structural analysis chapter 03.ppt

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Structural Analysis 7Structural Analysis 7thth Edition in SI UnitsEdition in SI UnitsRussell C. HibbelerRussell C. Hibbeler

Chapter 3:Chapter 3:

Analysis of Statically DeterminateAnalysis of Statically Determinate Trusses Trusses



Common Types of Trusses

• A truss is a structure comosed of slenderA truss is a structure comosed of slendermembers !oined to"ether at their end ointsmembers !oined to"ether at their end oints

• The !oint connections are usually formed by The !oint connections are usually formed by

boltin" or #eldin" the ends of the membersboltin" or #eldin" the ends of the membersto a common late called "ussetto a common late called "usset

• $lanar trusses lie in a sin"le lane % is often$lanar trusses lie in a sin"le lane % is often

used to suort roof or brid"esused to suort roof or brid"es

© 2009 Pearson Education South Asia Pte Ltd

Structural Analysis 7th Edition

Chapter 3: Analysis of Statically Deterinate !russes




• Roof TrussesRoof Trusses• They are often used as art of an industrial They are often used as art of an industrial

buildin" framebuildin" frame

• Roof load is transmitted toRoof load is transmitted tothe truss at the !oints bythe truss at the !oints by

means of a series of urlinsmeans of a series of urlins

• To &ee the frame ri"id % thereby caable of To &ee the frame ri"id % thereby caable of

resistin" hori'ontal #ind forces( &nee bracesresistin" hori'ontal #ind forces( &nee braces

are sometimes used at the suortin" columnare sometimes used at the suortin" column







• Roof TrussesRoof Trusses







• )rid"e Trusses)rid"e Trusses• The load on the dec& is *rst transmitted to The load on the dec& is *rst transmitted to

the strin"ers +, -oor beams +, !oints ofthe strin"ers +, -oor beams +, !oints of

suortin" side trusssuortin" side truss• The to % bottom cords of these side trusses The to % bottom cords of these side trusses

are connected by to % bottom lateralare connected by to % bottom lateral

bracin" resistin" lateral forcesbracin" resistin" lateral forces







• )rid"e Trusses)rid"e Trusses• Additional stability is roided by the ortal %Additional stability is roided by the ortal %

s#ay bracin"s#ay bracin"

• In the case of a lon" san truss( a roller isIn the case of a lon" san truss( a roller isroided at one end for thermal e/ansionroided at one end for thermal e/ansion







• Assumtions for Desi"nAssumtions for Desi"n• The members are !oined to"ether by smooth The members are !oined to"ether by smooth

insins

• All loadin"s are alied at the !ointsAll loadin"s are alied at the !oints• Due to the 0 assumtions( each trussDue to the 0 assumtions( each truss

member acts as an a/ial force membermember acts as an a/ial force member






Classication of Coplanar TrussesClassication of Coplanar Trusses

• Simle ( Comound or Comle/ TrussSimle ( Comound or Comle/ Truss• Simle TrussSimle Truss

• To reent collase( the frame#or& of a truss To reent collase( the frame#or& of a truss

must be ri"idmust be ri"id• The simlest frame#or& that is ri"id or stable The simlest frame#or& that is ri"id or stable

is a trian"leis a trian"le







• Simle TrussSimle Truss• A simle truss is the basic 1stable2 trian"leA simle truss is the basic 1stable2 trian"le

element is A)Celement is A)C

• The remainder of the !oints D( E % 3 are The remainder of the !oints D( E % 3 areestablished in alhabetical se4uenceestablished in alhabetical se4uence

• Simle trusses do not hae to consist entirelySimle trusses do not hae to consist entirely

of trian"lesof trian"les







• Comound TrussComound Truss• It is formed by connectin" 0 or more simleIt is formed by connectin" 0 or more simle

truss to"ethertruss to"ether

• 5ften( this tye of truss is used to suort5ften( this tye of truss is used to suortloads actin" oer a lar"er san as it isloads actin" oer a lar"er san as it is

cheaer to construct a li"hter comoundcheaer to construct a li"hter comound

truss than a heaier simle trusstruss than a heaier simle truss







• Comound TrussComound Truss• Tye 6 Tye 6

• The trusses may be connected by a common The trusses may be connected by a common

!oint % bar !oint % bar• Tye 0 Tye 0

• The trusses may be !oined by bars The trusses may be !oined by bars

• Tye Tye

• The trusses may be !oined #here bars of a The trusses may be !oined #here bars of a

lar"e simle truss( called the main truss( haelar"e simle truss( called the main truss( hae

been substituted by simle truss( calledbeen substituted by simle truss( called

secondary trussessecondary trusses







• Comound TrussComound Truss







• Comle/ TrussComle/ Truss• A comle/ truss is one that cannot beA comle/ truss is one that cannot be

classi*ed as bein" either simle or comoundclassi*ed as bein" either simle or comound







• DeterminacyDeterminacy• Total un&no#ns 8 forces in b no. of bars of Total un&no#ns 8 forces in b no. of bars of

the truss 9 total no. of e/ternal suortthe truss 9 total no. of e/ternal suort

reactionsreactions

• 3orce system at each !oint is colanar %3orce system at each !oint is colanar %

concurrentconcurrent

• Rotational or moment e4uilibrium isRotational or moment e4uilibrium is

automatically satis*edautomatically satis*ed






• DeterminacyDeterminacy• Therefore only Therefore only

• )y comarin" the total un&no#ns #ith the)y comarin" the total un&no#ns #ith the

total no. of aailable e4uilibrium e4n( #etotal no. of aailable e4uilibrium e4n( #e

hae:hae:





ateindeterminstatically 2edeterminatstatically 2

jr b jr b

>+=+

∑ =∑ = 0 and 0 y x F F



• StabilityStability• If b 9 r ; 0! 8, collaseIf b 9 r ; 0! 8, collase

• A truss can be unstable if it is staticallyA truss can be unstable if it is statically

determinate or statically indeterminatedeterminate or statically indeterminate• Stability #ill hae to be determined eitherStability #ill hae to be determined either

throu"h insection or by force analysisthrou"h insection or by force analysis







• StabilityStability• E/ternal StabilityE/ternal Stability

•A structure is e/ternally unstable if all of itsA structure is e/ternally unstable if all of its

reactions are concurrent or arallelreactions are concurrent or arallel• The trusses are e/ternally unstable since the The trusses are e/ternally unstable since the

suort reactions hae lines of action that aresuort reactions hae lines of action that are

either concurrent or aralleleither concurrent or arallel







• Internal StabilityInternal Stability• The internal stability can be chec&ed by The internal stability can be chec&ed by

careful insection of the arran"ement of itscareful insection of the arran"ement of its

membersmembers

• If it can be determined that each !oint is heldIf it can be determined that each !oint is held

*/ed so that it cannot moe in a 1ri"id body2*/ed so that it cannot moe in a 1ri"id body2

sense #rt the other !oints( then the truss #illsense #rt the other !oints( then the truss #ill

be stablebe stable

•A simle truss #ill al#ays be internally stableA simle truss #ill al#ays be internally stable

• If a truss is constructed so that it does notIf a truss is constructed so that it does not

hold its !oints in a */ed osition( it #ill behold its !oints in a */ed osition( it #ill be

unstable or hae a 1critical form2unstable or hae a 1critical form2







• Internal StabilityInternal Stability• To determine the internal stability of a To determine the internal stability of a

comound truss( it is necessary to identify thecomound truss( it is necessary to identify the

#ay in #hich the simle truss are connected#ay in #hich the simle truss are connected

to"etherto"ether• The truss sho#n is unstable since the inner The truss sho#n is unstable since the inner

simle truss A)C is connected to DE3 usin" simle truss A)C is connected to DE3 usin"

bars #hich are concurrent at oint 5bars #hich are concurrent at oint 5







• Internal StabilityInternal Stability• Thus an e/ternal load can be alied at A( ) or Thus an e/ternal load can be alied at A( ) or

C % cause the truss to rotate sli"htlyC % cause the truss to rotate sli"htly

•3or comle/ truss( it may not be ossible to3or comle/ truss( it may not be ossible to

tell by insection if it is stabletell by insection if it is stable

• The instability of any form of truss may also be The instability of any form of truss may also be

noticed by usin" a comuter to sole the 0!noticed by usin" a comuter to sole the 0!

simultaneous e4ns for the !oints of the trusssimultaneous e4ns for the !oints of the truss• If inconsistent results are obtained( the truss isIf inconsistent results are obtained( the truss is

unstable or hae a critical formunstable or hae a critical form







Classify each of the trusses as stable( unstable( staticallydeterminate or statically indeterminate. The trusses aresub!ected to arbitrary e/ternal loadin"s that are assumedto be &no#n % can act any#here on the trusses.

Example 3.1Example 3.1






3or <a=(• E/ternally stable

• Reactions are not concurrent or arallel

• b 8 6>( r 8 ( ! 8 66

• b 9 r 80! 8 00• Truss is statically determinate

• )y insection( the truss is internally stable

SolutionSolution






3or <b=(• E/ternally stable

• b 8 6?( r 8 @( ! 8 >

• b 9 r 8 6> ,0!

• Truss is statically indeterminate


SolutionSolution






3or <c=(• E/ternally stable

• b 8 >( r 8 ( ! 8

• b 9 r 8 60 8 0!

• Truss is statically determinate


SolutionSolution






3or <d=(• E/ternally stable

• b 8 60( r 8 ( ! 8 B

• b 9 r 8 6? ; 0!

• The truss is internally unstable

SolutionSolution






• Satisfyin" the e4uilibrium e4ns for the forcesSatisfyin" the e4uilibrium e4ns for the forcese/erted on the in at each !oint of the trusse/erted on the in at each !oint of the truss

• Alications of e4ns yields 0 al"ebraic e4nsAlications of e4ns yields 0 al"ebraic e4ns

that can be soled for the 0 un&no#nsthat can be soled for the 0 un&no#ns

The Method of JointsThe Method of Joints






• Al#ays assume the un&no#n member forcesAl#ays assume the un&no#n member forcesactin" on the !oints free body dia"ram to beactin" on the !oints free body dia"ram to be

in tensionin tension

• umerical solution of the e4uilibrium e4nsumerical solution of the e4uilibrium e4ns#ill yield ositie scalars for members in#ill yield ositie scalars for members in

tension % ne"atie for those in comressiontension % ne"atie for those in comression

• The correct sense of direction of an un&no#n The correct sense of direction of an un&no#n

member force can in many cases bemember force can in many cases be

determined by insectiondetermined by insection







• A 9e ans#er indicates that the sense isA 9e ans#er indicates that the sense iscorrect( #hereas a e ans#er indicates thatcorrect( #hereas a e ans#er indicates that

the sense sho#n on the free+body dia"ramthe sense sho#n on the free+body dia"ram

must be reersedmust be reersed







Determine the force in each member of the roof truss asDetermine the force in each member of the roof truss assho#n. State #hether the members are in tension orsho#n. State #hether the members are in tension or

comression. The reactions at the suorts are "ien ascomression. The reactions at the suorts are "ien as

sho#n.sho#n.

Example 3.2Example 3.2






5nly the forces in half the members hae to bedetermined as the truss is symmetric #rt both loadin" %"eometry(

SolutionSolution




)(93.6

030cos8 ;0

)(8030sin4 ;0

A,Joint

0

0

T kN F

F F

C kN F

F F

AB

AB x

AG

AG y

=

=−∑ =+

==−∑ =↑+



SolutionSolution




)(50.6

030sin38 ;0

)(60.2

030cos3 ;0

G,Joint

0

0

C kN F

F F

C kN F

F F

GF

GF x

GB

GB y

=

=−−∑ =+=

=−∑ =↑+



SolutionSolution




)(33.4

093.660cos60.260cos60.2 ;0

)(60.2

060sin60.260sin ;0

B,Joint

00

00

T kN F

F F

T kN F

F F

BC

BC x

BF

BF y

=

=−++∑ =+=

=−∑ =↑+



• Truss analysis usin" method of !oints is Truss analysis usin" method of !oints is"reatly simli*ed if one is able to *rst"reatly simli*ed if one is able to *rst

determine those members that suort nodetermine those members that suort no

loadin"loadin"

• These 'ero+force members may be These 'ero+force members may be

necessary for the stability of the truss durin"necessary for the stability of the truss durin"

construction % to roide suort if theconstruction % to roide suort if the

alied loadin" is chan"edalied loadin" is chan"ed• The 'ero+force members of a truss can The 'ero+force members of a truss can

"enerally be determined by insection of the"enerally be determined by insection of the

!oints % they occur in 0 cases. !oints % they occur in 0 cases.

ero!"orce Mem#ersero!"orce Mem#ers






• Case 6Case 6• The 0 members at !oint C are connected The 0 members at !oint C are connected

to"ether at a ri"ht an"le % there is noto"ether at a ri"ht an"le % there is no

e/ternal load on the !ointe/ternal load on the !oint

• The free+body dia"ram of !oint C indicates The free+body dia"ram of !oint C indicates

that the force in each member must be 'erothat the force in each member must be 'ero

in order to maintain e4uilibriumin order to maintain e4uilibrium







• Case 0Case 0• Fero+force members also occur at !ointsFero+force members also occur at !oints

hain" a "eometry as !oint Dhain" a "eometry as !oint D







• Case 0Case 0• o e/ternal load acts on the !oint( so a forceo e/ternal load acts on the !oint( so a force

summation in the y+direction #hich issummation in the y+direction #hich is

erendicular to the 0 collinear memberserendicular to the 0 collinear members

re4uires that 3re4uires that 3D3D3 8 G8 G

• Usin" this result( 3C is also a 'ero+forceUsin" this result( 3C is also a 'ero+force

member( as indicated by the force analysis ofmember( as indicated by the force analysis of

!oint 3 !oint 3







Usin" the method of !oints( indicate all the members ofthe truss that hae 'ero force.

Example 3.$Example 3.$

© 2009 Pearson Education South Asia Pte LtdStructural Analysis 7th Edition




e hae(

SolutionSolution



0

00 ;0

0

0sin ;0

,Joint

=

=+∑ =+

=

=∑ =↑+

DE

DE x

DC

DC y

F

F F

F

F F θ



SolutionSolution



0 ;0

G,Joint

0 ;0

!,Joint

0 ;0

",Joint

=∑ =↑+

=∑ =↑+

=∑ =+

GA y

HB y

EF x

F F

F F

F F



• If the forces in only a fe# members of aIf the forces in only a fe# members of atruss are to be found( the method oftruss are to be found( the method of

sections "enerally roide the most directsections "enerally roide the most direct

means of obtainin" these forcesmeans of obtainin" these forces

• This method consists of assin" an This method consists of assin" an

ima"inary section throu"h the truss( thusima"inary section throu"h the truss( thus

cuttin" it into 0 artscuttin" it into 0 arts

• $roided the entire truss is in e4uilibrium($roided the entire truss is in e4uilibrium(

each of the 0 arts must also be ineach of the 0 arts must also be in

e4uilibriume4uilibrium

The Method of SectionsThe Method of Sections





• The e4ns of e4uilibrium may be alied to The e4ns of e4uilibrium may be alied toeither one of these 0 arts to determine theeither one of these 0 arts to determine the

member forces at the 1cut section2member forces at the 1cut section2

• A decision must be made as to ho# to 1cut2A decision must be made as to ho# to 1cut2the trussthe truss

• In "eneral( the section should ass throu"hIn "eneral( the section should ass throu"h

not more than members in #hich thenot more than members in #hich the

forces are un&no#nforces are un&no#n






• If the force in C is to be determined(If the force in C is to be determined(section aa #ill be aroriatesection aa #ill be aroriate

• Also( the member forces actin" on one artAlso( the member forces actin" on one art

of the truss are e4ual but oositeof the truss are e4ual but oosite• The un&no#n member forces( 3 The un&no#n member forces( 3)C)C( 3( 3ICIC % 3% 3I3I3

can be obtained by alyin" the can be obtained by alyin" the

e4uilibrium e4nse4uilibrium e4ns






• hen alyin" the e4uilibrium e4ns(hen alyin" the e4uilibrium e4ns(consider #ays of #ritin" the e4ns to yield aconsider #ays of #ritin" the e4ns to yield a

direct solution for each of the un&no#n(direct solution for each of the un&no#n(

rather than to sole simultaneous e4nsrather than to sole simultaneous e4ns






Determine the force in members C3 and C of the rooftruss. State #hether the members are in tension orcomression. The reactions at the suorts hae beencalculated.

Example 3.%Example 3.%





The free+body dia"ram of member C3 can be obtained byconsiderin" the section aa(

SolutionSolution



)(#3.$

0)3$.2(50.$)4(30sin

0%e,asmomentscloc&'iseantiit*

.sim+licityor- +ointtoslideis

iility,transmisso /rinci+alA++lyin

0a++lyin yotained ecan1orsoltiondirectA -1

C kN F

F

M

F

M

CF

o

CF

E

CF

E

=

=+−

∑ =+

∑ =



The free+body dia"ram of member C can be obtained byconsiderin" the section bb(

SolutionSolution



)(#3.$

0)4(30sin#3.$)4()3$.2(50.$


.sim+licityor- +ointtoslideis

*a%e'e-, +ointtolidin.andn&no'ns

t*eeliminateorder toinA +ointaotsmmed e'illoments

T kN F

F

M

F

F F F

GC

o

GC

A

CF

CF CD HG

=

=−+−

∑ =+



Determine the force in member 3 and D of the truss.State #hether the members are in tension orcomression. The reactions at the suorts hae beencalculated.

Example 3.&Example 3.&





The distance E5 can be determined by roortionaltrian"les or reali'in" that member 3 dros ertically @.? 8 6.?m in m.

Hence( to dro @.?m from the distance from C to 5must be >m

SolutionSolution





The an"les 3ID and 3I3 ma&e #ith the hori'ontal aretan+6<@.?J= 8 ?.o

tan+6<@.?J>= 8 0.o

SolutionSolution



)(83.#

0)3(#)6(6.26sin


. +ointtoslideis

0

a++lyin ydirectlydetermined ecanG1inorce7*e

C kN F

F

M

F

M

GF

oGF

D

GF

D

=

=+−

∑ =+

∑ =



SolutionSolution


)(80.$

0)6(3.56sin)6(2)3(#

0%e,asmomentscloc&'iseantiit*. +ointtoslideis

0

a++lyin ydirectlydetermined ecanGinorce7*e

C kN F

F

M F

M

GD

o

GD

O

GD

O

=

=++−

∑ =+

∑ =

structural analysis chapter 03.ppt

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