BY,

GROUP -6

AIM:To determine the young’s modulus of elasticity of

1. Mild steel

2. Brass

3. Aluminium

By beam deflection method

APPARATUS Beam deflection apparatus

Dial gauge arrangements

Screw gauge

Weights

Hangers

THEORY Bending may be accompanied by direct stress,

transverse shear or torsional shear, however for convenience; bending stresses may be considered separately .

In order to separate the stresses it is assumed that the loads are applied in the following manner:

loads act in a plane of symmetry, no twisting occurs, deflections are parallel to the plane of the loads, and no longitudinal forces are induced by the loads or by the supports

ELEMENTS OF A BENT BEAM : the fibres form concentric arcs, the top fibres are compressed and bottom fibres are elongated

“Deflection” of a beam is the displacement of a point on the neutral surface of a beam from its original position under the action of applied loads . Before the proportional limit of the material, the deflection, Δ, can be calculated using the moment of inertia, modulus of elasticity along with other section properties that will depend on the given situation imposed on the beam. The position of the load, the type of load applied on the beam, and the length of beam are examples of section properties that depend on the situation.

The Euler-Bernoulli equation for the bending of slender ,isotropic ,homogenous beams of constant cross-section under an applied transverse load q(x) is

EId^4w(x)/dx^4 = q(x)

where E is the Young’s modulus, I is the area moment of inertia and w(x) is the deflection of the neutral axis of the beam

Deflection is a measure of overall stiffness of a given beam and can be seen to be a function of the stiffness of the material and proportions of the piece . Deflection measurements give the engineer a way to calculate the modulus of elasticity for a material in flexure. The stiffness of a given material is calculated using the following equation:

Stiffness = Wa (3L ^2-4a^2)/24EIwhere: P = load, (N) Δ = deflection, (mm) Stiffness (N/m)

FAILURE:A beam may fail in any of the following ways:

A beam may fail by yielding of extreme fibbers, in long span beams compression fivers act like those of a column and fail by buckling, in webbed members excessive shear stress may occur and stress concentrations may build up in parts of beam adjacent to bearing blocks

SCOPE AND APPLICATION:The scope and applicability of the bending tests are defined

as:a. Used as a direct means of evaluation behaviour under

bending loads, particularly for determining limits of structural stability of beams of various shapes and sizes.

b. Made to determine strength and stiffness in bending.c. Occasionally made to get stress distribution in a flexural

member.d. May be used to determine resilience and toughness of

materials in bending.e. Uses simple and inexpensive apparatus.f. Used as control test for brittle materials and not suitable

for determining ultimate strength of ductile materials.

PROCEDURE1) Draw the sketch of the apparatus and the loading

arrangement 2) Measure the breadth and depth of the beam at a few

locations and obtain the average breadth and depth3) Measure the span of the beam and mark the loading

points on the beam 4) Set the dial gauge at the span and note the initial reading5) Put 1/4kgf loads at the loading points and observe the

deflection the dial gauge6) Repeat the experiment with ½ kgf, ¾ kgf and 1 kgf loads

for mild steel , brass and aluminium beams7) 7. Plot a graph between W and stiffness and calculate the

value of E from the graph

OBERVATION TO FIND THE CROSS SECTIONAL DIMENSION

Material Width of beam ‘b’ Depth of beam ‘d’ Moment of inertia I =bd^3/12

A B C Mean A B C Mean

Mild steel 1 1 1 1 1 1 1 1 0.083

Brass 0.99 0.98 1 0.99 1 1 1 1 0.0825

Aluminium 0.96 0.95 0.98 0.96 0.95 0.94 0.95 0.94 0.0664

DETERMINATION OF E FOR MILD STEEL

Sl. No Material Load

distance

from

support ‘a’

Load ‘w’

kgf

Measured

deflection

‘mm’

E

N/mm^2

Mean E

1 MILD

STEEL

30 ¼ 0.3 165.29 1551.03

2 30 ½ 0.7 771.38

3 30 ¾ 1.12 1851.32

4 30 1 1.55 3416.13

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Lo

ad i

n (

kg

)

Deflection in (mm)

Load vs Deflection Graph for Mild Steel

DETERMNATION OF E FOR BRASS

Sl. No Material Load distancefrom support ’a’

Load ‘w’ kgf

Measureddeflection ‘mm’

EN/mm^2

<E

1 BRASS 31 1/4 0.83 464.320 3675.42

2 31 ½ 1.70 1902.05

3 31 ¾ 2.63 4413.87

4 31 1 3.54 7921.47

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Lo

ad i

n (

kg

)

Deflection in (mm)

Load vs Deflection Graph for BRASS

DETERMINATION OF E FOR ALUMINIUM

Sl. No Material Load distance from support ’a’

Load ‘w’ kgf Measured deflection

‘mm’

E N/mm

Mean E

1 ALUMINIUM 31 ¼ 1.4 630.35 4962.91

2 31 ¼ 3.79 3412.92

3 31 ¾ 4.25 5740.73

4 31 1 5.59 10067.66

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6

Lo

ad i

n (

kg

)

Deflection in (mm )

Load vs Deflection Graph for ALUMINIUM

PRECAUTIONS1) Handle the dial gauge with great care and take the

readings very carefully.

2) See that the loading is done exactly at the points chosen previously

3) The dial gauge should be kept exactly at the mid span