A5_ELEC462
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University of British Columbia *
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Course
462
Subject
Mechanical Engineering
Date
Oct 30, 2023
Type
Pages
3
Uploaded by BaronMorning113
Assignment 5
October 20, 2023
Problem 1
(a)
Given the formula for the stiffness
k
of a cantilever beam subjected to lateral
forces:
k
=
3
EI
L
3
(1)
For 4 beams in parallel, their combined stiffness
k
total
is:
k
total
= 4
×
3
EI
L
3
=
12
EI
L
3
(2)
This is the stiffness of the entire structure for lateral forces, assuming the
beams are fixed at the bottom, free at the top, and perfectly rigidly connected
to the plate on top.
(b)
Given a single beam’s stiffness:
k
beam
=
3
EI
L
3
(3)
For four beams in parallel:
k
parallel4
= 4
×
k
beam
=
12
EI
L
3
(4)
For two beams in parallel:
k
parallel2
= 2
×
k
beam
=
6
EI
L
3
(5)
For the entire configuration in series:
1
k
total
=
1
k
parallel4
+
1
k
parallel2
(6)
k
total
=
4
EI
L
3
(7)
1
(c)
Given the stiffness of a single beam:
k
beam
=
3
EI
L
3
(8)
Stiffness of 4 beams in parallel (Flexure A):
k
A
=
12
EI
L
3
(9)
Stiffness of 4 beams in parallel (Flexure C):
k
parallel4
=
12
EI
L
3
C
(10)
For two sets of parallel-configured beams in series:
1
k
C
=
2
L
3
C
12
EI
(11)
k
C
=
6
EI
L
3
C
(12)
For the stiffness of Flexure C to be the same as Flexure A:
6
EI
L
3
C
=
12
EI
L
3
(13)
From which the modified length of beams in Flexure C is:
L
C
=
L
×
1
2
1
3
(14)
Problem 2
Given:
F
= 1
µ
N
L
= 100
µ
m
H
= 500
µ
m
w
= 10
µ
m
E
= 150 GPa
ν
= 0
.
3
1. Calculating the applied torque
T
:
T
=
F
×
L
= 1
µ
N
×
100
µ
m = 100
µ
N
·
m
(15)
2
2. Determining the shear modulus
G
using the relation between
E
and
ν
:
G
=
E
2(1 +
ν
)
=
150 GPa
2(1 + 0
.
3)
=
150 GPa
2
.
6
≈
57
.
69 GPa
(16)
3. Polar moment of inertia
J
for the rectangular cross-section:
J
=
w
×
H
3
3
=
10
×
500
3
µ
m
4
3
=
1250000000
µ
m
4
3
= 416666666
.
67
µ
m
4
(17)
4. Using the formula for the angle of twist
ϕ
:
ϕ
=
T
×
L
G
×
J
ϕ
=
0
.
1
µ
N
·
m
×
100
µ
m
57
.
69
×
10
9
Pa
×
416
,
666
,
666
.
67
µ
m
4
ϕ
≈
1
.
04
×
10
−
10
rad
3
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PART 1
1. Use the graphical method to construct the shear-force and bending-moment diagrams for the beam shown. Let a=11 ft, b=5 ft and w = 11.5 kips/ft.
IMAGE*
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120LEI
Wo
15 x 107 N
L = 2m
%3D
max
m
E = 7 × 1011 N
m2
I = 6 x 10-4m
Use the Newton-Raphson Method to determine the x location along the beam where the
deflection is y(x) = 0.14 m. Use a starting guess of xo
1.2 m. Be sure to show your
%3D
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?
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X
Figure Q1
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O
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(a)
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