Related to SOM

1. JAHANGIRABAD INSTIUTE OF TECHNOLOGY
BARABANKI
Department of Mechanical Engineering
Spring
RAVI VISHWAKARMA
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 1

2. Introduction
Springs are elastic bodies (generally metal) that can be twisted, pulled,
or stretched by some force. They can return to their original shape
when the force is released.
At their most basic definition, springs are devices that store mechanical
potential energy. Springs are incredibly common, and can be found in
virtually every industry. While helical (coiled) springs are typically
what comes to mind, springs come in many shapes and materials, such
as a wooden bow (used with arrows) or serpentine springs (made of
wire and used in furniture).
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 2

3. Deflection by Strain Energy Method
The concepts of strain, strain-displacement relationships are very useful
in computing energy-related quantities such as work and strain energy.
These can then be used in the computation of deflections. In the special
case, when the structure is linear elastic and the deformations are
caused by external forces only,(the complementary energy U * is equal
to the strain energy U ) the displacement of structure in the direction of
force is expressed by
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 3
j
j
P
U
∂
∂
=∆

4. This equation is known as Castigliano's theorem. It must be
remembered that its use is limited to the calculation of
displacement in linear elastic structures caused by applied loads.
The use of this theorem is equivalent to the virtual work
transformation by the unit-load theorem.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 4

5. Close – coiled helical spring subjected to axial torque T
or axial couple
In this case the material of the spring is subjected to pure bending
which tends to reduce Radius R of the coils. In this case the bending
moment is constant through out the spring and is equal to the applied
axial Torque T. The stresses i.e. maximum bending stress may thus be
determined from the bending theory.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 5
3max
32
d
T
Π
=σ

6. Laminated Spring
A leaf spring is a simple form of spring commonly used for the
suspension in wheeled vehicles. Originally called a laminated or
carriage spring, and sometimes referred to as a semi-elliptical spring or
cart spring, it is one of the oldest forms of springing, dating back to
medieval times.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 6

7. Columns and Struts
Structural members which carry compressive loads may be divided into
two broad categories depending on their relative lengths and cross-
sectional dimensions.
Columns:
Short, thick members are generally termed columns and these usually
fail by crushing when the yield stress of the material in compression is
exceeded.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 7

8. Struts
Long, slender columns are generally termed as struts, they fail by
buckling some time before the yield stress in compression is reached.
The buckling occurs owing to one the following reasons.
(a) The strut may not be perfectly straight initially.
(b) The load may not be applied exactly along the axis of the Strut.
(c) One part of the material may yield in compression more readily than
others owing to some lack of uniformity in the material properties
through out the strut.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 8

9. Buckling and stability
Stability –
a state of a secular equilibrium of a structural member.
Buckling –
a law of a structure movement.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 9

10. Slenderness ratio (k)
It is the ratio of unsupported length of the column to the minimum
radius of gyration of the cross sectional ends of the column. It has no
unit whatsoever.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 10

11. Euler’s Theory
The following assumptions are made while deriving the Euler’s
Formula:
1.The column is initially straight and of uniform lateral dimension.
2.The compressive load is exactly axial and it passes through the
centroid of the column section.
3.The material of the column is perfectly homogeneous and isotropic.
4.Pin joints are frictionless and fixed ends are perfectly rigid.
5.The weight of the column itself is neglected.
6.The column fails by buckling alone.
7.Limit of Proportionality is not exceeded.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 11

12. Euler’s Formula
Euler’s formula is used for calculating the critical load for a
column or strut, and is as follows:
Where, P=Critical load,
E=Modulus of elasticity,
I=Least moment of inertia of section of the column, and
le =Equivalent length of the strut.
10/06/17 Ravi Vishwakarma ,Assistant Professor JIT 12
2
2
e
Euler
l
EI
P
Π
=