structure analysis ii. s tructural a nalysis ii ce 1352 by r.revathi, m. tech., structural lecturer...
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Structure Analysis II
Structure Analysis II
STRUCTURAL ANALYSIS II CE 1352
By
R.REVATHI, M. Tech., STRUCTURAL Lecturer
Department of Civil Engineering PITS
VI Semester 2011-2012
OBJECTIVES• This Course aims at teaching the students the
concept of analyzing indeterminate structure using classical and up to date methods.
• It provides students with an understanding of the methods of analyzing indeterminate structure:– The force method of analysis– The Displacement method
• Slope deflection • Moment distribution• Stiffness Method (An Introduction to The Finite
Element Method)
UNIT I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES
Equilibrium and compatibility – Determinate Vs Indeterminate structures–
Indeterminacy – Primary structure – Compatibility conditions – Analysis of
indeterminate pin-jointed plane frames, continuous beams, rigid jointed plane frames
(with redundancy restricted to two.)
UNIT II MATRIX STIFFNESS METHOD
Element and global stiffness matrices – Analysis of continuous beams – Co-ordinate
transformations – Rotation matrix – Transformations of stiffness matrices, load vectors
and displacements vectors – Analysis of pin-jointed plane frames and rigid frames.
UNIT III FINITE ELEMENT METHOD
Introduction – Discretisation of a structure – Displacement functions – Truss element–
Beam element – Plane stress and plane strain Triangular elements.
UNIT IV PLASTIC ANALYSIS OF STRUCTURES 9
Statically indeterminate axial problems – Beams in pure bending – Plastic moment of
resistance – Plastic modulus – Shape factor – Load factor – Plastic hinge and
mechanism – Plastic analysis of indeterminate beams and frames – Upper and lower
bound theorems
UNIT V SPACE AND CABLE STRUCTURES 9
Analysis of Space trusses using method of tension coefficients – Beams curved in
plan Suspension cables – Cables with two and three hinged stiffening girders.
L: 45 T: 15 Total: 60
TEXT BOOKS
1 .Coates R.C., Coutie M.G. and Kong F.K., “Structural Analysis”, ELBS and
Nelson, 1990.
2 .Negi, L.S. and Jangid, R.S., “Structural Analysis”, Tata McGraw-Hill
Publications, 2003.
REFERENCES
1 .Ghali, A., Nebille, A.M. and Brown, T.G., “Structural Analysis” A Unified
Classical and Matrix approach”, 5th Edition, Spon Press, 2003.
2 .Vazirani Vaidyanathan, R. and Perumail, P., “Comprehensive Structural
Analysis
–Vol. I and II”, Laxmi Publications, 2003
INTRODUCTION What is statically DETERMINATEDETERMINATE structure?
When all the forces )reactions( in a structure can be determined from the equilibrium equations its called statically determinate structure
Structure having unknown forces equal to the available equilibrium equations
No. of unknown = 3
No. of equilibrium equations = 3
3 = 3 thus statically determinate
No. of unknown = 6
No. of equilibrium equations = 6
6 = 6 thus statically determinate
INTRODUCTION What is statically INETERMINATEDINETERMINATED structure
Structure having more unknown forces than available equilibrium equations
Additional equations needed to solve the unknown reactions
No. of unknown = 4
No. of equilibrium equations = 3
4 3 thus statically Indeterminate
No. of unknown = 10
No. of equilibrium equations = 9
10 9 thus statically Indeterminate
INDETERMINATE STRUCTURE
Why we study indeterminate structure Most of the structures designed today are
statically indeterminate Reinforced concrete buildings are considered in
most cases as a statically indeterminate structures since the columns & beams are poured as continuous member through the joints & over the supports
More stable compare to determinate structure or in another word safer.
In many cases more economical than determinate.
The comparison in the next page will enlighten more
CONTRASTDeterminate StructureIndeterminate Structure
10
Def
lect
ion
PP
3
48
PL
EI
3
192
PL
EI
4
1
Str
ess
PP
4
PL8
PL
2
1
Considerable compared to indeterminate structure
Generally smaller than determinate structure
Less moment, smaller cross section & less material needed
High moment caused thicker member & more material needed
CONTRASTDeterminate StructureIndeterminate Structure
Sta
bilit
y in
cas
e of
ove
r lo
ad
PP
Plastic HingePlastic Hinge
Support will not develop the horizontal force & moments that necessary to prevent total collapse
No load redistribution
When the plastic hinge formed certain collapse for the system
Will develop horizontal force & moment reactions that will hold the beam
Has the tendency to redistribute its load to its redundant supports
When the plastic hinge formed the system would be a determinate structure
CONTRASTDeterminate StructureIndeterminate Structure
Tem
pera
ture
PP
Dif
fere
nti
al
Dis
pla
cem
ent
PP
No effect & no stress would be developed in the beam
No effect & no stress would be developed
Serious effect and stress would be developed in the beam
Serious effect and stress would be developed
Distinctive Features of Suspension Bridge
•Major element is a flexible cable, shaped and supported in such a way that it Major element is a flexible cable, shaped and supported in such a way that it transfers the loads to the towers and anchoragetransfers the loads to the towers and anchorage
•This cable is commonly constructed from High Strength wires, either spun in situ This cable is commonly constructed from High Strength wires, either spun in situ or formed from component, spirally formed wire ropes. In either case allowable or formed from component, spirally formed wire ropes. In either case allowable
stresses are high of the order of 600 MPAstresses are high of the order of 600 MPA
•The deck is hung from the cable by The deck is hung from the cable by HangersHangers constructed of high strength ropes constructed of high strength ropes in tensionin tension
•As in the long spans the Self-weight of the structures becomes significant, so As in the long spans the Self-weight of the structures becomes significant, so the use of high strength steel in tension, primarily in cables and secondarily in the use of high strength steel in tension, primarily in cables and secondarily in
hangers leads to an economical structurehangers leads to an economical structure..
•The economy of the cable must be balanced against the cost of the associated The economy of the cable must be balanced against the cost of the associated anchorage and towers. The anchorage cost may be high where foundation anchorage and towers. The anchorage cost may be high where foundation
material is poormaterial is poor
Distinctive Features of Suspension Bridge
•The main cable is stiffened either by a pair of stiffening trusses or by a system of The main cable is stiffened either by a pair of stiffening trusses or by a system of girders at deck levelgirders at deck level..
•This stiffening system serves to (a) control aerodynamic movements and (b) limit This stiffening system serves to (a) control aerodynamic movements and (b) limit local angle changes in the deck. It may be unnecessary in cases where the dead local angle changes in the deck. It may be unnecessary in cases where the dead
load is greatload is great..
•The complete structure can be erected without intermediate staging from the The complete structure can be erected without intermediate staging from the groundground
•The main structure is elegant and neatly expresses its functionThe main structure is elegant and neatly expresses its function..
•It is the only alternative for spans over 600m, and it is generally regarded as It is the only alternative for spans over 600m, and it is generally regarded as competitive for spans down to 300m. However, shorter spans have also been competitive for spans down to 300m. However, shorter spans have also been
built, including some very attractive pedestrian bridgesbuilt, including some very attractive pedestrian bridges
•The height of the main towers can be a disadvantage in some areas; for The height of the main towers can be a disadvantage in some areas; for example, within the approach road for an AIRPORTexample, within the approach road for an AIRPORT
Components of a Components of a Suspension BridgeSuspension Bridge
•Anchor Block: Just looking at the figure we can compare it as a dead man having no function of its own other than its weight.
•Suspension girder: It is a girder built into a suspension bridge to distribute the loads uniformly among the suspenders and thus to reduce
the local deflections under concentrated loads.•Suspenders: a vertical hanger in a suspension bridge by which the road
is carried on the cables•Tower: Towers transfers compression forces to the foundation through
piers.•Saddles: A steel block over the towers of a suspension bridge which
acts as a bearing surface for the cable passing over it.•Cables: Members that take tensile forces and transmit it through
saddles to towers and rest of the forces to anchorage block.
Suspension BridgeSuspension Bridge
Suspension BridgeSuspension Bridge
Suspension BridgeSuspension Bridge
The deck of a suspension bridge is usually suspended by vertical hangers,
though, some bridges, following the example of the Severn bridge, use inclined
ones to increase stability. But the structure is essentially flexible, and great effort
must be made to withstand the effects of traffic and wind. If, for example, there
is a daily flow of traffic across a bridge to a large city on one side, the live load
can be asymmetrical, with more traffic on one side in the morning, and more
traffic on the other side in the evening. This produces a periodic torsion, and the
bridge needs to be strong enough to resist the possible effects of fatigue .
We can list the main parts of Suspension bridge
Two towers
Suspended structure
Two main cables
Many hanger cables
Two terminal piers
Four anchorages
5. ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD
5.1 ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD - AN OVERVIEW
5.1 ANALYSIS OF INDETERMINATE STRUCTURES BY FORCE METHOD - AN OVERVIEW
5.2 INTRODUCTION 5.3 METHOD OF CONSISTENT DEFORMATION 5.4 INDETERMINATE BEAMS 5.5 INDETRMINATE BEAMS WITH MULTIPLE DEGREES
OF INDETERMINACY 5.6 TRUSS STRUCTURES 5.7 TEMPERATURE CHANGES AND FABRICATION
ERRORS
5.2 INTRODUCTION
5.2 Introduction While analyzing indeterminate structures, it is necessary to
satisfy )force( equilibrium, )displacement( compatibility and force-displacement relationships
)a( Force equilibrium is satisfied when the reactive forces hold the structure in stable equilibrium, as the structure is subjected to external loads
)b( Displacement compatibility is satisfied when the various segments of the structure fit together without intentional breaks, or overlaps
)c( Force-displacement requirements depend on the manner the material of the structure responds to the applied loads, which can be linear/nonlinear/viscous and elastic/inelastic; for our study the behavior is assumed to be linear and elastic
5.2 INTRODUCTION (CONT’D) Two methods are available to analyze indeterminate
structures, depending on whether we satisfy force equilibrium or displacement compatibility conditions - They are: Force method and Displacement Method
Force Method satisfies displacement compatibility and force-displacement relationships; it treats the forces as unknowns - Two methods which we will be studying are Method of Consistent Deformation and (Iterative Method of) Moment Distribution
Displacement Method satisfies force equilibrium and force-displacement relationships; it treats the displacements as unknowns - Two available methods are Slope Deflection Method and Stiffness (Matrix) method
5.3 METHOD OF CONSISTENT DEFORMATION
Solution Procedure: )i( Make the structure determinate, by releasing the extra
forces constraining the structure in space )ii( Determine the displacements )or rotations( at the
locations of released (constraining) forces )iii( Apply the released (constraining) forces back on the
structure )To standardize the procedure, only a unit load of the constraining force is applied in the +ve direction( to produce the same deformation(s) on the structure as in )ii(
)iv( Sum up the deformations and equate them to zero at the position)s( of the released )constraining( forces, and calculate the unknown restraining forces
Types of Problems to be dealt: )a( Indeterminate beams; )b( Indeterminate trusses; and )c( Influence lines for indeterminate structures
5.4 INDETERMINATE BEAMS
5.4.1 Propped Cantilever - Redundant vertical reaction released
(i) Propped Cantilever: The structure is indeterminate to the first degree; hence has one unknown in the problem.
)ii( In order to solve the problem, release the extra constraint and make the beam a determinate structure. This can be achieved in two different ways, viz., )a( By removing the vertical support at B, and making the beam a cantilever beam )which is a determinate beam(; or )b( By releasing the moment constraint at A, and making the structure a simply supported beam )which is once again, a determinate beam(.
5.4 INDETERMINATE BEAMS (CONT’D)
(a) Release the vertical support at B:
The governing compatibility equation obtained at B is,
fBB = displacement per unit load (applied in +ve direction)
x
y
PP
BC
L/2L/2L
C
B= +B B
RB
BB=RB*fBB
B + '
B B = 0
BBBB
BBBB
fR
fR
/
0)()(
F r o m e a r l i e r a n a l y s e s ,
)/)(48/5(
)16/()24/(
)2/()]2/()2/([)3/()2/(
3
33
23
EIPL
EIPLEIPL
LEILPEILPB
)3/(3 EILf BB
PEILEIPLR BB )16/5()]3/(/[)]/)(48/5([ 33
Applied in +ve direction
G o v e r n i n g c o m p a t i b i l i t y e q u a t i o n o b t a i n e d a t A i s , )()( AAAA M , AA = r o t a t i o n p e r u n i t m o m e n t
AA
AAM
F r o m k n o w n e a r l i e r a n a l y s i s , )16(
2
EI
PLAA [ u n d e r a c e n t r a l c o n c e n t r a t e d
l o a d ])]3/()[1( EILAA
T h i s i s d u e t o t h e f a c t t h a t + v e m o m e n t c a u s e s a – v e r o t a t i o n
PL16)/(3
EI)]L/(3/[EI)]/(16PL[M 2A
5.4 INDETERMINATE BEAM (Cont’d)
5.4.2 Propped cantilever - Redundant support moment released
L
PL/2
(b )Release the moment constraint at a:
A B
A
=
A BP
Primary structure
+ BA
MA A=MAAA
Redundant MA applied
To recapitulate on what we have done earlier,I. Structure with single degree of indeterminacy:
(a) Remove the redundant to make the structure determinate (primary structure)
(b) Apply unit force on the structure, in the direction of the redundant, and find the displacement
(c) Apply compatibility at the location of the removed redundant
A BRB
A BBo
fBB
5.4.3 OVERVIEW OF METHOD OF CONSISTENT DEFORMATION
B0 + fBBRB = 0
P
P
5.5 INDETERMINATE BEAM WITH MULTIPLE DEGREES OF INDETERMINACY
(a) Make the structure determinate (by releasing the supports at B, C and D) and determine the deflections at B, C and D in the direction of removed redundants, viz., BO, CO and DO
AB C D E
RB RC RD
B0 C0D0
w/u.l
)b (Apply unit loads at B, C and D, in a sequential manner and determine deformations at B, C and D, respectively.
AB C D E
fBBfCB fDB
1
AB C D E
fBCfCC fDC
AB C D E
fBDfCD fDD
1
1
(c ) Establish compatibility conditions at B, C and D
BO + fBBRB + fBCRC + fBDRD = 0
CO + fCBRB + fCCRC + fCDRD = 0
DO + fDBRB + fDCRC + fDDRD = 0
5.4.2 When support settlements occur:
Compatibility conditions at B, C and D give the following equations:
BO + fBBRB + fBCRC + fBDRD = B
CO + fCBRB + fCCRC + fCDRD = C
DO + fDBRB + fDCRC + fDDRD = D
AB C D E
B C DSupport settlements
w / u. l.
5.5 TRUSS STRUCTURES
(a))a (Remove the redundant member (say AB) and make the structure a primary determinate structure
The condition for stability and indeterminacy is:r+m>=<2j ,
Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2j =) 2*4 or 9 > 8 i = 1
C
80 kN
60 kN
A B
D
C
80 kN
60 kN
A B
D
1 2
Primary structure
5.5 Truss Structures (Cont’d)
(b)Find deformation ABO along AB:
ABO = )F0uABL(/AE
F0 = Force in member of the primary structure due to applied load
uAB= Forces in members due to unit force applied along AB(c) Determine deformation along AB due to unit load applied
along AB:
(d) Apply compatibility condition along AB:
ABO+fAB,ABFAB=0
(d) Hence determine FAB
AE
LABu
ABABf
2
,
(e) Determine the individual member forces in a particular member CE by
FCE = FCE0 + uCE FAB
where FCE0 = force in CE due to applied loads on primary structure )=F0(, and uCE = force in CE due to unit force applied along AB )= uAB(
5.6 TEMPERATURE CHANGES AND FABRICATION ERROR
Temperature changes affect the internal forces in a structure
Similarly fabrication errors also affect the internal forces in a structure)i( Subject the primary structure to temperature changes
and fabrication errors. - Find the deformations in the redundant direction
)ii( Reintroduce the removed members back and make the deformation compatible
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