engr221 lecture 13
TRANSCRIPT
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Trusses and Machines
ENGR 221
February 26, 2003
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Lecture Goals
7-2 Plane Trusses
7-3 Space Trusses
7-4 Frames and Machines
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Example ProblemThe axis of the three-hinged arc ABC is a parabola with
vertex B. Knowing that P = 20 kN and Q =0 determine a) the
reactions at C, b) the components of the force exerted at on
segment AB.
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Example Problem
What is a hinge?
At a hinge the moment is
zero and can translate theforces across the point.
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Example Problem
Draw the free-body
diagram of the arch.
x Ax Cx
y Ay Cy
Ay Cy
A Cy
Cy
Ay
0
0 20 kN
20 kN
0 10 m 20 kN 3 m
6 kN
14 kN
F R R
F R R
R R
M R
R
R
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Example Problem
Draw the free-body
diagram of the arch at B.
x Ax Cx
B Ay Ax
Ax
Ax
Cx
0
0 5 m 20 kN 2 m 2.5 m
2.5 m 14 kN 5 m 20 kN 2 m
12 kN
12 kN
F R R
M R R
R
R
R
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ZeroForce members
When only two members form a non-collinear truss joint
and no external load or support reaction is applied to the
joint then the members must be zero-force members.
When three members form a truss joint for which two of the
members are collinear and the third is forms an angle with
the first two, then the non-collinear member is a zero-force
member provided no external force or support reaction isapplied to the joint, The collinear members carry equal
loads.
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ZeroForce members
Is there a zero force
member in the truss?
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ZeroForce members
Is there a zero force
member in the truss?
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Example Problem
The awning structure
is supported by ball
and socket joints at A
and C and by a short
link at B. The 750 N
force is parallel to the
z-axis. Determine the
support reactions andthe force in each
member of the space
truss.
EGAD!
3-D Trusses!
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3-D Trusses
The difference between 2-D and 3-D is that there
are three more equations we will be concerned
with and will require a bit more bookkeeping.
Instead of just
use the 6 sets of equations
x y z0 0 0F F M
x y z
x y z
0 0 0
0 0 0
F F F
M M M
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Simple Truss
A truss is analysis by using m=3*j-6, where m is
number of members, j represents the number of
joints and 6 represents the external supportreactions.
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Simple Truss
If m< 3j-6, then the truss is unstable and will
collapse under load.
If m> 3j-6, then the truss has more unknowns
than know equations and is an indeterminate
structure.
If m= 3j-6, ensures that a simple space truss is
rigid and solvable, it is neither sufficient nor
necessary to ensure that a non-simple plane truss
is rigid and solvable.
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Example Problem
The awning structure
is supported by ball
and socket joints at A
and C and by a shortlink at B. The 750 N
force is parallel to the
z-axis. Determine the
support reactions andthe force in each
member of the space
truss.
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Example Problem
Draw the free body diagram
A C B
x Ax Cx
y Ay Cy B
Ay Cy B
z Az Cz
Az Cz
0 750 N
0
0
0 750 N
750 N
F R R R j k
F R R
F R R R
R R RF R R
R R
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Equilibrium ProblemExample
Take the moment about A.
A AW 1 AB B AC C0M r W r R r R
1
1
AW
AW 1
4.0 m 5.0 m 0.0 m
4.0 m 5.0 m 0.0 m
0 0 750 N
3750 N-m 3000 N-m 0
r i j k
i j k
r W
i j k
AB
AB B
B
B B
4.0 m 0.0 m 4.0 m
4.0 m 0.0 m 4.0 m
0 0
4.0 m 0 4.0 m
r i j k
i j k
r R
R
R i j R k
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Equilibrium ProblemExample
Take the moment about A.
A AW 1 AB B AC C0M r W r R r R
AC
AC C
Cx Cy Cz
Cz Cy
8.0 m 0.0 m 0.0 m
8.0 m 0.0 m 0.0 m
0 8 m 8 m
r i j k
i j k
r R
R R R
i R j R k
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Equilibrium ProblemExample
Using the summations of moments about A and look at the
components.
B B
Cz Cz
B Cy Cy
component
3750 N-m 4.0 m 0 937.5 N
component
3000 N-m 8.0 m 0 375 N
component
4.0 m 8.0 m 0 468.75 N
i
R R
j
R R
k
R R R
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Equilibrium ProblemExample
Using the summations of forces about
A and look at the components. The
results are
Unfortunately, RAx and RCx could notbe computed.
Ay
Az
component
468.75 N
component
375 N
j
R
k
R
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Equilibrium ProblemExampleUsing the method of joints look at joint A
A AB AB AD AD
x Ax AB AD
y AD
AD
z AB
AB
Ax
0
0 0.7071 0.6247
0 468.75 N 0.7809
600.3 N
0 375 N 0.7071
530.33 N
750 N
F R T T
F R T T
F T
T
F T
T
R
So the bars are in compression
multiply by the unit vector to get the
final set of forces.
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Equilibrium ProblemExampleUsing the method of joints look at joint C and the
values will be same due to symmetry.
C CB CB CD AD
x Cx CB CD
y CD
CD
z CB
CB
Cx
0
0 0.7071 0.6247
0 468.75 N 0.7809
600.3 N
0 375 N 0.7071
530.33 N
750 N
F R T T
F R T T
F T
T
F T
T
R
So the bars are in compression multiply by the unit
vector to get the final set of forces.
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Equilibrium ProblemExampleUsing the method of joints look at joint D
z DB
DB
0 750 N 0.6247
1200.6 N
F T
T
Look at the y component of joint D
DB
2 2 2
DB
DB
0 m 5 m 4 m
0 m 5 m 4 m 6.403 m
0 m 5 m 4 m6.403 m 6.403 m 6.403 m
0.0 0.7809 0.6247
r i j k
r
i j k
i j k
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3-D Trusses - Class Problem
The space truss is supported by ball and socket joints at D and
by a short link at A and C. Determine the support reactions
and the force in each member of the space truss.
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Frames -Definition
Frames are structures that
always contain at least one
member acted on by forces
at three or more points.Frames are constructed and
supported so as to prevent
any motion. Frame like
structures that are not fully
constrained are called
machines or mechanisms.
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Frames
In order to determine the
internal forces holding the
various parts of a frame
together we mustdismember the frame and
draw a free-body diagram
for each of the component
parts.
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FramesExample
For the marine crane, which is
used in offshore drilling
operations, determine (a) the
force in link CD (b) the force inbrace AC and ( c) the force
exerted at on boom AB
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Frames
Example
Draw the free-body diagram
x Ax
Ax
3 2
y Ay
Ay
0
0 N
0 90 x10 kg 9.81 m/s
882900 N or 882.9 kN
F R
R
F D R
D R
RAxRAyD
W
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Frames
Example
Take the moment about A
A
Ay
0
15 m 882.9 kN 38 m
2236.68 kN
R 3119.58 kN
M
D
D
RAxRAyDW
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Frames
Example
Cut along CD
A
CD
CD CD
0
15 m 882.9 kN 38 m
2236.68 kN 2236.68 kN T
M
T
T T
TCD
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Frames
Example
Look at joint A
FACFAB
1 o
1 o
18 mtan 50.19
15 m
58 mtan 55.4135 m
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Homework (Due 3/5/03)
Problems:
7-58, 7-60, 7-66, 7-68