University of Nigeria Research Publications

MBAJIORGU, Magnus Sylvanus

Weniteafa Aut

hor

PG/M Eng/84/2369

Title

A Higher Order Theory Applied to Beams Resting on Elastic Foundations

Facu

lty

Engineering

Dep

artm

ent

Civil Engineering

Dat

e

July, 1986

Sign

atur

e

A HIGHER ORDER THEORY APPLIED TO BFAMS

RESTING ON ELASTIC FOUNDATIONS

MBAJIORGU, MAGNUS SILVANW WENITEAFA

(PG/M. Eng/84/ 2369)

A THE3IS SUBMITTED '1'0 THE CIVIL ENGINEERING

L)EPARTMENT, UNIVERSITY OF NIGERIA, NSUKKA,

IN PARTIAL FLJL,FILMENT OF THE RE?UIPEMENTS

FOR THE MASTERS IN ENGINEERING DEGPEE I N

CIVIL ENGINEERING

JULY 1986

Ngozika Faustina I j eoma Chiakwa

CONTENTS

2.2 Beams Resting on Elastic Foundations -------------- 5

2.3.2 Elastic Continuum Foundation Model ---------------- 10

2.3.3 Two-Parameter Foundation Wdels ------------------- 11

2.3.4 Three-Parameter Foundation Model ------------------ 18

2.3.5 Prediction of the Foundation Parameters ----------- 19

3 ' - A HIGHER ORDER THEORY FOR BEAMS TI-IROUGH A VARIATIONAL ~ p p ~ 1 , ~ m .......................................... 24

3.1 Variational Principles ............................ 2 4

3.2 Development of a Higher Order Theory for Beams ---- 2 8

3.2.1 Displacement and Stress States .................... 2 8

5.2 Simply Supported Beams Resting on Elast ic Founda- t ions by the F i r s t Order Theory --------------

ABSTRACT

The mixed var ia t ional pr inciple of Reissner (1950) was used here

t o derive higher order equations for beams. The s t r e s s and disnlnccmcnt

s t a t e s of the c lass ica l theory were modified t o include two h ighe r ordcr

e f fec t s ( t h i s was cal led the t h i rd order theory). These s t r e s s rmd tlis-

placement s t a t e s were used in the var ia t ional pr inciple . Thc p m - n i n g

equations and boundary conditions appeared in such a form t h a t the

symmetric and antisymmetric pa r t s were uncoupled and hence c o u l d hc con-

sidered separately.

The th i rd order theory was used for the solution of simply supnorted

beams carrying uniformly dis t r ibuted and concentrated loads. lics111 t s

were presented and compared with resu l t s from other theories l i ke the

c lass ica l , the f i r s t order and the Method of I n i t i a l Functions (MIF) .

The t h i r d order theory proved superior t o the other theories and enabled

the solution f o r the beam problem t o be extended t o beams with dep th t o

length r a t i o s of greater than 0.25.

The present theory was a l so used t o analyse three s e t s of free-

ended s t e e l beams res t ing on e l a s t i c foundation tha t Vesic and Johnson

(1963) presented experimental r e su l t s for . The resu l t s of the analyses

were plot ted together with Vesic and Johnson's and conventional analyses

resul ts . As the beams became more r ig id the advantage of the nresent

theory was most evident. b

ACKNOWLEDGEMENT

The author wishes t o express h i s profdund grat i tude t o h i s

supervisor, D r . V. K. Sebastian, f o r h i s guidance, suggcs t ions and

cr i t ic isms during the course of t h i s work.

The contributions of my colleagues T.S. Koko and E. C. Oguej io for

a r e acknowledged. M r . V.O. Ekechukwu and F.O. Okafor were of immense

help during the computer calculation of the numerical resu l t s .

The author is a l so grateful f o r the m r a l supnort o f t h e rncmbcrs

of h i s family and h i s chr i s t i an friends. Finally, Mrs l i . C. Chinwuba,

is being thanked f o r the good work she d id i n typing t h i s thesis .

LIST OF TABLES

Table

Recommended Procedures for Analysis ------------

Convergence Patterns of Moment and Deflection a t Midspan .....................................

Values of Maximum Non-Dimens ionalised Deflect ion and Moment for different Depth t o Length Ratios

Values of M a x i m ox/q for different h / t values

Values of Maximum -r /q for different h/R values xy

Maximum values of v, v* and v by the Third Order for different h/R valueg -----------------

Convergence Patterns of Moment and Deflection a t Midspan .....................................

Values of Maximum Non-Dimensionalised Deflection and Moment for different D e ~ t h to Length Ratios

Simply Supported Beam on Elast ic Foundation carrying a Uniform Load, q .....................

Simply Supported Beam on Elast ic Foundation Point Load, P .........................

Beam Characteristics ------------------.---------

L;ad;ng 3&in,tiuns ,n B G m ------------------

Values of am and A,,, apnearing i n Qm ------------

Results from different Analyses fo r the Wide Flange Beam ....................................

Results from different Analyses fo r the Flat Beam ...........................................

Results from different Analys& for the Channel Beam ...........................................

v i i

LIST OF FIGURES

Figure

Beam Resting on an Elastic Foundation -----------

Characteristics of the Winkler Foundation Model -

Observed Displacement Patterns under Loading for most Foundations .................................

The Filonenko-Borodich Foundation Assumption ----

The Hetenyi Foundation Model .....................

The Vlasov-Leontiev Foundation Assumption --------

Values of the Foundation Modulus for Different Subgrades ----------------------------------------

Fletcher and H e r m m Curves for the Detcrminat ion of Foundation Parameters -------------------------

General Loading on Beam ..........................

Simply Supported Beam Problem --------------------

Variation of ox/q across the Depth of Beam ------

Variation of -r /q and a /q across the Depth of Beam --------- 3 --------- YYYyY- - - - ---------------

Beam resting on an Elastic Foundation -----------

Location of Point Loads -------------------------

Results for Wide Flange Beam with Single Load ---

Results for Wide Flange Bean with Two Loads -----

Results for Wide Flange Beam with Three Loads ---

Results for Flat Beam with Single Load ----------

Results for Flat Beam with Two Loads ------------

Results for Flat Beam with Three Loads ----------

viii

5.9 Results for Channel Beam with Single Load -------- 7: i

5.10 Resul LS fur Channel Beam with Two Lodds .--- ---- - - 79

5.11 Results for Channel Ream with Three Loads --------- X O

LIST OF SYPBOLS

Cross-sectional Area

Series coefficients

Width of Beam

Constants

Youngs Modulus of Elast ici ty

Youngs Modulus for the foundation material

Modulus of Rigidity

Depth of Beam

Moment of Inert ia

Foundation llodulus ( f i r s t foundation naramc t c r 1

Second foundat ion parameter

Length of Beam

Coefficients appearing i n ax

Coefficients appearing i n -r xy

Coefficients appearing i n a Y

Foundation Reaction Pressure

Concentration load on Beam

Distributed load on Beam

Coefficients of the Fourier Series expansioll of 1 oad function

Pastenak's second foundation parameter

Vlasovfs second foundation parameter

Filonenkofs second foundation parameter

Displacements in the x,y,z directions respcc-t i v r l v

Coefficients appearing in u

vO,v1,v2 - Coefficients appearing in v

Depth t o length r a t io (h/R)

Poisson's r a t io

Poisson's ra t io for the foundation material

Constant

mT

Characteristic length of a Beam on Elastic 1 :o~1r~l ;1 t i on

Constant

Beam Function

Coefficients appearing in the Beam Function i ( ; , , I

a m R I

INTRODUCTION

I. 1. GENERAL BACKGROUND

The c lass ica l theory of beam bending has f o r a very l o n ~ I rrrrcx

been the bases of the solution of many engineering problcm~ . I 11 is

theory due t o its simplifying assumptions, takes the longi t ~ r t l i I ] < ! l

normal s t r e s s t o be l inear ly distributed accross the depth o l I l i ( s

beam while the shear s t r e s s assumes a parabolic d i s t r ibu t io11 I!), i t l l c

transverse normal s t r e s s is neglected. The deflection i s ~ I I v ~ ! il\cw

as the deflection of the middle plane of the beam while the. (.I 1 1 t

of shear deformation on bending is neglected. This result., 1 1 , I

fourth order d i f fe ren t ia l equation i n terms of the def1cc.t I 01 I 1 1 1 ~ 1

loading.

The r e su l t s obtained from t h i s theory become qui te i n , \ ( i I I I

for d v beams and a t the immediate v ic in i ty of concentr.irc>(l I . i t l i .

On the other hand, the exact solution f o r any engineering 11 I ( ) I I I 1 1 1 ' i C,

obtained through the theory of e l a s t i c i ty . For one t o o h t ; ~ i 11 I 1 (

theory of e l a s t i c i t y solution of a problem, he w i l l have to >o I

s ix f i r s t order l inear pa r t i a l d i f fe ren t ia l equations of ~ Y \ I I I 1 1 1 ! I ran,

s ix second order l inear p a r t i a l d i f fe ren t ia l equations of L-OIII[);I~ I -

b i l i t y and s i x stress-strain relationships with s t r e s s nntllor ( ! I . n

rnent components subject t o appropriate boundary conditionh. 1 1 1 o

these'equationsane w i l l have t o obtain f i f teen unknowns ( s i \ 1 1 1 ~ l t 1 1

dent s t r e s s components, three displacement components aid G > I , I I , I 111

components) . This approach of seeking exact solutions i:? I I ~ . I I, I I I

very d i f f i c u l t and f o r some problems have remained irrrpossil)lt~.

To bridge the gap between what is expected and what c.,111 I ) , oi,t : I ~ I K Y I ,

especially what can be eas i ly obtained, simplifying assump-t 1oi1~ , I I (' i n-

troduced t o reduced the basic three dimensional problem t o ; I I \ , ( ; ~1irncr1-

sional one (as in the c lass ica l p l a t e theory) and fur ther t o ;I O I I ~ , cli~ncn-

sional problem (as i n the c lass ica l beam theory) . By as sun^ i I]!? ( 1 r v . l PI, r ic~tc.

displacement and s t r e s s s t a t e s it is possible t o t r e a t the I Y \ < I I ~ , I 0 1 ) l c>m as

a simplified two dimensional problem. This way one is ablc 1 o ~ l i r I i r i ; I

theory higher and be t t e r than the c l a s s i ca l beam theory yCt 1 1 0 1 1 ) ( r r l l l ;I.;

tedious as the exact solution.

The var ia t ional o r energy methods become very handy f o r I f f m o w .

'Ihis is bccause, given proper def ini t ions or s t r c s s :~nd t l r * 1 1 l ' ' 1 1 I. l r 1 ( o r

s t ra in) s t a t e s the governing d i f f e r en t i a l equations Cor sly I ) I t I t 1 1 1 . I I cx

eas i ly obtainable through one of the var ia t ional methods.

The conventional approach t o the solution of beams on ( , I . I , I I I I o~lntla-

t ions has been t o modify the c lass ica l governing equation Toi I > ~ , I I I 1)cntling

t o incorporate the e f f ec t of the foundation react ion pressu r c.. I 1 I ( ' s t rcrqths

and weaknesses of the c lass ica l beam theory w i l l therefore al t t . c ~ ~ ~ ~ l t s

obtained by the conventional approach of solution f o r beam (In P I ; I ' T i C-

foundations. This implies t ha t a be t t e r theory f o r beam beritl i l i t 1 <>1lo11 Id

improve resu l t s obtained f o r e l a s t i c foundations.

Another factor that w i l l a f fec t r e su l t s obtained for hc;l111~ I c 3 t t i nlg OII

e l a s t i c foundation is the mathematical model used t o represcnt 0 1 t)rcdict

the response of the actual foundation. The conventional app i ~ r c . 1 r~cn t ionecl

e a r l i e r uses the foundation model proposed by Winkler (1867) . I I I ( $ \ Y i nk I e r

mde l is equivalent t o a s e t of c losely spaced independent i i ri( :I i I jr in::s

that deflect ver t ica l ly i n response t o vert ical ly applied p r ~ ; s t i I-: , , This

foundation model though mathematically simple gives erroneous 1 P ~ > L I i ts i f

the beam is not suff icient ly long and outside the loaded port I O I ! ( 1 1 the

beam. According t o Vesic (1961), the fundamental fallacy ol t ! ~ c onvcn-

t ional computations fo r beams resting on e l a s t i c foundations I i c. i :I thr

application of the elementary theory t o short beams. A lot of- \ , ( ~ I - L I N S

however been done in e l a s t i c foundation modelling giving risc. I o lil.lll)j two

parameter fokdat ion models and some multi-parameter models.

1.2 SCOPE OF WORK

The work done includes a concise ?resentat ion of avail :h 1 (, 1 i I I r . : ~ I tire

on the various aspects of the problem in chapter 2. Chapter ~ l ~ ~ v ( ~ l o p s

a higher order theory fo r beams based on the variational pr i I 1 1 I ( 0 1

Reissner (1950) . This theory called the "third order" theory O,; I I . o r 1

proper truncation a lower theory that was termed the " f i r s t ortic-1'' ! l~coty,

The theories were applied t o simply supported beam problems i n t I : ~ l j t ; . r * 4

while i n chapter 5 they were applied t o beam problems on ellist I ( t x u ~ t l : ~ - .

tions. In each case numerical resul ts were compared with ot1lt.1 I \ I 1 :ill I c

experimental or analytical resul ts .

The accuracy of a solution one obtains for the problem of a beam

resting on an e l a s t i c foundation depends on the beam theory used, the

method of incorporating the ef fec t of the foundation and the ab i l i ty of

the mathematical foundation model t o predict the response of the real

foundation material. A l o t of materials appear in l i t e ra tu re in each

of these aspects of the problem 3f beams on e l a s t i c foundations. A

review of these materials is presented in t h i s chapter.

The classical theory ol: beam bending has been used lo r a very long

time for the solution of many practical problems. In these problems,

the classical theory gives acccptable estimates inspitc ol: i ts simple

mathematics. The governing different ial equation for t h i s theory is

d4v E l p = q ........................ 2 . 1 m

where v is the deflection of the middle plane of the beam.

The effect of shear s t r e s s on the deflection of beams was considered

by Timoshenko (1930). He gave the ef fec t as an increase i n the curvature

of the bent beam. The governing d i f ferent ia l equation comes out as

where

A is the cross sectional area of the beam

G is the modulus of r ig id i ty

5.

a is a constant equal t o 312 f o r rectangular beams md

4/3 fo r c i rcu la r beams

Investigators have considered higher order e f fec t s such as the

effects of transverse shear and normal s t resses with maim of improving

on Timoshenko's resul ts . Karman (1927) has given a more general

expression f o r curvature than tha t of Timoshenko. Seewald (1927) using

an integral transform technique; Boley and Tolins (1956) using n stcr,

by s tep solution; and Donne11 (1952) have a l l derived expressions fo r

s t resses and displacements in thick beams.

Soler (1968), using a Legrendre polynomial expansion f o r s t resses

and displacements, has derived a higher order theory fo r beams. Vlasov

(1957) by expanding the unknowns i n Maclaurin sgr ies in the thickness

coordinate has given an exact formulation (known generally :IS the Method

of I n i t i a l Functions, MIF) f o r two dimensional problems. Iyengar e t a1

(1974) using the same method gave some numerical resu l t s f o r thick rec-

tangular beams. Das and Set lur (1970) extended Vlasor's formulation t o

both plane s t r e s s and plane s t r a i n two dimensional elastodynamic problems.

Bahar (1972) combined the method of i n i t i a l functions with the integral

transform method suggested by Sneddon (1951).

2.2 REAMS RESTING ON ELASTIC FOUNDATIONS

A loaded beam w i l l def lect and when t h i s beam i s res t ing on a foilntla-

t ion .the deflection w i l l exert a pressure on the foundation. 'I'he founda-

t ion reaction w i l l therefore be t o produce continuously dis t r ibutcd forces,

p, opposing the deflection of the beam (Fig 2.1)

Fig 2 . 1 Beam Resting on an E la s t i c Foundation

The e l a s t i c property of the foundation is ref lected i n the rorm of

p however the exact form of p i s subject t o the e l a s t i c foundation modcl

adopted. The Winkler foundation model takes p t o be d i rec t ly proportional

to the deflection a t any point , i . e .

p = kv = bkov ............................. 2.3

where the modulus of the foundation, ko[kN/d] , chnractcri sr:; tllc cl :,st i -

c i ty of the foundation. Applying the c l a s s i ca l theory f o r bc;un Iwnd ins ,

the governing equation f o r t h i s problem becomes

Hetenyi (1946) used eqn 2 .4 t o solve qu i te some problems on berms oS

i n f i n i t e and semi-finite lengths. Using an end conditioning technique hc

obtained r e su l t s f o r f i n i t e beams from the r e su l t s of the i n f i n i t e beams.

Lee e t a1 (1961) derived the slope def lect ion equations of r i n i t e beam

columns on e l a s t i c foundation from the d i r ec t solution of the govcrninp

d i f f e r en t i a l equation. S to l l e (1 962) described a method that m a k c . s ilsc\

of t h e solutions f o r a beam-column with simply supported ends ; ~ n c l has

tabulated the solut ions f o r several loading conditions.

Timoshenko and Cere (1961) used trigonometric s e r i e s Ibr tlw s o l u t i or1

of simply supported beams. Iyengar and Anantharamu (196.3) used the s c r i c ~

of character is t ic functions representing the normal modes of tr:msverse

vibrations to analyse beams with and without axial loading. Ilctcnyi (1971 1

also used ser ies solutions while Reaufait (1977) used numerica 1 1 ys is .

Rao e t a1 (1971), Heyashi (1921), Wolfer (19711, e t c have a1 l worked on

beams on e l a s t i c foundations.

Vesic (1961, 1973) and Vesic and Johnson (1963) carricil out oxperi-

rnental studies on beams and slabs resting on e l a s t i c foundat ions. They

compared the i r resu l t s with calculated resu l t s using Winklcr'.; : ~ s s u q t i o n s

and Ohdels (1942) isotropic e l a s t i c so l id foundation reprcscntntion. l k

Beer (1948), and Thomas (1960) have also done experimental works whilc

Drapkin (1955) and Hetenyi (1946) solved problems on gr i l lage beams on

e l a s t i c foundations.

Work has also been done on beams rest ing on non-Winklcr Sounclations.

Biot (1937) derived expression fo r the moment dis t r ibut ion f-or thc case

of an i n f i n i t e beam carrying a concentrated load resting on an isotropic

e l a s t i c solid. De Beer (1948, 1948), Habel (1938), Ohde (19112) Ik Hccr-

and Krsmanovitch (1951), Kany (1959), e t c presented approximate s o l ~ ~ t ions

for par t icular cases of beams of f i n i t e lengths on the e l a s t i c solid

foundation loaded by one or several concentrated loads. IIarr c t a I [ 1909)

worked on beams on a two parameter foundations model.

2.3 ELASTIC FOUNDATION MODELS

In Civil Engineering, the foundation material is most c o r ~ n ~ ~ ) n l y thc.

s o i l which unlike e i ther s t ee l or concrete has properties and cJla~.actcristics

that vary widely with location and time. Also, the corrcctnc..;~ of' thc

assumption tha t the foundation is e l a s t i c depende on the lontlinp, and his-

tory of the par t icu la r s o i l in question. These f ac t s makc thc jot) ol

gett ing a mathematical model which w i l l predict the response (I( ' t i \ ( \ r c ~ ~ 1

foundation a very d i f f i c u l t task. Reviewers have thercforc prormwtl rrnny

foundation models i n the bid to gct bc t tc r results'. Thc prol) I ( V I i t 8 how-

ever not completely solved by get t ing a foundation model I'or ccirc;l l l y

important a l so i s the prediction of the foundation parmct i IK-o ~ - p c ~ I-:rt c d

t o every model.

2.3.1 ~ i n k l e r 's Foundation Model

The simplest and e a r l i e s t representation of a continuo~ls cl;~.;t i c

foundation was proposed by Winkler (1867) . He .assumed the So\lntl;t t i on I)asc,

t o consis t of closely spaced independent l inear springs that c1t.T 1 c.c. t vcrt i -

cal ly i n response t o ver t ica l ly applied pressure. Such a Sour~l:lt ion is

equivalent t o a l iquid base. The foundation is therefore char;~ctcr-iscd h v ,

( i ) a react ion pressure d i rec t ly proportional t o the tlc rl CY- t ion

occuring a t the point (Fig 2.2a) , and

( i i ) a foundation tha t deforms only along the portion tl i r vc t l y

under the applied loading (Fig 2.2b) . 92

7 , * tlYfv'"'/ P 2

P1 (a) (17)

fig 2 .2 - Eharacterist ics of the Winkler Foundation Modcl

?he re la t ion between the f o ~ d a t i o n reaction pressure, p , ;111tl rlic

deflection, v, a t any point i n question is then given hy

where ko is the foundation modulus representing the spr i IIV, ( - o ~ l ~ - . l - : l n t

i n the spring i l l u s t r a t i o n and thc spec i f ic weight o r t l ~ . I r c 1 1 1 1 l l i r r

the l iquid base i l l u s t r a t i on . The behaviour of the loiulck~t I O I I 1 ' ;

therefore described by only one parameter, ko, hence the iVi 1 1 h 1 , (mrnda-

t ion model is a l so known a s the One-Parameter foundation n~oc!c~l. h t )

has the dimension of F L - ~ . However for a bkam of constant i rm.; sc i t i on

and constant width, b, supported on the Winkler foundation, wc, c-an

represent the foundation reaction as

where k = kob includes the e f f ec t of the width of the hc;m :~l\r! I S tlrc

dimension of F L - ~ .

I t can be seen from Fig 2 . 2 t ha t fo r the foundation n ~ o t l ( ~ I , thc d i s -

placements of the loaded region w i l l be constcant whethcr t l i c , l - o ~ m ~ l : r t i o n

is subjected t o a r i g i d stamp (concentrated load) or a m i I on.. l o;itl.

Also for the two types of loading, the displacements out:; I ( 1 1 ' t I i c . I O;I t l ( ~ l

region will be zero. I t was observed, however, t h a t lo I I I 'O. 1 I I I 1 1 t ' r i $ 1 l '

the displacements of the foundation surface a r e as shown i 11 I I 0 i . 3 b

(Kerr, 1964).

Fig 2.3 Observed Displacement Patterns under Loadirlp for most Foundations.

Therefore, though the Winkler foundation has the atl\l;~nt t l l ( x ol'

simple mathematics, i ts predictions are not sui table f o r ; I 1 I ! I;~sTic-

foundation problems.. lletenyi (1946) has pointed out ~ I I ; I I i I , ! ' ~ , ; I I I I

networks, (as i s character is t ic in the construction of 1'1 I , ,; t - ( w

for ships, buildings and bridges) and in thin she l l s of' I-,.! I : : 1 i orls

(as i n pressure vessels, boi ler , reinforced concrete !);I 1 I :; ; ! ! , I ! t lor~vs 1

the predictions of the Winkler foundation a re met more I. i I 1 I I v

than as in s o i l supported s t ructures .

2 . 3.2 Elas t ic Continuum Foundation Model

In the bid for a foundation model tha t w i l l predict 111( I {\.;pon:;c

of the foundation be t te r than the Winkler's model, thc I o ~ I I : ~ ' 1 1 1 o r 1

was considered as a semi-infinite e l a s t i c continuum. 'l'hc. I \ I i ; \ l c ~ r l ol

bending of beams rest ing on an isotropic e l a s t i c sol id c.i1:1 I . : ! ' t rr.i :'(:(I

by a Young's modulus, ES, and a Poisson's ra t ion, vs, I I t vr:~tcd

by many authors. Boit (1937) considered the problem 01' 1 ) O I ! C I 1 ! : < I ~o~1c.r.

a concentrated load of a f lexible beam of in f in i t e ler-l;r~ll i ! : I s ( ~ i -

in f in i t e , homogeneous, e l a s t i c and isotropic sol id. V(,c- r c , I ) l !

carried out a continuuation of Biot 's work while Ile Rec 1 i ' i , I lahc 1

(1938) , Ohde (1942), Kany (1959) and others solved difl'c r i I 01) 1 (mi

considering the foundation as an e las t i c continuum.

In addition t o the increased mathematical difficu 1 1 i c . 1 i I ! I ) I I I I ~ ~ ' r ( d ,

it soon became obvious that for some materials, thc P o c l ~ r t l I t i t , I 1 1 ) ~ ( Y - I

to loads behaved differently than predicted by the thcor-y o I ' I 15 r i c-

solids. I t was found, for example, that for so i l s thc . ; I I I I c l i L ; -

placements away from the loaded region decreased more r . , l l ~ I ( 1 I ' 1 !1:1r1 I ) rc -

dicted by .the theory (Foppl , 1909). I t is also question;~ll l c I S0:11il

rubber-like materials with a relatively large void ra t io k i l l ' ~ l h ; ~ v c l i k c

a homogeneous isotropic medium (Kerr , 1964)

2.3.3 Two-Parameter Foundation Models

There i s a large class of foundation materials occ~r r I 1 1 I 11 [)r.:ic--

t i ce the behaviour of which can neither be represented by ;I 1: 1 ~ ~ 1 I c r rotlntla-

tion nor by an isotropic continuum. In an attempt to Sir l t l I I ) I I \ < ; i c : ~ l l y

close and mathematically simple representation for these I I I < I ~ ( > ! I 11'; a t thc

contact area, one may proceed in ei ther of two ways. 'I'hc. 1 1 i L; t o

s t a r t from the continuum representation and introduce s irnl) l I i I 11v

assumptions with respect to the expected displacements m~l / ) I I I cSCjL;c\s

and the second w i l l be t o s t a r t with the Winkler foundat iorl ~r i n order

to k i n g it closer to rea l i ty assume some kind of interac-t i or ' 1 t w c m

the spring elements. These turn out t o be two-parameter I'or 1 1 1 I, 11 ion

models and most of them were postulated on intui t ive basih. i l l 1 , 111oclcls

suggested in available litearature w i l l now be cons i flerctl s t< i r t i 1 1 ~ from

the group derived by modifying the Winkler foundation.

2.3.5.1 I:ilonenko-130rodichf s Ikmxlat ion Modcl ---- -

To achieve some degree of in teract ion between thc. 11 I I I I ' , 1 1 1 i 110

elements, Filonenko-Borodich (1940) assumed tha t thc tol) ( I [ I N ,

springs a re connected t o a stretched e l a s t i c membrane scdl I , ' 1 t o : I

constant-tension f i e l d , T, as shown i n Fig. 2.4.

Fig 2.4. The Filonenko- Borodich Foundat ion A~sumpt I c 11 1

1 l e condition of equilibrium i n the v e r t i c a l dircc-t I I , I rtsr4~rme

element gives the foundation reaction pressure, p, i n 1 t 1 1

p(x,z) = kv - T V ~ v ....................... Z a 7 , l

where, k is s t i l l the Winkler parameter, T is the appli(.'l 1 1 :11u1

d is the Laplace operator i n x and z. For a beam prol-, 1 c.11 , (~111;11 1 on

reduces t o

From equations 2.7 it can be seen tha t the interaction ( ) I I 1 I ? 1 ) ) I

elements is characterised by the in tens i ty of the t e n s i o ~ ~ ( I I , I ,

t h ~ membrane and t h i s becomes the second foundit ion pa I . 11111' I I I I I ( <

tension f i e l d has the physical e f f ec t of reducing thc I ( ~ ~ I I I ~ I I I I c ' I , t inn

pressure.

Schiel (1942) suggested a foundation model consi > . I I I I I C - I \ ,V

l iquid with surfdice tension. This i s however basically , I \ I \ ) (

Filonenko-Borodich foundation.

2.3.3.2 He teny i ' s Foundat ion Mode 1 - -------

Hetenyi (1946 and 1950) introduced p a r t i a l continu i I v ! I1 i n k l c r

foundation a s an improvement. He assumed a continuous ~ ( . : I I I I I t l ( - t l i n

the material of the Winkler foundation. Therefore when L ) ( ~ i 11 I I I ( ! : I ~ ion

is loaded by a d i s t r ibu ted load over a shor t sect ion, t h r , I 1 i O I I undcr

the load w i l l consis t of a discontinuous par t and a cont i nclc 1r.c-tion

curve (Fig 2.5)

Fig 2.5. The Hetenyi Foundation Model

v1 is the discontinuous def lect ion and v the contrihut I I 11 1 2 , r u l? i 111 IOLIS

deflection curve.

In the two-dimensional case, the foundation model i ~ , I I I,cd I,\

imbedding an e l a s t i c beam and i n the three-dimcnsioml ( 1 1 1 ( % i l )

the material of the Winkler foundatibn. I t is assumed t h : ~ t I ' "1111 or

p l a t e deforms in bending only. The re la t ionship between t l c I t I , ~ l iotl

.where the second foundation parameter, D, is the flexura 1 t I I ) I ' t I N ,

p la t e (Kerr 1964).

2.3.3.3. Pasternakts Foundation Model ----------

Pasternak (1954) assumed the existence of shear interactions

between the Winkler spring elements. He assumed t h i s t o be accomplished

by connecting the ends of the springs t o a beam o r plate consisting of

incompressible vert ical elements which arc deformed only by transverse

shear. Considering the vert ical equilibrium of a "shear layer" element

cut out from the top of the foundation, he got for a beam problem

and fo r a p la te problem

where the second terms on the r ight hand side of the equations are the

effects of the shear inteaetions of the springs. S the second parameter

is the shear modulus.

Based on the works of Ratzersdorfer (1929 and 1936) as well as the

consideration of the mechanical behaviour of model and real medium,

Kerr (1964) argues that the Pasternak foundation is the most natural

extension of the Winkler model for homogeneous foundations. By adding

linear viscous elements t o the e l a s t i c elements of the foundation model

he extended the concept of the Pasternak foundation for cases of visco-

e l a s t i c derorms. Thus he was able t o obtain a model t o represent a

semi-infinite snow base (Kerr, 1961)

2.3.3.4 "Generalized" Foundation - - - - - - -- Galletly (1959), Sokolov (19521, Urbanowski (1956) ,and others hnvc

also suggested a foundation model tha t apart from the Winkler

pressure a t each point of contact, there ex is t s an applied moment

proportional t o the angle of rota t ion a t t ha t point. Analytically

t h i s is described by

where k and kl are the corresponding proportionality factors. Usinp

the re l i t ionship between p and rn, the two equations c:m hc rcnlnccd

where k and k are the two foundation parameters. 1

2.3.3.5 Vlasov-Lemtiev Foundation Model ----------

Unlike the preceeding two-parameter models, Vlasov (1'34!)) m d

Vlasov and Leontiev (1960) s ta r ted with the foundation as a semi-

i n f in i t e e l a s t i c continuum. They considered shear interactions jn

the foundation md using the pr inciple of v i r t ua l work formulated

the i r problems by a var ia t ional method. Considering :m el a s t i c con-

tinuum foundation of thickness H, res t ing on a r i g i d base; a p l a t e of

f lexural r i g id i ty , D, lying upon the upper surface of the loundation

and i s subject t o ver t ica l load q(x,z) , (see Fig 2.6) they obtained

. pa r t i a l d i f f e r en t i a l equations f o r v and @.

Fig 2.6. The Vlasov-Leontiev Foundation Assumption

Imposing restr ic t ions [u=w=o) upon the possible deformation

of an e l a s t i c layer, they obtained, for one s e t of assiunpt ions, :I

load-deflection relat ion in the form,

Based on experimental evidence, they assumed the function @(y) i n the

form

where 11 is an unknown constant determining the variation, w i t h tlcnth,

of the vert ical displacements.

Jones and Xenophontos (1'3771, stressing the import:ulcc of t h c

correct choice of the vertical deformation nrof i le , @ ( y ) , i nprovcd I h(.

1 7

Vlasov-Lemtiev foundation by providing a theoret ical 1)as i s Tor thc

form of $( y) . They used an approach s imi la r t o V1;lsov's Ixit thcb

formulation was based on the p r inc ip le of t o t a l s t r a i n cner-py. l h c )

e t a1 (191) suggested t ha t f o r e l a s t i c foundations o r f in i tc*

th ic laess f ixcd on a r i g i d basc, $ (y) coulcl bc takcn :IS,

while fo r an e l a s t i c layer r e l a t i ve ly th ick o r of i n f i n i t c thicknces

2.3.3.6 Reissner Foundation Model ---------

Reissner (1958) startct l from the cqunt i ons 01' ;I c - O I I ~ ~ I H I I I I I ~ .

Assuming t ha t thc in-pkme s t r c s sc s throughout the l ' o ~ ~ i ~ l : ~ t r o r1 1 : 1 \ c . r

are negl igibly small (ox = oz = T = 0 ) ) ,and tha t thc h o r i x m t ; ~ l xz

clisplacements ( r l and w) a t the uppcr and lowcr surracc.:; ol the. Totnd;i-

t ion layer a r e zero, he obtained f o r the e l a s t i c casc thc relation, ,-

b

where v is the def lect ion, q is the d i s t r i bu t ed loading act ing on the ronnclat i on si~rl';~c.cs

E,G are the e l a s t i c constants of the foundation

and I I is the thickness of the foundation layer.

I t is noteworthy that fo r a constant or

the Reissner foundation model gives the same

model i f the constants are redefined so that

l inearly v a ~ . y i t l y , ( 1 ,

equations ;I..; t l v I'astc~nr;~lt%

C =k and (:?=!;. 1

2.3.3.7 The Generalized Two-Parameter Foundation - - - - - - - - - - - -- --

The equations for a l l the two-parameter foundati on> ; I 1 c h .; i r n i 1 ; I r

md mathematically equivalent. The only difference i s i r l t l ~ x tlc. l ' i r i i t i o n

of the parameters. In solving problems mathematically, , ~ t c.11 t ion rwtd

not be paid to these parametric differences. 'I'he two-]):I I ;wc..t c , r. l i ) t ~ r ) t l ; i -

model can then be represented as

where k and kl are the f i r s t and second parameters o l ti^ 1011ncl:1t ion

mode1 .

2.3.4 Three-Parameter (Fletcher) Foundation Model --------------

. Fletcher and Hermann (1971) as an extension of thc W i t)l. l c r f o i u l c l ; ~ - .

t ion included terms involving the derivatives of the d c . l - l ~ ~ - t ion. 'I'hc

foundation reaction was assumed t o be a l inear

'deflection and its derivatives, i .e.

1 ' )

Using an approximation of the same order as the beam t l l c ~ o r v , on1 y

terms with n < - 4 were retained to give

p(x) = k v + kl v' + k2 v*'+ k3 v"' + k4 v I V -------- i 2 .18 )

By considering inf in i te beams on foundations whose propcrt rc .5 arc

not functions of x, they got

and

where kl,k2 and k4 are the foundation parameters and h is I t l w l1:11 1 -.

width of the beam.

They derived formulae relat ing the model parametc.~.~, t 1 , I IKI c B l :s:;t i c -

properties of 'the supporting medium and also producctl ( - 1 1 1 1 I 1 \ ) I tlw

mdel parameters for inf in i te and semi-infinite foundat i o ~ ) ~ , .

2.3.5 Prediction of the Foundation Parameters ------------- The accuracy with which any of the foundation nmlc 1'- n m t- i oiled

will predict the unknowns of a given problem depends t o ;I : , I I ( - , I I vxtent

on the values assigned to the foundation parameters. 'I i111o4u I I ~ O :~nd

Krieger (1959) are Of the opinion that the numerical v ; ~ l ( I ( I I hck

foundation modulus (i .e. the parameter of the Winklcr- rrn~t i t . l ! q ) ~ \ r ~ d ~ i

largely on the properties of the subgrade on which t h ~ I X S ; I I I ~ I ( %.;t :; .

?hey gave a chart fo r the determination of the foundat ion ~ K > ~ ! L I l tv; , k,

for different so i l s based on the Casagrande c lass i f ica t i O I I ol w i lL;

(Fig 2.7)

Table 2 . 1 Recommended Procedures for Analysis

Class af Beam

Long Beams

Moderately Long Beams

Cri ter ion for.. Distinction

Moderately Short Beams

'Treat a s per ;md sirnilar

beam

. Recommended Procedure

For Rough For Refined Estimates Analysis

X R > 5.00

2.25<XR4.00

Short Beams

Scott (1981) updated a l o t of works in t h i s area including

Conventional Analysis using k as given i n eqn 2 . 2 0

- do -

0.80<XR<2 .25

XRd.80

Barden's (1963, 1963) expression fo r the Winkler model ' s k as

1-v; where J = -

'IT

Conventional Analysis

In the two parameter foundation model proposed by Vlasov and

Rigorous Analysis by L k Beer's, Ohde's

Leontiev the expression f o r the f i r s t foundation parameter, k, was

p t i o n a l l y obtained (e-g. 2.12b) . If $(y) i s taken in the Porn of

eqn 2.13 where u i s a constant of dimension L-l, k i s given as

( i ) for the plain s t r a in case

( i i ) for the plane s t ress case

Es v where Bo = --

s

- - v0 = 1 -v S

For resul ts close to those from more rigorous methods, p has to be

between 1 and 2. The second foundation parameter, S, is given for

the plain s t r a in case by

For the plane s t r e s s case, the form is retained but E and v are S S

replaced with Eo and vo respectively.

Fletcher and Hermum (1971) gave curves for the selection of k

for a Winkler foundation model and k and S for a two parameter founda-

tion model as shown in Fig 2.8. Their resul ts apply to cnscs i n

which the r a t io of E t o E is less than 0.01 S

Fig 2.8 Fletcher and Herrmann Curves fou the Determination

of Foundation Parameters.

A 11IGlER ORDER TIIEORY FOR H E N 6

THROUQ1 A VARIATIONAI, APPROACH

3.1 VARIATIONAL PRINCIPLES

The theory of e l a s t i c i t y solution t o problems requ i rc. i n I I ( . ~ Y I 1

the solution of six f i r s t order l inear pa r t i a l different ia 1 (YIIIXI I $11.; 01'

equilibrium, s ix compatibility re la t ions ( in the C o n of scCorlil-1 1 , &Ic. I ,

l inear pa r t i a l d i f fe ren t ia l equations), and s ix s t r e s s - s t r :~ i l l I ( 1 1 i o n s

with s t r e s s and/or displacement components subject t o a~pror7ri < 1 t 1 ( ) L I I ~ : I r y

conditions. The d i rec t approach of seeking exact solution\ t o 1 , I /)I-o-

blem is usually d i f f i c u l t . Consequently solutions a re o f t c ~ ~ ~ o l l ~ ~ r I ) \ /

a l ternat ive methods and frequently a var ia t ional approach i h I I ( ( 1 ilitx

variational (or energy) principles have some character is t ics t I L I I '~1\(,

them very useful, namely

( i i )

( i i i )

( iv)

work,

they enable simplified derivation of the governing d i i ' l c s r ( , I I 7 I I l

equations for the par t icular problem

they a s s i s t the choice of appropriate boundary condit ion<,

they are ideal ly suited for use with approximation rnctl~otl~~

they enable cer ta in features of the solution of a prol~lcw I I

be deduced without actually completing the solution.

The major principles of these methods are the princip 0 1 1 I t (1 .1 I

the principle of v i r tua l s t r e s s and the mixed virtua 1 .,I I ( ~~i 1.1 ~ I ; I I

s t r a i n principle. In the principle of v i r tua l work (or v i r t I I : I l ( 1 I ; I 1:lc.e-

ment) the potent ia l energy of the body is used and the varini j o n I taken

where

u = rrr (0 6 + 6y~0z6z~~xy*rxy+~xz~xz~~yz~yz) dv v X X O ;

W = 1 /(Xu+Yv+Zw)dV C t rr (%I+Yv+ZW) CIA

*1

TI = (U-W) is the potential energy of the system.

U = is the potent ia l energy of deformation.

-W represents the potent ia l energy of the external forces act ing on thc

body i f the potent ia l energy of these forces fo r the unstressed con-

d i t ion (u=v=w=o) is taken a s zero.

Al is tha t par t of the bountlary srlrrace on which tlisplaccmmts arc not

described

o , ~ are s t r e s s components

6,y zre s t r a i n components

u,v,w are displacement components

X,Y,Z are components of the body forces while - - - X,Y,Z are surface t ract ions .

This principle iden t ica l ly s a t i s f i e s the compatibility conditions while

the var ia t ional approach t r i e s t o s a t i s f y the equilibrium rcclui remcnts

as w e l l a s possible within the framework of the approximation being used.

In the pr inciple of v i r tua l s t r c s s , the complementary energy o r the

systcm is used and the variation is carr ied out on the s t resses . In

t h i s case the equilibrium conditions a re ident ical ly s a t i s f i ed and the

principle t r i e s t o sa t i s fy the compatibility conditions as well as

possible. The variational equation i s given as ,

where

n* is the complementary energy of the system

A2 i s that par t o r the boundary on which thc surf;rrc forces

are not prescribed.

The mixed vir tual s t ress-vir tual s t r a i n pr inciple i s the most

recent and it was suggested by Reissner (1950). He pointed out that

it may on occasion be advantageous t o t r e a t the equations of compatibility

and equilibrium on a more equal footing. The var ia t ion i s thcrelorc

carried out on both the s t resses and s t ra ins . . The Reissner variational

equation is given as

where

- - Strain energy density function 1

The following a1 ternative form of thc Reissncr variational cquatiori

can be obtained i f the s t ra ins a re expressed in terms of the displacc~ment.;

u,v,w and thc f i r s t var ia t ion carried out.

+(- aw + - av - -- 2 ( 1 + v ) ~ ) 6r 1 dxdydz ay a z E YZ. yz.

I t can be seen tha t eqn 3 .4 yields the equilibrium arid strcss-

s t r a i n equations of e l a s t i c i ty . The boundary conditions arc also

given direct ly . This form of the variational equation i s c-orivcnicnt

for use in develq ing a higher order theory fo r beams.

3.2 DEVELOPMENT OF A HIGHER ORDER THEORY FOR BEAMS

Consider the beam loaded generally by transvcrsc loads

with coordinate di rect ions as shown i n Fig. 3.1

Fig. 3.1. (kncral J ~ a d i n g on Ream

3.2.1 . - I)isp&cemcnt ,and Strcss S ta tes --------

Assuming a Jisplacement s t a t e i n the Corm of ~ ~ ~ I N T I - ortlcl-

pol.ynomials in the depth coordinate, y

where u 0'

. . . , v2 a r e coef Cicients which am T i m t i on^, o l

.x only.

A stress s t a t e consis tent with eqn 3.5 above nrtl..:t i t 1 so h

obtained. This can be achieved by modifying the c- l ;I..:<; i (-,I I So I - I ~ ;

* by assuming higher order polynomials i n y , the coc i- l ' i L- i c w t s o I'

these polynomials being a rb i t ra ry functions of x. '11~ ' po 1 ynomi :I 1s

should be such t ha t the resu l t ing s t r e s s s t a t e comp 1 y \V i t 11 tlx'

de f in i t ion of c l a s s i c a l s t r e s s resu l tan t s while s a t is l'y i rip t h r

boundary conditions on the top and bottom of the beam i . v .

'ox dy = N

'yox dy = M

T = O a t y = + h/2 ---------------- 3 . 0 v CI y = -q a t y = -h/2

o = -p a t y = h/2

This gives the s t ress s t a t e as

where the coefficientsN,M, ..., T are functions of x on ly .

I f the l a s t two terms are deleted from eqns 3.5 and 3 . 7 the. followinq

dkplacement and s t r e s s s t a t e s which correspond to t h e c. 1 c w n t n r v

f o m are obtained

I f they are regarded as the f i r s t order assumptions, ~ h c . 1 1 t.r'<crlt

assumptions containing two higher terms (one odd and onv c.\ t 11 t ( ' 1 - n )

are then the third order assumptions,

3 . 2 . 2 Derivation of Governing Equations

The governing equations f o r the problem are o b t ; ~ i r ~ c ~ ~ l I ] \ ~;III)-

s t i tu t ing the displacement and s t ress expressions (eqns .;. ' I , I I K \

3 . 7 ) into the two dimensional version of the variation;^ l ( . ~ I I I:I t i orl

3.4 and integrating with respect to z and y. Grouping I l r c I t . c ~ ~ i ions

gives the following :

I t follows t h a t each o f the expressions in bracket is zero,

which y ie ld the p v e r n i n g equations m d boundary c o n d i t i o ~ l s Por thc.

problem. 'fie expressions appear i n such a Corm t h a t the synmtr ic-

and antisymmetric p a r t s are uncoupled rmd hcncc can bc considcrcd

3.2.3 Antisymmetric Equations - - - - - - - . - -

The equations governing the antisymmetric (bending) problem

for a beam arc 3s follows:

The associated boundary conditions a re ,

M = M o r u is prescribed 1

b

P = o r u i s prescribed 3 ............................ 3.11

Q = 0 or vO is prescribed

= i? o r v is prescribed 2

h Defining v* = v + - 0 20 v2 and eliminating u and u rcduccs cqn 3 . 1 0

- 1 3

to the following equations

d412 6(v+2) (q-p) + 280(~+2)T 8 4 0 P F = F ---Zli-r - - r ; T

Eliminating v2 and T gives

Differentiating twice to eliminate P and twice again t o c.1 imin:~tc

M gives an eight order differential equation in v*,p a n d ( 1 : IS

where V* = Vo h2

+ m v 2

All the other coefficients of the displacements and st r ~ s ~ ; c . ~ ; ; I rc. ( m i -

sequently expreesable in terms of v*, p and q as f o 1 1 . o ~ ~

3h2 dv* + 12(l+v)Q + 4 (l+v) B '-$"zo=-z r ---TIT- - -------- 7 - 2 3

3.2.4. Symmetric Equations

Equation 3.9 also gives the governing differential c , r l r l , l t ions

and their associated boundary conditions for the symmct r i c t>(~:t~n pro-

blem. The governing equati~ns are as follows,

'2 - 60vS - 180R d x - 737-

The boundary conditions are

N = fi or u is prescribed 0

A = A or v is prescribed 1

The s i x f i r s t order d i f ferent ia l equation and one algebraic equa-

t ion of eqn 3 . 2 4 can be reduced to a s ix th order equation i n u*, p

and q , where u* is deffned as

Eliminating v from eqn 3 . 2 4 gives b

1

1:limincrting S ;md d i f r c r e n t i a t i n g oncc t o eliminiltc N g ivcs

'fie other c o e f f i c i e n t s of displacements md s t r e s s e s b e i n g ;

3.3 FIR5T ORDER TlIEORY

Equation 3.8 s ta tes the displacement and s t ress s ta tes cosrespond-

ing t o the f i r s t order theory. The governing equations are obtained

from those of the third order theory (eqns 3.10 and 3 .24) by deleting

the terms introduced by the higher orclcr components of s t rcsscs and

displacements. The resulting equations are rour f i r s t orclcr tl i rfcrcn-

t i a l eqyations for the antisymmetric case and two for thc symmetric.

These give a fourth and a second order d i i fe rcnt ia l equations for thc

antisymmetric and symmetric cases respectively.

Equation 3.34 is equivalent t o Timoshenko's equation Lor the

governing equation for the deflection curve of a beam i n bending

with the shear deformation taken into account (see eqn 2 . 2 ) .

CHAPTER 4

SIMPLY SUPPORTED BEAMS IN BENDING

In th i s chapter the f i r s t and third order theories arc applicd

to transverse bending of beams with simply supported cclgcs ( F i q 4.1)

Fig 4.1 Simply Supported Beam Problem

This problem is a special case of the general beam prohlc~n I'or which

governing equations were derived in Clapter 3. Mrikinl: thc sul )s t i t~~t ion

p = o in equations 3.14 to 3.31 yields the equations for t h i s pro-

blem.

4.1 ANALYSIS

4.1.1 -- Third Order lheory

The governing d i f fe rent ia l equations fo r the anti syrmnctric 'md

symmetric problems by the third order theory are

where

The solutions of these equations are taken in t11c. 1'0 r v ~ o f

inf in i te ser ies as:

- where v* = Ev* -- A = - EA

so'? Y

qoQ

- Eu* u* = - EB Y

j j s - qoR 90

and % = fourier ser ies expansion of the loading fiincti(irl.

substituting eqns. 4.3 t o 4.5 in to eqns. 4 .1 and 4 .2 m t l c . r l ry ing

out the differentiations gives the cdeificients A and 1! , I L .

- c2 = 1/45 ; C3 = 1 2 ; where, z = h i , C1 - 3960;

All the.other unknowns are obtained as follows by subst i tut ing f o I for v", u* and q i n cqns 3.15 to 3 . 3 3 .

-- q m s i n ax ---- = rCm z5a6+n z3n4+n3za2)r\-(12 +I) z2a2+D z*a4]~ -- -- 4.8

q$l 1 2 4 5 6 qo 9.

1 1 . 1 where Dl = 4x2F ; D2 = To , D3 = - 6v+l3

1 2 ; D 4 = m-

P E [(-n7z 'a6-D8z 3 a 4 + ~ za2) 8- ( D ~ ~ - D ~ ~ z ~ ~ ~ - ~ ~ ~ z ~ ~ ~ ~ ] - 'i, s i n x- wi 9

---4.11 90

n cos a x - --.q . I 7 = 1 [(II~Z~CX~+TI Z'~~~-T$Z~~~)~\+(D a-D z3a3-~)12~iasfl - - Y,$ 8 10 11 qo I!

99v2+352v+210 . - 784080 (

- 99v7+247v+30 where 1)20 = v+Z) ' 8910(v+2)

N 9, s i n cxx - = ~ ~ ( f ~ z ' a ~ + P z 2 a 3 +a)B- ( F ~ + F ; ~ ~ ' ~ ' + F z'a')] - --- -------

2 5 9 4 . 1 4

3 90

26 2 7 s, - (F24-F25~2a2-~ z4a4+F z6a6)l- Sin -- ax ------ 4.18 90 R

4 .1 .2 F i rs t Order Theory ----

The governing different ial equations based on the f i I.~; t ordc r

assumption of s t resses and displacements are deduced Srorrr cyrls. 3.34

and 3.37 as

but v = v 0

therefore taking the ser ies solutions as

- ax v = ~ii Sin - R

s, C1X q/qo= E - Sin - 90 R

E A Ev a n d & - where = - qoQ 'LQ

substitution into eqn 4.19 gives A as

Accordingly the moment and shear are given by

2 3 - 01 x 1 ' ; (v+2)ia]% . cos - ---------- 1 2 . A - -lo, R %h

4.1.3. Loading

Two loading conditions, a t o t a l uniformly d i s t r i h 11 c \ , ! i ) ; ! ( I in)!,

and a central point load are considered. For the un-ifo!-u l I i s t r i -

buted loading, the fourier coefficients, %, are given I,,

4q0 = 1- Sin mRr % ma -

R

for m = 1,3,5, . . . For a central point load, P

. - - P - 2 Sin m 0

2-

b

4.2 NUMERICAL RESULTS

Numerical resul ts were obtained with the aid of c-t m > ~ i I a I.:-; I'or

each of the two loading conditions and for different t l ( ~ r a . ~ i t o

l cng th values. Prior t o t h i s , the convergeme pat terns o r the

n x r r n m t and tlc f lect ion werc examined by cornput ing thrv for d i f Ter.cnt

n1urr1w1.s of' tcnns ol- tho scrjes summation.

4.2 .1 Uniformly I l is t r i butcd Loading ---

The numeric 11 resu l t s fo r simply s~ipported be'ms under a uni-

formly dis t r ibuted loading, q , over the whole span a re prcsentecl 0

in l'ahl c s 4 . 1 t o 4.5 and Fi pyres 4 . 2 and 4.3. I 1 4 . I shows

the convcrgencc patt-crn Tor M and v both by thc f i r - s t and t h i rtl 0

order theories lor two depth t o lcngth ( h / ~ , ) ra t ios . Tab le 4 .5

cornpares the values the t h i rd order analysis gives for def'lection a s

dcf ined by

h h ' (i) v 6t y = h/Z) = v0 + ~ v ] + -v 4 2

( i i ) v* - h2 - vo + 7j-jv2

(iii) the midplane deflection = v 0

Tn Table 4.3 coml,nrizon is made betwen the maximum values of non-

climcnsionnlised momcnt and dcflection obtained by the c lass icn l ,

f i r s t order, t h i rd order theories. Some r e su l t s prcsentcd by 0 j iako

( I !jH5 ) who usclcl tlic Method ol' I n i t ial Furrctions (MTF) :I rcs ;I] so

inclutlcd whc3re the h / R r a t i o pcrnlits. Figs 4 . 2 and 4 . 3 show thc

h r i : ! t i o n s of the strcsscs across the d e p t h oT t l ~ a ])e;,m ~ u h i ] ~ ~ ~ h l p ~

3 . 3 and 4 . 4 show t h c n l : l s i r n m v :~ lucs 01- these. st!-cssc.:;.

Table 4.1: Convergence P a t t e r n s of Moment and k f l e c t i o n

a t Midspan

(a) h/R = 0.01 - I

- Third Order Theory

156284.34 156283.61 156283.64

0.1251 0.1250 0.1250

F i r s t Order Theory

0.1251 0.1250 0.1250

.. , .

No. of Terms 5 10 1.5 20 30 - I I 'I'hircl Order Theory

I I F i r s t Order 'I'heory

1 H

T;itle 4.2 Values of Ivkxirum Non-Dimensionaliscd 1kTlcctior~ ; I I ~

Wment for different Ikpth t o Length Ratios.

(a) Deflection, - Ev

'l'heory w Classical

Third Order

05. Difference

Fi rs t Srder

% Difference 0.022

MI F I - .% 'Difference

(b) Moment, - M qOe

I c lass ica l 0.125 0.125

Third Order 0.1250 0.1250

I d Differenccl - I I - .-I Fi r s t Order / 0.i25 / 0.:25

I % Difference

(h) h/11=0.05

Thi rd orcler Thcory "

F i r s t Ordcr/Clsssicrrl "ihcor i c s

F ig 4 . 2 Var i a t i of' I.my,-i t,uclin:\l Nor-rlnl S t r c s s , ox/q, ac ross t h e l h t h o r P,c;un

SO.

I '

[a) h/R=0.01

b i g 1.3 Variation of Shear Stress, T /q, m d Transverse Nonr!:~l Stress, al,/q, XY

1 1 .

Table 4.3 Values of Maximum 0.-/q for cli TTerent h/l(, v a l t ~ c s .

Classical

Depth to Length Ratio 0.01 0. 05 0.1 I 0.1s '

Third Order

% Difference

Fi rs t Order

1 % Difference

7500

I MIF

I % Difference

300.00 --

75.00 1 33.33

Table 4.4 Values of Maximum T /q for diffkrent h/% values xy

- - -

, Depth t c

0.01 1 0.05

Classical

Third Order

% Difference

Fi rs t Order

% Difference

MI F

% Difference

Table 4.5 M a x i m values o f v , v* and v by t h e Third O r d e r Theory 0

f o r d i f f e r e n t h / ~ va lues

11

v* 0

v 0

Depth t o Length Ratios 0.01

156280.03

1 156282.92

156383.64

0 .3

6.764

0.05

1256.00

0.4

3.160

6.883 3.262

3.263 I

6.894

-.

0.1

159.244

0.2

21.015

1256.58 159.541 I

21.177

1256.73 I

159.609 21.204

4.2.2 Central Point Loading

Numerical r e s u l t s f o r the loading case of a point load located

a t the cent re o f the beam a re presented i n Tables 4.6 and 4.7. 'I'ablc

4.6 showsthe convergence pa t t e rn while Table 4.7 shows thc rlnximum

values o f de f lec t ion and moment by t h e c l a s s i c a l , f i r s t . and t.hird order

Table 4.6 Convergence Pat terns of Merit and Deflection a t Mid span.

(a) h / ~ = 0.05

(b) h/Q = 0.1

I

F i r s t Order Theory

30

20L.20 1 LOl . i . lO

Third Order 7'heor-y 1 256.472 256.433 256.355

I

I NO. of rrcms

F i r s t Order ~heor-yl I

10 5

M

1s

0.2399 0.2449 1 0.2466

v . 0

Third O r c r Thcory

LO1.5.0'3 2013.18 0 1 2 . 5 7 , I

54

Table 4.7 Values of Maximum Non-Dimemionalised Deflection and Momc~n t for

Di f fe rmt Depth t o Length Ratios.

Ev (a) Deflection, - P

Class ical ( 250000

% Difference Third Order

I F i r s t Order 250061.03 2013.19 % Difference 0.02 1 / 0.66

M (b) hment , - PR

Classical 0.25 0.25

Third Order 0.2449 0.2449 % D i f f e r e n c e l 2 . 0 7 2.04

F i r s t Order 0.2449 0.2449 %Dif fe rence 2.04 1 2.04

The comparison of the maximum values of deflection and mori~ent obtained

from the c l a s s i ca l , f i r s t order, t h i rd order and MIF theories (Table 4 .2)

show good agreement fo r values oi- h/R < 0 .2 . 'lhe d~f fe rences beinp generally

within 2%. The other theories gave higher values of dcflection than clocs

the c lass ica l theory. Using thc t h i rd ordcr tllcury as thc clat~~m, the

classica 1 theory unclerest~mtcs deflection, thc f l r s t orclcr thcory s 1 iy,I\t 1 y

ovcrestimatcs lt whilc the b \ 1 1 : yives the highest values for dcl-lecL ion. I:os

depth to length values greater tlxin 0.2, a t which range the c1:lss icLi 1 theory

sei zos t o give acceptable r v s u l ts i n cornparLon w i t h cr net- mcnt:rl r - ~ u l t s ,

the r e s ~ ~ l t s ohtnjnccl through the other tlrcor-jes d i f f cr from t h c ~ l : i s s i ~ : ~ l

resu l t s by 1~p t o 10';. Thc MlF :md the f l r ~ t orclcr Jicvry :I~:I'CT w 1 111 t lx?

t h i rd order thcory even a t t h i s r~vlge of hcnni th'ich1,c.';~.

The ;lrtssicnl and f i r s t order theorjcs cive t11c stltrlc valucc; anl.[

t r ibut ions f o r the longitudinal normal s t r e s s , a a r d shear st rt1:,s, .I . x' X)'

The nuximum values oP u m d -r gi ven by the th i rd orcler thcoly w e 35; X XY

greater and 10% l e s s r c s ~ e c t i v c l y than then the values from the c lass ica l

or f i r s t order theory. The c l a s s i ca l theory neglects the transverse normal

s t ress , a while the MIF, the f i r s t and the th l rd order theories Fiyree Y'

in their d i s t r ibu t ion of a Y'

In 'Table 4.5, the comparison of the maximum values of v, v* and vo

show tha t these def ini t ions of def lect ion generally give values close t o

&ch other. However v* gives a closer estimate of v than vo. A

replacement of v with v* w i l l give deflections within 1hccu r ; l cy for

beams tha t are not too thick.

The second loading case of a central point load gave resul ts that

generally followed the patterns observed for the uniformly d is tr ibutcd

loading case.

CHAPTER 5

BEAMS ON ELASTIC FOUNDATIONS

5.1 GENERAL

The bean theories developed i n chapter 3 arc apnl i d i n

t h i s chapter t o the solut ion of bending problems of hc:ms r - c s t - ; t i n?

on e l a s t i c foundations. The general problem of a 1)c:m rest in!>

on an e l a s t i c foundation i s as shown in Fig. 5.1

Fig. 5.1 Beam rest ing on an E la s t i c Foundation

Because the beam i s continuously rested on the foundnt ion ~l~, l tcr i i l l ,

the loading p w i l l be induced on the beam as shown. In thi.; case,

therefore, p i s the foundation reaction pressure. Thc I'o r-IT\ of p

is dependent on the e l a s t i c foundation model adopted, as cliscussed

in section 2.3. Generally thb one and two parameter fo~lniI;~tion

models can be mathematically represented as

where eqn 5.1 gives the one parameter (Winkler) Foundat ion

when kl= 0. Recalling the governing d i f f e r en t i a l rquo-

t ions f o r the f i r s t and th i rd order beam theories.

F i r s t Order Theory Equatiom:

'Ihird Order Theory Equations :

For the f i r s t order theory v=v therefore ? becomes 0

?his enablesthe antisymmetric par t of the governing eqwt i c , ~ ; ~

to be modified to include the ef fec t of the foundation I - ~ Y I ~ T I O I I .

In the case of the third order theory, defining v ;I\ I!I( clc-

flection a t the bottom of the beam ( i .e . y = h/2) givc.5

Introducing th i s expression into eqns. 5.4 and 5.5 w i l l m i x

symmetric and antisymmetric expressions moreover the m:lt I l cm:~ I i c:;

length of the cxpress i orls 1.1 I r - w i l l become impossible given the

v v1 and vZ. 0'

Due t o th i s diff icul ty, a s

considering only the antisymmetr

v a s

implification i s introt111c.c~ 1 ' ) \ I

i c equation (cqn 5 .4 ) ; i r l r l t l ( . l i n i np

v = v* ......................................... 5.8

The resul ts presented in Table 4.2 jus t i f ies th i s simp1 i l'i c , ~ t ion

- because of the closeness of the values of v and v* for t lw 1 x ~ m

depth to length rat ios considered.

5.2 SIMPLY SUPPOl?TED BEAMS RESTING ON ELASTIC FOUNDATIONS

BY THE FIRST ORDER THEORY

Simply supported beams resting on e las t i c foundnt i (XI:; : I I ~

loaded by ( i ) uniformly distributed load, q,-, and ( i i ) cc.rr1 r-:I l p o i n t

load, P, are analysed by the f i r s t order theory. Thc r e s ~ l l t ~ , ;ire

compared with Hetenyi (1946) resul ts f o r the same prohltm. lk-

cal l ing eqns 5.2 and 5.6 and modifying f o r a beam of width b ,

(where v = v ) gives the governing equation a s 0

The moment comes out as

For a simple support problem, t h ~ solut ion is taken i n the Corm

of an i n f i n i t e se r ies . Following the s teps taken i n sect ion 4.1.2

% - = C - Sin ax 9/90 , , q -

0 R

whcre a = m %

- E A and A = --

qoa

all tlic o the r expressions a r e obtained i n sequence.

Taking the loading functions a s in sec t ion 4.1 . ?, Sor ;I

m i f~onn ly d i s t r i b u t e d load over the whole span and ; I ucnt rnl

point load.

7 2 I:l a s t i c Modulus, I:, of 1x10 kN/m . 'I'hc f ' i r s t 1'ound;rt ion p : ~ r:l-

4 2 mctc%r, k,of 1x10 KN/m was usccl whilc thrcxc v:lluc\s 0 1 - t h c sc.contl

louncht ion p a r m u t e r , kl ( lW0. S(X1 :md O kV) were t is(~1. 'I'lir

k = (1 option representing a Winklcr Soumd:ltion nlodel . '1'c.n tcrrn:, 1

were used i n thc s c r i c s stlmmat ion. Rcsul ts wcrc corrrp;lrc\tl ~ v i t h

vr111l.s~ obtained Prom expressions s iven by l lctcrlyi ( 1 9 4 0 ) , So r

d i f f e r e n t depth t o length r a t i o s a s shown i n Tables 5.1 xncl 5 . 2 .

7"n ccxprcssions givcn by lletcnyi a r c a s follows :

1. Sinply supported 13eam on E l a s t i c Foundation carryinc: a

IJniPonnl y Distributccl I,ond, q

but X Q = 4 E

where z = h/R

b. Maximum moment ( a t midspan)

Sirih XR Sin XR

c. Maximum shear (a t suwor t )

= q Sinh X R + Sin XR 2 1 rosh XR + Cos XR

2. Simply Supported Beam on Elas t ic Foundation carrying

, a Central Point Load, P

a. Maximum deflection ( a t midspan)

v = PA Sinh X R - Sin X R

Cosh X R + Cos X Q

moment (at midspan)

Sinh 1% + Sin XR Cosh XR + Cos XR

1 Sinh XR + Sin XR M = - PR 4X R Cosh XR + Cos XR -------------- 5. l o

c. Maximum shear (a t support)

Cosh XR Cos X R 7 T- L ,Q = P L

Cosh X R + Cos XR

Table 5.1 Simply Supported Beam on E la s t i c Foundation carrying a Uniform

Load, q

(a) h /R=0.0667 , XR = 3.32, k = l O O O O k N / m 2

1- I Max Deflection I I

v -' iletenyi' s Results 1 86.470

I Fi r s t Order Ann 1 ys i s I kl = 15OOm 1 71.121

( ",)if fe rence f 17.75

kl = 500 kN

$ Difference

(b) h/k = 0.1, Ap, = 2 .21 . k = 10 I - -7- ( t ie tenyi ' s Ilesul ts I I F i r s t Order Analysis I kl= 1500 W

I % Difference I 1 k l = 5mkN

I m i f i e r e n c e I I k l = O k N

I % D i f Ierence I

M a x Moment M $'

5.3 JrtU :I:- F-NDED BEAMS ON El. A?' TONS BY TIE TI-IIW @P?ER THEORY

Vc~;ic r m d Johnson (1.903) rcoortecl rcsults Exnm an clahorntc experi-

Y' Fig 5.2 Location of Point Loads

The three se ts of loading on each beam are shown i n Table 5.4 where

the unit load P = 18.37 kN

Table 5.4 L,oading Combinations on Beams

The beams were resting on a foundation of compacted m~c-:rccorls s i l t

of E = 8218.58 kN/m2 and v = 0.29. The foundation constants used i n S 5

the analysis for a Winkler and a two-parmeter foundat icm r-t.p-resentn t ion

i ) Winkler foundation model 0.65 6

k = S

9-"f

= 41 70.03 M/ma fo r the Channel Beam

- 3310. SO k~/rn' for the Wide-Flange Beam

( i i ) Two parameter foundation model (using F l e t c h e r ' s curves,

Fig 2.8)

5.3.2 Solut ion -a-

liec?lling governing equation f o r the antisynunctr i c problem hy

the t h i r d order theory (eqn 5.4) and modifying t o incluclc t h e

CouncL7tion react ion pressure a s dcrined i n eqns 5.1 and 5 . 8 , I'or ;I

beam of width b , gives

The moment expression (eqn 3.15) is a l s o modified t o give

The solution of the governing equation f o r the problcm (ccp 5.18)

is taken as a s e r i e s of the charac te r i s t ic function r-oprcrxmting

the beam's mode of vibration. For a f ree ended bctlrn tho c-1i:lrxctcr-i s t i c -

function i s given by

where the values of u and h for dif ferent nl valuci; ; ~ r c j:ivcn i n m

Table ,5. 5. Values of % and Am appearing in

Therefore, *

C1 X C1 X m u x X nl m ~ i n " r n ~ ;* = 1 [X (cos~$- - A sinh-) + $ (Cos - , - - 1 - 5.21 n~ m m Q, m 1,

Am and ljm being coefficients of the deflection funct ion wit I lr

qn/qo is the non-dimensionalised fourier coeificient~ o l ~ h o

load function.

Substituting equations 5.21 and 5.22 into the governing c.cli~:~t ion

8 gives % and Ern as

where

Equation 5.19 gives

where

The expression fo r the contact pressure becomes

The fourier coefficients of the loading function are ca lc~r la ted

noting that fo r a single point load, P , located a t a d i s t a~)ce x/R=n

on the beam

lh dx 6 S n = c-a

n-c m

= E[(COS R %n + Sinh sn)-Am(Cosh amn+Sin an;J

For a s e t of three point loads, P1, P , P3, located : i t * i n , d y I 3 2

respectively

where R. = - 1 P

P = hit load

5.3.3 Numerical Results -.------

The expressions for deflection, moment and cont ; I L I I I l x > , s l r c .

were computed using 20 terms of the ser ies summation. 1 1 l : i h l c a \

5.6 t o 5.8 the m a x i m values of moment and deflcctiorr 1' i the

Winlcler and two-parameter foundation models were conrl),~ I l w i t h t-hc.

experimental and conventional analysis resul ts prcsc-n t c , 1 hy

Vesic and Johnson (1963). Figures 5.3 through 5.11 show t 1 I 1) l (it

of two-parameter foundation resul ts by the ?resent an, t l L i n

comparison with the experimental and conventional rcL I I 1 t

The curves fo r the conventional resul t s are omittcd wrtl.cS tllcv

cannot d is t inc t ly be included.

Table 5.6 Results from different Analyses for the Wide I:lanlr,c

Beam.

la) Maximum Bending Moment (KN-m)

LOADING CONVENT I ONAI, SOLUTION

20.033 18.624 16.778

-- -- EXPERIMENTAL RESULTS

19.529

-12.853

4.156

-3.826 -

(b) Maxim Deflection (mm) --

I -

2P

1 . 5 P 3 P

4P

-

-

2P

1.5P r

14.28

16.03

29.08

12.83

12.83

19.56

r

13.86

14 .97

26.00

15. 20

17.20

33.10

Tabie 5 , 7 R e s u l t s from d i f f e r e n t Analyses fcr t:ie F l a t Ceam

(aj Maximum Bending Moment (kii-m)

Table 5.8 R e s u l t s from d i f f e r e n t Analyses f o r t h e Channel Remm

I LOADING

(b) Maximum D e f l e c t i o n (nun)

(a ) Maximum Bending Moment ( H - m ) -- . - - -

P1 I

/ - -

1, l .5P

-

EXPER IMEN'l'AL RESULTS

- - -- - ---

PRESENT RESULTS

F'DN

(b) Maxim D e f l e c t i o n (mm)

I 0 .88 16.51 25 .20 - -- -. - i

CONVENTIONAL SOLUTION

1.51' r P2

2 1 1 . - -

.

I PRESENT RESULTS I PJ 2-PARA FlN WINKLER 1 SOLUTION ESUCI'S

- 3

P 3.638

-3.197

2P -

3P

3P

.

4.094

-2.176 4.858

- 2 . 3 ) 5 - - -- - - -

8.72

14.74

24.11

-

P

1.5F

1 P

4.406

-2.153

I . - ---

4.703 I 4 . 180

-2.124 -2.339

-- - . - --

14.73 14.71

22.96 20.57 -- 25. 80

7.210

-5.621 4.362

1.5P 4.837

-2 .04!) -

.

r n N -- . 7.437 7.626 6.937

-5.970 -6.350 A -- -- --

3.014 -

4.201

-2.546

-

-3.897 -4.531 I 1.5P

P

3P -3.025

Fig. 5.5 Iiesults for. Wide Fl z?~:;o !:cnm Fig. 5.6 Kes~llts for F l a t Re<m *

w i t h Three Loads. w i t h One Load

I . 5.8 Resu l t s Cor F l a t Beam

wi th Three Loads.

6 I i

.-- .. i

" , ~ i B i i j /A

\\. ' .

5 i j ,/:,;/y.:.::-., ,,.,,, 1 .,* - ' , -". ,,.'. ...- .. I... \

c. . ,: ' , /$/' '%>:, , . 14 *\ t ,/,+ ," , , i . ' .,,,.

rn CI +\ ! ; ,<;,

*,; ,., / / ;; G ",! ?\$ 1 , 1' & * , ; ':? i. i *? .

i s . . I 1 ~:csults Tor ( lemncl k : m i w i t h i'hrec Loads.

5.3.4 Discussions

The three beams analysed f a l l under different classes based on

the i r character is t ic lengths (see Table 2.1) while the Wide Flange

and channel beams are a t the two ends of the class of moderately short

beams (0.80 < X R < 2.25), the Flat beam is i n the class of moderately

long beams (2.2 5 < X R < 5.00) . The analyses for the Wide Flange beam

show, for a l l the three loading cases, a concave parabolic pressure

distribution with concentration of pressure a t the edges. This i s i n

agreement with experimental resu l t s and i n contrast w i t h the convcntionnl

resul ts . While the conventional analyses predicted a uniform d i s t r i -

bution pressure and deflection under the beam, the prescnt resul ts show

that the maximum dcfl ection occur a t the midpoint ( o r a t the cdgcs for

loading 2) while the maximum pressure occur a t the cdges.

The f l a t beam being very f lexible (XR=3.96) gave prcssure d i s t r i -

butions that in a general manncr followed the distributions of the

applied loads. Due to the f l ex ib i l i t y of the beam, thc rcsul ts by the

present analyses are in agreement t o a good extent with the rcsul ts of

the conventional method of analysis. There were noteablc differcnccs

only when the beam was loaded with three point loads.

The channel beam by i ts re la t ive f l ex ib i l i t y represents an inter-

mediate case between the other two beams. Although the pressurc dis-

, tr ibutions I'ollow, to a ccrtain extent, the dis t r ibut ion of cxtern:~l

loads on the beam, there is also some pressure concentration a t the

edges even i n the case of the central point load. Again there arc

variations between the conventional analysis results and the

results of the present method. These are however moderate for the

loading cases of a central mint load and two end loads.

O T E R 6

ONCLUS ION

The variational principle of Reissmr has been usetl to obtain

equations for a higher order theory for beams. The s trvss :mi dis-

placement s ta tes of the classical theory were modi rictl 1 o i ncludc

two higher order effects . The theory was applied to t l ~ ~ lwrding of

transversely loaded beams and beams resting on e l a s t i c f'ormdations.

Results were compared with resul ts from other sources.

The f i r s t and third order theories were found to irwrove

resul t s of the elementary theory when depth t o length t-itios are

high. The th i rd order theory proved superior t o thc ot1lc.r theories

and with it solutions could be extended for beams oP dwth to length

ra t ios of greater than 0.25.

The third order theory with a simplified definition of deflec-

t ion was used fo r the analysis of a s e t of free-entletl I K am5 resting

on a Winkler and a two parameter foundation models. Qc.;t~l ts for the

two parameter foundation model were plotted against cxncl-i~nental and

conventional analysis resul ts presented by Vesic and Jol~ilson. Again

the usefulness of the higher order theory was seen, nos t cspecial ly,

for short beam (beams of lower X R values).

The strength of the approach used i n t h i s work l i e , , i n thc iact

that the variational theorem of Reissner can be uscd c-oriveniently

to derive governing equations for a theory when the str.ct\s :mcl di5-

placement states can be initially assumed. Future work can extend

this ap~roach . - to shells, domes, etc.

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