Chapter 2
Analysis of Stress
2.1 Introduction
A body under the action of external forces undergoes distortion, and the effect due to
this system of forces is transmitted throughout the body developing internal forces in
it. To examine these internal forces at a point O in Fig. 2.1a, inside the body, consider
a plane MN passing through the point O. If the plane is divided into a number of
small areas, as in Fig. 2.1b, and the forces acting on each of these measured, it will
be observed that these forces vary from one small area to the next. On the small area
A at point O, there will be acting a force of F as shown in Fig. 2.1b. From this, it
can be understood the concept of stress as the internal force per unit area. Assuming
the material is continuous, the term “stress” at any point across a small area A can
be defined by the limiting Equation as below.
Stress = lim
A→0
F
A
(2.1)
where F is the internal force on the area A surrounding the given point. Stress is
sometimes referred to as force intensity.
2.2 Notation of Stress
Here, a single suffix σ notation, like σ x , σ y , σ z , is used for the direct stresses and
double suffix τ notation is used for shear stresses like τ xy , τ xz , etc. τ xy means a
stress, produced by an internal force in the direction of y, acting on a surface, having
a normal in the direction of x.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_2
7
Analysis of Stress
2.1 Introduction
A body under the action of external forces undergoes distortion, and the effect due to
this system of forces is transmitted throughout the body developing internal forces in
it. To examine these internal forces at a point O in Fig. 2.1a, inside the body, consider
a plane MN passing through the point O. If the plane is divided into a number of
small areas, as in Fig. 2.1b, and the forces acting on each of these measured, it will
be observed that these forces vary from one small area to the next. On the small area
A at point O, there will be acting a force of F as shown in Fig. 2.1b. From this, it
can be understood the concept of stress as the internal force per unit area. Assuming
the material is continuous, the term “stress” at any point across a small area A can
be defined by the limiting Equation as below.
Stress = lim
A→0
F
A
(2.1)
where F is the internal force on the area A surrounding the given point. Stress is
sometimes referred to as force intensity.
2.2 Notation of Stress
Here, a single suffix σ notation, like σ x , σ y , σ z , is used for the direct stresses and
double suffix τ notation is used for shear stresses like τ xy , τ xz , etc. τ xy means a
stress, produced by an internal force in the direction of y, acting on a surface, having
a normal in the direction of x.
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_2
7