Chapter 3
Analysis of Strain
3.1 Deformations
In Chap. 2, the effect of externally applied forces on the body was discussed in terms
of stresses developed within the material concerned. Usually, these stresses may also
cause the body to transform into a new configuration. Hence, the change in the shape
of the body from initial and final configurations is termed as deformation.
3.2 Displacement and Strain
Deformations occur mainly due to relative displacements between the points in a
body. Suppose, P and Q are two points in a body at a distance L o apart before
deformation, as shown in Fig. 3.1. Now let this body be acted upon by a combination
of external forces causing the body to take up the new position in which the points
P and Q move to P
and Q
. Here, the distance PP
through which P has moved is
called the displacement of P, and Q Q
is the displacement of Q. If P
Q
is parallel
to PQ, then the deformation in this case has been translation only. But, if points P
and Q have moved to P
and Q
in such a way that P
Q
is not parallel to PQ, then
the deformation includes both translation and rotation.
Further, if the distance L 1 between P
and Q
is equal to L o then the line PQ has
been rigidly moved and no straining has taken place in the line element PQ and thus
also does not affect the stresses. However, if the distance L 1 is not equal to L o , then
there has been displacement of Q relative to P and a state of strain has been set up.
The position and displacement of any point in the body can be specified with
respect to any co-ordinate system such as the Cartesian set x, y, z or the cylindrical
set r, θ and z. Thus, in the two-dimensional case as shown in Fig. 3.1, the point
P has co-ordinates (x P , y P ). The components of the displacement of P to P
can
be represented by u P and v P along the co-ordinate directions x and y, respectively.
© 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_3
55
Analysis of Strain
3.1 Deformations
In Chap. 2, the effect of externally applied forces on the body was discussed in terms
of stresses developed within the material concerned. Usually, these stresses may also
cause the body to transform into a new configuration. Hence, the change in the shape
of the body from initial and final configurations is termed as deformation.
3.2 Displacement and Strain
Deformations occur mainly due to relative displacements between the points in a
body. Suppose, P and Q are two points in a body at a distance L o apart before
deformation, as shown in Fig. 3.1. Now let this body be acted upon by a combination
of external forces causing the body to take up the new position in which the points
P and Q move to P
and Q
. Here, the distance PP
through which P has moved is
called the displacement of P, and Q Q
is the displacement of Q. If P
Q
is parallel
to PQ, then the deformation in this case has been translation only. But, if points P
and Q have moved to P
and Q
in such a way that P
Q
is not parallel to PQ, then
the deformation includes both translation and rotation.
Further, if the distance L 1 between P
and Q
is equal to L o then the line PQ has
been rigidly moved and no straining has taken place in the line element PQ and thus
also does not affect the stresses. However, if the distance L 1 is not equal to L o , then
there has been displacement of Q relative to P and a state of strain has been set up.
The position and displacement of any point in the body can be specified with
respect to any co-ordinate system such as the Cartesian set x, y, z or the cylindrical
set r, θ and z. Thus, in the two-dimensional case as shown in Fig. 3.1, the point
P has co-ordinates (x P , y P ). The components of the displacement of P to P
can
be represented by u P and v P along the co-ordinate directions x and y, respectively.
© 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_3
55