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.jpg


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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.jpg


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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.jpg


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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.jpg


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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.jpg


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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.jpg


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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.

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Chapter 5 Analysis of Structure