70
3 Analysis of Strain
3.14 Mohr’s Circle for Strain
The Mohr’s circle for strain is drawn and that the construction technique does not
differ from that of Mohr’s circle for stress. In Mohr’s circle for strain, the normal
strains are plotted on the horizontal axis, positive to right. When the shear strain is
positive, the point representing the x-axis strains is plotted at a distance
γ
2
below the
ε-line; and the y-axis point a distance
γ
2
above the ε-line; and vice versa when the
shear strain is negative.
By analogy with stress, the principal strain directions are found from the equations
tan 2θ =
γ xy
ε x − ε y
(3.29)
Similarly, the magnitudes of the principal strains are
ε 1,2 =
ε x + ε y
2
±
ε x − ε y
2
2
+
γ xy
2
2
(3.30)
3.15 Equations of Compatibility for Strains
Expressions of compatibility have both mathematical and physical significance. From
a mathematical point of view, they state that the displacements u, v,w are single
valued and continuous functions. Physically, this means that the body must be pieced
together.
It is to be noted that, if the three displacement components u, v and w are given as
the continuous functions of the co-ordinates X, Y, Z then the six strain components
can be uniquely obtained from the strain–displacement equations given below.
ε x =
∂u
∂ x
, ε y =
∂v
∂ y
, ε z =
∂w
∂z
γ xy =
∂v
∂ x
+
∂u
∂ y
, γ yz =
∂w
∂ y
+
∂v
∂z
, γ zx =
∂u
∂z
+
∂w
∂ x
However, if a strain field in terms of six strain components is arbitrarily specified in a body, then the displacement components cannot be uniquely determined
easily. This is because; there are six equations for the three unknowns u, v and w.
Therefore, if the values of the displacement components are to be single valued and
continuous, then certain interrelationships between six strain components must exist.
These interrelationships between the strains represent compatibility equations.
Now consider a body with a triangle PQR before straining as shown in Fig. 3.5a.
The same triangle may take up one of the two possible positions as shown in Fig. 3.5b
3 Analysis of Strain
3.14 Mohr’s Circle for Strain
The Mohr’s circle for strain is drawn and that the construction technique does not
differ from that of Mohr’s circle for stress. In Mohr’s circle for strain, the normal
strains are plotted on the horizontal axis, positive to right. When the shear strain is
positive, the point representing the x-axis strains is plotted at a distance
γ
2
below the
ε-line; and the y-axis point a distance
γ
2
above the ε-line; and vice versa when the
shear strain is negative.
By analogy with stress, the principal strain directions are found from the equations
tan 2θ =
γ xy
ε x − ε y
(3.29)
Similarly, the magnitudes of the principal strains are
ε 1,2 =
ε x + ε y
2
±
ε x − ε y
2
2
+
γ xy
2
2
(3.30)
3.15 Equations of Compatibility for Strains
Expressions of compatibility have both mathematical and physical significance. From
a mathematical point of view, they state that the displacements u, v,w are single
valued and continuous functions. Physically, this means that the body must be pieced
together.
It is to be noted that, if the three displacement components u, v and w are given as
the continuous functions of the co-ordinates X, Y, Z then the six strain components
can be uniquely obtained from the strain–displacement equations given below.
ε x =
∂u
∂ x
, ε y =
∂v
∂ y
, ε z =
∂w
∂z
γ xy =
∂v
∂ x
+
∂u
∂ y
, γ yz =
∂w
∂ y
+
∂v
∂z
, γ zx =
∂u
∂z
+
∂w
∂ x
However, if a strain field in terms of six strain components is arbitrarily specified in a body, then the displacement components cannot be uniquely determined
easily. This is because; there are six equations for the three unknowns u, v and w.
Therefore, if the values of the displacement components are to be single valued and
continuous, then certain interrelationships between six strain components must exist.
These interrelationships between the strains represent compatibility equations.
Now consider a body with a triangle PQR before straining as shown in Fig. 3.5a.
The same triangle may take up one of the two possible positions as shown in Fig. 3.5b