M6.1 Torsion concepts
Series 1 - internal torques: 4 correct out of 4 problems
Series 2 - shear stress: 3 correct out of 3 problems
Series 3 - twist angle: 3 correct out of 3 problems
In total, you have answered 10 of the 10 problems
correctly.
Pl

Example
General Combined Loading Problems
1. Find critical cross-sections:
Wind exerts a force
perpendicular to the
sign as shown.
cross-sections on which V, M, and/or T reach their largest values, and thus
cause large stresses (scan along the length of

EMA 303, Fall 2011
Midterm #1, closed book/notes, 90 min.
Name: _
Instructions
Write your name on EVERY sheet.
Show your work.
To maximize opportunities for partial credit, solve problems symbolically to the extent
possible and substitute numerical values

EMA 303, Spring 2011
Midterm #1, closed book/notes, 90 min.
Name: _
Discussion Section:_
Problem 1 (20 points)
A rectangular bar having a width w = 6.00 in. and thickness t = 1.50 in. is subjected to a tension
load P as shown in the figure below. The norm

Discussion Section
EMA 303 Final Quiz - Semester II, 2007-08
Each problem is equally weighted. You may use three pages of notes (8.5" x 11") and a
calculator. All work must be shown, including FBDs.
(CQ
1) [50 pnts] A rigid halfspace has a square hole w

EMA 303, Fall 2014
Exam #2
Name: _
Discussion: _
INSTRUCTIONS:
The instructors and the University of Wisconsin expect the highest standards of honesty and
integrity in the academic performance of its students. Any student caught cheating in this
course wi

EMA 303, F2112014 Name: K E 1
Exam #2
Question 1
Construct the shear and bending moment diagrams for the beam shown below and answer the
questions on page 4. On the diagrams below, note x-coordinates where the shear is zero. Also
note shear and moment val

Name 50 Law Discussion Section
EMA 303 Quiz #1 - Semester II, 2014-15
You may use a calculator. Each problem is equally weighted. All work must be shown,
including FBDs.
la) [60 puts] A winch consists of a horizontal bar BCD that is supported by a A

EMA 303, Summer 2014 Name: KEY
Exam #2
Question 2 I
In fusion reactors, structural materials must simultaneously be strong, able to withstand high heat.
and able to transport that heat away. Below is the cross section ofa beam that might be used in
such a

Name _
Discussion Section _
EMA 303 Quiz #1 - Semester II, 2014-15
You may use a calculator. Each problem is equally weighted. All work must be shown,
including FBDs.
grades
1
2
3
1a) [60 pnts] A winch consists of a horizontal bar BCD that is supported by

EMA 303, Fall 2014 Name: K E 1
Exam #1
Question 1
Determine the llet radius r of the bar
shown so that the maximum normal stress at
the llet and at the hole is the same for any P
load P that does not cause yielding.
Jc :- \2 mm .
Stress concentration fact

EMA 303, Fall 2011
Midterm #2, closed book/notes, 90 min.
Name: _
Question 4 (20 points)
Mohrs Circle is shown below for a point on a physical object that is subjected to plane stress.
(a) Determine the stresses sx, sy, and txy and show them on a stress e

EMA 303, Fall 2014
Exam #1
Name: _
INSTRUCTIONS:
The instructors and the University of Wisconsin expect the highest standards of honesty and
integrity in the academic performance of its students. Any student caught cheating in this
course will receive an

Buckling
A simple model problem
Straight members
subjected to
compressive axial
loading are susceptible
to buckling.
L
AB and BC are rigid bars with a
M t + P sin
=0
MB = 0 : cfw_
cfw_
torsional spring at B. The spring
2 for
=k 2
<1
produces a moment

Strain Gages
If interested, see
13.3 of text
for derivation
of eqns below.
Strain Gage Rosettes
Strain gages measure in-plane normal
strains only. Out-of-plane and shear strains
must be calculated.
Strain gages indicate
normal strain through
changes in el

Beam Deflection Coordinates
Radius of Curvature
The radius of curvature, , was introduced in
Chapter 8 (Bending) to analyze flexural strains:
x: longitudinal coordinate of the undeformed beam
(positive to the right)
v: displacement of the neutral surfac

VM Revisited
Changing M
Recall from VM
diagrams that
vertical loads
create internal
shear forces
(V) in beams
Where V is not
zero, the
moment M
changes across
the beam
Lets take an easier example and consider the
section from B to C:
Shear Stresses in B

Mohr's Circle
Recall:
procedure for drawing Mohrs circle
x + y x y
+
cos2 + xy sin2
2
2
y
= x
sin2 + xy cos2
2
x# =
x #y #
This is a parametric equation: given x, y and xy, a plot of x
and xy for 0 180 produces a circle. These equations can be
rearrang

Combined Axial and Torsional Loads
Combined Axial and Torsional Loads
Using these tools effectively requires simultaneous
pushing and turning, creating stresses from a
compressive axial load and from torsion.
Example
Example
A solid 1.50-in.-diameter sh

Impact Loading
Roadmap on methods to be discussed:
1) Complete energy transfer limited to uniform axial bars
under axial impact.
2) Equivalent static load applicable to more general geometries
and loading (e.g., axial and bending deformation).
Assumptions

Combinations of Stresses
Review
F
Recall from the 2nd week of class:
Recall from the 5th week of class:
For uniaxial loads, the maximum
normal stress is in the longitudinal
direction and the maximum shear
stress is on planes at 45 to the
longitudinal dire

Mohr's Circle for stresses in 3-D
Consider the a state of principal stress:
Procedure for drawing Mohrs circle:
1) Draw a Mohrs circle for the 1&2 surfaces.
2) Draw a Mohrs circle for the 2&3 surfaces.
3) Draw a Mohrs circle for the 3&1 surfaces.
Definiti

Pressure Vessels
Designing Complex Systems that Use
Pressure Vessels
absorber
Sandia test engineers will pressurize a
1/5-scale model of a pressure vessels
lower-head assembly (the inverted steel
dome that would hold the damaged
uranium core in a meltdown

The total work done by P for a displacement from 0 to final value
1 is
total strain
= 1
energy for a bar
U U = P d
Strain Energy
cfw_0
=0
The strain energy is the energy required to deform a structure.
Units for U
=0
SI:
N m = joule (J)
US:
in. lb or ft

Superposition for Statically
Indeterminate Beams
Example
Determine the reaction at the roller support.
vB
(vB)RB
vB
Replace redundant supports with forces and/or
moments
Determine the deflection that each load would
cause, including the new replacement

Method of Superposition
Using Tables
Principle of superposition: Total deformation = sum
of deformations from individual loadings. In fact:
Caution 1: Watch the boundary conditions!
Caution 2: Watch your variables!
Caution 3: Watch sign conventions!

EMA 303, Summer 2014
Exam #2
Name: _
Question 2
In fusion reactors, structural materials must simultaneously be strong, able to withstand high heat,
and able to transport that heat away. Below is the cross section of a beam that might be used in
such a de

EMA 303, Fall 2011
Midterm #2, closed book/notes, 90 min.
Name: _
Question 4 (20 points)
Mohrs Circle is shown below for a point on a physical object that is subjected to plane stress.
(a) Determine the stresses sx, sy, and txy and show them on a stress e