Strength of Materials 17ME34
Module 2
Compound Stresses
Objectives:
Derive the equations for principal stress and maximum in-plane shear stress and calculate their
magnitude and direction. Draw Mohr circle for plane stress system and interpret this circle.
Learning Structure
• 2.1 Introduction
• 2.2 Plane Stress Or 2–D Stress System Or Biaxial Stress System
• 2.3 Expressions For Normal And Tangential Components Of Stress On A Given Plane
• 2.4 Mohr’s Circle
• 2.5 Problems
• 2.6 Thick Cylinders
• 2.7 Thin Cylinders
• Outcomes
• Further Reading
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 45
, Strength of Materials 17ME34
2.1 Introduction
Structural members are subjected to various kinds of loads. This results in combination of
different stresses which changes from point to point. When an element (considered at any point)
in a body is subjected to a combination of normal stresses (tensile and/or compressive) and shear
stresses over its various planes, the stress system is known as compound stress system. In a
compound stress system, the magnitude of normal stress may be maximum o n some plane and
minimum on some plane, when compared with those acting on the element. Similarly, the
magnitude of shear stresses may also be maximum on two planes when compared with those
acting on the element. Hence, for the considered compound stress system it is important to find
the magnitudes of maximum and minimum normal stresses, maximum shear stresses and the
inclination of planes on which they act.
2.2 PLANE STRESS OR 2–D STRESS SYSTEM OR BIAXIAL STRESS SYSTEM
Generally a body is subjected to 3–D state of stress system with both normal and shear stresses
acting in all the three directions. However, for convenience, in most problems, variation of
stresses along a particular direction can be neglected and the remaining stresses are assumed to
act in a plane. Such a system is called 2–D stress system and the body is called plane stress
body.
In a general two dimensional stress system, a body consists of two normal stresses (fx and fy),
which are mutually perpendicular to each other, with a state of shear (q) as shown in figure.
Further, since planes AD and BC carry normal stress fx they are called planes of fx. These
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 46
, Strength of Materials 17ME34
planes are parallel to Y–axis. Similarly, planes AB and CD represent planes of fy, which are
parallel to X–axis.
2.2.1 PRINCIPAL STRESSES AND PRINCIPAL PLANES
For a given compound stress system, there exists a maximum normal stress and a minimum
normal stress which are called the Principal stresses. The planes on which these Principal
stresses act are called Principal planes. In a general 2-D stress system, there are two Principal
planes which are always mutually perpendicular to each other. Principal planes are free from
shear stresses. In other words Principal planes carry only normal stresses.
2.2.2 MAXIMUM SHEAR STRESSES ANDITS PLANES
For a given 2–D stress system, there will be two maximum shear stresses (of equal magnitude)
which act on two planes. These planes are called planes of maximum shear. These planes are
mutually perpendicular. Further, these planes may or may not carry normal stress. The planes
of maximum shear are always inclined at 450 with Principal planes.
2.3 EXPRESSIONS FOR NORMAL AND TANGENTIAL COMPONENTS OF STRESS ON A
GIVEN PLANE
Consider a rectangular element ABCD of unit thickness subjected to a general 2-D stress
system as shown in figure. Let f n and f s represent the normal and tangential components of
resultant stress ‘R' on any plane EF which is inclined at an angle ‘?' measured counter clockwise
with respect to the plane of f y or X–axis.
fy
fx fx fn
fs fx
fy fy
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 47
Module 2
Compound Stresses
Objectives:
Derive the equations for principal stress and maximum in-plane shear stress and calculate their
magnitude and direction. Draw Mohr circle for plane stress system and interpret this circle.
Learning Structure
• 2.1 Introduction
• 2.2 Plane Stress Or 2–D Stress System Or Biaxial Stress System
• 2.3 Expressions For Normal And Tangential Components Of Stress On A Given Plane
• 2.4 Mohr’s Circle
• 2.5 Problems
• 2.6 Thick Cylinders
• 2.7 Thin Cylinders
• Outcomes
• Further Reading
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 45
, Strength of Materials 17ME34
2.1 Introduction
Structural members are subjected to various kinds of loads. This results in combination of
different stresses which changes from point to point. When an element (considered at any point)
in a body is subjected to a combination of normal stresses (tensile and/or compressive) and shear
stresses over its various planes, the stress system is known as compound stress system. In a
compound stress system, the magnitude of normal stress may be maximum o n some plane and
minimum on some plane, when compared with those acting on the element. Similarly, the
magnitude of shear stresses may also be maximum on two planes when compared with those
acting on the element. Hence, for the considered compound stress system it is important to find
the magnitudes of maximum and minimum normal stresses, maximum shear stresses and the
inclination of planes on which they act.
2.2 PLANE STRESS OR 2–D STRESS SYSTEM OR BIAXIAL STRESS SYSTEM
Generally a body is subjected to 3–D state of stress system with both normal and shear stresses
acting in all the three directions. However, for convenience, in most problems, variation of
stresses along a particular direction can be neglected and the remaining stresses are assumed to
act in a plane. Such a system is called 2–D stress system and the body is called plane stress
body.
In a general two dimensional stress system, a body consists of two normal stresses (fx and fy),
which are mutually perpendicular to each other, with a state of shear (q) as shown in figure.
Further, since planes AD and BC carry normal stress fx they are called planes of fx. These
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 46
, Strength of Materials 17ME34
planes are parallel to Y–axis. Similarly, planes AB and CD represent planes of fy, which are
parallel to X–axis.
2.2.1 PRINCIPAL STRESSES AND PRINCIPAL PLANES
For a given compound stress system, there exists a maximum normal stress and a minimum
normal stress which are called the Principal stresses. The planes on which these Principal
stresses act are called Principal planes. In a general 2-D stress system, there are two Principal
planes which are always mutually perpendicular to each other. Principal planes are free from
shear stresses. In other words Principal planes carry only normal stresses.
2.2.2 MAXIMUM SHEAR STRESSES ANDITS PLANES
For a given 2–D stress system, there will be two maximum shear stresses (of equal magnitude)
which act on two planes. These planes are called planes of maximum shear. These planes are
mutually perpendicular. Further, these planes may or may not carry normal stress. The planes
of maximum shear are always inclined at 450 with Principal planes.
2.3 EXPRESSIONS FOR NORMAL AND TANGENTIAL COMPONENTS OF STRESS ON A
GIVEN PLANE
Consider a rectangular element ABCD of unit thickness subjected to a general 2-D stress
system as shown in figure. Let f n and f s represent the normal and tangential components of
resultant stress ‘R' on any plane EF which is inclined at an angle ‘?' measured counter clockwise
with respect to the plane of f y or X–axis.
fy
fx fx fn
fs fx
fy fy
DEPARTMENT OF MECHANICAL ENGINEERING, ATMECE, MYSURU 47
