# CIVE 302 San Diego State Torsion of Member with Circular Cross Section Lab Report

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Engineering

CIVE 302

San Diego State University

CIVE

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CIVE 302

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Statement of Objective
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.

Procedure
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 crosssections, 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 T₀.

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

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

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

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 geometri...

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