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