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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Axial Deformations 
 
Introduction 
Free body diagram - Revisited 
Normal, shear and bearing stress 
Stress on inclined planes under axial loading 
Strain 
Mechanical properties of materials 
True stress and true strain 
Poissons ratio 
Elasticity and Plasticity 
Creep and fatigue 
Deformation in axially loaded members 
Statically indeterminate problems 
Thermal effect 
Design considerations 
Strain energy 
Impact loading 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Axial Deformations 
 
Introduction 
Free body diagram - Revisited 
Normal, shear and bearing stress 
Stress on inclined planes under axial loading 
Strain 
Mechanical properties of materials 
True stress and true strain 
Poissons ratio 
Elasticity and Plasticity 
Creep and fatigue 
Deformation in axially loaded members 
Statically indeterminate problems 
Thermal effect 
Design considerations 
Strain energy 
Impact loading 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Axial Deformations 
 
Introduction 
Free body diagram - Revisited 
Normal, shear and bearing stress 
Stress on inclined planes under axial loading 
Strain 
Mechanical properties of materials 
True stress and true strain 
Poissons ratio 
Elasticity and Plasticity 
Creep and fatigue 
Deformation in axially loaded members 
Statically indeterminate problems 
Thermal effect 
Design considerations 
Strain energy 
Impact loading 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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 
be considered for free body diagram. 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Axial Deformations 
 
Introduction 
Free body diagram - Revisited 
Normal, shear and bearing stress 
Stress on inclined planes under axial loading 
Strain 
Mechanical properties of materials 
True stress and true strain 
Poissons ratio 
Elasticity and Plasticity 
Creep and fatigue 
Deformation in axially loaded members 
Statically indeterminate problems 
Thermal effect 
Design considerations 
Strain energy 
Impact loading 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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 
be considered for free body diagram. 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
Axial Deformations 
 
Introduction 
Free body diagram - Revisited 
Normal, shear and bearing stress 
Stress on inclined planes under axial loading 
Strain 
Mechanical properties of materials 
True stress and true strain 
Poissons ratio 
Elasticity and Plasticity 
Creep and fatigue 
Deformation in axially loaded members 
Statically indeterminate problems 
Thermal effect 
Design considerations 
Strain energy 
Impact loading 
 
 
 
 
 
 
 
 
 
 
 
 
 Strength of Materials Prof. M. S. Sivakumar  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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 
be considered for free body diagram. 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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  
 
 
 
 
 
 
 
 
 
 
 
 
 Indian Institute of Technology Madras 
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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