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8/13/2019 Chapter 5 Analysis of Structure
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VECTOR MECHANICS FOR ENGINEERS:
STATICSCHAPTER
5Analysis of structure
-Trusses
-Frames
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It is expected that students will be able to:
Analyze the equilibrium of structures made
of several parts, using the concept of theequilibrium of a particle or of a rigid body, inorder to determine the forces acting onvarious parts.
Topic Outcomes (TO)
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Contents:
1. Definition of a truss.
2. Examples of trusses.
3. Simple trusses.
4. Analysis of trusses by the method of joints.
5. Analysis of trusses by the method of sections.
6. Frames - Structure containing multiforce
members.
7. Analysis of a frame.
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1. Definition of a truss.
A truss is a structure consisting of stra ight members
connected at their extremities (the outermost or farthest
region or point) only.
a truss may thus be assumed to consist of pins and two-force
members.
two-force member pinpin
Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
http://upload.wikimedia.org/wikipedia/en/5/57/Bow-string-truss.jpghttp://upload.wikimedia.org/wikipedia/commons/c/c5/RRTrussBridgeSideView.jpg8/13/2019 Chapter 5 Analysis of Structure
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Truss Bridge
Truss roof of Cluny abbey,France
Pre-fabricated steel bow stringroof trusses built in 1942 for war
department properties in
Northern Australia.
http://upload.wikimedia.org/wikipedia/en/b/ba/Truss_Dachstuhl.jpghttp://upload.wikimedia.org/wikipedia/en/5/57/Bow-string-truss.jpghttp://upload.wikimedia.org/wikipedia/commons/c/c5/RRTrussBridgeSideView.jpghttp://upload.wikimedia.org/wikipedia/commons/3/3f/Below_Auckland_Harbour_Bridge_Hossen27.jpghttp://upload.wikimedia.org/wikipedia/commons/2/22/Auckland_Harbour_Bridge_Watchman.jpg8/13/2019 Chapter 5 Analysis of Structure
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Old Little Belt Bridge in Denmark Vierendeel bridge at GrammenBelgium
Auckland Harbour Bridge,
Auckland, New Zealand
http://www.wordiq.com/definition/Image:The_Little_Belt_Bridge_%281935%29.jpeghttp://upload.wikimedia.org/wikipedia/commons/5/5c/Grammene-vierendeelbridge_20030618.jpghttp://upload.wikimedia.org/wikipedia/commons/5/5c/Grammene-vierendeelbridge_20030618.jpghttp://upload.wikimedia.org/wikipedia/commons/2/22/Auckland_Harbour_Bridge_Watchman.jpghttp://upload.wikimedia.org/wikipedia/commons/3/3f/Below_Auckland_Harbour_Bridge_Hossen27.jpghttp://upload.wikimedia.org/wikipedia/commons/2/22/Auckland_Harbour_Bridge_Watchman.jpghttp://upload.wikimedia.org/wikipedia/commons/3/3f/Below_Auckland_Harbour_Bridge_Hossen27.jpghttp://upload.wikimedia.org/wikipedia/commons/5/5c/Grammene-vierendeelbridge_20030618.jpghttp://www.wordiq.com/definition/Image:The_Little_Belt_Bridge_%281935%29.jpeghttp://upload.wikimedia.org/wikipedia/commons/3/3f/Below_Auckland_Harbour_Bridge_Hossen27.jpghttp://upload.wikimedia.org/wikipedia/commons/2/22/Auckland_Harbour_Bridge_Watchman.jpghttp://upload.wikimedia.org/wikipedia/commons/2/22/HK_Bank_of_China_Tower_View.jpghttp://upload.wikimedia.org/wikipedia/commons/3/38/HK_HSBC_Main_Building_2008.jpg8/13/2019 Chapter 5 Analysis of Structure
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HK HSBC Main Building HK Bank of China Tower
http://upload.wikimedia.org/wikipedia/commons/2/22/HK_Bank_of_China_Tower_View.jpghttp://upload.wikimedia.org/wikipedia/commons/3/38/HK_HSBC_Main_Building_2008.jpghttp://upload.wikimedia.org/wikipedia/commons/f/f6/Elledningsstolpe2_lund.jpghttp://upload.wikimedia.org/wikipedia/commons/0/07/Large_Timber_Howe_Truss.jpg8/13/2019 Chapter 5 Analysis of Structure
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A large post and beam style
Howe truss cantilevered well
away from the main building.
Transmission tower
http://upload.wikimedia.org/wikipedia/commons/f/f6/Elledningsstolpe2_lund.jpghttp://upload.wikimedia.org/wikipedia/commons/0/07/Large_Timber_Howe_Truss.jpghttp://upload.wikimedia.org/wikipedia/commons/d/d2/Pylon-gorai.jpg8/13/2019 Chapter 5 Analysis of Structure
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Four tetrahedons form each the
two lower base structures of this
power pylon
Diagram of a planar space
frame such as used for a roof
http://upload.wikimedia.org/wikipedia/commons/a/ad/SpaceFrame02.pnghttp://upload.wikimedia.org/wikipedia/commons/d/d2/Pylon-gorai.jpg8/13/2019 Chapter 5 Analysis of Structure
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A truss is said to be r ig id if designed that it will notgreatly
deform or collapse under a small load.
A triangulartruss consisting of three members connected at
three joints is clearly a rigid truss.
A
B
C
3. Simple trusses
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A truss obtained by adding two new membersto the first one
and connecting them to a new joint (D) will also be rigid.
Trusses obtained by repeating this procedure are called simpletrusses. We may check that in a simple truss the total number
of members is m= 2n- 3, where nis the total number ofjoints.
A
B
C A
B
C
D3. Simple trusses
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A
B
C
The forces can be determined by the
method of jo in ts.
The free-body diagram of each pin is drawn,
showingthe forces exerted on the pin
by the members
The force exerted by a member on the pin is directed along that
member, themagnitude of the force is unknown.
In a simple truss to draw the free-body diagrams only 2 unknownforces are included in each diagram.
These forces can be obtained from the 2 equilibrium equations or
if only three forces are involved - the force triangle.
4. Analysis of trusses by the method of joints.
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A
B
C
Tension (T)
Compression (C)
C
T T
C
Force exerted by a member on a pin is
directed toward that pin, the member is in compress ion;
if it is directed away from the pin, the member is intension.
The analysis of a truss is sometimes expedited by first
recognizingjo in ts under spec ial load ing cond it ions.
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AB
C
D
E
G
P1 P2 P3
n
n
Writing SME= 0, we determine the magnitude of FBD, which
represents the force in memberBD.
A positivesign indicates that the member is in tension; anegativesign indicates that it is in compress ion.
AB
C E
P1 P2
FBD
FBE
FCE
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B
C E
P1 P2
FBD
FBE
FCE
The method of sections is useful in the analysis of compound
trusses(not from the basic triangular but by several simple trusse
If the component trusses have been properly connected(e.g., 1 pin) and if the resulting structure is properly supported
(e.g., 1 pin and 1 roller), the compound truss is
stat ically determ inate, r ig id, and completely constrained.
The following necessary - but not
sufficient - condition is then
satisfied: m+ r= 2n,
mnumber of members,
r number of unknowns
representing the reactions at the
supports, and nnumber ofjoints.
S f
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Frames and machines are
structures which contain mult i force
members, i.e., members acted
upon by three or more forces.
Frames are designed to support
loads and are usually stationary,fully constrained structures.
A B C D
M
Machines are designed to transmit or modify forces and
always contain moving parts.
6. Frames Structure containing multiforce members.
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To analyze a frame, we
first consider the entire
frame as a free body and
write three equilibriumequations.
If the frame
remains rigid when
detached from its supports,
the reactions involve only
three unknowns and may be determined from these equations.
On the other hand, if the frame ceases to be rigid when detachedfrom its supports, the reactions involve more than three
unknownsand cannot be completely determined from the
equilibrium equations of the frame.
A B C D
M
EyEx
Dy
Dx
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We draw the free-body diagram of each of
the multiforce members, noting that when two
multiforce members are connected to the
same two-force member, they are acted uponby that member with equal and oppos i te forces o f unknown
magni tude but know n d i rec t ion. When two multiforce
members are connected by a pin, they exert on each other
equal and opposi te forces of unknown direct ion, which
should be represented by two unknown components.
Then dismember the frame and identify the members as either
two-force members or multiforce members; pins are assumed to
form an integral part of one of the members they connect.
A
B
C D
M
EyEx
Dy
Dx
B
B
B
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The equilibrium equations obtained from the free-body
diagrams of the multiforce members can then be solved
for the various internal forces.
The equilibrium equations can also be used to complete
the determination of the reactions at the supports.
Actually, if the frame is stat ical ly determ inate and rigid,the free-body diagrams of the multiforce members could
provide as many equations as there are unknown forces.
However, as suggested above, it is advisable to firstconsider the free-body diagram of the entire frame to
minimize the number of equations that must be solved
simultaneously.