Possible questions for Exam 2
Nonsymmetrical Bending of Straight Beams
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Derive
sigma = My/I
and show clearly all assumptions, all limitations of
the formula, and show that all equations of
equilibrium are satisfied.
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Use Mohr's circle to determine principal
directions and magnitude of the moments of
inertia of the cross-section shown.
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Given the cross-section and applied bending moment
shown, determine the direction of the neutral axis.
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Given the beam and cross-section shown, determine
the maximum tensile stress and the direction and
magnitude of the maximum deflection.
Shear Center for Thin-wall Beam Cross Sections
Beginning with the flexure formula, derive
q = VQ/I
and list all assumptions and limitations.
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Given the three cross-sectional shapes
shown, estimate the location of the shear
center. Show clearly on which side of the
line "a-a" on each figure the shear center
is located.
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Given the cross-sectional shape shown,
sketch the direction of
shear flow and plot its magnitude along
each part of the cross-section. Clearly
show, using a free-body diagram, how you
determined the direction of shear flow.
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For the multiply connected cross-section
shown, determine the magnitude of the
shear flow in each of its walls.
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For the following cross-section with the
shear flows shown, calculate the location
of the shear center.
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For the following cross-section with the
shear flows shown, is the axial displacement
single valued? Explain.
-
Derive the equation that is necessary to
insure single valued displacements in the
axial direction of a beam in shear.
Curved Beams
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Derive the equation for the flexural stress in a curved
beam stating clearly all basic assumptions.
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For the beam shown, sketch the stress distribution
over its cross-section.
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For the curved, wide-flange beam shown, the upper flange is made of
a material with an E that is twice the E of the material
of the lower flange. If the web is neglected, determine
the location of the NA and the stress in the top flange
for a unit moment.
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Given the cross-sectional shape shown drawn to scale,
along with the location of the axes of beam curvature,
estimate the location of the N.A. and the maximum flexural
stress per unit of Moment.
Beams on Elastic Foundations
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Derive the governing equation for a beam on
an elastic foundation.
Define a Winkler foundation.
State clearly the necessary and sufficient
boundary conditions for this problem.
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Given the finite difference molecule shown
for a beam on an elastic foundation, set up
the governing matrix equation for the deflection
of the points shown.
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Energy
-
Derive the expression for strain energy in terms of the
bending moment in a beam.
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Use the following approximation for the deflection of the
beam shown and setup the governing equations for the
undetermined parameters using Ritz's method.
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State the principle of stationary potential energy and outline
its derivation.
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Plasticity
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Define the following yield criteria: J2, Octahedral
Shear Stress, Radius of Von Misses circle, Maximum
Shear Stress Theory (Tresca's theory), and Maximum
Energy of Distortion.
-
If during a one dimensional tension test
it was determined that the yield stress, sigma_y,
was 35000 psi, plot in stress space on the paper given
the Von Mises yield surface and the Tresca yield surface.
Clearly mark the units on the graph.