PROBLEM 6.1
Using the method of joints, determine the force in each member of the truss shown. State whether each member is in tension or compression.
SOLUTION
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Frames
They may look like trusses, but at least one member is subjected
to more than two forces.
a)
b)
c)
d)
Dismember the frame
Draw different free body diagrams for each member
Indicate internal for
Statically Indeterminate Problems
We must use information about deformation
in order to determine forces
Example 1:
Brass core
E = 105 Gpa
Aluminum shell E = 70 GPa
300 mm
The composite rod is subject
Poissons Ratio
y
P
x
A
y z 0
x
x
Ex
y z x
A
z
P
lateral strain
axial strain
x
Poisson' s ratio
For almost all engineering materials
0< <
Homogeneous - Isotropic materials E x = E y = E z
genera
Structures
Point
Rigid Body
Structures
Three different types of structures:
a) Trusses
b) Frames ( at least one multi force member)
c) Machines
Trusses-by joi
Jacob Y. Kazakia 2
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Trusses
Simple tru
Distributed Loads - beams
p(x)
We want to replace the distributed
load by a single force equivalent to it.
x=0
dx
x=a
x
The magnitude of the resultant:
x a
W p( x) dx
x 0
The location of the resultant
Machines: Example 1
4000 lb
12
40
12
4000 lb
Determine the forces at C and E
B
A
4000 lb
64 in
30 in
D
C
C
50 in
50 in
E
E y = 4000 lb
4000(57) = C(50)
C = 4560 lb
E x = 4560 lb
8000 lb F
Ex
50 in
M
Internal Forces
The forces required to balance a virtual piece of a rigid body, which is created by
an imaginary cut of the body.
F
F ( internal axial tension force)
cut
cut
F
F
cut
F
cut
F
Here the i
Plastic Deformations
Under loading
B
Y
For small strain
=E
( < Y )
C
A
Plastic Defor
For larger strain
= Y for all
D
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Example 1
The element AB is elastoplastic with
E = 200 Gpa ,
Strain under axial loading
denotes the deformation under the
applied load P.
L
P
P
lim
x 0
x
P
x+x
x+x+
Strain (axial)
x
if the area is constant , then the
stress and strain are constant
L
Jacob Y
Shear and bending-moment diagrams
Positive moment:
positive shear:
x
x
Negative moment:
negative shear:
x
Shear and be
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Example 1
9kN
2kN
9kN
B
A
C
Reactions: R A = R B =
(1/2 )
Torsion
y
Circular shaft under torque T
T
x
T
z
F
T
A
Shear stress
2c
0 <= <= c , F = A
the summation of all moments of
forces must equal to T
T = (over area) dA
In general :
circular shaft:
Torsiona
Bending Deformations
M
y
PURE BENDING
A
M
y
C
A
M
M
C
B
x
B
z
M
x
pressure
tension
Bending Defo
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Example 1: Pure bending
F
A
F
C
a
B
b
a
M
F
B
C
a
Bending Defo
By = F
At any point
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: November 20, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for s
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: October23, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for sol
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: October23, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for sol
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: November 13, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for s
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: December 2, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for so
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: September 30, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: September 11, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for
Fundamentals of
Engineering Mechanics
Lehigh University
MEMORANDUM
DATE: September 18, 2016
TO: Professor Vermaak
FROM: Student Zhijian Yang
CC: Bashar Attiya
SUBJECT: Summary and supporting work for
Machines - Second lecture Ex1
30o
0.04 m
Determine the magnitude of the forces exerted
on the rod and the force exerted by the pin at
A on portion AB of the pliers.
A
60o
Moments about A:
F(.04) 400(.
Trusses. Analysis by sections
An example:
Determine the forces in members CE and DE
A
16 kN
B
16 kN
3m
C
16 kN
E
D
16 kN
F
3m
G
H
3m
We draw a section line through the
sides involved
Method 1: We firs
Design of Transmission Shafts
: angular velocity
measured in radians per second
F
r
t
F
= t : angle of rotation ( in radians)
( t is the time )
s=r =rt
: arc length
Work done: F * s = F r t = T t
(
Friction Example on Wedges
s = 0.3
s = 0.6
15,000 lb
Q
P
10 0
concrete
between steel surfaces
between steal & concrete
Determine the force P needed
to raise the I beam and the
corresponding force Q on
Forces in Plane - Definition
Force: action of one body on another
Point of Application
Magnitude (must specify units)
Direction
Mech 003 Forc
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Forces in Plane - Vectors
Symbols for
Equilibrium in 3D: Example 1
y
.32m
C
Find the tension
in the wires:
AB, AC, AD
.45m
.36m
D
AB .45 i .6 j 0 k
B
.5m
z
~
x
.6m
~
~
AC 0 i .6 j .32 k
~
~
~
AD .5 i .6 j .36 k
A
W=1165 N
~ AB
~ AC
A ( 0,
Equilibrium of a particle
F 0
~
~
We must draw the free body diagram
( disconnect the particle from other bodies and
place forces instead of contacts)
Newtons Third Law of Motion: action reaction
For
Two force body equilibrium
For balance
We must have
2 & 3 force b
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Three force body equilibrium
RB
100 N
RC
For equilibrium
the two reactions
and the force of
100 N must
intersect