influence lines value;

Equation 1 for load position 40 to 260 mm

Mx = (0.44 – 0.04) (0.3) – 1(0.3 – 0.04)

0.44

= 0.013 Nm

Equation 2 for load position 320mm to 400mm

When x = 0.32 m

Mx = (0.32) (0.14) – (0.32 – 0.3)

0.44

= 0.082 Nm

PART 2

F1 = 100g

= 100 x 9.81

1000

= 0.981N

F2 = 200g

= 200 x 9.81

1000

= 1.962N

F3 = 300g

= 300 x 9.81

1000

= 2.943N

0.981 1.962 N 2.943 N

x1

x2

x3

y1 y2 y3

Moment influence line for cut section

*For location 1,

Experimental moment at cut section (Nm) = Digital force reading x 0.125

= 2.1 x 0.125

= 0.263 Nm

Moment at cut :

∑Mx = 0

Mx = 1(0.3)- x (0.3) – 1 (0.3-x)

0.44

= 0.3 - 0.3x – 0.3 + x

0.44

Mx = 0.318x

When x = 0.3

Mx = 0.318x

= 0.318 (0.3)

= 0.095 Nm

Use interpolation to get y1,y2 and y3

y1, 0.095 = y1

0.3 0.04

0.3y = 0.0038

y1 = 0.013 m

y2, 0.095 = y2

0.3 0.1

y2 = 0.032 m

y3, 0.095 = y3

0.3 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

0.981 N 1.962 N 2.943 N

x1

x2

x3

y1 y2 y3

*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

2.943 N 1.962 N 0.981 N

x1

x2

x3

y3 y2 y1

*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

2.943 N 0.981 N 1.962 N

x1

x2

x3

y3 y1 y2

*For location 4,

Experimental moment (Nm) = 0.4125 Nm

When y1 = 0.082m , y2 = 0.027m , y3 = 0.019m

F1

a b

cut

RA = =

RB

x

L

Theoritical moment (Nm) = 0.190 Nm

9.0 DISCUSSIONS

PART 1

1. Derive equation 1 and 2.

ΣFx = 0

ΣFy

= RA + RB – 1 = 0

RA + RB = 1

RA( L ) – 1( L – x ) = 0

RAL = 1(L- x)

RA = 1( L – x ) L = 1 - x LRB = 1 – (1 – x) = x L L

Equation 1 ; 0 ≤ x ≤ a-Mx + RA(a) – 1(a - x) = 0

Mx = (1 – x/L)a – 1(a - x)

= (L – x)a – 1(a - x) L

Equation 2 ; a≤ x ≤ bMx – RB(b) + 1(x - a) = 0

Mx = RB (b) – 1(x - a)

= x/L (b) – 1(x -a)

= xb/L – 1(x -a)

2. On the graph, plot the theoretical and experimental value against distance from

left and support. Comment on the shape of graph. What does it tell u about how

moment varies at the cut section as a load moved on the beam?

G R AP H E X P E R IME NT AL VAL UE (Nm) VE R S US T H E OR E T IC AL VAL UE (Nm) VE R S US D IS T ANC E (m)

0

0.05

0.1

0.15

0.2

0.25

0.0

4

0.0

6

0.0

8

0.1

0.1

2

0.1

4

0.1

6

0.1

8

0.2

0.2

2

0.2

4

0.2

6

0.3

0.3

2

0.3

4

0.3

6

0.4

DIS T ANC E (m)

MO

ME

NT

(N

m)

Theoretic al V alue

E xperimental V alue

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 0.770.363 0.366 0.820.263 0.26 1.150.2 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.

10.0 CONCLUSION

As a conclusion, both objectives were achieved. Moment influence line could be

plot and the influence line can be use to 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.