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ME 211 R
EVIEW
© M. D. Thouless (August 2001)
Use equilibrium to calculate
reaction forces, moments, torques
Draw freebody diagram with
cut at section of interest
using correct
sign convention
for forces, torque and
moments
Use equilibrium to calculate
normal force, shear forces, torque
and moments at section
Calculate
all
(3) normal stresses
and (3) shear stresses from the forces,
moments and torque and internal pressure
Use stress transformation
(Mohr’s circle)
to find
principal stresses
Use in design calculations
for strength (ME 382)
Given applied
displacement
or twist
Solve inverse problem to find
load / moment or torque for
given displacement / twist
Calculate displacements
or /and twists
Use in design calculations
for stiffness
Given strains
from strain gauges
No
Remove redundancy
Yes
Determine if structure is statically determinate
Any structure subjected to arbitrary
loading of forces, torques, moments,
internal pressure (and temperature)
Draw freebody diagram
with supports removed and replaced by
appropriate
reaction forces, moments and
torques
Calculate stresses
from
3D Hooke’s law
End of ME211
Sketch diagram of problem.
Include:
(i) Appropriate
righthanded coordinate system
(ii)
Direction and magnitude of loads
(iii) Location and nature of supports
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1. C
O

ORDINATE
S
YSTEMS
A righthanded coordinate system and a
consistent sign convention
for forces,
moments, torques, displacements, twists and stresses must be established.
One of the
most useful system for ME211 and ME382 is Cartesian coordinates.
1.1 Cartesian coordinates
The directions in which the axes point can be chosen to be whatever is most
convenient.
However, as shown below, it is very important to get the relative
orientations of the three directions correct in a “righthanded sense.”
All of the axes
shown below are identical righthanded Cartesian systems in that they follow the same
pattern on
x
→
y
→
z
.
(i)
z
x
y
(ii)
(iii)
Figure 1.1
Righthanded Cartesian coordinate systems.
12
1.2 Converting between axes in different directions
Note that if an equation of interest is given in terms of a set of righthanded axes
in one orientation, it is very easy to convert the equation for use with another set of right
handed axes.
A simple set of substitutions is made:
(i)
(ii)
(i)
(iii)
(i)
righthanded system (
a
,
b
,
c
)
x
→
y
or
x
→
z
or
x
→
a
y
→
zy
→
xy
→
b
z
→
xz
→
yz
→
c
Notice that the sequences always follows the righthanded convention of the form
x
→
y
→
z
.
For example, the equation for the bending stress in a beam lying along the
y
axis is given
by
σ
yy
xx
xx
zz
zz
Mz
I
Mx
I
=−
+
.
The equivalent equation for a beam lying along the 
x
axis is
xx
zz
zz
yy
yy
My
I
I
+
.
The equivalent equation for a beam lying along the 
z
axis is
zz
yy
yy
xx
xx
I
I
+
.
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This note was uploaded on 09/08/2011 for the course MECHENG 382 taught by Professor Thouless during the Fall '08 term at University of Michigan.
 Fall '08
 Thouless

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