Advanced Structural Analysis a K Jain

(1)

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iPjq,r't:2.-3

are registe¡ed rrade ma¡ks

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PTry*t

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COPYRIGHT

@

NEM

CHAND

&

BROS, ROORKEE, 1996

í

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of.this publication may bq Rrinted or reproduced

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Published by

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t

a

(4)

U

{")

t-,'

Ç)

O

{.f

{-.'1

(iv)

of

ùe

two

prograæ

ue

providcd so tlrat the r€ader may have a. _{direct cxpos¡re to}_the

computcr spplicdioûs and develops confidence. Earlier

it

was planned that the sourr¿

listing of thc

nropÐgams srAP-3D

and CABLE-3D

will

be included in the

book,

but

later

it

v¡Es dccided to make

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available on a floppy so that the ¡r¿der does not have to

go through

tte

&udgery of feeding into

a

computer and thcn checking the same

till fi¡ll

c¡nfidenec

is

dcveloped.

Those

who have gone through this exercise

wilt

appreciate

ttb

&cisioa.

The

floppy

is available

with

the publisben

_of

_{this book}

_at

_{a nominal}

price.

.!

U

o

i)

ü

t-i

f,-ì

(_'

(, _;

ü

(,,

(,_

₎

t-,

(i

Ahttottgh

written

mginly.for

the

undergraduatc students, the practicing engineers

will

find

it

e4ually useñrl.

The

inticacies of

structural analysis have Ueeñ explaineA

wiú

the help of over one hundred and

fifty

solved examples.

over

one hr¡ndred ana

nry

problems are included at the end of thc chapters and answers of most of them are given at

the end ofeach chúþter.

I

am

graæñrl

_to

my students

who

we¡p

taught from

the manuscript over the

past fifteen years

for

inadveræntly

proriding

heþfiri

comments,

for

including

_Orpir"i

examples

and

improving

its

contents.

The

critical

comments

offerpd

uy

- -

*y

colleagrres have been utilized in the

ftrat

text.

Dr.

p.w. co¿uole

deserves ã

speciat

_¿,

mention who gladly provided

_I

copy of

hii

cable program for use in this

book.

rñ"n¡.s

- r

to shri

l.P.S.vcrma

for meticulously

and

neatly drawing the

tacings.

I

am

especially

,

grateftl

_to

my

wife

saríta and children

payal and Gaurav

_{foi oeir}

_patience

_anã

:;

encouragement _{in Writing another last book.}

PREFACE December

I,

1995 /&1:

(_

e

a'

Chapter

1.

BASIC CONCEPTS

i.:

l*i"ojìiln

"*inate

vs. rndeterminare Stn¡ctures

1.3,

Flexibiliry

Method

1.4

Stiffiress Method .

1.5

Sysüem _{Approach vs. Element Approach}

1.6

Choice of a Method

1.7

Degree of Static Indeterminacy

1.8

Degree of Kinematic Indercrminacy

1.9 IllustativeExamples

Ashok K.

Jain

.

2.

MATRTX ALGEBRA

2.1 Intoduction

2.2

Definitions

2.3

Matrix Algebra

2.4

Application of a Work Sheet

t

_2.5

_Solution_{of Linea¡ Simulaneous}

Equations

PART

t

:

TLEXIBILITI,METHODS

3. METHOD

OF

CONSISTENT DEFORMATIONS

t

_3.1

Introduction

\-/ ¡¡¡svggvf¡9¡¡

3.2

Choice of Redundants

3.3.

Beams with One Redundant

3.4

Beams _{with Two}_or_More_Redundants

J.5

Reactions due to

yielding

of Supports

':'r

_r,

_3.6

_Frames

3.7

Trusses Problems

CONTEI\ilTS

Page

1 -25

I

4 7,

l0

ll

t2

l3

26 - ¡¡e

26 26 29-36 38

41 -91

4l

42 42 53 66 69 79 o.t

(5)

,j

l

Ë.,

The

aim

of

this book

is to

present

up-to-ãate

methodologies-in

the

arylvs]s

oj

statically

indeterminate

structurìs.

Thisbookis,a

companign

of.my

earlier

boók

entitled

Elementary

Structural

Analysis

which

deals

with

statically

determinate

strucû¡res.

Thus

this

set

of

two

books

completes

the

wide spectrum

of

structural

anaþsis. The methods

of

sÍuctur.al. analyses

a¡e

classified

into the

flexibility

methods

and stifftress methods.

In

the

present'

book, both

the

classical methods and

matrix

based mçthods

are

discussed

in

detail.

The attention

is

devoted entirely to

develop understanding

of

the behaviou¡

of

statically indeterminate sfuctures' There is

sfrong emphasis throughout the text on the use of computers'

Each chapter begins

with the

introduction and

develops the algorithms along

with

suiøble

sign convãntion

which'is

peculiar

to

eqch classical method. The examples are

chosen

anã

solutions arranged

so that the fure

points

of

structural

.analysis are

clearly brought

out.

They selve to

ampliff

and supplement the theory'

Chapter

I

provides an

overview

of

basic concepts

fo¡

the

analysis

of

statically

indeteräinate

ìttt"tut"t.

Chapter

2

deals

with

the

elements

of

matrix

algebra'. Now

onwards the book is

divided

in

two

parts

:

Flexibility

methods and stiffrress methods.

Part

I

covers

the

method

of

consistent deformations, three moment equation, strain

energy method, column analogy method, influence cogffrcient method

,

influence

lines

and ãiches. The last

two

chapters do

not form part

of

the

flexibility-

methods

but

are

included here

for

the

sake

ofcompleteness.Part2coverstheslope

deflectionmethod,

moment distribution method, di¡ect stiffiress method :

2'D

elements, anü 3-D elements ,

and salient features

of

STructural Analysis Program

Sfnp-¡O.

A

key

feature

of

this

book is a comprehensive fieatment

of

non-linear analysis of structures

covering

theory

of

plastic

análysis, material as

well

as geometric non-linea¡ problems. The concepts

of

nysieresis models, unbalanced load vector, updating

of

stiffrress

matrix,

ductility

and

incremental and iterative methods of solution are introduced. The sequence of formation

of,

ptrastic

hinges

is

explained throu.eh

exarnples.

The

salient

features

of

C*lil,n-¡O

program are also discussed. Several listings of sample input data and ouÞl¡t

Preface

(6)

\")

{)

{J

{"}

{.)

(vi)

4.

THREE.

MOMENT

EQUATION

¡

4.1

Infioduction

f

4.2

Derivation of Th¡ee

'

Moment Equation

4.3

Beams

i:.1 ïffi::"'due

to Yielding orsupports Problems

ii

_l.

srRA,ru

ENERoY

Tl,t",on?o0.,.,,""

L]/

û

a

r)

(,,ì

{,,

L

5.2

Work and ComPlementary Work

5.3

Strain EnergY

5.4

EnergY Theorems

5.5. Beams' Illustative

ExamPles

5.6

Frames - Illustrative Examples

Problems

6.

COLUMN A:NALOAY METHOD

J

6.1

Introduction

6.2

Stess in a Column

6'3

DeveloPment of the Method

6.4

Sigrr Convention

6.5

Analogous Column Sections

6.6

Fixed End Moments in Beams

of

Uniform Cross-Section

6.7

Stiffiress and

Carry

Over Factors

6.8

Beams with Va¡iable Cross-Sectíon

6.9

Portal Frames with One Axis

of

SymmetrY

6'10

lii;|rff'swith

one Axis

of

6.1I

Pqrtal Frames with no SYmmeûry Problems CONTENTS t_.'

(i

92 -

106

92 92 96

r0l

'

t02

105 "

to7

-

156

107 t08 109

ll3

ll6

t22

149 154

157 -192

157 t57 158 160

l6l

163 168

l7l

177

(-r

(.';

i

l*

/

i-2.

wrueucl

coEFFløENT METHoD

(

:

_it

$ä'#h""

7.3

Force Diagrams \l

,,)

a-'

t

(

J.

AROHES COì¡TENTS

8.3 IllustativeExamPlês

Problems

S.INFLUENCE

LINES

9.1 Intoduction

9.2 Two-HingedArch

9.3

IllustrativeExamPles

9.4

Fixed Arch

9.5

Symmetrical Fixed Arch

9.6

Elastic Centre

9.7

lllustr¿tive ExamPles

9.8

Influence Lines for a Hinged Arch

g.g

Influence Linês

fora

Fiied

Arch

Problems

'7.4

Graphical Method of Integration

7.5

IllustativeExamPles

PART

2 : STIFFNESS

METHODS

SLOPE.

DEFLECqION

METHOD

l0.l

Introduction

lO.2

DevelopmentofSlope'

Deflection

Problems

10. Ll

R7

Introduction

M¡¡ller - Breslau Princiole

Equatiòns

10.3

Equations of Equilibrium

l0'4

,. Beams

10.5

Frames :

No

Side SwaY

'

10.6

Frames :

\Vith

Síde SwaY

10.7

Frames

with

SloPing

tegs

10.8 Flexibility

and Stiffiress

Matices

Problems

11. MOMENT

DISTRIBUTION

METHOD

l8l

184 190

'(vii)

235

24t

24¡'F278

243 244 247 252 254 257 259 272 274 276

193 - 231

193

t94

195 197 198 229

J

ll.l

DevelopmentoftheMethod

343'

ll.2

Distribution

Factors

345

I1.3

Sign

Convention

347

I1.4

Be'ams and Frames with no Side

Sway

347

I1.5

Beams with Uneven Support

Settlement

361

ll.6

Frames

with

Side

SwaY

3&

'

_I1.7

_Frames

_with

Uneven Support

Settlements

372

ll.8

SYmmeryandAnti-SYmmetrY

375

I1.9

Comments on the Moment Disüibution

Method

396

Problems

397

2'¡2 -

242

232 232

12.

DIRECT SNFFNESS

METHOD.

z

D

ELEMENTS

l2.l

Developmentof.stifhess

Matrices Tn¡ss element

279 -

942

279 279 282 283 290 295 315 331 341

343 -

397

398 -

¿r78 398

(7)

LJ

*

{.)

,,J

{,,)

*

t:)

{:

t;

ü

t,

i-')

(viii)

Effect of node numbering

12.6

IllushativeExamples

12.7

Boundary Conditions

12.8

Support Reactions

.

12.9 Incllred

Roller Support

12.10

Summary of Direct Stiffness Method

l2.ll

fllustrativeExamples

12.12

Comparison of

Flexibility

and Stiffiress Methods

Problems

13.

DIRECT STIFFNESS

METHOD. SD

ELEMENTS

t2.2

t2.3

12.4

t2.5

CONTENTS

Properties of Stiftress Matrices

Transformation of Coordinates

Element Load Vector

Assembly of Global

Matices

Stiffiress matrix Load vector

i..,

(-(,

l-r

(ì

405 407

4ll

4t2

13.l

Stiffiress Matrix - Truss

Element

479

13.2

Stiffiress Matrix - Beam

Element

4g0

13.3

Stiftress Matrix - Grid

Element

4g3

13.4

Stiffiress

Matix

- Shea¡

WallElement

_4g4

13.5

Stiftress Matrix - Beam with Rigid

Ends

496

13.6 Stçped

Members

'

488

13.7

Transformation Maûix - 3D Truss

Element

495

13.8

Transformation Matrix - 3D Beam

Element

4gg

14.

STAP.SD COMPUTER PROORAM

423 430 434 434 435 437 476 477

1

5.

NON-LINEAR

ANALYSß :

_MATERTAL_{NON;,LINEAR|Tí}

l5.l

Inhoduction

lS.2

SFess-Süain Curve of Steel

15.3

Theory of plastic Analysis

15.4

plastic Hinge and Mechanism

15.5

Moment - Curvatu¡e Relation

15.6

Plætic Analysis Statical method Mechanism method l5 7 llhrctmtiwe Þv¡-^l.o

l4.l Inûoduction

SO2

14.2

Salient

Features

₅₀₂

14.3

Adding'New Elemenrs ûo the

Library

506

14,4

User's

Instuctions

₅₀₆

14.5

Illustrative

Examples

_Sl2

15.8 l'5.9 15.

l0

l5.ll

15.12 Modification

of the Strucural Stiffiress

Matrix

15.13

Incremental Displacementand

Load Vector

15.14

Unbalanced Load Vector

15.15

Step-by-Step Incremenral Analysis Method

15.16 Ductilþ

'

_15.17

_Illusûative_E-xamples

16.

NON-LINEAR A,NALYSIS

:

OEOMETRTC NON-L|NEARìT1

16.l

Introduction

16.2

Geometric Stiffiress

Maüix

-2D Truss Element

16.3

Non-Lineqrity of Cable Suspension Systems

16,4

Non-Linea¡ Solution Algorithms Iterative method

Incremental method

Incremental cum Iterative method

(

16.5

Convergence Criteria

16.6 CABLE-3D

program

16.7

User'slnstruciions

16.8 IllustativeEiamples

APPENQ'X

A-

aROORAMS FOR SOLUT//ON

OF

L//NEAR

SIMULTANEOUS ÊQUATIONS

A.l

Gauss Elimination Method

4.2

Gauss-Jordan Method

4.3 Cholesþ

Method

A'.4 Successive Over Rela¡<ation Method

APPENDIX B

-

SLOPES

AND

DEFLECTIONS

479 - 501

CONTENTS

Non-Linear Stiffness

Marix

Analysis Iterative method

Incremental method Hysterèsis Loops Assumptions

Member Stiffiress

Matrix

2D Beam element 2D Truss element

so2.542

569

(ix)

5¡ß

-

589

543 543 544 546 547 548 573 576 576 579 579 580 583 583 585

APPENDIX

C. FIXED END

ÙNOMENTS

T$TÐEX

590 -

622

590 591 594 595

5n

598 600 602

623 -

634

623 625 627 631

635 -

636 6t7

-

6¡t8

639 -

6¡tO

(8)

{J

o

ü

(,;

(*,

(_;

(:

O

C

(,j

(l

(-r

(*)

L (

1.I

INTRODUCTION

A

framed structure is a

network of

a number

of

units

known

as members or elements;

A floor

may consist

of

beams and

slabs. A

building

may

consist

of

beams, columns,

floors

and

ioundations.

Sim

ilarly

a

roof

truss is made up

of

top

chords,

bottom

chords; diagonat members,

purlins. ties

and cover

shcets.

A

bridge

may

consist

of

longitudinal

Ueartrs, transverse bàams, deck slabs and even

cables.

Such cornponents are.referred

to

as nembers or elements.

A.beam

or

a çolumn

element,

and

a

truss

or a

cable

elem.ent are

most frequently

employedlin framôd

sfuctures.

These are one

dimensional

(f

-D)

elements since

their'

cross-sectional dimensions are

very

small as compared to

theil

lengths.

The scope of

this.

book

is restricted to the study

of

various analytical rnethods employed

for

the analysis

of

various

2-D

and

3-D

framed structures consisling of

these

l.-D elements. There are three

basic requirements

for

a.unique structural analysis :

BASIC

CONCEPTS

CHA.PTER

one

l.

2. à

stress-strain relationship of the materi¿ils in the structure,

equationi of

static

equiiibrium,

and

conditions

of compatibility

or liinematics.

ifhes,e were discussed in

detail

in chapter 2

of volume

1 of this book.

STATICALLY

DETERMINATE

VS.

INDETERMINATE STRUCîURES

(9)

BASIC CONCEPTS

rioith¡hl.{ o,'only axiál members, that is, pin-connected bars. A structure wlrethera truss, a

çutìlinuoufi beam

or a

rig-id frame

is

either

.

stable

ór

unstable. and either sratically

dglgrminate or statically indetennínate depending upon the number and ariangement

of

¡t'tçrnl)cls, internal

joints

and . _extemal

_supports.

_A

_structure

_uray

_be

_extêrnally

illrlr-'lr'nrtirli¡te.

or

internalþ indeternrinate

or both.

Idealization

of

internal

joints

and

tflitii[D¡11 $upports, deteÛirinacy and stability of structures were discussed in volunie

I of

lhil

br¡ok, Figure

l.l

shows typical statically determinate beams rvhile Fig. 1.2 shows t),picn I statically índeterminate beams.

Sirniltrly, typical statically detenninate and statically indetermínate franres are slrown

irr

liigs,

Li

and 1.4 respectively; and rypícal statically deterrrrinate änd indeterminate

Ilrss"r

,u'b shown

ín

Figs.

l.5a and

1.56 respectively.

The

stafically determinate

stlucturcs can be

fully

analyzed by usìng the equations ofstatic equilibriurn:

I

'(o)

SlMPLE

BEAM

ÐF*= 0,

X-Fy:0.XM.

=-g

I

_lr,''0.

Ð

Fr:0,

ÐF,=0.

E

M-=0,E Mr=0,

ÐM,:0

(2-D

structure)

( l. I a) (3-D

structure)

(.1. I b)

'f _ltc_{statically indeterminate structures cannot}_be_analyzed_directly

and need additional

rrclr¡ntions due

to

conditions of conrpatibílity.

(b)

BEAM WITH

OVERHANGS

(c

₎

CANf

ITEVER

BEAM

(d)

BEAM

WITH INTERNAL HINGE

Fig.

L

l

Statically determinate beams

STA1'ICALLY DETERMINATË

VS,

INDF,]'ERMINATE STRUCTURES

(q

₎

3 SPAN

C0NTINUOUS BEAM

(b)

_ç0NTINU0USBEAM

Fig. 1.2 Statically indeterminate beams

(o)

P0RTAL

FRAME

Fig. 1.3 Statically determinate frames

Fig.

t.4

Statically indeterminate frames

L-I I

lt

(b) (o) ':

(b)

CANIILEVER FRAME

(10)

T.-j

*4

*

d]

o

'í-,j

{_}

,{"-)

{i

f;

Ç

(.._i (--, ( -,,

(.)

(_'.

(',

(b)

FiE.

t.5

(a)

Statically dcterniinate

truss

(b) Statically indeterminate truss

ln

practice.staticätty inaeièrminate structures are frequently encountered because

of-better itrength, stiffness and economy. Therefore, it ís essentiai to study the development

of various methods of analyzing such structures. One of the important conditions for the

analysis

of

structures is that the principle

of

superposition is

valid. lt

means that the

structure is linear and deflections are

small.

The stresses in the structure are below the

proportional

limit.

The

first

fourtqen chapters áre devoted

to

the analysis

of

linear

statically

indeterminate stn¡ctures.

Practical

considerations

require

that

there

is

considerable strength

in

the

nonlinear region

of

a

stiucture

which

should

be fully

exploited.

The non-linearitl, may be due

to

material or geometry.. Material nonlinear

analysis is discussed in chapfer 15, while geometrical nonlinear analysis is discussed''in

chapter 16.

I.3 FLEXIBILJTY

METHOD

The most powerful method for the analysis of staticälly.firdeterminate structures is the

method ofionsisteht deformations.

It

is also called as the

_flexibititlt

melhod

or

lhe _force

method

or

the compatibility method. The application

of

the

flexibility

method requires

that

fìr}t

the structure be reduced to a stable, statically determinate system. The problem

then reduces

to

establisþing a set

of

independent; simultaneous equations in-terms

of

unknown forces or.redundant actions so that they may be evaluated

and

anaþsis

ofthe

indeterminate sfiucture completed.

When the statically determinate

or

the released stiucture is subjected

to

the applied

loads,

it

will

uridergo deformations

that are

inconsistent

with

the

behaviour

of

the

original structure..

However,

by

applying

forces equal

to

the

released actions, the

deformations

of

the released structure are made consistent

with

those

of

the original

struiture. In analyzing the released Structure, the displacement at the point of application

and

along

the

line

of

action

of

each redundant must

be

êvaluated.

Although the

rnagniiude and direction of the redundant actions are unknown at this stage, the reledsed

structure can

be

analyzed due

to

the

application

of4n

assunled

unit

value

ofeach.

redundant, successively. Finally, considering displacement

of

released structure at the

point

of

application

of

each redundant action, due

to

both the applied loads and the

individual unit values of the redundants, a set of compàtibility equations can be written.

These equations/ describe the actual unknown displacement ofthe origînal structure at

BASIC CONCEPTS

t)

ï

(

Qnch,'point

of

the application'of the

redundant

force.

The

solution

of

these linear

sirnulta¡reous,equations provide the magnitude and direction of each redundant necessary

to

maintain cornpatibility

of

the

system.

Finally; the

member

end

forces,

end displacernents and support reactions can be determined.

Consider

a

threè span continuous beam shown

ín

Fig. 1.6a.

The

total

number

of

reactions is

six.

Theref'<lre, the degree

of

statical indeterminacy is

three.

The structure

can be nrade statically determinate in several ways:

¡'ì

FLEXIBILITY

METHOD

(q)

ACTUAL STRUCTURE

AND

L,OADS

(b)

RELEASED STRUCTURE

{

c)

DEFORMATI0NS DUE

T0 UNIT

FORCE

'M4

Ø¡e

(d)

OEFORMATIONS

DUE TO UNIT

FORCE RB

I

R^=

I

1

tn=

t

( e

)

DEFORMATI0NS

DUE

T0 UNIT

FORCE R.C

Fig.

1.6

Developmênt

of

flexibility

method

(11)

t_i

{}

{"}

{)

,r.-:'1,, [Iy

renroving the supports at B. C and D.

it

reduces to a deteiminate cantilever

.r,,,

.

hë$tn.

,

'."å,

.

By rcuroving the support

A,

it reduces to an unstable continuous beam, hence

it

ls

not

a

feasible solution.

"ì,

ßy rernoving the restraint at A, and the supports B and C, it reduces to a simply

supportcd and stable beam.

l.ct

rrs aclopt the

third

option as shown

ih

Fig.

1.6b.

The simply supported beam

Unclerg,ocs rotation

0oo

at

A,

and deflections

Âro

and

A.o

at

B

and

C,

respectively..

Therc are three unknown actions or redundants, Mn, Rn arfd

(..

'lheir

magnitudes and

diroctions are not

known.

Let us assume that the moment

Mo

is anti-clockwise and the

¡S¿rçtions Ro and

R.

are acting

upward.

Let us apply unit redundants on the $tructure,

one by one, and d€tennine the deformations at the points of application and direction

of

each redundant as shown in Figs. 1.6c,

l.6d

and 1.6e. The slopes and deflections can be

de(erntined

by

using

any

one

of

the

sevéral rnethods

_,(viz.

mornent-area method,

conjugate beam me-thod, or unit load rnethod) as discussed in volume

I

of this

book.

The

number of equations is equal to the.number of redundants.

tlt)

t")

\";-.::. lt.

',tJ

..'{-., .)..,:4.,

,{_:

'{),

{.)

{.,j

r\

\a.-,)

t)

(. .", (...,

{",:

lr-('.

\.. I

(,)

(.,

BASIC CONçEPTS

'l'he _{compatibility}_{condition iequires that}

0e =

0, À, :0,and

Â.:0

that is,

Ooe

+

_ôaaMa

+

0^reR,

+

OocR.:0¡:

-

Âso

*

Âa¡-M¡

*

Ass

Rs

+

Ânå Rc

-

Âs

:

and

.-

Aco

*

Àco

Mo o

.Âcs

Rs

*

Âcc Rc

:

Ac

:

0

in rnatrix notatbn

fe^o

I

fooo

öou- oo.1

[vtn

I

f

r^'Ì

(-)J^Bol

*looo

Âss

o'.

lJ

*'

f

=lo"f

=o

[AcoJ

LÂ.o Âcn

Acc

j

_IRc

_J

_[^cJ

:.

ì-

or

..:

or where {4....ì

:

' \ ¡('l

.::.i

'

_tFl

'

lRl=

tÂ \ s t air, STIFF}{ESS METHOD

ln case vector

_{4.

_}

is zero. Eq. 1.5 can be written as:

tFl

{R}:

{^,o }

ot

FR:

À

This equation represents the defonnation

-

foròe relationship. For a single elemept. the

term F represe nts deforntation per unit force and is called as

_flexibility.

Knowing Å and

F,

ti

can be evaluated which leads to the solution of the c-omplete structure.

The ¡nethod

of

consistent defonnation is one

of

the earliest methods in vogue (Circa

1860).

Theothermethodsthatcantàll

inthiscategoryarêthreemotnentequation,strain

energy rnêthod, column analogy method and influence coefficient method. These

will

be

discussed in detail in the subsequent chapters.

I.4 STIFFNESS METHOD

Another approach for the analysis of statically indeterminate*structuies is fhe stffie.ss

metlzod.

tt

is also called as tl'te tlisplacement method orthe

equilihriuh

method.

Letvs

reconsider the three span continuous bearn

which

was used

to

develop the

flexibility

mèthod. The application of the stiffness method requires that a staiically indeterminate

structure

be first

reduced.

to

a

kinetnaticalb) determinate

system.

A

kinematically

deterrrrinate iystem is the one whose end displacements are

known.

Kinematics relates

the deformations and displacements

of

elements. The

first

step is

to

identify the dègree

'of kinemetic indeterrninacy and. there-fore. the unrestrained

joint

displacements. Now.

the correspondi4g artificial restraint5 must be introduced so as to make it a kinematically

determinate struclure, that is, a reslrained sffucture.

This

structure

must now be analyzed for the

aitual

loading imposed on the original structure. The suoport reactions of the restrained stluctüre for any loading condition are

sirnþly the action required

to

constrain the various

joint

displace¡nents.

At

each

joint,

each restrai¡ring reaction is equal to the sum

of

(a)

the fixed end forces required to constrain the ends of the members that franìe

i¡rto

thatjoint,

and

'

(b)

the action equal and opposite to the force acting directly at

tllejoint.

The restrained structure also needs to be analyzed for displacements of lhe

artificially

restrained

joints.

But

magnitude and direction

of

dísplacements

of

the unrestrained

joints

are.

unknown,

It

is, therefore. convenient

to

assume unit .value

of

each

of

the

artificially

restrained

joint

displacements and analyze the reStrained structure for'each

displacement

individually.

Since. the artificial restraining support actions do not exist in

tlre original structure. a set

of

independent linear sirnultaneous equations can be written

irr terms

of

unknown

joint

displacernents.

The

number'of

equations

is

equal

to

the

number

of

constraints required

to

make the displacements

zero.

The solution

of.this

systetn

of

equatíons

gives

the

mâgnitudd

ahd direction

of

the

joint

displacements

neçessar.y

to

maintain equilibrium

of

the

systern.

Finally.

member end forces and

"$upport reactions can be deterrnined.

deformation vector due tô the actual applied loads; the

first

subscript

'i'

represents the location

of

the redundânt where the displacement is

computed and the second subscript 'o' represents the applied loads.

the

elements ₄

_¡

of

this

rnatrix

represent the displacement Â, in the

ditection

and

at

the

location

of

the

redundant

_\

caused by a unit

action redundant

R,

acting at the location

j.

This

is

called

_flexibility

natrix.

redundant foröe vector

total known

displacernents

at

the

location

and

direction

of

the

redundants in the actgal structure. Some of these displacements may

not be zero as'in the case ofsettlenìent ofsupports.

(1.2) 0

o

(t.3)

-

{^rq}

+

[F]{R}

=

{^r}

-À.o* FR:

Â,.

(t.4)

( t.5) ( l.6a) ( r.6b)

(12)

ionsider

the three span continuous beam shown

in f:ig-

t.lu.

The

unrestrained

clisplacernents are shown in Fig. l .Tb and the corresponding restraiúed structure is shown

in Èig.

t.2".

The unrestrained displacements are

0u,0.

and

eo llt

fix-ed end actions

due ¡o the applied loading on the original structure are shown

in

Fig.

l.7d'

These are

required to constfain the

joint

displacements, and the net support reactions are shown in

Fig. 1.7e.

Now

remove the applied loading and-impose unit

joint

displacement,_successively,

and evaluate the support reactions as shown in Figs.

l3f'.l.7gand

l:1h'.

_[,

,

represents

moinent at

joint

i due to a unit displacement imposed at

jointi.

Thus,

Ki,

is force per unit

displacement and

is

called

stifftess or

stifness

coeficient'

Similarly'

R'

_j

rep-resents

vértical reaction at

joint i

due

to

unit displacement'imposed at

joint

j.

The equilibrium

equation can be written at each

joint,

that is. BASIC CONCEPTS

Z'Ms

=

0, tMc:0

or, Mso

+

KsB eB

+

KBc

ec Mco

*

Kcs, eB

+

Kcc

ec

and

Moo

+

Kps 0É

+

Koc

0c in matrix notation i l' I 1..:

i.:

LI

{.' _,:, ä₃ Ë :l; ç iã

Ê

.11 t' ti1 rli

IM"o]

lr"'

1M.o

l*l

K.u

Ivr*J

LKo"

unq

EMD:

0

+

Kro

0o:

0

+K.o0o:0

*Koo0o=0

or

{P}

+

[K]

{^'}

:

{0}

or,

{P}

:

tKl

{^}

if

^

:

-

Â' for conveniencè

()

(,.,

It

or,

p=KÂ

(l.lOb)

where,

_{P}

:

equivaient nodal load vector due

to

the actual applied loads;

the f,rrst'subscript represents location

ofthejoint

and second

subscript

'o'

represents the applied loads.

tK]

:

elements of this matrix k,

'

represent the force.M, at

joint

i

in

the direction

of

the conitraint

due

to a

unit

displaôement

.

applied at joint

j.

This is called

stffiess

matrix.

{^}

:

unknswn disPlacement vector

The pquation

P

:

KÂ

represents the force-deformation relationship.

For

a

single

element. the term

K

represents

_.þrce

per

unit

deform.ation and is.

called

as. stffiess.

Knowi¡rg P and K, À can be evaluated which leads

to

solution of the complète structure.

The.slope-defleition method and rnoment distribution method äre in vogue since

l9l

5

and 1930, respectively. The

other

method

that

falls

in

this category

is

the direct

stiffness

method.

A

relatively recent method,

it

has revolutionized

the

concept

of

K".

Kcc

Kp.

r"o'l Ie'l

*."

_llo.

_l=9

KooJ

_leoj

(

{

(

SYSTEM APPROACH VS. ELEMENT APPROACH

(1.7)

{o)

ACTUAU STRUCTURE ANO LOADS

-

roe

\oc

too

. (1.8) .

(b)

UNRESTRÂINEO DISPLACEMENIS

(l.e)

^ÐL-<ø

e,

e41)=.--)-4,

ffi&,

(l.l0a)

(c)

RESTRAINED STRUCTURË

I

1 1-

11 id),FIXED

ACTIONS DUE TO APPLIED LOADS

I I Í-L

_"'ry+ry

t

_frrín.+,

'$r

_'Rro

Mao

l-1¡

'R

_eo

M

eo

{-,

_'Rco

Mc9

r.f¡

'RooMoo

(e)

REACTI0NS DUE TO APPLIED LOADS

l^

øe=

.tLítl

,

_W

{¡ xae

t-1.,

Kea

-1.'

Kca

_'1,

Koa

'Rle

'ReB

'RCg

'ROe

(fI

REACTIoNS DUE

f0

UNIT RO¡ATIoN SB

rþxac

_'Rac

*î,

_^r,

Rec

(9)

REACTI0NS DUE

Jrrro

'J-r

xeo

{-,1.n {,

xoo

'R¡D

'ReD

'

Rco

'Roo

(h)

REACTTONS DUE TO UNIT ROTATION 9o

Fig. 1.7 Development of stiffness

¡iethod

lw

\+,

_^cc 'Rcc TO .UN

IT

ROIATION

q

Kco CD ec

r-.oD=

1

(13)

€"-J

*

r.)

{:)

*

{-)

t0

etfUçtufttl nnalysis

due

to

the

simultaneous development

of

powèrful

eléctronic içif,pUi*,,*, I'ltesc methods w,ill be discussed in detail in the subsequent chapters'

i¡#

SYSTEM APPROACH

VS.'ELEII{ENT

APPROACH

Vnrious methods

of

structural analysis weie developed and perfected over a number

ol. ycnrs during the mid-nineteenth

'century.

There were no. computers

at

that timg,

lh(ìreforc, methods of analysis had to be basLd on the physicál reasoning.of the structure'

nfit

Ough the concept of matrix algebra was very well known, but it could not atfract the

oite,rtioi of the analysts for obvious reasons. Methods such as the consistent deformation

tngth6d, strain energy method, slope-deflection method and moment distribution method

considered the structure as a whole.

w¡th the advent of digital computers around the middle

of

twentieth century' matrix

approach became attractive and

the

stiffness methods

and

flexibility

methods were

alvefop"O.

The F and

K

matrices

for

the entire structure were developed through the

,.rp."iiu.

rnatrices

for

the constitutive.elements. This approach came

to

be known as

nfu)uunt ctpproach.

lt

is tnore appropriate for an automatic analysis by a digital computer'

Having understood the concept

of flexibility

and stiffness methods. the analysts took

another look at the conventional methods. tt was a pleasant surprise to discover that the

earlicr methods could be classified either as a

flexibility

or as a stiffness

method'

Since

they considered the structure as a whole, this approach was called as system approach'

]-hus tt¡e classification

of

system or eletnent approach is a relatively new tenn and quite

convett ient.

The

flexibility

method

is

also known as the force method because

the

forces are

treated as unknowns. Since the condition of compatibility of displacements is'imposed to

generare the final equation,,,

it

is also known as the compatibiliry method. Simi{arly. the

ititfn.$

nrethod is also known as the displacement method because the displacements are Ireared as unknowns. Since the condition of equilibrium of forces is imposed to generate

thc

fi¡ral

equation.

it

is

also

knôwn

as the equilibrium

method.

lt

is

necessary to

understand that both these approaches make use of equilibrium as well

as

compatibility

conditions.

It

iS convenient

to

write a general purpose

"oniput",

programme using the

stiffness

nretlrod

for

static or dynamic analysis of any framed

structure-

lt

is also very efficient

for íhe non-linear analysis ofsuch stiuctures.

lt

is referred to asthe usetfriendly method.

l"lre flexibility

nlethod cannot.be generalised due

to

several possibilities

of

selecting

piOper reduniants and the associated difficulties in

writing

simple and straight forward

aigåritlrms. l{ence,

it

is not much in use. Nevertheless.-the classical

flexibility

methods

prõvide n deep insight into the physical understanding of the structural behaviour and are

ncccssary to study.

I.6 CHOICE OF

A

METHOD

A

structure can be analyzed using the force approach or the displacement approach'

The question is how to decide

whjc

method is bctter. As discussed earlier, the number

\)

{¿,

iJ

t)

{)

t:

BASIC C()NCEPTS (

(

(,,

(,)

/. _-,1

(.,

(. _,ì

\.)

t,,

(,'

(-of

equations

in

the force method

is

equal

to

the nr,¡mber

of

redundants, whereas, the

numher

of

equations

in

the displacetnent method is equal to the number

of

constraints

required to make the displacernents

zero.

ln other words. size of the

flexibility

matrix is

equal to the degree of statical indeierminacy (qs), whereas, size of the stiffness matrix is

equal to the degree

of

kinematic índeterminacy

(a*).

Obviously, a method which leads

to a fewer number of equations. is preferable.

Thus.

first step is to determine the degree

of

statical indeterminacy as

well

as

the

degree

of

kinematic indeterminacy and the

method

of

analysis

can

be

selected accordingly keeping

in

víew the

convenience

involved. It is also possible to adopt a mixed approacå in certain types of problems. The

mixed approach is beyond the scope

ofthe

present text book.

I.7 DEGREE

OF STATIC

INDETERMINACY

lf

all

unknown reactions

in

a structure can be uniquely determined with the help

of

equations

of

static equilibrium, the reactions of the structure are referred to as statically

deternlinate. Otherwise.

the

reactions

of

the

structúre

are

referred

to

as

statically

indeterrninate.

The

degree

of

indeterminacy

is

equal

to

the

number

by

which

the

unknowns .exceed

the

available equations

of

statics.

.

The

total

_.degree

of

static

indeterminacy

of

a

structure

is

considered

as sum

of

the following two

types

of

indeterm.inacies:

(?)

degree ofexternal indeteiminacy

and (b)

degree

ofinternal

indeterminacy

'The external indeterminacy is_ related to the number and type

of

supports.,, The intÞ¡nal

indeterminacy is concemed

with

the determination

of all

member forces knowing'the

support reactions. In a pin-jointed truss.

if

the number

bf

members rneeting at a

joint

is

just suffìcient to preserve its geometry. the truss is internally determinate, otherwise

the

-truss

is

internally

indeterminate.

Similarly,

a

rigid- jointed frame

is

internally

determinate

if

its members forrn an open configuration. that is.there are no closed cells,

otherwise

the

rigid-jointedtframe

is

internally

indeterminate.

Alternatively,,if

more

member actions

are

presen{ than can be solved for from statics . the frame is internally

indeterminarc

DEGRËE

OF

S'I'A'I'IC

INDETERMINACY

il

For

a

2-

Dpin-jointedtruss.

a,

=

(m

+r) -2j

For

a 3-D

pin-jointedtruss,

o,

=

(m+r)

_-3j

For

a !-D

rigid-jointedframe,

as:

(3m+r)

_-3j

For

a

3 -

D

rigid-jointed frame,

,

qs:

(6m+r)-6i

The total static indeterminacy is given by the above relations. The degree of external

indeterminacy

is

easy

to

calculate. The

total

number

of

external reactions minus the

Irumber

of

available equation

of

static equilibrium gives

the

degree

of:

external

indeterminacy. Thus, the degree

of

internal _{indeterminacy çan be}computed. Often a

$tructure

is

said simply

to

be

statically

indeterminate

without

stating whether

it

is

indeterminate internally, externally or in both manners.

(l.l

la)

,

(t.l

lb)

(t.t2a)

(14)

6'.--) ¿- ,\ t2

e)

*

()

t)

I.8 DEGREE

OF KINEMATIC

INDETERMINACY

when

determining deformations, there are no equations which are analogous to the

equations

of

static equilibrium. Therefore,

all

struòtures.

with

certain exceptions' are

kinematically indeterminate.

If

deformations Of the ends

of

a nrenrber are known' then

the deformation at any other point in the member can be easily deterrnined'

A

fixed-end

beam

is

kinematically determinate

and

a

simple

supported beam

is

kinematically

indeterminate..The degree

of

kinematic indeterminacy

of

a

structurc

is

equal

to

the

number

of

independent displacement components.

ln

a pin-jointed tl'uss, the ¡nember

of

independent- translations

at

each

joint

is' 2 or

3

depending upo¡ whether

it

is a plane

trúss

or

a

space truss.

'ln

a

rigid-jointed

frame,

the

number

of

independent

joint

Jirpluc"rn"nt, is 3 or 6 depending upon whether

it

is a plane frame or a'rspacc frame' The

number of independent

joint

displacements is also called as the degrce'olfreedom or the

joint

or of the node.

If

the number

of

fully

restraint

support

displacements is

S'

and'

ino*n

support dispiacements is.

S,

the

degree

of

can

be

computed by the following equations:

Ç

t)

L,{

{)

(,;

{.:") {_)

(-,;

(_

BASIC COI.ICEPI'S

For

a

2-D

Pin--jointedtruss,

qx:2i -

S'

_-St

For a3

-

D

pin-jointedtruss, cfk

:

3j

_-

Sr'-

Sz

For

a

2

-D

rigid-jointed frame,

or.:

3j

_-

S'

_-St

'

For

a

3-Drigid-jointedframe,

dr.:

6j-

St -S?

i_;

(,,,

f';

(--; (' '\. ,,'' \

(,.j

(l

r"\

(..

_l

('

ln most practical situations. the supports are unyielding and; hence'

St

:

Q'

Under certain conditions, in rigid-jointed plane as well as space frames,

it

is possible

to

reduce the degree

of

by

ignoring the axial dgformations'

irt"t,

¿"*p"tatiãns

in the classical methods

of

structural analysis wére simplified to a

/.

great extent. .

The number of independent

joint

translation in a plane frame is glven by tlre'equation:

ILLUSTRATIVE EXAMPLES

I.9 ILLUSTRATIVE

EXAMPLES

.

dku:2i

-

(2Sr+ 2Sh+Sr+m)

wherp,

S¡

=

number of fixed supports

-

S,,

:

numbenof hinged suPPorts

S,

:

number of roller suPPorts

m':

number of members

The

following

examples illustrate the concept

of

static and kinematic indeterminacy,

and the applications of stiffness.and

fleribility

methods in simpie cases. .

Example

l.l

.ì-Determine the degree of static indeterminacy

of

plane frames shown

in Fig.

1.4, and

the pir-jointed trùsses shown in.Fig. 1.5.

Soiution

lf

tlre number

ofjoint

rotations is

a*t

a

pl3nètifid-joiite.l

frame

':j"l T;.

(l.l3a)

(r.r3b)

(l.l4a)

(r.r4b)

Plctne

_frame

(Fig. l.4a)

Total number of independent external

reaction'i

r :

3

+

3

+.2 =

g

nurnberofjoints

j - 7

numberofmembers

_m

₌

6

using Eq.

l.l2a,

Total statical

indeterminacy

d, :

(3m + r)

_-

3j

:3x6+8-3x7=5

Degreeofexternalíndeterminacy

: r-3:3+3 +2- 3:

₅ Degreeof internal

indeterminacy *-

crs

-5 :5-5:0

Plane _frante

(Fig.

l.ab)

Total nurnber ofexternal reactions

r

=

2 x

3:

6

Total number

of.members

_m

₌

6

Total number

ofjoints

j

₌

6

the total degree of kinematic indeterminacy

of

cko

( l. r5)

Hence,

t3

Degreeofexternal

indeterminacy

: r-3= 2x3- 3:

Degreeof internal

indeterminacy

= o.

_{-3 :6-3}

=

3

Plane truss (Fig. l.5a)

Total members

m:21,

joints

j

=12,

¡svç¡itrs

r

=

3,

Eq.

I.llagives,.

:

.

d,

:

(m+

r)-2j= 2t+3'2r12:=

0

The truss is statically

determinate.

'

Plane Íruss

(Fig:

1,5b)

_{, ..}

.

Total memberr

r'=

41,

joints

j:

18, reac-tions

r.=

4

Total statical

indeterminacy

g.

:. 4l +

4

-

2

x

lB: .9

'

.

Degreeofexternal i¡determinacy

=

r:3:2+,

I+

l.-3=

I

Degreeof intemal indeterminacy

=

9-

I:8

(1. l 6)

{rr:3x6+6-3x6=

'.;...

I

'Þete¡ntin"

the dcgrees

of itatic

and kinematic indeterminacies

for

the multistoreyed

t.2

e shown

in

Fig. 1.8a. Assume that axial effects, that is, changes

in

member lengths

bo ignored.

6

(15)

l.Ël BASIC CONCEP'I'S

s

í.!r'. l:i: t::.' ii.:-¡1 ¡:r;. !:r:,r r,

ii¡.:

¡iì:r,

í

.!irj.r .

.:i:l

ll!::' (q) Solution

,{ilat ic i nde te r nt i nacY

Totalnumberofexternalreactions

=

3x

3=9

'..',

Degreeofexternal

staticindeterminacy

=

r- 3:9'3:6

.

.ltl

a structure is cut at any point, three forces or actions must' be

applied

to the left

,"i6oÈitt"

cut

and three equal and opposite member forces must be applied

at

the

.¿pîUti¡Ar¿nA

o{the

cut to maintain continuity. These three, forces are

:

an axial force. a

*hoâr.lbrçe and a bending momenl

ïhis,

is a convenient method to determine the degree of static indeterminacy. Simply

intr,Oduqe as many cuts as necessary to reduce a slructure to be,statically determinate but

ensure,that each cut portion is stable'

ln

the present frame, introduce a cut in each beam.

'Ihe

sturcture is reduced to three

¡n¿euerr¿eit vertical cantilevers. There are 20 cuts, and 3 unknowns per cut.

Thus'

total

;ü;Ëtt".

;*tic

indeterminacy is

20 x

3 j

60'

Therefore"the degree

of

internal static

indctciminacY, Ís 60

-

6:

54.

'l'he totål decree of indeterminacy can also be found using

Eq'

I ' l2a'

Total

ñiembers m

:

50,

Totaljolnts

j :

33î

Total reactions .r

5

9

.,:i.. , i ;

_.

cls:(3m+r)_3j

K i nena! ic i ndetcnn i nacy

Total number

ofjoints

Fig. 1.8

.

Degree

of

freedom

perjoint

-

3

Known zerosuppondisplacementsSl

=

3

x3 =9

.'.

Degree of kinematic indeterrninacy

cru:

3 x 33

_-

9

= 90

If

axia!

deformation is ignored,

Lateral displacement at each

floor :

I

IILI"JSTRATIVË

IiXAMPtES

.'.

Total lateral displacements

Vertical displacement at each

joint

=

0

Rotation at

eachjo¡nt

:

I

Total rotations in the

frame

30

x

I

:30

Total unknown displacements in the frame

:

l0

+

36 =

49

Degree of kinematic

indeterminacy

an

:

40

I

Thus, the degree-of kinematic _{indeterminacy cr*}is reduccd by

'

axial deformations in the members.

iÈ:;t.:r

r:,

_.

=

3x50+9_-3x33:60

=33

Alternatively,

Kinematic indeterminacy can atso be determined using

Eq.l.l6.

: l0xl:10

j:

dku 0te clk

i3,

Sr:3,

:

_2x

33-I5

=30

= dku+ Okg:

S¡,:0,

S, =

0.

m

:50

(2x3+0+0+50).:

l0

50

by

neglecting the

t0+30:40

o.

K.

o.

K. Po as in

(16)

'forePring

_t'

BASIC CONCEPTS

^Po

or

_:,

=K,

if

t'o

=

¡g

kN'

Kr

=

100

kN/m

and

Kr:

150 kN/m

"

_-r

n.

_='lo

=

o.lm

100 '--tor snrinsrfr,

net elongation in spring 2 is

(4,

-

Â¡)'

,.,:tl'

Po:

Kz(^r

-

A¡)

displacement

or

a,

=

_f,+a'

or

ar

=

jl'

*6'¡

= o'167m

î50

:

To obtain the equivalent rpring tii-ftn"ss, let ùs rearrange Eqs'

(i) and

(ii);

DÞ-P^

Lì.

=.ü.Ë,

uno.

^,

=

rî

P-

Pn

Po '

KrK,

(iiÐ

or,

_^"0

=

KlK,

r' '':

;ingie spring' whose stiffness'is given by

l'hus, ,pringt

I

and2 can be replaced with a s

Eq.(iii), which

*itt

gin" the same displacement'

ExtmPlc

1.4

.;t

.,;;'urings

_{system shown}in Fig.

l.lOa

is _{subjected !9}

_T

_Tiul

load

Po'

Determine

,fto'inttttit"äi.aã

i"

the two springs and displacement

ofjoint

B'

$oruui;

:

-¡,ett'forccs iróduceA in the

two springs-bef'

and

F''

lhe

f1e1'bodVdiagrams

of the

springs aro $hown

ir

nËr.ì.

iõu""¿

J.

rrr"ç,rotæiefoniation

relations can

be wrinen for

;ñ;

ü"

springs:

(i)

+e---{¡\¡\,\¡\d¡\'t\,t\y'e--:io+

TlJ

+A

'

(b)

F2

K2

F?

---:+

o---_-4Å,¡V\Ày'rl\ÁÄfu-<

+

ia

@,¡,

_tc'

Fig.

l.l0

á!s' F¡ = K¡

À

and

F2:

K2

^

The equilibrium

ofjoint

B gives,

Fr +

Fz:

Po ILLUSTRATIVE EXAMPLES

bru

A

(ii)

Fl

Ä!-+åe-v

Kt

..:,,.

lt

may be seen

that

spring

I

elongates by Â, whereas spring 2 shortens by the same

tf*o

_@

P

,,i,¡,.

_amount._{Substituting the values}_{of F,}_and_F,_{in Eq.}

(ii)

gives, ::ils;-l-r:

rç{ìr,

lK-+K:ìÂ:

p-Lii.

.t,,

'

o,

Á=

Po

.i,

K,+K,

r:i.i

jÉiriì

_¡f

Po

:

lOkN,

Kr

:

l00kN/m,

K,

:

l50kN/m,

Fl

(o)

,;'

,.ii

^=

lo

=0.Ò4m

$;¡i:,'

roo+

l5o

l7

:1,:..

;;

Fl

=

100

x

0.04 :

4

kN

tension

'i.ì, and

i'". and

F2

:

150 150

x

(-0.04): -6kN

comlcompression.

.-'

_,,.

Example

l.$

''

.

'A

four

springs systern shown

in

Fig:l.lla

is subjected

to

axial

loads

P,

and

pr.

.:

.

Determíne the displacements

À,

and, d,¡, and spring forces,

if

they are .connectéd through

',

_a_weightless_rigid_body_Q.

.j,

t

,o,u'on

ra

:,lrìfi

Since

the

springs

are

intercqnnected through

a rigid

body

.Q, the

compatibility

,,-:

c.ondition requires^that

^springs.l,

4

and

3

must elongate

by

Â,

each. The

free

body

,;ji-Ëdiugtutt

of each of the four springs are shown in Figs.

l.l

lb

to d.

ii;liã'., Fnr¡c-rlcfnmafinn ralqfinnc oir¡p

(i)

(iD

(17)

l8

BASIC CONCEPTS (.,.ì

(_;

(_ì

(_,;

l,

(--) ì

(:,

(.,'

(.'

(,

P2'aZ

I

F1e

F1

Fr

_r+

Fig'

l'll

Fr

:

Kr Lz,

Fr= Kz

Ap

F, = K,

Â'

'

F¿ = K¿

(^2

_^l)

'

_(i)

Joint equilibrium equations givei

ffrr¿¡arJe¿rl

lixanple

I

,ct

K,

*, .51¡

kN/rn. K,

= 75

kN/m,

K.,

:

50

kN/m'

Ko = 60 kN/m, P, = 20 kN and

l\

'.

40 kN.

There are six relaiions and six unknowns F1 , F2 ,

Fr,

Fo , A, and

A"'

Substituting the

values

of

Fr to F4 in Eqs.

(ii)

and

(iii)

give'

KzÂr-K4(42-À')=Pr

and

KlOl+Ko(42

-

Ar)+KiAt:Pt

'l

:-

Fr-Fo

:Pt

and

Fr+Fo+F,:P,

(Fig'l'lle)

Ëq.

(iv)

gives,

0r

(e)

Eos.

(i)

eive

1;1.:

:'r :r .

&rgmplc

l.6

Irrs -60lf^,ì

fzoì

L-uo

roojla,J:

iroJ

f

¡,

ì

[o.rt

tl

iorl

=

lo.:ozJ*

.

_'

Weightless bar

ABCD is

supported

on two

spiings as shown

in Fig.l.l2a.

The

*$ritrg flexibilities are

_f,

and fr. Delermine the spring foices and rotati<iñ of the bar

if

it

is

:tubl€ctËd to a load P at Point D.

{l}

{ri,ïi-[Fo

j

13.34 J

(ii)

(iii)

$olution ,[f ,bal AD '1i.. ,,1 ¡,,i¡:.:

,L/3,L/3,t/3,

f-T-î.---i

. (o)

(b)

Fig.1.12 P D

rotates by 0, (Fig.

l.l2b),

the compatibility,condition requires that

^:

Lo.

¡,

=

!o

and

L,,

=Lg

a -JJ

(18)

; :l.

t'

Substituting the values of À, and Â, from Eqs'

(i)

in Eq'

(iv)'

PL

=

Rr

l."rT

""'

₁₁

,' ';''

Let

P=20kN, L=6*'fr=0'005m/kN,ft:0'010m/kN'

"

e=0.05radian':2.86"

l'

, ,

À,

:0'l0nì'

az

:0'2oin'

Rr

:20kNarid

Rz:20kN

o-Lr-2Lz

' 3\'3f2

o=!9*L9

- 9fi_

9

rz o

_"

=

er'E

fll

(4fi

+

fr)

\

L/

Spimgs

I

and 3 compress by

Â,

and A, whereas, Springs

2

and 4 elongate by A, and Ân'

Tlíe.force-dcformation relations give,

,:

,.t.,1'.tta,,

Rl=Kl^l ,

R2:KzLz,

Rj:Kra,

and

Ro:K¿a¿

(ii)

whcro,

Kt

_,

K2, K3,

K4

:

spring stiffnesses

:,'r, \,

R¿

=

sPring forces

(iv)

Ar

=:0, t,

=

ã

=fe,

Âr=ae

and

Âo=Lo

(v)

,1,

"tl.

I

.1,'

rhl

Pl

A

,:i;¡-.,f1',.r

.

,,..,

fhe

momerti equilibrium equation gives,

'.i

r

Mo:

o

:a1.,

'.,l*.

or

PL:Rr*!+R2xfr"RoxL-2t

iîffiou'*uting

Eqs' (i) and

(ii)

in Eq'

(iii)'

(D

Ëîi.

,=[,()'.,.,(;)'.,(;)'..]*

R2r A2

R4, A4

Eianple

Let

K, :30kN1m,

Kr=

40

kNim,

Kr:50kN/m,

Ko

=

60kNim,

(ii i)

(19)

1l: [ìxnntplc 1.8

Apin.jointedthreebartrussisshowninFig.l.l4a.Determinethememberforcesand

loittt displacements. BASIC CONCEPTS

P

:

20

kN

and

l-=

6m

e :

0.033 radian

:

l.9lo,

R,

= -

l.485kN

comPression, R,

=

-'

7 .425 kN comPression,

î,:!,'

a'\:.';: ' :1?1. ' ,:

.

'

r"ì

t\

¡'--t:,' (''':,

A1,El

L1

Lt:

Rz=

R¿=

0.198 m 3.96 kN tension I 1.88 kN tension

A2rE2

.12

30"

iiC

li 450

,iL)

'i, ,--., |

\,,'

I

ic

I

i(.'

A3,E ₃ L3 lì

[,

Solution

The

joint

o occupies position o' under the application of vertical load W as showil in

Fig.

l.l4b.

Displacements are small.

I-et

o'o"

=

u,

oo" =

v

Considering the equilibrium

ofjoint

o'

(o)

EF*=

0,

T,

cosa

=

T¡

cosP

ÐFy:

0. T,

sino

Tz

TtsinP

:

Y

Eqs. (i) and

(ii)

can be arranged in the matrix form :

foì

l-coscr

0

tùl=l

sinc'

I

.or.

P:

RrT

where T :

{Tl

, Tz, T.,} is'the vector of internal forces

P :

{0,

W}

is the vector

ofjoint

loads

Rr :

equilibrium matrix

-

Fig.

l.l4

Pin-jointed three bar truss

There are thlee unknown foices but only two equilibrium equations, hence compatibility

do¡ditions are required to obtain a solution. Considering the compatibility conditions,