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# Axial Deformation- strength of material by IIT Madras GATE Notes | EduRev

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## GATE : Axial Deformation- strength of material by IIT Madras GATE Notes | EduRev

``` Page 1

Strength of Materials Prof. M. S. Sivakumar

Axial Deformations

Introduction
Free body diagram - Revisited
Normal, shear and bearing stress
Strain
Mechanical properties of materials
True stress and true strain
Poissons ratio
Elasticity and Plasticity
Creep and fatigue
Statically indeterminate problems
Thermal effect
Design considerations
Strain energy

Page 2

Strength of Materials Prof. M. S. Sivakumar

Axial Deformations

Introduction
Free body diagram - Revisited
Normal, shear and bearing stress
Strain
Mechanical properties of materials
True stress and true strain
Poissons ratio
Elasticity and Plasticity
Creep and fatigue
Statically indeterminate problems
Thermal effect
Design considerations
Strain energy

Strength of Materials Prof. M. S. Sivakumar

1.1 Introduction
An important aspect of the analysis and design of structures relates to the deformations
caused by the loads applied to a structure. Clearly it is important to avoid deformations so
large that they may prevent the structure from fulfilling the purpose for which it is intended.
But the analysis of deformations may also help us in the determination of stresses. It is not
always possible to determine the forces in the members of a structure by applying only the
principle of statics. This is because statics is based on the assumption of undeformable,
rigid structures. By considering engineering structures as deformable and analyzing the
deformations in their various members, it will be possible to compute forces which are
statically indeterminate. Also the distribution of stresses in a given member is
indeterminate, even when the force in that member is known. To determine the actual
distribution of stresses within a member, it is necessary to analyze the deformations which
take place in that member. This chapter deals with the deformations of a structural
member such as a rod, bar or a plate under axial loading.

Top

Page 3

Strength of Materials Prof. M. S. Sivakumar

Axial Deformations

Introduction
Free body diagram - Revisited
Normal, shear and bearing stress
Strain
Mechanical properties of materials
True stress and true strain
Poissons ratio
Elasticity and Plasticity
Creep and fatigue
Statically indeterminate problems
Thermal effect
Design considerations
Strain energy

Strength of Materials Prof. M. S. Sivakumar

1.1 Introduction
An important aspect of the analysis and design of structures relates to the deformations
caused by the loads applied to a structure. Clearly it is important to avoid deformations so
large that they may prevent the structure from fulfilling the purpose for which it is intended.
But the analysis of deformations may also help us in the determination of stresses. It is not
always possible to determine the forces in the members of a structure by applying only the
principle of statics. This is because statics is based on the assumption of undeformable,
rigid structures. By considering engineering structures as deformable and analyzing the
deformations in their various members, it will be possible to compute forces which are
statically indeterminate. Also the distribution of stresses in a given member is
indeterminate, even when the force in that member is known. To determine the actual
distribution of stresses within a member, it is necessary to analyze the deformations which
take place in that member. This chapter deals with the deformations of a structural
member such as a rod, bar or a plate under axial loading.

Top

Strength of Materials Prof. M. S. Sivakumar

1.2 Free body diagram - Revisited
The first step towards solving an engineering problem is drawing the free body diagram of
the element/structure considered.
Removing an existing force or including a wrong force on the free body will badly affect the
equilibrium conditions, and hence, the analysis.
In view of this, some important points in drawing the free body diagram are discussed
below.

Figure 1.1
At the beginning, a clear decision is to be made by the analyst on the choice of the body to
Then that body is detached from all of its surrounding members including ground and only
their forces on the free body are represented.
The weight of the body and other external body forces like centrifugal, inertia, etc., should
also be included in the diagram and they are assumed to act at the centre of gravity of the
body.
When a structure involving many elements is considered for free body diagram, the forces
acting in between the elements should not be brought into the diagram.
The known forces acting on the body should be represented with proper magnitude and
direction.
If the direction of unknown forces like reactions can be decided, they should be indicated
clearly in the diagram.
Page 4

Strength of Materials Prof. M. S. Sivakumar

Axial Deformations

Introduction
Free body diagram - Revisited
Normal, shear and bearing stress
Strain
Mechanical properties of materials
True stress and true strain
Poissons ratio
Elasticity and Plasticity
Creep and fatigue
Statically indeterminate problems
Thermal effect
Design considerations
Strain energy

Strength of Materials Prof. M. S. Sivakumar

1.1 Introduction
An important aspect of the analysis and design of structures relates to the deformations
caused by the loads applied to a structure. Clearly it is important to avoid deformations so
large that they may prevent the structure from fulfilling the purpose for which it is intended.
But the analysis of deformations may also help us in the determination of stresses. It is not
always possible to determine the forces in the members of a structure by applying only the
principle of statics. This is because statics is based on the assumption of undeformable,
rigid structures. By considering engineering structures as deformable and analyzing the
deformations in their various members, it will be possible to compute forces which are
statically indeterminate. Also the distribution of stresses in a given member is
indeterminate, even when the force in that member is known. To determine the actual
distribution of stresses within a member, it is necessary to analyze the deformations which
take place in that member. This chapter deals with the deformations of a structural
member such as a rod, bar or a plate under axial loading.

Top

Strength of Materials Prof. M. S. Sivakumar

1.2 Free body diagram - Revisited
The first step towards solving an engineering problem is drawing the free body diagram of
the element/structure considered.
Removing an existing force or including a wrong force on the free body will badly affect the
equilibrium conditions, and hence, the analysis.
In view of this, some important points in drawing the free body diagram are discussed
below.

Figure 1.1
At the beginning, a clear decision is to be made by the analyst on the choice of the body to
Then that body is detached from all of its surrounding members including ground and only
their forces on the free body are represented.
The weight of the body and other external body forces like centrifugal, inertia, etc., should
also be included in the diagram and they are assumed to act at the centre of gravity of the
body.
When a structure involving many elements is considered for free body diagram, the forces
acting in between the elements should not be brought into the diagram.
The known forces acting on the body should be represented with proper magnitude and
direction.
If the direction of unknown forces like reactions can be decided, they should be indicated
clearly in the diagram.
Strength of Materials Prof. M. S. Sivakumar

After completing free body diagram, equilibrium equations from statics in terms of forces
and moments are applied and solved for the unknowns.

Top

Page 5

Strength of Materials Prof. M. S. Sivakumar

Axial Deformations

Introduction
Free body diagram - Revisited
Normal, shear and bearing stress
Strain
Mechanical properties of materials
True stress and true strain
Poissons ratio
Elasticity and Plasticity
Creep and fatigue
Statically indeterminate problems
Thermal effect
Design considerations
Strain energy

Strength of Materials Prof. M. S. Sivakumar

1.1 Introduction
An important aspect of the analysis and design of structures relates to the deformations
caused by the loads applied to a structure. Clearly it is important to avoid deformations so
large that they may prevent the structure from fulfilling the purpose for which it is intended.
But the analysis of deformations may also help us in the determination of stresses. It is not
always possible to determine the forces in the members of a structure by applying only the
principle of statics. This is because statics is based on the assumption of undeformable,
rigid structures. By considering engineering structures as deformable and analyzing the
deformations in their various members, it will be possible to compute forces which are
statically indeterminate. Also the distribution of stresses in a given member is
indeterminate, even when the force in that member is known. To determine the actual
distribution of stresses within a member, it is necessary to analyze the deformations which
take place in that member. This chapter deals with the deformations of a structural
member such as a rod, bar or a plate under axial loading.

Top

Strength of Materials Prof. M. S. Sivakumar

1.2 Free body diagram - Revisited
The first step towards solving an engineering problem is drawing the free body diagram of
the element/structure considered.
Removing an existing force or including a wrong force on the free body will badly affect the
equilibrium conditions, and hence, the analysis.
In view of this, some important points in drawing the free body diagram are discussed
below.

Figure 1.1
At the beginning, a clear decision is to be made by the analyst on the choice of the body to
Then that body is detached from all of its surrounding members including ground and only
their forces on the free body are represented.
The weight of the body and other external body forces like centrifugal, inertia, etc., should
also be included in the diagram and they are assumed to act at the centre of gravity of the
body.
When a structure involving many elements is considered for free body diagram, the forces
acting in between the elements should not be brought into the diagram.
The known forces acting on the body should be represented with proper magnitude and
direction.
If the direction of unknown forces like reactions can be decided, they should be indicated
clearly in the diagram.
Strength of Materials Prof. M. S. Sivakumar

After completing free body diagram, equilibrium equations from statics in terms of forces
and moments are applied and solved for the unknowns.

Top

Strength of Materials Prof. M. S. Sivakumar

1.3 Normal, shear and bearing stress
1.3.1 Normal Stress:

Figure 1.2
When a structural member is under load, predicting its ability to withstand that load is not
possible merely from the reaction force in the member.
It depends upon the internal force, cross sectional area of the element and its material
properties.
Thus, a quantity that gives the ratio of the internal force to the cross sectional area will
define the ability of the material in with standing the loads in a better way.
That quantity, i.e., the intensity of force distributed over the given area or simply the force
per unit area is called the stress.
P
A
s = 1.1

In SI units, force is expressed in newtons (N) and area in square meters. Consequently,
the stress has units of newtons per square meter (N/m
2
) or Pascals (Pa).
In figure 1.2, the stresses are acting normal to the section XX that is perpendicular to the
axis of the bar. These stresses are called normal stresses.
The stress defined in equation 1.1 is obtained by dividing the force by the cross sectional
area and hence it represents the average value of the stress over the entire cross section.
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