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!
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
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
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
Name _
Discussion Section _
EMA 201 Final Quiz (Quiz 4) - Semester I, 2015-16
This quiz is closed book and closed notes. Each problem is equally weighted. Please ask a
proctor if you do not understand the calculator policy for quizzes. All work must be
sh
ME 240 / EMA 202, Fall 2016 Name: SatJlL ran 3
Midterm #1, closed book/notes, 90 min.
Instructions
Show all of your work.
To maximize opportunities for partial credit, solve problems symbolically t0 the extent possible and
substitute numerical values at t
EMA 202, Fall 2014
Midterm #1, closed book/notes, 90 min.
Name: _
Instructions
Write your name on every sheet.
Show all of your work.
To maximize opportunities for partial credit, solve problems symbolically to the extent
possible and substitute numerical
EMA 202, Spring 2015
Midterm #1, closed book/notes, 90 min.
Name: _
Instructions
Write your name on every sheet.
Show all of your work.
To maximize opportunities for partial credit, solve problems symbolically to the extent
possible and substitute numeric
Practice Exam 1
(1) A bomber ying with a horizontal veioclty
of 300 mm at an attitude of 18,000 ft. is to
score a direct hit on a robotic train moving
with a constant velocity at 90 mm in the
same crecon and same verticat piane.
Determine:
a) The angle 9
S.I. Practice Problems: Particle Dynamics
This is not a Sample exam; these are just practice problems formatted in
sequential steps
1. The water-park ride consists of a sled
of weight W, which slides from rest
down the incline and then into the pool.
If t
S.I. Practice Problems: Particle Dynamics
This is not a Sample exam; these are just practice problems formatted
in sequential steps
2. The bob of mass M of a
pendulum is fired from rest at
position A. If the spring is
compressed to a distance and
released
S.I. Practice Problems: Particle Dynamics
This is not a Sample exam; these are just practice problems formatted
in sequential steps
1. The roller-coaster car has mass M, including its passenger, and starts
from the top of the hill A with a speed ! .
When
Problem 1: [100 pts.]
The 2 [ kg ] block slides over the rigid arm AB,
which is rotating in the vertical plane at
2[ rad /s ] counterclockwise and slowing at the
rate of 1.5 [rad /s 2] . A radial Force F gives the
block a constant inward speed of 1[m/s] w
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