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
FBD Truss:
M B = 0: ( 6.25 m ) C y - ( 4 m )( 315 N ) = 0
Fy = 0: By - 315 N + C y = 0
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 solution to MECH003 HW 10.32
The following problem demon
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 forces
Write the equations of balance and solve them
Frame
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 subjected to a force
and it shortens by = 0.40 mm. What is
the
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
general strain
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3D Normal Stress
y
x
y
z
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
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Trusses
Simple trusses are composed by triangles
build on other triangles
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 is determined by ensuring that both
loads have the sam
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
Machines I
Ey
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25 + 32 = 57
1
Machines
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 internal force
is the same for all cuts.
x
M
T
Internal
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 , Y = 300 Mpa .
The bar PBC is rigid and pinned at P,
in
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
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Strain under axial loading cont.
L L0
L0
Shear and bending-moment diagrams
Positive moment:
positive shear:
x
x
Negative moment:
negative shear:
x
Shear and be
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x
1
Example 1
9kN
2kN
9kN
B
A
C
Reactions: R A = R B =
(1/2 ) ( 9 - 2 + 9 ) = 8 kN
V
Shear at a point between C and D
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:
Torsional Def
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Deformation under Torsion
Ev
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 solution to MECH003 HW 8.41
The following problem demonstr
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 solution to MECH003 HW 8.41
The following problem demonstr
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 solution to MECH003 HW 9.70
The following problem demons
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 solution to MECH003 HW 11.17
The following problem demons
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 solution to MECH003 HW 4.54
The following problem demon
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 solution to MECH003 HW 3.35
The following problem demon
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 solution to MECH003 HW 3.72
The following problem demon
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(.3) = 0
F = 3000 N
Hence
A x = F sin60 +400 =
2998N
&
A
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 between the two
forces, like point C, we have
pure bend
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 first calculate the
external reactions and then
analyze the
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
( note that the torque T = F r )
Power : work done per un
Examples in bending Ex1
0.5
0.5
0.5
0.5 0.5
1.0
M
0.5 0.5
All dimensions in inches
The beam shown is made of a material with
- 18 ksi <= all <= +5 ksi
Determine the maximum moment M that can be safely applied
Bending Exa
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a) locate the
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 the
pad ( welded on the beam).
N 1 = 15,000 lb
F1
P
F2
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 vectors : bold F
or
Addition : The parallelogram rule
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, -.6, 0 )
~ AD
Mech 003 Equ
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~
~
~
~
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 every action, there is an equal and opposite reaction.
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 at a
single point
RB
RC
100 N
2 & 3 force b
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