ES13 - Mechanics of Deformable Bodies
TRANSFORMATION OF
PLANE STRESS
General State of Stress
• At an arbitrary point in a rigid body, a general state of stress
can exist
• This state of stress consists of 6 components:
σx , σy , σz , τxy , τyz , τxz
• Equal and opposite stresses exist on the hidden faces
Plane Stress
• If the coordinate axes are rotated, a different state of stress
will exist on an element representing the same point.
• The stresses in the primed coordinate system can be related
to those in the unprimed coordinate system by developing
stress transformation relationships.
• If two of the faces of the cube are free from stress, these
relationships are simplified.
Plane Stress
• We assume that the stresses on the faces of the cube
perpendicular to the z axis are free from stress. This is known
as a state of plane stress:
• A state of plane stress occurs:
– When a thin plate is subjected to forces acting in the midplane of
the plate
– On the free surface of a structural member
Plane Stress
• The stress transformation relationships for plane stress are
developed based on rotations of the element about the z-axis
Plane Stress Transformation
• The relationships between the stresses in the primed and
unprimed coordinate systems are derived as below:
Equations of a Circle
• The transformation equations can be manipulated so that they
represent the equation of a circle:
– We define an average stress, σavg:
– The equation for σx’ is re-written as
– Squaring both sides and the one for τx’y’, we get
Equations of a Circle
– Adding the 2 equations, we get
• This expression represents a circle in the σ-τ plane. The circle
is centered at (σavg , 0) and has a radius R, where
Principal Stresses
• Principal stresses are the maximum and minimum stresses
that can exist on any plane passing through the point being
analyzed.
• The maximum normal stress occurs at point A, and the
minimum normal stress at point B.
Principal Stresses
• Principal angle θP, or the angle between the original element
and the plane on which the principal stresses occur, can be
found:
• Principal stresses can be expressed by
Maximum In-Plane Shear Stress
• Maximum in-plane shear stress occurs on a plane where the
normal stress is the average normal stress acting on the
element. This corresponds to the center of the circle.
• The magnitude of the maximum shearing stress is the radius
of the circle.
Maximum In-Plane Shear Stress
• The angle θS between the original element and the plane on
which the maximum in-plane shear stress occurs can be
found by
Maximum In-Plane Shear Stress
• The planes of maximum in-plane shear stress are 45°to the
principal planes.
Mohr's Circle for Plane Stress
• Mohr's circle is a graphical method of stress transformation.
• Important points in construction of Mohr’s circle:
1. An angle θ on an element is represented by 2θ on the
circle.
2. The planes perpendicular to the x and y axes are
perpendicular to one another. Therefore, on Mohr’s circle
they are diametrically opposite (180°apart).
3. For the state of stress below, the shear stress is positive
by convention.
Construction of Mohr's Circle
1. Set-up coordinate axes σ and τ and plot the 2 points (σx , -τxy)
and (σy , τxy). Connect the 2 points with a straight line.
We assume σx > σy > τxy and that τxy is positive
Construction of Mohr's Circle
2. The center of the circle corresponds to the average normal
stress, σavg. The radius of the circle can be determined from
simple geometry.
Construction of Mohr's Circle
3. Once the radius is known, the circle can be drawn. The
principal stress, maximum in-plane shear stress, and the
angle 2θP can be derived from the constructed circle.
Construction of Mohr's Circle
4. A counterclockwise rotation of the original element through
an angle θP exposes the principal planes. A clockwise
rotation through an angle θS produces the planes of
maximum in-plane shear stress.
Example
The state of stress at a point in a structural member is
determined to be as shown below. Using Mohr’s circle,
determine the principal stresses, the principal angle associated
with this state of stress, the maximum in-plane shear stress, and
the angle of maximum in-plane shear stress.
Absolute Maximum Shear Stress
• Consider the material to be subjected to the in-plane principal
stresses σ1 and σ2.
Absolute Maximum Shear Stress
• The material will have different values for maximum in-plane
shear stress in the y-z, x-z, and x-y planes.
• The largest maximum in-plane shear stress is the absolute
maximum shear stress.
Absolute Maximum Shear Stress
Hooke's Law for Plane Stress
• We make the following assumptions regarding the material
first:
1. The material is uniform throughout the body and has the
same properties in all directions (homogeneous and
isotropic material)
2. The material follows Hooke's law (linearly elastic
material)
Hooke's Law for Plane Stress
• The normal stresses σx and σy cause the material to
experience normal strains εx, εy, and εz.
• The shear stress τxy causes the material to experience shear
strain γxy, which is the decrease in angle between the x and y
faces.
Hooke's Law for Plane Stress
• The resultant normal strains εx, εy, and εz in the x, y, and z
directions can be expressed as:
1 1 v
x x v y y y v x z x y
E E E
• The shear strain γxy is expressed as:
xy
xy
G
Hooke's Law for Plane Stress
• Consequently, the normal stresses σx and σy can be
expressed in terms of the normal strains εx and εy as:
E E
x 2
x v y y 2
y v x
1 v 1 v
• The shear stress τxy can be expressed in terms of the shear
strain γxy as:
xy G xy
Example
Determine the strains corresponding to the state of stress shown,
knowing E = 200 GPa, G = 77 GPa, and v = 0.30.
