Solid Mechanics Lab Report (Torsion)
Lab 4. Torsion of Member with Circular Cross-Section: Principal Strains and Stresses
Thus far our only focus has been on axial deformations which cause elongation or compression
of a member along their axis of application. Other types of deformation exist, though, such as
shear deformations which translate or slide the face on which they act. To help visualize this
behavior, in this lab a solid steel shaft with circular cross-section is loaded under pure torsion.
Torsion is caused when a moment is applied in-plane with the cross-section of the member,
twisting it and causing it to deform in shear. A visualization of this is provided below in Figure
4-1.
Figure 4-1: Deformation of a body under shear conditions
Recall that the measurement devices used thus far only measure elongation or contraction
along their primary axis. Meaning that strain gages cannot directly measure these values.
Looking at more detail at Figure 4.1, it is clear that the shear strain, γ , is not a linear quantity
but an angle of distortion. We thus need additional techniques to be able to somehow
determine shear strain from linear-based measurement equipment.
For a bar in pure torsion, each section along the longitudinal axis rotates strictly in a plane; that
is, all points within the member follow a strictly a circular arc in the plane and do not translate
in either direction along the member axis. Note that this is not the case for non-circular cross-
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sections, as torsion develops warping deformations which displace points axially in addition to
rotationally. This behavior is shown below in Figure 4-2, where it can be seen that point B at the
outermost fiber of the cross-section travels along the circumference of the circle to B’ at an
applied torsion of To.
Figure 4-2. Member deformation from applied torsion To
Torsional loading at a section causes shear stresses that are zero at the center of the member
and increase linearly with radius r to a maximum at the surface of the cylinder at r = R. Shear
stress is found from the section torsion T, radius r, and polar moment of inertia J as
J
Tr (4-1)
For a circular section, the polar moment of inertia is equal to the sum of moments of inertia
about the x and y axes as
A B
To To
L
B’
3
244
444 Rrr IIJ yx
(4-2)
Shear strain is found by dividing the shear stress by the shear modulus G, where the shear
modulus is found for a given material from material testing as the slope of the shear stress
versus shear strain plot. Thus shear strain is found from
JG
Tr
G
(4-3)
Note the similarities in this equation to those seen in the previous labs. To calculate axial
stresses, we had a driving axial load P that was divided by the cross-sectional area – a geometric
property of the member. Then, to calculate strains, we divided this stress by the modulus of
elasticity – a material property of the section – and concluded that the deformation is a
function of the driving load P, the geometrical resistance of the member, and the material
resistance of the member. In the case of torsional strain, we see a similar pattern: a torque T
drives the deformation of the member, a geometric resistance forms via r and J, and a material
resistance forms via the shear modulus G.
Fixing a member on one end and applying a torsion To to the end free to rotate yields the
deformed shape given in Figures 4-2 and 4-3. At the left end of the member all points of the
cross-section are restrained from rotation and remain in the same place before and after
torsion is applied. At the right end of the member, however, torsional loading causes the cross-
section to rotate through an angle indicated by the movement of point B to B’. This point starts
at the top of the section but shifts along the perimeter of the cross-section. All points on the
surface at the right end of the member move through the same angle as point B, though,
generating a rotation of the full cross-section.
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For the example shown in Figure 4-3, the torsion is constant along the member length which
results in a linear variation of deformation from Points A to B. In other words, the amount of
rotation at half the member length is one-half of the maximum rotation at the right end of the
member. To demonstrate this, two lines are given in Figure 4-3 that were originally parallel to
the member geometry and have deformed from load application. A stress element shows that
the original rectangular shape has been deformed due to shear stresses; the length of the sides
remain unchanged, but the amount of deformation of the corners on the left face and the
corners on the right face differ in magnitude. Applied and reacting torsions are indicated with
double arrows: arrows pointing away from the member indicate positive torsion while arrows
facing toward the member indicate negative torsion, with the rotational direction following
that defined by the right hand rule. Positive torsion definition is given for any slice of the beam
in Figure 4-4 and the torsion diagram for the example provided in Figure 4-3 is given in Figure
4-5.
Figure 4-3. Side view of member under torsion with stress element indicated
A B
To To
L
B’
Stress Element
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Figure 4-4. Definition of positive torsion for a short length of the member (slice)
Figure 4-5. Torsion diagram over member length
When the torsion is constant along a member length as in Figure 4-4 the rotation at its right
end, called the angle of twist, is given as
JG
Tl (4-4)
TT +
T
X A B
To
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If the torsion varies along the member length, however, rotation needs to integrated over the
length of the member similar to how distributed loads must be integrated for beam
displacements. To do this the twist of the member, which is the rotation per unit length, is
calculated for a given cross-section
JG
T
(4-5)
The rotation at this cross-section is then calculated by integrating over the length
l
dx JG
T
0
(4-6)
An example of torsion that varies along the member length is given in Figure 4-6, with evenly
distributed torsion of To/L applied for the entire member length. There is no torsion in the
member at its right end and the torsion increases linearly to a maximum at the left of the
member. From statics, the torsion reaction at the fixed end is To and the torsion diagram can
be developed as shown in Figure 4-6. The torsion in the member at any section along its length
is given as
L
x Tx
L
T TT o
o o 1
(4-7)
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Figures 4-6. Distributed torsion along member length with torsion diagram
Torsion given as a function of x in Eq. (4-7) is substituted into Eq. (4-6) to find member rotation.
Thus
l ol
dx JG
L
x T
dx JG
T
00
1
(4-8)
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If the material is the same for the entire member and the cross-section is constant then J and G
are constant and can be moved out of the integral, as can the constant To, giving
JG
lT dx
L
x
JG
T o l
o B
2 1
0
(4-9)
This is half the rotation of a member that has constant torque along its length with the same
reacting torsion at its left end of To.
The stress element shown in Figure 4-3 distorts due to an upward shear stress on its left side
and, from statics, an equal and opposite shear stress on its right side (down) similar to how the
block in Figure 4-1(b) deforms. However, while this satisfies equilibrium of forces in the vertical
direction, equilibrium of the stress block as a whole is not satisfied: by themselves these shear
stresses form a couple and thus give a tendency for the block to rotate.
To correct for this, moments can be taken about one of the bottom corners of the stress block
to generate a balancing horizontal stress on the top face that imparts an equal but opposite
rotation to the vertical shear stresses. It is found that this horizontal shear stress is equal in
magnitude to the two vertical shear stresses, and equilibrium in the horizontal direction shows
that an equal but opposite shear stress forms on the bottom face. To help visualize this process,
an element is also shown at the top right of Figure 4-7 with these shear stresses indicated. The
stress element has dimensions of dx and dy, and when it has a square or rectangular shape it
vanishes to a single point.
With shear stresses defined at a stress element from torsion, it is possible to determine
maximum compressive and tensile stresses of the stress element that act on different faces of
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the element (at different angles). These are called principal stresses and can be determined
through the use of Mohr’s circle as shown in Figure 4-7. For a given set of shear and normal
stresses on two planes of a stress element, Mohr’s circle is constructed. The horizontal axis is
for normal stresses on a plane and the vertical axis is for shear stresses on a plane. X and Y
planes are defined by passing the x and y axes through the element and the faces that are
intersected are defined, as indicated at the bottom right of Figure 4-7. There are two X planes
and two Y planes, and either one can be used in the following discussion as they are completely
interchangeable.
Figure 4-7. Mohr’s circle with stress element and plane definitions
t
t
dx
dy
t
s
X (0, t )
Y (0, -t )
s 1s 2
f 1
f 2
x
y
x Planex Plane
y Plane
y Plane
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To construct Mohr’s circle an X coordinate and Y coordinate are plotted, representing the
normal and shear stresses on X and Y faces, respectively, of the stress element being
considered. On the X plane of our stress element there is a positive shear and no normal stress,
resulting in the X point given in Figure 4-7. As discussed before, the X plane on either left or
right sides of the element can be used to determine the shear and normal stresses to plot in
Mohr’s circle.
If the shear stress on the face being considered wants to turn the element in the clockwise
direction we call this positive shear. The stress element in Figure 4-7 shows that the upward
shear stress on the left of the element wants to rotate the element in the clockwise direction,
and so does the downward shear stress on the right of the element, thus they both give the
same positive shear stress and no normal stress. On the Y plane (top or bottom of the element)
the shear stress wants to turn the element in the counterclockwise direction and this is defined
as negative shear. There is no normal stress on the Y plane, yielding an X coordinate of zero
and a Y coordinate equal to the negative shear stress found from the torsion expression given
in Eq. (4-1).
Once two points are given on a circle then the entire circle is completely defined as shown in
Figure 4-7. In Mohr’s circle we are working in stress space where all values are stresses. Thus,
if a ruler is put on a graph of Mohr’s circle to scale a result, the values will be in stress units such
as psi or ksi. Only results that lie on the circle have meaning, providing normal and shear
stresses at all different angles within the stress element. One can imagine a plane cut through
the stress element at some arbitrary angle and asking what the shear and normal stresses are
on this exposed new plane. The construction of Mohr’s circles provides the answer to this
question in an elegant and efficient manner. It can be shown from solid mechanics that the
angle swept in Mohr’s circle is twice the angle within the element. Or, equivalently, the angle
within the stress element is one-half of the angle swept in Mohr’s circle. The direction of
rotation, clockwise or counterclockwise, is the same within the element and in Mohr’s circle.
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Maximum tensile and compressive stresses 1 and 2, called principal stresses, occur where the
shear stresses are zero, as indicated in Figure 4-7. The angle rotated to the principal stresses
from the X and Y points on Mohr’s circle is 90 degrees in the clockwise direction. Therefore the
angle rotated within the physical element from the X and Y planes to the principal planes is 45
degrees. Also from Mohr’s circle it is clear that for this case the magnitude of the principal
stresses is equal to the magnitude of the shear stresses found from torsion (radius of Mohr’s
circle).
A stress element showing the principal planes is given in Figure 4-8, occurring on a surface at 45
degrees from the element in pure shear that has normal stresses but no shear stresses. Figure
4-9 breaks this down mechanically, showing how tensile and compressive stresses develop
internally in an element to enforce equilibrium in the element. Note that rotating 45 degrees
clockwise from either X plane gives the same values on the principal tensile stress plane, and
rotating 45 degrees clockwise from either Y plane gives the same values on the principal
compressive plane. It is interesting to note that if the whole element is rotated 45 degrees to
the principal planes then only normal stresses are seen on the surfaces, with no shear stresses.
Thus, even though the element deforms in pure shear, pure tensile and compressive stresses
develop at an angle 45 degrees from this axis.
Figure 4-8. Element rotated to principal planes
45
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Figure 4-9. Stresses on principal planes from Mohr’s circle
The reverse is true when considering a wire in tension. From Mohr’s circle for a stress element
within the wire a normal stress is found in the vertical direction and no normal stress in the
transverse direction. A rotation of 90 degrees in Mohr’s circle shows a maximum shear stress
at this angle which is an element rotation of 45 degrees. Yielding of the wire in tension is
caused by shear of the crystals at 45 degrees resulting in Luder’s lines. This is clearly shown on
a polished coupon tensile test.
Experiments
A steel bar with ¾-inch diameter is tested in torsion. The steel material has modulus of
elasticity E and Poisson’s ratio of 30,000 ksi and 0.25, respectively. Two strain gages are
applied to the surface of the specimen one positioned in the principal tensile stress direction
and the other in the principal compressive stress direction. Thus they are both positioned 45
degrees from the longitudinal member axis and 90 degrees from reach other. The strain gages
utilize a half bridge.
t
t
dx
dy
x
y
x Planex Plane
y Plane
y Plane
t
t
45 x Planet
t
45 x Plane
t
t
y Plane
45
y Plane
t
t
45
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Apply a torsion of 1,000 lb-in. at intervals of 250 lb-in. and record the principal strains at each
load increment. Compare theoretical and measured principal strains. Principal strains are
related to principal stresses in Eq. (4-10) and Eq. (4-11). With measured principal strains, these
equations can be solved for the principal stresses.
EE 21
1
(4-10)
EE 12
2
(4-11)
Provide the following plots of theory versus measured (or calculated from measured):
Torsion diagram (torsion along member length)
Torsion versus principal tensile stress
Principal tensile stress versus principal compressive stress
Principal tensile strain versus principal compressive strain
Torsion versus shear stress
REQUIRED DISCUSSION:
How do the measured results in the above plots compare to theoretical results? Provide
detailed explanation for any variations.
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Also, if an axial load was applied to the rod would the orientation of the strain gages need to
change or would the orientation of 45 degrees still be valid? Justify your answer.
Definitions:
Shear Modulus of elasticity G (ksi). This is the initial, linear slope of the stress-strain diagram in
shear. In this range the material (and any structure that is made of this material) will be linear-
elastic, meaning that if the load is doubled the deformations will double (linear), and upon
removal of the load it will return to its original position (elastic) with no permanent
deformation.
Strain gage. An electrical strain gage is a resistor that changes its resistance as a function of
strain and temperature. By calibrating the strain gage (gage factor), this change in resistance
that is associated with strain in the gage is accurately measured as a voltage change and
converted to a strain value. The change in resistance of the strain gage due to temperature
effects is removed from the measured results by use of a Wheatstone bridge.
Polar moment of inertia. For circular sections the polar moment of inertia is equal to the sum
of the section moment of inertias about the x and y axes that run through the section centroid.