Post on 29-May-2018
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Equilibrium and TrussesEquilibrium and Trusses
ENGR 221
February 17, 2003
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Lecture Goals
Lecture Goals
6-4
Equilibrium in Three Dimensions 7-1 Introduction to Trusses
7-2 Plane Trusses
7
-3 Space Trusses
7-4 Frames and Machines
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EquilibriumEquilibrium ProblemProblemDetermine the reactions
at A and the force in bar
CD due to the loading.
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EquilibriumEquilibrium ProblemProblemDraw the free-body
diagram of the main
body.
1 o6 in.tan 26.56512 in.
U ! !
RAx
RAy
TCD
U
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EquilibriumEquilibrium ProblemProblemLook at equilibrium
o
x x
o
x
o
y y
o
y
cos 26.565 125 lb 0
cos 26.565 125 lb
sin 26.565 40 lb 60 lb 80 lb 0
sin 26.565 20 lb
F R T
R T
F R T
R T
! !!
! !
!
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EquilibriumEquilibrium
ProblemProblemTake the moment about A
RAx
RAy
TCD
U
o
A CD
o
CD
CD
0 cos 26.565 6 in. 40 lb 4 in.
60 lb 8 in. 80 lb 12 in.
cos 26.565 6 in. 320 lb-in
59.628 lb
M T
T
T
! !
!
!
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EquilibriumEquilibrium
ProblemProblemTake the moment about A
RAx
RAy
TCD
U
o
x
o
y
59.628 lb cos 26.565 125 lb
71.667 lb
59.628 lb sin 26.565 20 lb
6.667 lb
R
R
!
! !
!
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Equilibrium in 3Equilibrium in 3--DimensionsDimensionsIn two dimensions, the equations are solved
using the summation of forces in the x, y and z
directions and the moment equilibrium includes
moment components in the x, y and z directions.
x y z0 0 0 F F F ! ! !
x y z0 0 0M M M! ! !
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TrussesTrusses --DefinitionDefinitionTrusses are structures
composed entirely of two
force members . They
consists generally of
triangular sub-element and
are constructed and
supported so as to prevent
any motion.
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FramesFrames --DefinitionDefinition
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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TrussTrussPlanar Trusses - lie in a
single plane and all
applied loads must lie in
the same plane.
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TrussTrussSpace Trusses - are structures that are not
contained in a single plane and/or are loaded out
of the plane of the structure.
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TrussTrussThere are four main assumptions made in the
analysis of truss
Truss members are connected together at theirends only.
Truss are connected together by frictionless
pins.
The truss structure is loaded only at the joints.
The weights of the members may be neglected.
1
2
3
4
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Simple Truss
Simple Truss
The basic building block of a
truss is a triangle. Large truss
are constructed by attachingseveral triangles together A
new triangle can be added
truss by adding two members
and a joint. A trussconstructed in this fashion is
known as a simple truss.
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Simple Truss
Simple Truss
It has been observed that the analysis of truss
can be done by counting the number member
and joints on the truss to determine the truss isdeterminate, unstable or indeterminate.
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Simple Truss
Simple Truss
A truss is analysis by using m=2*j-3, where m is
number of members, j represents the number of
joints and 3 represents the external supportreactions.
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Simple Truss
Simple Truss
If m< 2j-3, then the truss is unstable and will
collapse under load.
If m> 2j-3, then the truss has more unknowns
than know equations and is an indeterminate
structure.
If m= 2j-3, ensures that a simple plane 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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Simple Truss
Simple Truss-- IdentifyIdentify
Determine type of simple truss is it
determinate, indeterminate or unstable.
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Method of Joints
Method of Joints --TrussTruss
The truss is made up of single bars, which are
either in compression, tension or no-load. The
means of solving force inside
of the truss use equilibriumequations at a joint. This
method is known as the
method of joints.
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Method of Joints
Method of Joints --TrussTrussThe method of joints uses the summation of
forces at a joint to solve the force in the
members. It does not use the
moment equilibrium equationto solve the problem. In a two
dimensional set of equations,
In three dimensions,
x y0 0F F! !
z 0F !
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Method of Joints
Method of Joints ExampleExample
Using the method of
joints, determine the
force in each member ofthe truss.
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Method of Joints
Method of Joints ExampleExample
Draw the free body
diagram of the truss and
solve for the equations
x x
x
y y
y
0
0 lb
0 2000 lb 1000 lb
3000 lb
F C
C
F E C
E C
! !
!
! !
!
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Method ofMethod of
JointsJoints ExampleExample
Solve the moment about C
C
y
0 2000 lb 24 ft 1000 lb 12 ft 6 ft
10000 lb
C 3000 lb 10000 lb 7000 lb
M E
E
! !
!
! !
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Method ofMethod of
JointsJoints ExampleExample
Look at joint A
y AD
AD AD
x AD AB AB
AB AB
40 2000 lb
5
2500 lb 2500 lb C
3 30 2500 lb
5 5
1500 lb 1500 lb T
F F
F F
F F F F
F F
! !
! !
! ! !
! !
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Method ofMethod of
JointsJoints ExampleExample
Look at joint D
y
x
4 4 4 40 2500 lb
5 5 5 5
2500 lb 2500 lb T
3 30
5 5
3 32500 lb 2500 lb
5 5
3000 lb 3000 lb
F F F F
F F
F F F F
F
F F
! ! !
! !
! !
!
! !
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Method ofMethod of
JointsJoints ExampleExample
Look at joint B
y
x
4 4
0 1000 lb5 5
4 42500 lb 1000 lb
5 5
3750 lb 3750 lb
3 305 5
3 32500 lb 1500 lb 3750 lb
5 5
5250 lb 5250 lb T
F F F
F
F F
F F F F F
F
F F
! !
!
! !
! !
!
! !
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Method ofMethod of
JointsJoints ExampleExample
Look at joint E
y
x
4 4
0 10000 lb5 5
4 43750 lb 10000 lb
5 5
8750 lb 8750 lb
3 305 5
3 33750 lb 3000 lb
5 5
8750 lb 8750 lb
F F F
F
F F
F F F F
F
F F
! !
!
! !
! !
!
! !
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Method ofMethod of
JointsJoints ExampleExample
Look at joint C to check
the solution
y CE
x CE CB x
40 7000 lb
5
48750 lb 7000 lb 0 OK!
5
305
38750 lb 5250 lb 0 0
5
F F
F F F C
! !
! !
! !
! !
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Method of Joints
Method of Joints Class ProblemClass Problem
Determine the forces BC,
DF and GE. Using themethod of Joints.
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Method of
Sections
Method of
Sections --TrussTruss
The method of joints is most effective when
the forces in all the members of a truss are to
be determined. If however, the force is only
one or a few members are needed, then the
method of sections is more efficient.
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Method of
Sections
Method of
Sections --TrussTruss
If we were interested in the
force of member CE. We
can use a cutting line or
section to breakup the truss
and solve by taking the
moment about B.
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Method of
Sections
Method of
Sections ExampleExample
Determine the forces in members FH, GH and GI
of the roof truss.
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Method of
Sections
Method of
Sections ExampleExample
Draw a free body diagram and solve for the
reactions.
RAx
RAy
L
x x
x
y
y
0
0 k
0
20 k
F R
R
F
L R
! !
!
!
!
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Method ofMethod of
Sections
Sections ExampleExample
Solve for the
moment at A.
RAx
RAy
L
A
Ay
6 kN 5 m 6 kN 10 m 6 kN 15 m
1 kN 20 m 1 kN 25 m 30 m
7.5 kN
12.5 kN
M
L
L
R
!
!
!
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Method of
Sections
Method of
Sections ExampleExample
Solve for the member GI. Take a cut between the
third and fourth section and draw the free-bodydiagram.
HI HI
HI
1 o
8 m 10 m8 m
15 m 10 m 15 m
5.333 m
8 mtan 28.1
15 m
ll
l
E
! !
!
! !
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Method ofMethod of
Sections
Sections ExampleExample
The free-body diagram of
the cut on the right side.
H GI
GI GI
1 k 5 m 7.5 k 10 m 5.333 m
13.13 k 13.13 k T
M F
F F
!
! !
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Method of
Sections
Method of
Sections ExampleExample
Use the line of action of the forces and take the moment
about G it will remove the FGI and FGH and shift FFH to the
perpendicular of G.
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Method ofMethod of
Sections
Sections ExampleExample
Take the moment at G
G
o
FH
FH FH
1 kN 5 m 1 kN 10 m 7.5 kN 15 m
cos 28.1 8 m
13.82 kN 13.82 kN C
M
F
F F
!
! !
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Method of
Sections
Method of
Sections ExampleExample
Use the line of action of the forces and take the moment
about L it will remove the FGI and FFH and shift FGH to
point G.
1 o5 mtan 133.25.333 m
F ! !
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Method ofMethod of
Sections
Sections ExampleExample
Take the moment at L
o
L GH
GH GH
1 k 5 m 1 k 10 m cos 43.2 15 m
1.372 k 1.372 k
M F
F F
!
! !
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Method ofSectionsMethod ofSections ClassClass
ProblemProblem
Determine the forces in members CD and CE using method
of sections.
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H
omework (Due 2/24/03)H
omework (Due 2/24/03)Problems:
6-34, 6-37, 6-38, 6-40, 6-45, 6-63
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TrussTruss Bonus ProblemBonus ProblemDetermine whether the
members are unstable,
determinate orindeterminate.
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TrussTruss Bonus ProblemBonus ProblemDetermine the loads in
each of the members.
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TrussTruss Bonus ProblemBonus ProblemDetermine the loads in
each of the members.
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TrussTruss Bonus ProblemBonus ProblemDetermine the loads in
each of the members.