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side figure shows the state of stresses when the element is rotated by an angle
φ
. In this case, the normal stresses are
σ
’
xx
and
σ
’
yy
, while the shear stresses
are
τ
’
xy
and
τ
’
yx
.
The relationship between the original state of stresses (
σ
xx
,
σ
yy
,
τ
xy
,
τ
yx
) and the
state of stress when the axes are rotated counterclockwise by f (
σ
’
xx
,
σ
’
yy
,
τ
’
xy
,
τ
’
yx
), can be represented graphically by the construct shown in the figure
below.
To construct Mohr’s circle we use a Cartesian coordinate system with the x-
axis corresponding to the normal stresses (
σ
), and the y-axis corresponding to
the shear stresses (
τ
). Locate the points A(
σ
xx
,
τ
xy
) and B
(σ
yy
,
τ
xy
), and draw
the segment AB. The point C where the segment AB crosses the
σ
n
axis will
be the center of the circle. Notice that the coordinates of point C are (½
⋅
(
σ
yy
+
σ
xy
), 0). When constructing the circle by hand, you can use a compass to
trace the circle since you know the location of the center C and of two points,
A and B.
Let the segment AC represent the x-axis in the original state of stress. If you
want to determine the state of stress for a set of axes x’-y’, rotated
counterclockwise by an angle
φ
with respect to the original set of axes x-y,
draw segment A’B’, centered at C and rotated clockwise by and angle
2φ