FINTTE ELEMENT TECHNTSUES

APPLIED TO THE ANALYSTS OF

BUS BODIES

by

A.F. CLOSE B.E. (Hons. )

A Thesis presented to the Faculty of Engineerinç; of the

University of Adelaide for the deglree cfMaster of Engineering

Civil En¡,¡ineeri-ng Department

Universlty of AdeJ-aide.OCTOBER, I9?5.

t

CONTEI.JTS

SUMMARY

ACKNI]!'JLEDGEI\lENT

SECTION 1:

SECTION 2:

Paoe

iv

V

2

4

4446

66

10

2,r.¿¿¿

TNTRODUCTTOI'I

REVTEI,,J OF LITERATURE

Introduction.1 .1 . Buses vuith stif f.I.2. ComposÍte buses.1.3. Integral buses

ethods ol' Analysis.1. I'lon-Crrmputerized Methods,2. Computerized methods

oading¡s and Dynamic Effects.1. Self-r,veight and Passenger Loads.2, Dynamic Loads Due to Uneven Road Surfaces.3. Acceleration and Bralcing Fnrces.4. Accident ProtectÍon Requirements

3.l. Investi-qation of Stresses in Steel- Beam urithCu'bouts.'

3.1.1. Description of beam and methocl of test:Lnq3.I.2. Finite Elernent Anal.yses3. 1 .3. Experimental results and com¡rarison vrith

Analyses3.1,3.1. Deflections3.1.3.2. Stresses at El-enrent centroi.ds3.1.3,3. Strains at points other than element

centroi-ds

3.2. Investlgation o1' Stresses in the Epoxy l,rlodsl.CornparÍson wi bh Finite Element Analysis

chassis

2.2. ¡,ri

2.22.2

2.4. Interpretation and Anaì-ysis of Fìesul-ts

SECTToN 3: INTTIAI- TNVESTTGATTONI 0F ACCURACY 0F FIIJITEELEÍ\/IENT AI'JALYSfS

19192A2324

25

2e

28

2,3. L2.32.3¿.Jt.)

2q32

33

33344I

\-t .I

.)

?_.I2.22'7

Descripti.on of beanr and method of testincrFinite El-err¡ent Analysj sBesults o1= tests and comparison v,rithAnalysis

3.2.3.1. Stress prcdictions at elemenl- centres3.2.3.2. Stress predj-ctions at surface of beam

42

4244

44

4545

55

55

56565959

SECTIOIi a: FINITE ELEI,iiENT ANALYS]S

4.I. Introductionrogramrne !cscription.1.. El-ement handl-ing and storage.2. Description of the Seven Subroutines4.2.2.I . Fornlation oF basic stiff¡ress matrices

4,2. P

4,24.2

ii.

Output ofl formed elementsf nput of' Formed elernentsEl-ement rotationAddition or combi-nation of elernentsReductionSolution

4.3. The Effectiveness of Super-elements in ImprovingProgramme efficiency

4.3.1. The Effects of the use of repeatedunreduced super-elenlents urpon the t:i.nrctal<en to form the stil'f'ness matr"ix oflar¡er structures

4.3.2. The Ef'fects of repeated reduced Supor-el-ements on Problem soluti-on time

4.3.2.I. Factors r,vhich reduce the eflficiencyof the use of reduced elements

4.3.2,2. The relationship between the efficiencyof the method and the percentage ofnodes retained in the reduction

4.3.2.3, The relationship between efficiencyand the number of reduced super-el-ements in the structure

4.3.2.4. Summary of the suitability of thereduced Super-element method

4.4. Other Applications of the Element Reduction Routine

5.6. Summary

STRAIN I'.,,IEASLJREÍ\4EI\TS TN BUS BODY

I ntroductionPosi-tioning of Strain Gauges

Computer Analysis of' the Static Tests

Static Tests

4 .2.2.2.4.2.2.3,4.2.2,4,

4 .2.2.2 .

4 .2,2.5 ,

4.2.2.6.

Page

616I6I61626?

79

81

81

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a2828591

91

91

96

96

IO?

LTz

114

114

114

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116

6B

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7L

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74

16

SECTIOI.I 5: DETATLED ANALYSIS AND TESTING OF CRTTICAL SECTIONS

Êr I ntroductionFinite Element IdealizationConstruction of Photo-elastic MocleI

Photo-elastic material-Fabrica'bionThe testing of the photo-elastic model

5.4. Discussion of Experj-mental and TheoreticalResults

5.4.1. Displacement of'points on model5.4.2. Strains at the Bosette Strain Gauge

Locations5.4.3. The Inclination of the Principal Stresses

and the Dj-flference .Ln magnj tude betrveenthe tvro Princi-pal Stresses

5,5. fnvestiç¡ation of the Accuracy of More DetaiLedElement [4eshes

E-¿té¿

5.3.

f:

E

3.1.3.2.3 .3.

I2.)

LI

6

6

tr

6

SECTTOII 6:

Jacl<ing the bus from the rearLoading behind the rear axleLoading between the two axlesLoading forward of the front axleDisplacing the wheels

6.5. Effect of the additÍon of extra stress panels tothe rear door pillars

6.6. Dynamic Testing6.6.1. Determination of the estimated static strain6.6.2. Dynamic strains measured when the bus was

driven over flat blocks6.6.3. Dynamic strains during normal running

conditions6.?. Summary

SECTfON 7: GOI{CLUSIONS

BÏBLIOGRAPHY

6.6.6.6.6.

4.I.4.2.4.3.4.4.4.5.

].LI.

Page

116t22I22131135

135

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141

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151

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158

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1V

SUfVIMARY

The airn of the investigations described in this thesis

was to examine the problems invol-ved j.n analysing bus bodies by the

finite element method. A number of facets of 1;he probl.em have been

examined. The improvenent in stress prediction with cl-oser el-ement

subdivision and the accuracy of stresses predi-cted at points other

than the centroicls of el-ements r¡¡ere investigated by comparing the

observed and predicted stresses in two simple experinrental tests.

Since the cornputinç¡ time tal<en to solve finite elennent

anal-yses increases dramatically as more nodes are included, tv'ro

possible urays of obrtaining greater accuracy without incurring the

extra penalty were examined. A computer programme uJas vuritten

nrith the ability to removb internal nodes from blocks of elements

that are repeated several times in the structure. The efficiency

of this programme in decreasing the computing time required for an

analysis wlthout reducing the accuracy of the resul-ts is discussed.

Experiments were carried out to test the suitability of

j-solating critical sections and analysing them in detail and a

commentary of the resul-ts is given. Finally,static and dynamic

tests were conducted on a partially finished bus in an effort to

determine if any correlation existed betv¡een static and dynanric

stresses that u¡ould enable allowance for dynamic loadings to be

made in static finite element analyses.

The finite element method appears suitable for analysing

bus bodies and it is likely that it vril1 supplant previously used

rnethods which r,vere based on large simplifying assumptions.

ACKNOWLEDGEMENT

The author wishes to acl<nowledç¡e the encou¡3gement,

advice and constructive criticism which he at all- times received

from Mr. G. Sved, Reader in Civil Engi.neerinq at the University

of Adelai-de, who supervised the project.

Vlr

UNITS

During the period in which the work f'or this thesis lvas

carried out, the laboratories vrere converted from the imperi-al system

of units to the metric S.1. system.

Some of the e><periments described in this thesis were com-

pleted before this changover too!< ple.cr:. Bather than convert all the

results to metric equival.ents, it has been decidecl to retain the

original units for these e><periments.

It

-@@tIIII II

TtT] TTT'I6¿K)

PLATE 1.1. ONE OF THE BUSES OPEBATED BY THE MUNTCTPAL TRAMU'AYS TRUST

't2.

1. INTRODUCTTON

Work on the analysis ol' bus bodies in the Civil Engineering

Department at the University of Adelaj-de was begun at the request of

the Municipal Tramways Trust. The Trust, which operates a large

fl-eet of buses, had discovered that craclcing was occurring in the

frame members of their l-atest series of bus. The cracking had been

first detected in buses brouqht into the v'rorl<shops 1"ollowing minor

collisions, but, on later j-nspection, it was found that all the buses

of this series which were checked, had developed cracl<s in their body-

worl<. These cracl<s, which were concentrated around the doors and

windows, were causing the rivets fastening the exterior body panels

to shear, enabling qui-te easily detectable movement to occur between

these panels, The cracking, although not immedlately endangering

the operation or function of the buses would obviously increase their

rate of deterioration and therefore decrease their life. The crack-

ing was especially disturbing sj-nce it had occurred during the busesl

fi-rst year of operation. There was therefore a possibility that it

might lead to progressive failure spreadj-ng throughout the body.

The Tramways Trust were at the time looking for a way of

remedying thÍs fault but it can be seen that the problem is more

easily and cheaply prevented than cured. The problem is one of

deriving a sr-ritable method of analysing the stresses which are likel-y

for any given design. Because of the difficulty in analysing, with-

out a computer, the highly redundant and complicated frame of the bus,

and the difficulty of determini-ng the type of l-oads that should be

used in any analysis, the design of bus frames has been more a process

of natural selection, where successful features have been copi-ed and

failures have been di-scarded, than of analysis. This process v¡i11

lead to a reasonable solution given a relatively stable bus design.

3.

However, changes have been taking place in the layout and styling of

buses, that have Íncreased the likelihood that existing designs lviIl

be unsuitabLe.

Ever since the first all--metal frames began replacing

wooden ones, it has been recognised that buses which were constructed

such that the body and the chassis together supported the loads, were

lighter and more efficient than those which relied on the chassis

alone for support. Thj-s has led to a steady reduction in the size of

the maln chassis members and a need therefore for a body with increased

strength. Also the desire to decrease the percentage of the weight

of the bus that was borne by the front axle and hence to reduce the

effort required to control the bus, has led to the engine being placed

behind the rear axl-e. This, too, has affected the distribution of

stresses.

In addition, styling factors, such as the demand for larger

windov¿s and doors and the removal of internal bulkheads have affected

the design.

It was found that at least one bodybuilding firm was basing

the selection of the size of the frame members for its latest rear-

engined bus, on calculations carried out twenty years earl-ier for a

mid-engined bus. It seems likely that this practice rvill- lead either

to a bus which is prone to structural failure and costly repairs or

to a bus which is unnecessarily heavy and hence more expensive to run

and to construct.

4,

2. HEVTEW OF LITERATUHE

2.I. Introduction

There have been three distinct approaches to the problem

of providing a bus with suitable strength and stiffness. These

three approaches are:

2.L.I. Buses wi-th Stiff Chassis

This is the original method of bus construction and u,ras

used before the introduction of metal bodies. The chassis and under-

body are the maln structural members of the bus while the body is

usually made of wood and is flexible. The chassis members are large

and stiff in bending but the bus has very little stiffness in torsion

unless suitable torsion boxes are incorporated in the underbody.

2.I.2. Composite Buses

The composite bus is ç¡enerall-y a product of tvro different

manufacturers, the bus manufacturer and the body-builder. In many

ways a product of the traditions of the bus-building industry, this

type of construction al-Lows a large range of different buses to be

constructed on the same basic chassis. The bus manufacturer produces

the engine, the controls and the bus chassj-s onto which the body-builder

constructs a body which satisfies the requirements of the bus operator.

Buses of this type are designed'so that the body and the chassj-s act

together to support the loads. The all metal body is consi-derably

stiffer than the chassis members, especially in torsion and necessarily

carries a large part of any loading. The main members in a composite

bus body are the cant rails, which run above the side vrindows, the

waist rarls which run below these windows and the sill- members which

are placed in the side wal1 at floor l-evel-. Discontinulties occur

in the waist rail at door openings and in the sills at wheel bays.

These areas are usual-ly strengthened by the addition of stress panels.

5

abcd

ef

I

Cant railWaist railSilI memberWindow pilIar (connected to outrigger and transverse

roof member)Main Ghassis memberOutrigger

ç

FIG. 2.T. GENERAL LAYOUT OF GOMPOSTTE BUS BODY.

View from below

()

The side-wal-Is are fastened to outriggers which are attached to the

chassis members. Normally the outriggers, window pillars and trans-

verse roof members are connected to form rj-ng structures around the

bus. The ç¡enerar layout of a composite bus is shown in figure 2.1.

2.L,3, Inteqral- Buses

Buses of integral construction are those in which the whole

body has been designed as a unit to support the bending and torsional

loads. Ful1y integrar buses are usuaLry constructed by a single

manufacturer and are chassis-less or have consj-derably srnaller chassis

members than those found in other buses. The advantage of integral

construction is that fuIl utilization i-s made of the stiffness of the

body in both bending and torsion. Because of improvements in the

corrosion protection of the body during construction, the outer panel-

ling of the bus can now be designed to carry load. This advance has

enab1ed very efficient bus body structures to be built.

As more of the bus body is util-ized to carry Loads, so more

complj-cated analyses of the strength of the designs must be made.

The various methods of analysis that have been used for bus body design

are discussed in the next section.

2.2. METHODS OF ANALYSIS

2.2.L. Non-Computerized Methods

Because the bus body is such a complicated and redundant

structure, desj-gners wishing to analyse the stiffness and strength of

their designs, have in the past been forced to make many simplications

and approximatj-ons in their analysis. These simplifications have

consj-derebly reduced the amount of calculatj-on required but have also

affected the val-ue of the results. A number of papers on the

?,

analysis of bus frames using non-computerized methods, have been

published in European ,journals.

Michelbernu"(f) based his work on the assumption that the

body of the bus above windovr l-evel- does not contribute to the strength

of the bus. The stiffness of the side-wal-I below the window was cal-

cufated and was incorporated in a grillage analysis. This anal-ysis

considered only the chassis members, side-walls and floor cross-members.

The stiffness of the side-wall at a door opening was set to zero and

the effects of the location of the door on the stress distribution

in the underbody of the bus was determined.

Another European researcher in this field u,r= E"r(2).

Althouqh his original paper could not be obtained, a description of it

was given by Tidburr(t). Erz developed equations to predict the

moments in the critical sections of an integral bus subjected to both

bending and torsÍon. The side-walls were assumed to support all the

vertlcal loads because of their greater stiffness. Al-so the side-

wall- beneath the window was assumed infinitely stiff when compared to

the beam elements surrounding the windows and doors. The bending

moment in the side-v¡aIl- at the centre of the door was assumed to be

fes¡gted by forces in the cant rail and the door siII and therefore

the force in the cant rail over the door was calculated. This force

was distributed back to the rigid section beLow the waist rail by

shear in the door and window panels. The shear ì-n any of the pillars

ìflas assumed to be in proportion to its stiffness when compared with

the sum of the stiffness of all the pillars on the same sj-de of the

door. The equations which he developed are shown in figure 2.2.

The maximum moment in any pillar was cal-culated as

Mpr kQpr h

t I ^ù - ,t- 4,.- I

+P

-T--

P _Jlra-

PM

I

Qulu

lu+[-

h

-t

J

Approximate sheardiagram.

0

f orce

I0

Approximate bending moment diagram I

hr i

upright'r

-'l

II

1I

I

r'--

Qu

L2

u

zQLQpr= It p

Ilrh

Mu =0 75Qu L2

Mpr=l onr n,

Approximate formulae developed by = r(") for determiningmaximum bending moment in window-pi11ars and shear forcein door si1ls. Drawing from TIDBURY(3).

¡

FrG. 2.2 Mr0r

Mr-

@

I

MddJ-

T= Md

Thickness tassumed constan

gb

M

T

Resisting shear f low q=å

T

frontL ln,

l:Qtr qL

lrsrde

E

vtr= $

Qhh¡

Torsion analysis of EHz(z) assumíng the coach to be athin-walled tube with a transverse axis'Drawing from TidburY(S).

Ii

Il

FrG. 2.3.

l_0.

where Mpr is the maximum moment in the pillar

QPr is the shear force

h is the height of the Pil1ar

and k is a factor which depends on the stiffness of the cant

raif and the cant rail-window pillar joint. A value or 2/s was given

to k after a comparison v¡as made r¡¡ith the extreme cases of the cant

rail--window pilIar joint being a pin, in which case k has a value of

1, and of the cant rail and cant rail-window pillar,ìoint being com-

pletely rigid. For this case k equals 1/2. An expression for the

bending moment in the door sill and the cant rail in terms of the

shear force at the centre of the door was also obtained, as was a

relation for the bending of the window pilLars due to torsional loads '

These equations are shown in figures 2'2' and 2'3'

Brzo=ka(4) puUlished a sophisticated analysis of bus struc-

tures in 195?. Using methods of analysis that had been developed

for aircraft fuselage deslgn, various integral structures were analysed'

The analyses vrere detailed and included the effects of non-linear stress

distributions in shel-l structures but ignored the effects of door

openings and of panelling at the front and rear'

2.2.2. C uterized methods of anal ìe

All of the computerized methods of analysls of bus bodies

that are mentioned belor¡¡ are forms of either the matrix force or the

matrix displacement methods of structural analysis. Both methods

involve subdividing the structure into members or finite elements for

which the structural properties are assumed. The boundaries of the

elements and the ends of the members are defined by nodes which are

placed arbitrarily throughout the structure. The difference between

the matrix force and the matrix displacement method lies in the choice

11.

of the unl.<nor,vn variables. The matrix displacenrent method involves

constructing a set of equations involving the stiffness of the elements

and members, the nodal dispJ-acements and the external nodal- Ioads t

and solving for the nodal displacements. The matrix force method

sol-ves a set of equations involving the member and element flexibili-

ties and the member and element forces. The member and element forces

are the unknovin variables for this method.

For both methods the stress and strain distribution of the

el_ements and the members in the structure are assJ-gned to be specific

functions of the displacements of the nodes connected to them. The

stiffnesses or the flexi.bilitÍes of the efements or the members are

then calcufated such that the requirement that strai-n energy be con-

served is satisfied within the constrictions of the assumed stress-

strain function in the element.

A large number of different elements and members can be

formed by varying the functions relating the stress-strain distributi-on

to the nodal- displacements, by varying the number of nodes in the el-e-

ment and by varylng the number of degrees of freedom of displacement

that are possible at the nodes.

The methods of solutlon in the analyses described beLorv are

basical-Iy the same. The major differences lie in the type of e]e-

ments that are used and in the assumptions that are made in modelling

the structure.

The matrix methods have been available for many years but

it has only been ruith the advent of computers that they have become

practical. yoshimiræ, Ito and A"ri(5, 6) describe a method deveJ-oped

by Suzuki in I92? lvhich has been adapted for computer solution.

Although developed for the design of railways passenger cars the method

can be used for motor bus design. All the load on the car was

L2.

assumed to be borne by the side-wal-Is. The side-f rame rlas replaced

in the analysis by a vierendeel truss whose members, the upper chord,

Ior¡ler chord and window piÌIar panels, had the same centroidal_ axis inthe analysis as they did in the cars. The flexibilities of the truss

members were calculated using the assumption that elastic deformati_on

could take place only along the length ofl the window and door openinos

for the upper and l-ov¡er chords, and only along the height of the windows

for the pilIars. A matrix force method was used for the solution.

ntrreu=on(7) analysed a rear-engined composite bus body in

1967 usinq the matri-x force method. The bus had two doors on the

left hand side, which were positioned just forward of the front and

rear axles and it had two bul-kheads stretching half way across the

bus on either: si.de of the rear door.

All major members of the body rvere represented in the ideal-

izatron although the element subdivision for the outer panelling and

for the chassj-s members was fairl-y coarse (".s. the stress panel

between the waist and cant rails, and the flanges and the web of the

chassis members, were represented by onÌy one element each per wj-ndou,l

¡ay). The curved roof and sides of the bus were assumed to be fl-at

surfaces which met at right angles and the front of the bus was de-

picted as a plane vertical surface.

Only hal-f the bus was considered at one time in order to

reduce the amount of computer storage required, and therefore the bus

was considered symmetrical. Two analyses were made, the first of a

bus wi-th no doors or rear bul.kheads, and the second, a modified analy-

sis of a bus which had doors and rear bull<heads on both sides.

Alfredson reasoned tlrat his experimentation with strain gauges had

sugoested that litt1e differenee vuouLd be observed between the two

different cases. This was partly because of the compensating effect

13.

of the rear bulkheads on the foss of stil=fness due to the rear door.

He stated that the higher of the two stresses obtained for each member

would usually be a safe value. The results of the theoreticaÌ analy-

sls were compared v¡ith tests carried out on a partly finished bus

using resistance strain gauges and a reasonable correspondence betrryeen

the two was obtaÍned.

There has been considerable work done on the analysj-s of

automobile bodies usi-ng finite element technj_ques. One of the earli-est cases was that o¡ tt¡.r"nn-(8) in 1961. He represented the body

frameworl< with geometrically accurate members but replaced the panel

sections witlr peripheral and diagonal beam members. He was abl_e to

predict the stiffness of the body to within 5 per cent but considerable

experimental worl< was required to determine and represent accurately

the stiffness of the panels.

Norville and Mills(9) .r"o analysed a complete vehic,l-e

body. A fairly coarse 104 node , I?O eLement idealizatj-on was used

and the caLculated stiffness was later compared with that obtained

in experiments on an actual- body. The effects of geometric inaccura-

ciesr unconforming elements, and experimental test techniques were

studied. fn particular the effect of replacing the curved roof fil3-et,see figure 2,4., with a right angle section was examined. The cal-cul_a-

ted stiffness of the idealization with a right ançrIe roof fillet was

founú to be 20 per cent greater than the stiffness of Lhe idealization

with a curved roof fiI]et. The idealization with the curved roof

fillet t¡ras itself 30 per cent stiffer than the stiffness measured v¡hen

an actuaf vehicle was tested. ft was concLuded that a coarse mesh

ideal-ization was only suitable for comparative qualitative analysis

because of the sensitivity of the analysis to geometrj-c lnaccuracies

i-n areas such as roof fillets and joints betv¡een beams and panels.

14.

FTÊ. 2.4. TI'JO FINITE ELEMENT REPRESENTATIONS OF A CURVED ROOFFILLET TI-IAT ìll/ERE EXAMINED BY NOBVILLE AND MILLS(g).

1"5.

It was claimed that the approximations of a coarse mesh analysis

could not be who11y justified until the approximations had been tested

in an otherwise accurate and geometrically true idealization because

of the possibility of mutuatly compensating errors'

The results of a number of other finite element analyses

of car bodies have been published by other researchers. okuba et aI

(10), r,,,too"=(11), prtu"=".,(12) and Kirioka et -t(rs) have all created

detailed finite element models incorporating from 250 to 3000 nodes.

Displacement predictions for these .n"tr="= agreed with experimental

test results to within 10 to 20 percent. The accuracy of the pre-

dictions of these detailed idealizations demonstrates that the results

of finite element analyses approach the real values as finer subdivi-

sions and more geometrically accurate models are used. The degree of

accuracy v¡hich can be obtained is only limited by the amount of time

required to set up a complex model and by the expense of the computing

time needed for its solutÍon.

Atthough some work is being done to automate the modelling

process and to provide checking systems to enable modelIÍng errors to

be detected easily, the process of idealization is still a manual and

time consuming one. For this reason and because of the fact that

solution times for finite element problems vary roughly in proportion

to the-cube of the number of nodes, complex finite element models are

expensive and are not well suited to the initial stages of design

lvhere many different basÍc designs might need to be checl<ed.

WardiIl(14) suggested that a series of graduated programmes

should be available for the automobile designer so that he could check

various parts of his desi-gn, starting with very simple idealizations,

and increasing the complexity and geometric accuracy of the modelling

16.

as the desj-gn advanced. A series of simp)-e programmes such as a

two dj-mensional anarysis of the sj-de frame and a grillage analysis

of the underbody frame lvere recommended. The stiffness and strength

of the design as obtained from these analyses should be compared r¡,rlth

the results obtained by similar methods for previous designs that had

proved satisfactory.

For similar reasons to those that prompted Wardill-, Tidbury(¡) developed empirical formulae that would enabl-e simple side frame

and grillage analysis to be used for composite bus body design. The

proportion of the bending loads carried by the chassis and the body

was estlmated so that side-frame anal-ysis could be made with confid-

ence. This proportion vlas calculated by comparing the stiffnesses

of the sj-de-frame and the chassis members. For touri-ng coaches with

no door openings rear of the front axIe, and an all-steel body, the

stiffness of the side-wall- below the windovu l-evel was calcul-ated and

a side-wal-1/chassis stiffness ratj-o of 4.2 to 1 was estimated. For

a street bus which has a door in between the tr¡ro axl-es the problem

was more complicated as the relative stiffnesses of the two different

si-des was requl-red. The sti-ff nesses of the two side-warrs of an

existing asymmetric bus were determined under pure bending loads

applied at the axl-es. The determinat j-on was made by testing pl-astic

models of the side-wal1s, by calculation using the formulae developed

øV erz(Z), and by a simple matrix analysis oF the side-wall. The

stiffness under pure bending was found tc bg.alfnOSt . the same for the

two sides. However the deflection of the side-walI with the door was

considerably larger than that of the opposite wall when the walls were

supported at the axLes and a single vertical- load was praced at mid-

span. The rati-o of the f lexibiliti-es of the two si-des for this l-oad-

ing case was found to be 3.15 to 1.

I7,

Formulae to enable the stiffness of the side-u¡alls to be

calcufated r,vere given. For the side without the door the stíffness

was the stiffness of the side-waIl below the window multiplied by an

empirical factor of l-.? to a1lotry for the contribution of the cant

rail-. For the side-walI with the doors, the formula was based on the

equations of Erz. The effect of the rings formed by the roof-bows

and the floor-members was studied and it vras found that their effect

r,vas to distribute the vertical loads more evenly betureen the side-

walIs.

The ratio of the load carried by the body to that carried

by the chassis was determined to be ?.4 to I. Because of the dis-

tribution of load by the rings, the side-urall lvith no doors carried

only L3 times more load than the other side. It was stressed that

other bus-chassis combinations should be analysed and that full-scale

vehicl-es should be tested before the empirj-cal factors given in the

paper v;ere used for design. A relation for the stiffness of the

body in torsion lvas determined using the assumptions that the roof'

floor and side-ural1s below t¡¡indow level vrere infinitely stiff int2)shear. Erz's\tJ formulae for bending and shear in the vrindow pillars

in a bocly subjected to torsion were used to calculate the deflection

of the pillars in the side-r,valls and in the front and rear sections.

For any of the methods of analysis described above it is

necessary to determine the types and slzes of the loadings that are

likety to be applied. Although the analyses described above are

only able to handre static loading, it v'ras generally admitted that

static loadings have produced only small- stresses in the vehicl-es

that lrave been tested.

]B

Therefore, a deterrnination of the dynamic loading effects

is required before these analyses can be used to predict operating

stresses.

19.

2.3. LOADINGS AND DYNAMTC EFFECTS

There are several- different types of loading which are carried

by the bus body. These loadings are the self-weiçJht and the r,veight

of passengers, the dynamic loads caused by the passage of the bus over

uneven road surfaces, the dynamic loads caused by mechanical vibration

of engllne and transmission and braking and accel-eration forces. A

special- case of the dynamic loadings is torsjon which is generated

v¿lren the dynamic loadj.ngs are not evenly applied

In addltion the bus must be capahrle of providing protection

for its passengers in case of accident and varying safety requirerncnts

for accidents such as overtùrning, frontal and side impact have been

recommended.

It is obvious that some of these loadings wil-1 have littl-e

effect on the bus brody. Horizontal accel-erations and decelerations

are not likely to be ì-arge and hence they are unlikely to cause large

forces in any rnembers other than those situated near the axles. How-

ever, other dynamic loadings cause quj-te considerable stresses and

some method of predicting these dynamic loads and incorporating them

into the design is necessary. The importance and the problems

associated with the different loads are discussed below.

2.3.I. Self-wei ht and ASSEN er loads

The self-vreight of the body and the maximum weight of the

passengers can be determined v¡ithout difficulty but the proportion

of the self-rr¿eight which will be carried by the chassis members in a

composite bus is not automatically determined. Unless the chassis

i-s .jaclced leveI prior to the fastening of the body the self-weight

of the chass j-s vuill be borne by the chassis al-one. ïf the chassis is

20.

impt"operly aligned when the body is attached to the chassis, or if i-t

contains an initial out-of-straightness and is jacked leve1, then the

bads carried by the chassis and the body wiJ-1 not be l-n proportion to

their stiffnesses. The jacl<ing procedures recommended by difl'erent

chassis manufacturers di-ffer. Both Volvo(rs) and Merceue= gunz(16)

specified that the chassis be aligned prior to the fastcninq of the

frame wl-rj.l-e teyfanU(1?) recommend that no attempt at alignment be made

to thei-r A.E.C. Swil't chassis except to lightly jact< the front riqht

hand corner. Thus the details of construction must be consi-dered

prior to analysis.

2.3.2. Dynamic Loads due to uneven road surfaces

Cluite large dynamic stresses are caused by the passage of

the bus over uneven road surfaces. These stresses can be larger than

the statÍc stresses, and I'atigue resulting from the dynamic stresses is

a maJor factor in the failure of bus bodies. ft 1s Lherefore jmport-

ant that somethlng be known about the size and the frequency of the

dynamlc loads and also about the ways in r¡¡hich their efl'ect can be

cal-culated.

A number of experimental- programmes have been completed 1n

whj-ch the dynamic stresses in a moving bus have been recorded.

Descriptions of these are given by Palm et al-(re), ntoyan(19),

Elizarde(20) and Alfreouon(?). ïn the main these experiments have

been to test the suitabil.ity of a given design and they have not been

extended to compare the effects of differino tyres, suspens:ì ons or

body stiffness on the dynamlc Ioads. Atfr"aton(?), horvever,

recorded the vertical accelerations at various sections of the bus

as lvell. The stresses at various locations 1n the bus had been

previously measured with strain gauges when the bus was sub,jected to

2I.

static l-oads applied to various parts of the bus. The peak dynamic

stresses at these locations were found to be roughly approxirnated by

summing the products of the mass of each section,times the peak

accel-eration recorded at that section, ti.mes the stresses obtained at

the gaug¡e locati-ons for a unit load at that section.

This method has the advantage that it may be used to deter-

mine torsional loads and that account can be made reasonably easily

of tyres, suspension and road surfaces by recording accelerations in

a bus of a similar type. The most commonly used method of account-

ing for dynamÍc stresses has been simply to multiply the static

stresses by a dynamic factor. This factor has been determined flrom

the results of experiments and from its success in previous designs.

Yosr,irin"(6), for example, uses a factor of 1.7 for railway passenger

cars and Michelberger uses a factor of 2 to 2.5 for the design of

motor buses.

Neither multiplying the mass of the bus and its passengers

by the vertical acceleration nor multiplying the static loads by a

dynamic factor will provide a real representati-on of the dynamic

loading and behaviour of the vehicle. Zienl<i-runi"r[21) has shown

that it is possible to extend finite element analyses into dynamic

problems. The basic set of simultaneous equatj-ons in the matrix

displacement method of structural analysis is represented by

EKI t 5l Ê . ..(r)

tKl

i r jfrl

IFJ

is the system stiffness matrÍx

is the nodal displacement vector, and

is the external force vector.

urhere

22

This set of equations uras extended to incl-ude an accel-eration term

so that:

tKl t5l + tMl fl!ì - irj . . .(z)ò t'

tM] is the system mass matrix.

This matrix can be formed j_n two ways. The first is to

distribute the mass of the elements evenly amongst the- nodes of the

elements. The matrix so formed j-s termed the lumped mass rnatri-x and

is diagonal. The other method, which is more correct, resurts in a

more reallstic distribution of lnerti-a forces but produces a system

mass matrix which occupies the same amount of storage as the system

stiffness matrlx.

The velocity of the nodes, # , was included as a variable

in the dynamic analysis and the formation of a recurrence relationship

was described that enabled the displacements and vel-oclties of the

nodes at the end of a tlme increment to be determined fronr their val-ues

at the start of that increment.

The relationship was of the form

ctBtt

2TB :rl i

cz

/\where lCJwas dependent upon the external forces applied during the

time increment and matrices IA] and tBl retained the sparsity of the

original stj-ffness matrix. ft can be seen that the solution of this

problem will require considerabÌe computation for each time increment.

23.

Anderson and Mi-1ls(22)

carry out a dynamic

using the recurrence

the natural modes and

removing the external

solving

[r*l LrJl rMn

used the finite element method to

analysis of a car chassis in L9?2. Instead of

relationship described above,they determined

frequenci-es of vibration of the chassis by

force vector Ip) r"or equation [2) and by

rù 0 . . .(s)

where C! is a natural frequency of the system.

The sol-ution to this equation was found by using a lumped

mass [rufl matrix and by manipulating the variables to reduce the pro-

blem to the classic eigenvalue problem. Using only a coarse mesh

idealization, the frequencies and modes of the natural vibrations

were predicted with considerable accuracy.

It is possible that methods of dynamic analysis such as

these could be used for determining the dynamic behaviour of bus

bodies. However, the large amount of computation reQuired to obtain

any useful results appears to limit the application of these methods

at this stage.

2.3.3. Acceleration and brakinq forces

since horizontal accelerations are likely to be small it

is unlikely that the stresses produced by these accelerations will

be 1arge. In certain eireas, however, problems have been detected

and the Vo1vo bodywork utorkshop bulletin(15) recommended that dia-

gonal members be placed in the chassis next to the front and rear

axles to withstand the longitudinal loadings. Allowance for accel-

eration and braking forces could be made in a static analysis wj-thout

cjifficulty if the maximum expected accelerations were l<nown.

24

2.3.4. Accident Protection requirements

The loadings applied to bus bodies as a result of traffic

accidents are both large and short-lived. Since it is not practical

to provide bus bodÍes with sufficient strength that they remain per-

fectly elastic after every possible accident, it is no longer possible

to use existing elastic analyses to predict deformation patterns.

In addi-tion, the behaviour ofl vehicle bodies under impact loading has

been found to be different from the behaviour of the same body under

a static load. This was shown by Lowe, Af-Hassani and Joþn=on(23)

r¡uho investj-gated head-on collisions of buses by using small scale

models. The effect of door openings, windows and wheel bays uras

examined. Impact testing was carried out by dropping a weight onto

the front of the model. The damage that occurued was compared with

the effects of static loads applied to the same position. Static

J-oadings were found to be no guide to either the crumpling loads or

crumpling patterns that were observed for impact loadings. The

damage that resulted from impact loadings was confined to the immed-

iate vicinity of the impact thus the effect of door openings and win-

dov¡s away from the impact area was smalI.

since it is ímpossible to predict accident damage using

existing static, elastic analyses' most accident study and safety

design is based on barrier tests in which full scale vehicles are

driven into barriers(Z¡). This is, of course' a very expensive

process and is better suited for testing existing designs than for

developing new ones. Results from tests such as these, hourever,

have shown that certain failure mechanisms are more acceptable than

others and the tests have made it possible to predict roughly the

safety of certain designs.

25.

Apart from colrisions, emphasis has been praced upon the

protection of passengers in the event of the bus overturning.

Rrro¡(24) describes a static load code that has been specified forcertaj-n school buses in the United States. To satisfy the code,the

deflections of the roof, side pillars and, floor centre must all_ be

l-ess than specified maximum deflections when the bus is subjected to

a test load. The test load is a weight, equal to the complete body

and chassis prus an overload factor, placed in a wooden roof rack.

rn addition all windows and doors must be operable when the bus i_s

fully loaded in this manner. This test, although not actuallytestlng the safety of the bus when overturned has been found toprovide adequate protection and has the advantage that it can be

carri-ed out without destroying the bus. The code al_so allows the

body-builder a certaj-n amount of freedom in his design. It uroul_d

appear to be relativefy easy to include specifications of this form

in a static analysis.

2,4. INTEHP RETATTON AND ANALYSTS OF RESULTS

Modern finite element analyses are capable of producing

vast amounts of results. Reactions, nodal- displacements, member

moments and forces, element stresses and member and element strainenergies can all be tabulated. For finite element models with large

numbers of nodes, the analysis of these results wj-]l be tedi-ous and

difficult. fn additlon, a decision will have to be made as to which

of the data is critical to the performance of the vehicle and on how

the design of the vehicle can be optimized. Al_though some programmes

now incorporate automatic plotting routines to plot deflections and

aperture dj-stortions and although more information 1s being presented

26.

in a form easier to comprehend, the determination of critical cri-

teria for deslgn presents some dì-fficulty.

The use of stresses as the critical criterion is made

difficult in a finite element analysis because the stresses in the

plate elements are accurate only at the centroid of the element.

This means that no accurate account of stress concentration is made.

¡¡oo"u(11) suggested that this problem could be overcome by isolating

the critj-cal areas, making more detailed idealizations of those

areas, and subjecting them to the forces calculated in the analysis

of the complete structure.

Anal-ysis with an emphasis on the stresses would enable

areas at which cracking was likely to occur to be detected.

Hov¿ever, it is possible that members that are highly stressed may

carry little load and could be omitted without significantly affect-

ing other members. ¡¡oo""(1I) proposed a method for optimizing

automobile bodies.

Deflections of nodes were used to detect any serious weak-

ness and also to measure the stiffness of the body. Since the

stiffness of the body and the distortj-on of particular parts of it

are related to noise, vibration, ulindscreen retention, shearing of

floor fasteners, vehicl-e suspension and sealing, the provision of

adequate stiffness was selected as the major design criterion.

When suitable stiffness was obtained, stresses in the automobile

were generally found to be smalI. fn order to determine rryhether

there vúere any members which could be omitted or modified,all members

which carried only small moments or forces were looked at individually

to see if they were necessary for some other locaf loading condition.

ff they were not they could be modified or efiminated. The calcula-

tion of the strain energy in each member was useful because those

2?.

members with high strain energì-es had a significant influence on

body stiffness and although the converse did not necessarily ho1d,

the cal-culation of strain energy enabled the effect of smal-I changes

in member properties on the overarl- stiffness to be determined

without rerunning the programme.

Although the process described above was suggested pri-

mari-ly for motor cars, it would obviously have relevance to bus body

design. The appearance of craclcing in bus bodies, however, suggests

that a certain amount of emphasis should be placed on the stresses in

the members. Because fatigue failure is dependent on stress con-

centrations and welding details, work has been done on fatioue

faiLures in differing types of members and vrelded joi nts. Rudnai

and Matol.=rIzs) and Atoyan et .t(ze) both tested various sections

commonly used j-n bus bodies. Fatigue failures were found to begin

in the tension flange of members at the end of seam wel_ds. ft was

advised that seam welds should be avoided at highly stressed points

where butt welds are preferable. It was demonstrated that consid-

erable increases in fatigue resistance could be achieved by careful

detailing of the joints.

28.

3. INITTAL INVESTTGATION OF AGCU RACY OF FÏNTTE ELEMENT ANALYSES

Before proceeding with the assembly of a general finite

element program, the accuracy that could be obtained in the calcula-

tion of the stress concentrations in a structure, using the finite

element method, was examined. To do this, two relatively simple

models, one constructed of steel and the other of photo-el-astic

epoxy, were obtained and they were tested using electrical resistance

strain gauges and photo-elastic methods respectively.

Both of the models were beams that contained cut out

sections. Models of this pattern were chosen because of their

relationship to the sidewalls of the bus body which have cut-out

sections at the doors and windows.

3.1. TNVESTIGATION OF STRESSES rN THE STEEL BEAM WITH CUT-OUTS

3. Ì.1. Description of the steel beam and of the way it ¡¡qstested

The test beam was constructed from a 22't x 4rr section of1l-/ 2tt ¿¡i"k steel plate and Ít contained four 3rr x 2rr cut-outs along

its length. The dimensions of the beam are shown on figure 3.1.

Twenty eight electrical resistance strain gauges were glued to the

surface of the specimen, including sì-x rosette strai-n gauges. The

positioning of these gauges is also shown on fj-gure 3.1. The beam

was símply supported at the ends and was loaded from the top in a

Mohr Federhaff testing machine. Plate 3.1. shows the beam being

loaded. The deflection of a point on the top surface of beam was

measured with a dial gauge and the strain gauges were read with a

BLH-1200 Digital Strain Indicator.

29

o

PLATE 3.1. LOADING OF THE STEEL BEAM WITH CUT-OUTS

PLATE 3.2. LOADED EPOXY BEAÍ\4 IN POLARISCOPE

10"

1

27 24

2,(4)1

25

315)

4'

A,B 25

FIG. 3.1. LAYOUT OF STEEL BEAM WTTH CUT-OUTS. HALF VIEW. STBATNGAUGE NOS. GTVEN. GAUGE NCS. TN BRAG<ETS ARE ON ßEAR

. suBFAcE 0F BEAM. THTcxNESS = å rrucn.

26

Lto

3r

(e)

(c)

FIG. 3.2. STEEL BEAM WI'TH GUT-OUTS - COARSE ELEMENT MESH -b. Rectangular lj_near element -4 nodes.e. Rectangular quadratic element -B nodes.

FTG. 3.3. FINE ELEMENT MESH

FIG. 3.4. Coarse Element Mesh - Triangular elements.

(b)

!l

!

32,

3.I.2. Finite E lement Analvses

Four different finj-te element nrodels were developed for the

analysis of this test. Three different element meshes and three

difl'erent element types were used in these model-s. The first model

consisted of the coarse element mesh shown in figure 3.2.(") and used

the rectangular linear dispS-acement el-ements depi-cted in figure 3.2.

(¡). The second modef used similar elements but had the finer ele-

ment mesh shown in figure 3.3. The thÍrd analysis again used the

coarse element mesh shown in figure 3.2.(") but the elements used

were the rectangular quadratic displacement efements with mid-side

nodes that are shown in figure S.Z.(c). Fina1ly, a modeL using tri-

angular constant strain elements, in the coarse element mesh of

fi-gure 3.4., was tried. Because of the planar nature of the test'

only two dimensional, plane stress elements were needed in the analy-

coc

The stresses in finite elements are usually calculated only

at the centroid of the element. Since we expect larger stresses at

the boundaries of the elements, the stresses at locations on the

boundary of some of the elements, were calculated in addition to

the stresses at the centroids. These predicted stresses were then

compared with measured stresses to determine their accuracy. For

any displacement-type finite element, a shape function is assumed

that relates the displacement of all the points in the element to

the displacements of the nodes. Thus the strain, and hence the

stress, anywhere within the element, is defined in terms of the dis-

placement of the nodes. For simple triangular elements the shape-

functÍon ic such that tha stress is constant throughourt the element.

For all other elements the stress vrill be different at different

33.

points in the element. Thus stresses can be calculated anywhere

in an element. The stresses, perpendicular to the boundary of two

adjacent elements, wiII generally be different in each of the two

elements at points on the boundary. At these points the mean of

the stresses in adjacent elements was used'

Althoughthere].evanceofatestassimpleasthis,to

the behaviour of bus side-uual-ls may appear limited, it should be

stated that many of the previous finite element analyses of vehicle

side-wal1s (refs. 7, g,11, 1/t) have been as simple as the ones used

to analyse this test.

3.1.3. BesuI ts of the test on the steel beam and theircomp SON W th e finite e ment analvses

Some of the results of the tests on

the comparison between them and the resul-ts of

analyses are shown graphically in figures 3'5'

the steel beam and

the finite element

to 3. I0.

3. 1.3.1 Defl-ection of a oint on the to surface ofam

The vertical displacement of point D on figure 3.1. was

measured with a dial gauge as the beam was loaded. Point D was

clrosen because it coincided with a node in all of the finlte element

idealizations and because the dial gauge could not be placed any

closer to the centre of the beam. The displacement of this point

under load is recorded in figure 3.5. some hysteresj-s is evident

in the plot of the observed displacement but there is a reasonable

correspondence between the exper\mental results and the predictions

of the more complex f inite 'element models. The j-dealization wj-th

quadratic rectanguLar elements and the fi-ne mesh idealization with

l-inear rectangular elements were both about zg,l' too stiff , while

the coarse mesh idealization with linear rectangular elements was

about 3Cp¡å too stiff . The fourth idealization, incorporating

34

simple trj-angular elements, produced results which urere 451/o too

stiff . These results highlight the propr,r''t¡z of the finite element

method that the finite element solution converges to the exact

theoretical solution as finer element meshes are used. The results

i11e and Milts(9) urho

found that a coarse mesh model with lj-near rectangular elements

resulted in predictions for the stiffness of a car body that vrere

between 20 and 6O?ä too large.

3.1.3.2. Stresses at the centroids of elements

Six rosette strain gauges were placed on the steel beam

at locations corresponding to the centroids of elements in the

coarse element mesh that lvas used in tu;o of the finite element

analyses. Turo pairs of guages vuere placed at the same locations

but at opposite sides of the beam and the other tt,ro tnrere located at

different positions on the same side. From the strains recorded at

these gauges, it lvas possible to check the accuracy cf the predictions

of stress at element centroids. The gauge positions did not coincide

r¡¡ith the element centres in either the triangular element or the fine

elenlent mesh models. For these tv;o cases, holvever, the gauge posj--

tion was directly between two adjacent element centres, and so the

mean of the stresses, at these two points, was used. Figure 3.6 is

a plot of the predicted horizontal stresses at positions {" below the

top surface of the beam for a load of 5 kips. The horizontal stress

preclicted by the triangular, constant strain eLementsr and the rectan-

gu1ar, Iinear elements vras constant along the centres of the panels

above the cut-out sections. Vihen rectangular quadratic elements

rnrere used the predictcd horizontal stress varied in these ârEâS.

FIG. 3.5. STEEL BEAM IVITH CUT-OUTS.TOP SURFACE WITH LOAD.

35

DISPLACEMENT OF POINT ON

.5atr

cde

O l'lserveclCoarse nlesh - quadratic rectangular elementsFj-ne mesh - linear recbangular elementsCoarse mesh - linear rectangular elementsCoarse mesh - triangular elements

a.4

bca

0)t{+0)E

.r{r{rl.r{

=

z.oH,-c)Ld.JlJ-lr,cl

.l

2

.1

d

e

050 10

L0AD (riloruewtons)15 20

FIG . 3.6.

X experimental values.

âr

b.G.d.

HOBIZONTAL STRESS 0N SECTI0N 12.5mm. BEL0W TOp 0FBEAM FOH 22 KILONEWTON LOAD.

Coarse mesh - rectangular quadratic elementsFj-ne mesh - rectangular linear elementsCoarse mesh - rectangular linear elementsGoarse mesh - triangular eüement s

LOADED HERE T

36

x

125

100

75

50

a

I

//i

,b

c

/xa-ar{(0oorúo-dÞ¡0)

=

aØUJ(ft-aÍ)

t--z.oNHGc):E

ItltlI

:l

t-J_l

I d

itt'tl

Í,

x/

/,l

l//

,!/I

I

rII

I25

00 50 100

DISTANCE FROM SUPPORT

150

(tvtittimetres)200 250

37

FIG. 3.7. OBSEBVEO AND PREDTCTED STRESSES AT GAUGE NO. 24,

a. observed strainsquadratic elementsfine mesh - linear rectangular elementscoarse mesh - linear rectangular elements

10

bcd

-1000

- 900

- 800

- 700

- 600

a

bc

dzHcÊt--(.t)

ocEclH=

- 500

- 400

-300

_ 200

- 100

050

LOAD KN15 20

1 000

900

800

700

400

300

200

100

FIG. 3.8. OBSEHVED AND PHEDICTED STRESSES AT GAUGE NO. 26.

El . Observedquadratic elementsfÍne mesh - linear elementscoarse mesh - linear elements

10

LOAD (t<ltoNewtons)

ba

38.

c

bcd

d

600

050

zH(rFg)

E(JH=

050 15 20

FIG. 3.9. OBSERVED AND PREDICTED STBESSES AT GAUGE No. 25.

a. observedb. quadratic elementsc. fine mesh - linear elementsd. coarse mesh - linear elements

39

200

175

150

125

100

75

25

b

aI

I

f

z.H(rt--U)

oEclH=

x

c

,

I

50

0d

5 10

L0AD (t<llotrtewtons)-25

15 20

40,

125

100

75

50

25

-25

-50

-75

- 100

- 125

- 150

B. observedb. quadratic elementsc¡ fine mesh - linear elementsd. coarse mesh - linear elements f

a

b

/

f.

50

zH(Et-U'

E(JH=

LOAD (Kitolrtewtons)

10 20

d

15

c

- 175FÏG. 3.10. OBSERVEO AND PBEDICTED STHESSES AT GAUGE NO. 2?.

4I.

The predicted values of the horizontal stress shown in

1=igure 3.6 agree quite wetl urith the values observed at the three

rel-evant rosette strain gauge locatj-ons. ft appears that the

quadratic elements best represent the actual beam as all the observed

vaLues are predicted to within 13/". The coarse and fine mesh models

using rectangular linear elements predicted stresses that r¡rere almost

identical and that vrere only slightty less accurate than those pre-

dicted by the quadratic elements. The stress predicti'ons of the

analysis using triangular elements give poflrr:l' ¿:luÏ'c1ern{:rll'b than tllu

others with all the observed values being under-estimated by bett'reen

2oil. and 3tli1.

3.1 .3.3 . Strains at Points othe r than element centres

BecauseonlySinglestraingaugeswereplacedattheother

locations, it is only possible to compare the predicted and observed

strains at these points. The method used to predict the strains'

at locations other than the efement centresr lvas to form a strain

matríx, for these locations, from the initiaf assumed shape function

oftheelement.Forlocatj-onsthatareontheboundaryoftt¡lo

el-ements, a value for the strain was determined from each element

and the mean of the two used '

Figures 3.?, 3'8, 3'9 and 3'10 are graphs of strain

VerSuS load at gauges 24, 26, 25 and 2? respectively. The horizontal

strainsinthesteelsectionsaboveandbelowthecut-outsarea

combinationoftvlocomponents.Thesecomponentsarethestrain

causedbythecurvatureofthebeamasawholelandthestrain

causedbytheverticalshearingforceintheindividualsections.

ïrese two components combine at gauges 24 and 26' but counteract

atgauges23and2?'Thepredictionsforthestrainatgauges

24 and 26 ate very good, with the predicted values being

42.

withln 1ü/o of the observed values, for the analysis using quadratic

elements, and the anal-ysis with the fine linear element rnesh, and

being within L3y' to 25o$ for the analysis with the coarse rinear

element mesh. At gauges 25 and 27, r¡rhere the two strain components

counteract, the strain predictions are poor. The straÍns predicted

at both of these locations by the analysis using the coarse linear

element mesh, are opposite in sign to the observed values. Siml1-

arly at gauge 2? the fine mesh ideal-ization predicté a strain with

the opposite sign. The analysis using rectangular quadratlc finite

elements, however, sti1l predicts strains that are within 23[ of the

observed values.

From these results, it appears that, the finite element

strains at positlons other than the centroid do not reliably repres-

ent the actual strains in those positions when rectangular linear

elements are used. Holvever, the fÍnite element strains in rectan-

gular quadratic elements appear to be more dependabLe in the pre-

diction of the actual strains.

3.2. TNVESTIGATTON OF STHESSES TN THE EPOXY MODEL AND THECOMPABISON OF ÏHE EXPERIMENTAL RESULTS \¡TTH F]NTTE

ELEMENT ANALYSTS PREDTCTIONS

3.2.1. Descriotions of the eooxy beam and the method oftesting

The test beam was made of epoxy resin and was cast in a

rubber mould. The dimensions of the beam are sho',vn in figure 3.11.

As can be seen, the cut-out sections are comparatively larger than

those in the steel beam described in Section 3.I. The beam was

supported at points A and B (see figure 3.11) and loaded by means

of suspended weights at poÍnts C and D. The loading was carried

out in the polariscope shov,rn in plate 3.2. The isoclinj-cs were

43,

15,5"

2" .5"

FIG. 3.}1. DIMENSTONS OF PHOTOELASTTC MODEL.

15" 1.5" 't.5"

LOADING ABRANGEMENT USED FOR THE DETERMTNATTON OF THEFBINGE CONSTANT OF THE PHOTOELASTTC MATERIAL.

tt

.37"

5.5

5"

FfG. 3.12.

WW

A

D

B

44

photographed on black and white film when the beam uias subjected

to a 51bf load. The isochromatic lines were photographed on

colour film for loadings of 251bf and 401bf v¡hen the beam was

il-l-uminated with circularly polarized lì-ght.

The fringe constant of the material was determined by

testj-ng three pieces of the same material- that had been cast at

i-he same time as the test beam. These were loaded in the manner

sholvn in fj.gure 3.12. The fringe constant was determined by com-

paring the number of fringes in the specimen with the predicted

maximum horizontal stress due to bending. A value of 101 lbfín,

order. was obtained. Thus, for a thickness of 3/8" th" fringe

stress was 270psi.

3.2.2. Finite element anal eì e

Only one finite element analysis was made and this used

only linear rectangular efements. The element mesh that was used

is shown in figure 3.I3.

3.2.3, Results of the tests and their comoarisons withthe fi-nite element anal s15

The resul-ts of the photo-elastic testing have been com-

pared with the results of the finite element analyses at the element

centroids and at the surfaces of the top and bottom chords. Graphs

of the maximunl shear stress ( o;- 6." J were plotted by using the known

values ofoi-o-" at the isochromatic fringes, and determining, from

the photographs, the points at which these fringes intersected the

surface of the test piece and the lines connecting the centroids of

the finite elements. This enabled cr 6r to be plotted at various

positions on the beam. These plots can be seen on figures 3.14. to

3.19. SÍnce there is no stress perpendicul-ar to the surface at a

free boundary the horizontal stress i-s equal to oî- o-r at the

surfaces ofl the top anci bottom chords.

45

3.2.3.I. Accuracy oi stress predictions at elementcentres

As can be seen from fi-gures 3.I4. to 3.17. the predicted

values of or- o-2 are only about 60% of the observed values in most

cases. In addition, although the shape of the plot of the pre-

dicted values is approximately the same as that of the plot of

the observed values for most of the beam, at the joints, where a

rapid variation in the stress occurs the finite el-ement mesh is too

coarse to permit the stress to be predicted accurately,

3.2.3.2. curac of stress redictions at the surfaceo the beam

Since the stresses at the boundaries of the top and bottom

chords are horizontal, the observed stresses plotted in graphs 3.1S.

and 3.19. are compared with the predicted horizontal stresses.

The surface stresses in the top and bottom chord members were pre-

dicted by extrapolating linearly across the member, the predicted

val-ues of the horlzontal stresses at the centroids of the upper and

lower elements. For l-inear rectangular finite elements this pro-

cedure resuLts in the same answers as would be obtained by calcula-

ting the strain in individual finite elements as was done in the

analyses of the steel beam. From the graphs it can be seen that

the surface stresses predicted are, on the who1e, quite accuratet

although in some places the predicted stresses are only between

5Oo/o and 60'/ of the observed values. This result concurs with the

similar results that were obtained when linear extrapolation of

stresses in linear rectangular elements was used to predict surface

stresses on the steel- beam. (See Section 3.1.3.3.).

Finally it can bc secn that if no extrapolation v¡as used

and the stress values predicted at the element centroids were used

as the basis for design, then a design factor in the vicinity of 2

to 3 wouLd be required fo¡anelement mesh of the form used here.

46

FTG. 3.13. FTNTTE ELEMENT MESH FOR TYPICAL BAY.

b100

200

-coc-f{

of{(tJooß{0)ooEcfoÈ

aUJ(Jz.H 3oot¡lI!l!Hoat' 400cEFU)

X0

X

ê. observed valuesb. predicted values

500

OBSERVED AND PREDICTED VALUES OF THE DIFFERENCE BETWEENPRINCIPAL STRESSES AT SECTTON X-X. LOAD 201bf. Þ

60

FrG.3.14.

600

FIG. 3.15. OBSERVED AND PRED]CTED VALUES OF THE DIFFERENCE BETV'EEN

PBINGIPAL FTRESSES AT SECTION X-X. LOAD 201bf.

b

,(

roc'rloSr6fctof{0)eaEcfoÈ

IlJCJztrJ(rLrJl!lJ-HoU)U)UJCEl-U)

s00

400

300

200

100

ê. observed value5b. predicted values

a

0

Âæ

x+

!

!

I

!

x

X

b

0

100

200

soc.rl0)f{(tfctoÍ{ooo!cfoÈ

llJ

2_ 3ooccIJl¡lJ-Hog 400lljcEFaJ)

êr. observed valuesb. predicted values

a

500

OBSERVED AND PREDTCTED VALUES OF THE DIFFEBENCE BETWEENPRTNCTPAL STRESSES AT SECTTON X-X. LOAD 201bf. À

(.o600

FïG. 3.16.

600

300

200

100

FÏG. 3.L7. OBSERVED AND PREDICTED VALUES OF THE DIFFERENCE BETITEENPBTNCTPAL STRESSES AT SECTION X-X. LOAD 201bf.s00

400

-oc.-l

0)S{dJo'oß{ooUIEcfo(L

b.JC]zUJEUJlJ-lJ.HoaØIJ(Et-U)

ab

x

ã. observed valuesb. predicted val-ues

0

X

I

I

TI

I

I

urc

+X

- 900 a

b

c

FIG. 3.18(a). HORTZONTAL SURFAOE STRESS - TOP SUBFACE OF BOTTOIvI

ct-ÐBD. ulAD 201bf .Ioc'rl(D

tdfoa¡{ft)oaEcJoo.

6Ú)IJ(rt-a

- 600

-300

ì a. observed horizontal surface stressb. predicted horizontal surface stress

-G. predicted horizontal stress at element centroids

e

b

c

0

rJì

¡

+300

X

coc.rl0)Íi(úfgof{0,o_

aEc5oo-

U)ú)IJ(rt--ú)

X

FrG. 3.18(bJ. HORIZONTAL SURFACE STRESS .BOTTOM SURFACE OF BOTTT]M CI.ÐRD. L0A0 201bf.

õr. observed horizontal surface stressb. predicted hori-zontal surface stressc. predicted horizontal stress at element centroids

a

-400

-200

0

400

)(

c

200

a

a

(¡f\,

b

b

c

b

c

600

-900

+2 00

FIG. 3.1s(a). äflåËÉ:NrAL

suRFAcE STRESS - roP suRFAcE oF TUP

{x

- 600

xx

-300

-cotr

-F{

otrqtluaf..fl¡oalfcf,oo-

aU)IJÊ.¡-U)

x

ê. observed horizontal surface stressb. predicted horizontal surface stressc. predicted horizontal stress at element

centroidsxí

x a

0

\cb

XX

(tlG)

x

\\---f

x

\

- 600

- 200

200

400

FIG. 3.19(b). HoRIZONTAL SURFACE STRESS BOTT0T'.4 SURFACE OF

BOTTOM CHORD.a

b

-400

c

0

-cfJc.rlft)fr6ftt(n

tr0¡oaEcfoÈ

ü)a' L¡JGF-U)

a

b

c

f

I

â¡ observed horizontal stressb. predicted horizontal surface stressc. predicted horizontal stress at nearest

.element centñoids

{x

600

(¡À

EÃ

4, FINITE ELEMENT A YSTS

4.I. IntroductÍon

The examples described in the previous sections(refs. ?, 9,

10, fl, LZr 13) demonstrate that it is possible, usì-ng existing finite

el,ement routines, to analyse complex vehicle bodies with computers.

With complex flinite element analyses of structures as large as bus

bodies, however, the amount of computi-ng time required to obtain a

solution becomes very large and expensive. In additÌon, lonç¡ pro-

grammes which require large amounts of central memory and auxiliary

storage, as is the case with finite element programmes, have low pri-

ority on some systems and the processing of them can tal<e an excessive-

ly long time, especially during busy periods. These factors are

especially lmportant if it is necessary to run a programme more than

once, as would possibly be the case in the design stages of a bus.

There is, therefore, considerable incentive to improve the efficiency

of the existing programmes.

Since it was observed that buses are constructed from a

series of almost identlcal sections or modul-esr it was declded that

various techniques that took advantage of this fact should be investi-

gated to determine whether any overall efficiency resulted from their

use. It was thought that if the stiffness matrix of a section that

is repeated in the structure was formed from basj-c finite elements

only once, thus forming a rrsuper-elementr', then the computation

required to form the overall- stiffness matrix of the structure would

be reduced. In addition it was decided to investigate the effect

that removing unwanted internal nodes from some of these super-

e1.ements, had upon the efficiency of the analysi-s.

56.

A programme using the finite eLement disptacement method

was wri-tten rvhich incorporated the formatÍon of super-el-ements and

the ability to reduce them by the removal of unwanted nodes.

4.2. Proq ramme Descripti.on

The programme consists of seven subroutines a1l of which

are summoned by an appropriate command in the problem data. The

seven operations of these subroutines are:-

1. Formati-on of the stiffness matrices of basic elements.

2. Output of the stiffness matrices of formed elements.

3. Input of previo¡.rsly formed or experimentally determined

stiffness matrices.

4. Rotation of elements.

5. Addition or combination of elernents.

6. Reduction.

2. Solution.

All of these subroutines operate on one, or in the case of

the Addition routine, on two elements or super-el-ements.

4.2,I. Element Handlinq and Storage

Each element has an identifying number, IDEN' attached to

it. The data necessary to locate and define any element is stored

in a two dj.mensional array in core L (fOeru, n).

Information stored j-n this array is:-

1. The number of nodes in the element '

Z. The location of the element matrices on the disc file.

3.Thetypeofelement-Thisparameterindicatesthe

format i n which the element is stored '

5?.

(l) Coded element - The stiffness matrix of this type

of eLement is sparse and the non-zero sub-matrices

are stored in coded format.

(Z) UncoOed element - The stiffness matrix of this kind

of element contains no zero sub-matrices and it is

stored in sequence without codes.

4, The number of positions in the element, other than the

retained nodes, at which the element may be loaded - Tfri-s witt be

zero unLess the element, or part of it, has been previously reduced

and removed node load matrices, necessary to distribute l-oads from

nodes that were removed by the reduction, to those that were retained,

have been calculated and stored.

5. The number of positions for whlch the stress matrices

are stored.

The stiffness matrices plus codes, stress matrj-ces and

load distribution matrices for removed nodes are stored on a disc

file in unformated binary blocks. L (fOEru, Z) lists the number of

records prior to the beginning of the element data for the element

with the identifying number IDEN.

For both coded and uncoded elements, the stiffness matrix

is divi-ded into sub-matrices. The size of these sub-matrices is

dependent upon the number of degrees of freedom of displacement per

node that are used in the problem. Thus, the size of the sub-matrices

in a problem with n degrees of freedom wilÌ be n x n. In the

following discussion, the terms row and column will refer to a row or

column comprised of these sub-matrices. Each such row and column

correspnnds to a single node of the element.

59.

For codetl elements, the first data blocl< or binary record

on the disc file, contains the row codes whj-ch designate the number of

non-zero columns in each row. Each succeeding record contains a rourl

o1' the stiffness matrix and its correspondj-ng column codes. A

similar arrangement is made wj-th stress matrices and with the load

distribution matri-ces for removed nodes.

For uncoded elements, the stiffness and stress matrices

are stored successively with one record corresponding'to one row of

the stiffness matrix or to one stress position. The load distrlbu-

tion matrices for removed nodes are stored in the same manner as is

used for coded elements.

For both coded and uncoded elements, only the sub-matrices

above or on the main diagonal of the stiffness matrix are storedt

since stiffness matrices are symmetrical-.

When elements are being handled in core the sub-matrices

are stored in a two-dimensj-ona1 matrix A (NDF' MAX) where NDF is the

number of degrees of freedom associated with each node for the parti-

cular problem. MAX is a dimension large enough to handle all the

data required during the various operations and it can be set so that

all the core is used. A1I codes are stored in an array M while

they are in core

Usually only one row of the stiffness matrix is in the core

at one time. Advantages from handling el-ements in this way are that:-

If) efficient use is made of core storaoe.

(Z) nffowance is made for matrix symmetry.

(s) Onfy non-zero sub-matrices are handled.

Disadvantages inherent wÍth this method are that¡-

Ão

[f) n large amount of shunting of data is done between disc

and core.

(Z) n certain amount of searching is required during the

addition or combination of two elements because only the sub-matrices

above the diagonal are stored, and only one row is in core at any one

time.

4.2.2. Description of the Seven Subroutines

4.2.2.L Formation of the stiffness matrices of the basicel-ements.

The stiffness matrices of two basic types of element and

that of prismatic members can be formed by this routine. The form

of these elements can be seen in figures 4.01 to 4.03, The parameters

required for the calculation of stiffness matrices are also shown on

these f igures. The el-ements and members can be for:med with from tvuo

to six degrees of freedom of displacement at the nodes depending upon

the type of problem to be solved. The fini-te elements that are

formed r¡rhen two degrees of freedom of displacement per node are speci-

fied, are simple linear displacement plane stress rectangles and simple

plane stress triangles. Shel1 elements with six degrees of freedom

per node have the same planar stiffness as the two degree of freedom

element but have bending¡ stiffness calculated using the methods des-

cribed by Zienki"*i.=[21) for rectangular and triangular e]-ements with

corner nodes. The el-ements have no in-p1ane rotation stiffness.

To avoid any difficulty arisi-ng from this zero stiffness during the

decomposition of the stiffness matrix, the matrix element in the lead-

ing diagonal- is checked in each line before that line is operated

upon. If the element is found to be zero then the line is ignored.

The effect of making the in-p1ane stiffness of the el-entents very larqe

was j-nvestigated in the analysis of the experiment described in

Section 5.5. The predictions were poorer than those that were made

X

FIG. 4.01. REGTANGULAH ELEMENT

3Xr,!r, O)

2( Xr,!a, o)

v

60,

Acl ditional parametersThiclcnessfvloclulus of elasticitYPoissons ratio

Additional pergrnglersThicknessModulus of elasticitYPoissons ratio

v

-tr]J!fIJ0)5{

-(-f

t

z

z

v

( o,o,o) X

FIG. 4.02. THIANGULAR ELEMENT

Additional etec onal area

Area resisting shear in Y directÍonArea resisting shear in z directionTorsional moment of inertiaMoment of inertia about Y axisMoment of inertia about z axisModulus of elasticitYPoissons ratio

le rhX

FIG. 4.03. PRTSMATIG MEMBER

61.

when there was zero in-plane stiffness (see figure 5.1?). This resuLt

agrees with the findì.ngs of Greene, Strome and Wei-l<et(so). 0n1y the

upper half of the stÍffness rnatrix is stored. Matrices for the deter-

mination of stress within the element from the nodal dlsplacements are

formed and stored if required.

4,2.2.2. Output of formed elements

This subroutine uras included to enable a programme to be

stopped and rerun and al-so to enable the stiffness matrix of any super-

element to be stored and used again without repeating its formation.

Matrices can be stored on punched cards, magnetic tape or permanent

files.

4.2.2.3. Input of previously formed or experimentallydetermined stiffness matrices

This routine is the consequence of the previous one and

aÌlows el-ements aì-ready stored to be read back into the system.

Matrices determined by experimental means and expressed in the proper

format can afso be read by this routine.

4 .2.2.4 . Element rotati-on

This subroutine calculates the transformed stiffness matrix

of an element after it has been rotated. The stress matrices of the

rotated element are also calculated.

4.2.2.5. ddition or combination of elements

The additlon routine forms a new element from the combina-

tion of two old el-ements. It also all-ows a string¡ of simj-1ar el-ements

to be combined. The combination is specified by listing the node num-

bers of the first element at which the second element is to be joined'

and listing the corresponding node numbers in the second element.

62.

The nodes of the new element are numbered in such a way that the

nodes forming part of the first named element, keep thej-r original

numbers, and the nodes formed by the addition of subsequent elements

are numbered in the same order as they were in the parent efement.

This results j-n a reasonably well sequenced numbering¡ pattern.

If matrlces have been attached to the original elements

that enable the stresses at various positions in thcse elements to be

calculated from the nodal displacements, then aLÌ or any of these

matrices may be retained in the new element. In addition' íf there

are matrices attached to the original element that enable the external

Loads, acting at nodes that have previously been removed, to be dis-

tributed, then these too, may be retained.

A disadvantage of this particular subroutine is that only'

two different elements can be joÍned together in any one operation.

Thls means that when a number of different elements or super-elements

are joined together, a fair amount of double-handling tal<es place.

This could be remedied if a more standard stiffness matrix formation

method were used which included the ability to use and to form super-

e lements .

4 .2 ,2.6 . Beduction

The reduction routine will reduce the number of nodes in

an element without affecting the overall- stiffness of the element.

Any number of nodes may be removed. The equations used for calcula-

ting the stresses within the el-ement can be converted so that they

are expressed in terms of the displacements of the retained nodes

and also, j-f required, in terms of external loads at the removed

nodes. In addition, matrj-ces that enable external loads on nodes

that are removed by this routine, to be distributed to the nodes that

are retained, can be formed if required.

63.

To illustrate the procedures followed in this subroutine'

consj-der an elenrent, [basic or built up), that stil1 has sets of

nodes called sets I and 2, but had the set of nodes, desig¡nated 3,

removed in a previous calculation. Assume that the set of nodest

numbered 2, are to be removed in the next operation. The reLation-

ship between the displacements, loads and properties of the el-ement

are expressed at this stage by the relatj-onshlp

liisJ2i [6J

+ ß { P3l["Jt'J'lts'l

I P3]

...[i)

= [eri+ [rr] t trt f'.1...[2)

. . .(s)

Eit

rr]

zlù

P1

to qive

is the stiffness matrix, the elements of

which are generally themselves matricest

are the deflections and

are the external foads at the sets ol'

nodes I and 2.

are the matrices which distribut" Itrl,the loads at node set 3, to node sets l-

and 2 respectively.

,.] ['.1 - ['.rl t t 'ì ]

j

.]'- [tt

Itr)

-[tr']

The equations are modj-fied by substitutinç¡

-1

I r'¡Itr)

F"l'[t

[-'tß"1

{'J

ß'']

Isrlt-"1F, r] [-rr]

F"l-1

[t-'']-1

tr-J

[tr] -( r.]

Is

or

[orttl

t

. . .(¿)

*.\

[trrl t PcJ ...(s)rl =( F"l t lP 2

where

[*rr]

,ù

-1and

In addition, the stress at a particular point

expressed by

S a t.il

where

and

where

["]'['t['.J

X

or

S

['r]*

F'r-l*

[ttu].

lT',1Ttrl

64.

. . .(o)

. . .(z)

. . .(e)

. . .[g)

-1

. . .(rz)

(r¡)

. . .(r¿)

,Å

. . .[io)

...(u)

[o'r] n

["r]*[tr.]* It'.]

Ir[*'i--1,r)

[*rr]t-

[*'']

[*'á

Itt.]

-1[orr]

E

in the element will be

t ) I ro]+[sr]tsrì

+Isrl ["iare the relationships between the stress and

the displacement of node sets I and 2

is the relatÍonship between the stress

the external loads on the former node set 3.

Substituting equation (S) i-nto equatl-on (S) yields

*I P2] i'J+ Ftr]

*[.,]* i5,l

['] ["][-['rr]-'-t

3l

- ["]

| ".I

+

anu It. ),

* [p.ì+

if +["']

ti [rr] [*rr]-t t*rr]- -l-

,z)

-l- [..] [*rrl'[rrJE

t'á t-["J

[*'r]*

To produce the stiffness matrix of the reduced el.ement,

must be calculated. If it is required to determine the

65.

stresses in the reduced element then [Srll *r=t be found and if theLII

element is to be toaded at removed nodes tr,en [rrr]*,[tta|,[tar]*ano l-rs^Jt tr=t be f ormed.L{

Thenewmatricesareformedwiththeaidofthenormal

solution routine described later on in section 4.2.2.?. To illus-

trate the way in which the solution routine is used, consj-der the

element viith the properties and characteristi'cs given above in equa-

tions (f), (Z) anU (9), anU consider that the nodes in'set 2 are to be

removed. Assume the displacements of node set l- are fixed and that

and ] ."= specifJ-ed. If the svstem of equations (f)' (Z)

was solved by the solution routine, then the unl<nowns at the start

oftheoperationandthesolutionsattheendwouldbe and

The stres= it) would also be calculated. The solution for

[S) wouro be sj-milar to equations (a) and [r¡)

F" 2Å-1

(t.i rr)

It,.JfIrJ{ rrj and

("i - [*r,] [*r,]-' [*,,ltJJ * [*,,][r'']ot']

[o

[*rr]-'[")['.)+ t-t

. . .[rs)

[.J

Ifitisnotrequiredtoloadtheelementatremovednodes

then the new matrices are formed by soÌvins NSt loading cases where

NS'eQualsthenumberofnodesinsettmultipliedbythenumberof

degrees of freedom per node. [S J is an NSt x NS, matrix and itt -rJ ".0 [tr] are setis set equal t. [f] the identitv matrix' ( F

[='t

equal to [0]the zero matri-x. The solutions obtained by the solution

routine can be found by substituti'ru [5] = [{, (pr\ = fo]anu

[o.J = [o]into equations [rs) and (re).

=þrrl

=Etrl

torrl

,I[orrl*

[.rlu

66.

.. .(rz)

...(re)

. . .[:-g)

. . .[zo)

The results obtained are

{'J '[rJ,

Ie

t

- [o,.rl Frrl-t- [trl ßrrl-t [ot

If it is required to load the element at removed nodes

then the previous operati-on is omitted and NS, sets of loadinll cases

are solved tryhere NS, eQuals the number of nodes in set 2 multiplied

by the number of degrees of freedom per nocje. I tr) is set equal to

ti] .nufeal rno [ 6 J u"" set equal. to [o ] . ïhe result of the solution

routi-ne is:

,1, = [orr] [orrl-t

)r = [trl lorrl-tS

= ltrrl*= þsrln

[or.]* is formed by post multiplyins (rs) uy Ir<r1J and subtractì-ng the

resurt f'rom[Krr_],.nd [ar_]* by post multiplying (zo) ¡v IKzr] and

subtracting the result t'rom[Srl. [tra]* rnu [rsal* are also ca]cu1-

ated fronr equations Iis) and (zo).

The normal solution routine was used in the formation of

the rnatricr:n for the nevi reduced element because:

(i) The solution routine took into account the syrnmetry

and the sparseness of the original stiffness matrix during decomposi-

tion and Forward and backu¡ard substÍtution.

(ii) fne use ol' the soLution routlrre reduced pro.q¡rarnminçt

time.

67.

(iii) The slmilarity betl'¡een the operations involved in

reduction and solution suogested that it was the most efficient method

available.

Certain factors altowed unnecessary computation to be

reduced. The l<nowLedse of the exact form of the initial loads and

displacements, for instance, enabled the number of operations in the

forward substitution process to be reduced by Stf,/o.

r,.2.2.?. Solution

The solution routine calcufates the deflectÍons of nodes,

the reactions at the supports, and the stresses at the required stress

positions for a structure, represented in the programme by a built up

element, subjected to nodal loads and prescribed displacements of the

supports. The supports are nodes of the element and they may be

either wholIy or partiatly restralned. The sol-ution is dependent

upon loads at nodes that have been removed by the efement reduction

routine, if the appropriate load distribution matrices have been found

and stored.

The solution method is based upon the method described by

Melosh and Bamfora(z?). This method, which has been calfed wavefront

analysis, takes advantage of the fact that, during the decomposition

of the stif,fness matrix, no non-zero terms can occur 1n a cofumn of

the decomposed matrix, in rows prior to the occurrence of a non-zero

term in that column of the stiffness matrix. Thus, the first appear-

ance of a non-zero term in a column of the stiffness matrix during

decomposition, causes the addition of that column to the wavefront

and causes the effect of that column on the following rows to be con-

sidered. Becarise no column is operated upon until- the appearance of

a non-zero term, the amount of storage required to store the wave-

front, is never greater and is oflten l-ess than the storage used in

68.

the bandwidth method. Also fewer calculations on zero terms are

rnade. The method involves storing the stiffness matrix, row by row,

on auxiliary storage and bringing the rows, one at a time, into the

central- memory of the computer. In all, the rows of the matrix pass

through the core three times, the first during decomposition, and the

other two times when the rows of the decomposed matrix are used 1n the

forward and backward substitution processes.

4.3. The Effectlveness of Super-elements in fmproving ProgrammeEfficiency

In order to check the efficiency of the use of super-elements

in finite el-ement programmes, several trials of the nev/ programme were

made on similar problems in which the size, the percentage of nodes

removed and the number of super-elements were varied. The basic

problem that was examined was that of a hollow tubc with eight iden-

tical units along its length, supported at two nodes at each end and

loaded at two nodes 1n the centre. A hollow tube was chosen because

of its similarity to a bus body. A diagram of the basic structure

is shown in figure 4.1.

The criterion that was used for the determination of

r¡effj-clencyt¡ was the elapsed Central Processor or C.P. time. The

elapsed C.P. tj-me was printed every time one of the seven subroutines

was call-ed. The C.P. time is a rather unsatisfactory criterion

because it is not totally dependent upon the number of computations

carried out, and vuil1 not necessarily have exactly the same value

after the completion of the same operation when done at different

times, since it is affected by other programmes being run simultan-

eously. It does, howeverr provide a guide to the efficiency of the

different methods.

69.

FrG. 4.1. BASIC FINTTE ELEMENT ARRANGEMENT . HOLLOW ruBE WITHB IDENTICAL SECTIONS

(a)

(b)

(c)

The three super-elements used in conjunction withthe basic structure.

{ { { { \ J\ {

\

I

70

4.3.1 The Effects of the use of repeated unreduced super-el-ements u on the ti-me taken to form the stiffness

ma Y er structures.o

The formation of super-elements from basic elements and

their use in the subsequent formation of the stiffness matrix of larger

structures was thought to provlde a means of reducing the computation

required for matrix formation. This was bel-ieved because it would

not be necessary to repeat the formation of the stiffness matrices

of the basic elements. To check this supposition, the stiffness

matrix of the structure shourn in figure 4.1-. (using the bay structure

shown in figure 4.IIb)) was formed both by using unreduced super-

elements and by using the AGES programme(28) urf,:-"h cal-cul-ates the

stiffness matrix of each element. The programme using the super-

elements took 10 C.P. seconds whiLe the ACES programme required 12

C.P. seconds to form the stiffness matrix of the structure. Ït

shoul-d be noted that the routine which formed the stiffness matrix

of the super-element was not very efficient since it could only add

two dlfferent elements together in one operation. The time taken by

the ACES routine to form the stiffness matrix of one super-element

was l-.5 C.P. seconds compared with the 4.6 C.P. seconds required by

the new programme. It appears therefore that if a more standard

method ulere used to form the stiffness matrices of the super-elements,

then the use of unreduced super-elements would be an advantage in

stiffness matrix formation.

The time spent in forming stiffness matrices j-s unlikely to

be a large proportion of the solution time and generally the need to

facilitate the.preparation and ValidatiOn ; of data wj-l-l be of

glreater i mpnrtance, In this regard the use of super-elements might

prove to be both easier to prepare and to check because fev¡er node and

element specifications would be necessary.

?T

4.3,2. The Effects of repeated lqduced super-elements ono robLem solution time

The reduction or condensation of a block of el,ements to

form a single super-element with fewer nodes is not a new principle.

A descri-ption of the method of reduction is given by Zi-enlci.*i.=(21)

and a recent exampl-e of its application is in the modetJ-ing of shear

panels in muLti-storey buildings(Zg). The removal of nodes from

the element, and hence fronr the structure, will result in there being

fev,,er sj-mul-taneous equations to be solved but rryi11, of course, reduce

the amount of information that is obtained from the solution.

Although decreasing the number of nodes in a structure will generally

decrease the time required for solution there are other factors which

will reduce the effj-clency of thÍs reduced super-element method.

4.3.2,L, Factors whiqh reduce the efficiency of theuse of reduced elements

The stiffness matrix of a super-element before reduction

will usually be sparse sínce each node will be connected to relatì-vely

few other nodes. After reduction, however, the stiffness matrix of

the super-element v,lilI contain few, if any, zero sub-matrices and

each node will be linked to every other node. Although the effects

of the sparsity of the initial stiffness matrj-x wil1 normally be re-

duced after the decomposition of the stiffness matrix during solution,

some penalty will be involved due to this loss of sparslty. This

factor lvj-l1- sometimes cause the use of super-elements, with only a

smafl percentage of nodes removed, to be less efficient than unreduced

elements.

Another penalty j-nvolved with the use of reduced elements,

arises irl tlre calculaLion oF Lhe stresses. The stresses at the cen-

troids of the basic elements that make up the unreduced super-element,

ere determined from the calculated displacements of the nodes of those

?2.

basic element's. fn the reduced super-element,

l'lill be calculated from the displacements of all-

those same stresses

the nodes in the super

element. Thus

the greater will

stresses.

the more nodes retained in the reduced super-element,

be the penalty attached to the calculation of the

The other major factor infl-uencing the efficiency of use of

reduced elements, is the time required for the reduction. The reduc-

tion routine¡as has been described earlier, uses the solution routine

to f'orm the reduced stiffness matrix, The number of solutions re-

quired, for a given reduction, is equaL to the numl¡er of columns in

the new stiffness matrix, if the matrices to distribute loads from

removed nodes are not formed. Thus the time taken for reducti-on will

be dependent upon the number of nodes that are retained. The relat-

ionship between the C.P. time taken to reduce a trial super-element

and the number of nodes that were retained in it, is shown in figure

4.2. The super-el-ement used was a. section of holl-ow tube. Graphs

(a) and (¡) in figure 4.2. are the relationships of the reduction

time versus number of nodes retained for the super-element when it

was constructed with rectangular shear elements, which have three

degrees of freedom of displacement per node, and when 1t u,ras construc-

ted with shell elements which have six degrees of freedom of dispface-

ment per node.

When the matrices that distribute the loads from removed

nodes are calculated as we11, the amount of computation becomes almost

independent of the number of nodes that are retained as can be seen

from fisure 4.2., graphs (c) and (u).

FrG.4.2.

73,

SUPER-ELEMENT REDUCTION TIME VERSUS THE NUT\4BER OF I

NODES RETATNED.

eir NDF = 6.Load distribution matrices not formed

b. NDF = 3. Load distribution matrÍces not formed

c¡ NDF = 6.Load distrÍbution matrices formed

d. NOF = 3. Load distribution matrices formed

N.0.F. - No. of Degrees of freedom per node.

20

15

10

c

a

z.oHt-(J=otlJEzHoUJttfÍnoz.o(JIJU'

o-a

cl

d

5

b

00 510

NUMBER OF NODES BETAINED IN 16 NODE ELEMENT

15

74,

4 .3 .2.2. The Relationsh betuieen the efficienc of thel-method a he percen e of the nodes retained

e reciuction.n

In addition to the factors mentioned alreadyr the corres-

pondence between the number of nodes removed fram the elements and

the number of nodes removed from the structure implies that the

efficiency of the use of reduced super-elements is heavily dependent

upon the percentage of nodes that are retained in the reduction.

Figure 4.3. is a plot of the ratio of the G.P. tlme tat<en for the

solutlon of a series of problems when reduced super-elements urere

used, over the G.P. time tal<en for the same problems v'rhen there vJaS

no reduction. This ratio is plotted against the percerltage of nodes

retained in the reduced super-element. The basic structure on whlch

these tests were made i-s shou¡n in figure 4.1. The number of nodes

and basic e1ements in the original super-ei-emen1- r¡¡as varied and al-I

tests were made with both three and six degrees of freedom of dis-

placement per node. The solution times included the C.P. time

necessary to form the stiffness matrlx and the time taken to reduce

the super-element.

When onJ-y one quarter of the nodes were retained, the use

of reduced super-elements decreased the amount of G.P. time required

by up to sixteen times. When two thirds of the nodes v¡ere retained

the tv,ro methods took approximately the same time.

4 .3,2,3. The relationshi between effi lenc and theS er-e ements in then ber of reduce

ture.

In order to examine how the efficiency of the reduced

super-element method varies with the number of super-elements in a

problem, the two different methods were compared for a series of

problems in which the number of super-elements was varied from one

75

FIG. 4.3. SOLUTTON EFFICTENCY vs. PERCENTAGE OF NODES BETATNED

r NDF - 3r 16 node element

X NDF - 6, 16 node element

O NDF - 3r B node element

* NDF - 6, B node element

100

90

80

10

NDF = [rls. of degrees of freedom per node.

50 100Percentage of nodes retained in reduced super-element

a

Oor{x

fl)

.5 zo+rco.il

+rf

iso0)I

U'

T]

3solE0)Srtr=.Ë

qo#co'.{+J

.l 30oo'0)Ia

E20oJEo(t

I

x

o

*a

xa

X

00

?6.

through to eight. The super-element was a sectj-on of hollolv tube

r¡rhich originally contained 16 nodes but was reduced to B nodes by the

reducti-on routine (see figure 4.f(.)). The results of these tests are

shown on figure 4.4. The solution time used in plottlng this graph

included the time taken to form the stiffness matrix and the time

taken for the reduction.

The ratio of the solution times for the two different methods

rapidly reaches a relatively constant value, decreasinq sliqhùJy as

the time tal<en for the reduction of the super-eJ-ement is distributed

more widely. If the reduction time is ignored, the ratio of the

solution times is remarkably constant. This suggests that the

effici-ency of the reduced super-element method is not greatly depend-

ent upon the number of reduced super-elements in the structure.

Hovrever, in the case where a larger proportion of nodes are retained,

the time taken for reduction will lncrease, and, since the savings

from using reduced elements wilI be less, the number of repetitions

will be critically important to the suitability of the reduced super-

element method.

4.3.2.4. Summar of the suitabilit of the reducedSU er-e ement method

It appears that the primary factor j-nfluencing the vafue

of the reduced super-element method is uhe percentage of nodes that

are removed from the super-elements. For the case that has been

discussed in which eight super-elements are involved, the removal of

one third of the nodes appears to be the least that is profitable.

As a greater percentage of nodes are removed the efficlency of the

method increases rapidly. The efficiency of the method was found

to be not greatly affected by the number of super-elements in the

1,2

1.0

77.

FIG . 4.4.

VARIATTON OF EFFICTENCY WITH THE NUMBER OF SUPER.ELEMENTS IN STRUCTUHE

Matrix reducti-on time included in solutlon time

a. NDF = 3.b. NDF = 6.

Matri-x reduction time excluded from solution time

c. NDF = 3.d. NDF = 6.

246NUMBER OF SUPER-ELEMENTS IN STRUCTURE

aoIo

Eft)fJfEû)hcf

-tr+r.-l3ft)E.rl+)

co.rl#f,rloU)

a0)I

út

'tl0)o5TI0)f{

-g.lJ'r'{3oE'rl.p

tro.rl+rfrlo

U)

IO

6a

4

a

b

c

d

tl

0I0

/ó

structure, althoug.Jh as more super-eì.ements are incl-uded the time taken

to reduce the super-eLement is more widely distributed. Another

factor v'rhich affects the effj-ciency of the reduced super-element

method is the sj-ze and shape of the super-element. This is the

result of such faotors as the loss of sparsity in the sti-ffness matrix

which r¡ri11 af fect some super-elements more than it will others.

Although there is a desire to reduce the number of nodes

that are retained in the super-elements, there are severaÌ limitations.

The first is that, for a correct solution of the problem, all nodes

on the boundaries between two super-elements, nìust be retained.

0therwise the displacements of adjacent elements will not be contin-

uous. The second arises because, if a node is removed, its displace-

ment is no longer calculated. Therefore, the necessity to cal-culate

the dispJacement of certain points may require that nodes be retained

for this purpose. However, it is unlikely that the displacement of

all the nodes v'rill be required as many nodes wil-l have been included

merely to represent the structure. Since there is no difficulty in

removing internal nodes, it would appear that the reduced super-

el-ement method r^rould be well suited to analyses using quadratic and

cubic elements, althoug;h this supposition has not been tested.

When matrices to distribute loads at removed nodes are cal-

culated, the time tal<en for reduction is increased as j-s the time

tal<en to form the stiffness matrix of the complete structure and to

calculate tlre solution. The inclusion of the facility to load the

removed nodes increased the time necessary to reach the solution to a

problem, involving the structure in fÍgure 4.!., by ?G/ when half the

nodes in the super-element were removed. The time taken for this

79.

case was stitl about one half that required to solve the problem

rvithout reduced super-elements. Thus the inclusj-on of this facility

decreases the efficÍency of the reduced super-element method although

the penalty involved will decrease as a smaller percentage of nodes

are removed.

A .4. Other Applications of the Element Reduction Fìourtine

Although the element reduction routinc was desiç¡ned to

be used v,rith repeated super-elements, j-t was found to have other

applications. In problems where several different support condi-

tions are defined for the same structure, the v¿hole structure can

be reduced, retaining supported nodes and those nodes necessary for

describing deflection. In this way the number of computations

required for individual load cases with different support conditions

is drast j-call-y reduced. An example of this is an idea]ization of a

structure involving 350 nodes. Three different loading cases were

required involving three different support conditions. The unre-

duced structure took zuo c.P. seconds per loadlng case. $jhen the

structure r¡Jas reduced to 5 nodesr an operation taking 300 C'P' seconds'

the individual load cases took 25 seconds each '

Another appli-cation of the element reduction routine rr¡ould

be in the case where it is desired to alter various members whilst

retaining most of the surrounding structure. such a case could

arise if it were desired to determine the effects of altering the

door and window columns of a bus body. The entire structure minus

the columns could be represented, the stiffness matrix reduced and

stored, leaving nodes at the supports ancl at the top and bottom of

80.

the door and wj-ndow columns. This would enable there to be a large

reduction in the amount of computation necessary to carry out a

serj-es of analyses involving a number of different designs.

91.

5. DETAILED ANALYSIS AND TESTING OF CRITTCAL SECTTONS

5.1. Introduction

several arthors(11t14) have suggested that one way of

deùermining the stress concentrations in the critical parts of a

vehicle, rvould be to isorate these areas, model them in detail with

f inite elementsr afld apply loads, calculated from a less detailed

analysis of the contplete structure, to the boundaries of these iso-

lated sections.

One of the locations at urhich cracking had been discovered

on the buses operated by the Municipal Tramurays Trust was in the

vicinity of the rear door. At the rear door the cant rail had been

mede discontinuous (see Figure 5.I) in order to l-ocate the door open-

ing equipment. It was thought that this discontinuity vrould produce

large stress concentrations.

A section of the bus adjoining the rear door (see Figure

5.1) v¡as chosen for investigation; and it r¡ras model-1ed vrj-th fj-nite

elements. In additionra full-scale photo-elastÍc model was built

of epoxy resin in order to determine whether the finite element mesh

that vras chosen, \¡/as sufficiently detailed to provide useful values

for the stress concentrations.

5.2. Finite Element Idealization

Because of the complexity of the shape chosen, only the

simplest model of finite elements that could adequately describe the

geometry rvas considered. Even so, a model incorporating 326 nodes

and 355 elements was required. A view of the element mesh used to

model the section is shown in Figure 5.2. The elements used r,¡ere tri-

angular and rectangular linear shel1 elements. SÌnce it was required

that the finite element analysis r¡iould exactly match the tests on the

model, the ends of the major l-lox sections vJere connected u¡ith

e2

very stiff beam elements to a sjngle node. This arrangement of stiff

beams, shovrn in Figure 5.3., enabled the stiFf end blocks in the

photo-el-astic model to be represented, and also al]owed pin support

conditions to be specified at these points. Because a number of diff-

erent support conditions were used durlngl the testing of the plroto-

tllastic model-, the stiffness matrix of the model- vras reduced by the

nrethods descrrbed in Section 4.2.2.6, to that of a structure vuith 5

external nodes, and 7 internal- nodes. The internal nodes were used

to conrpare the observed and the calculatecl displacements. This

operation enabled the series of tests to be analysed efficiently.

5.3. Construction of Photo-el-astic model.

5.3.1. Photo-elastic material

The photo-elastic model was made from an epoxy-resin,

EPIREZ 135. This material was sel-ected because jt is clear, has g¡ood

photo-eIastì-c properties, can be cast with a clean smooth surface and

is quite strong. At room tempprature the ultimate tensile strength

of the Epirez will rise fairly gradually to around 40 MPa at 15 days

after casting. After this time the strength remains almost con-

stant. The strength can be s j-gnif ica¡tly increased il" the material-

is then post-cured. The post-curi-ng process j.nvolves heatlng the

cast material in an oven to 25oG, slowly raisinçJ the temperature to

BOoC, holdinq it at this level for two hours, and then sl-oi^r1-y 1-otnrer-

ing it again. The post-curing process increases the ultimate tensj-le

strength of the resin to betrueen 65 and 75 MPa.

Two specialty prepared tension speci-mens r¿vere tested -i n

order to determine the fringe constant of the Epj'rez. The fr-Lncrr:

constant was found to be 14.6 N/mm, fringe.

83,

r - - - - - --l

L__

I

I

J

FIG. 5.1. BUS BODY FBAME\]ïJOBK ABOUND REAR DOOR SHO\}JING SECTTON

THAT WAS MODELLED AND TESTED.

Extremely stiff beam members

FIG. 5.3. VIE|ÂJ OF BEAM ARRANGEMENT USED TO MODEL ENt] PTECES.

84

280mm

x

200 200

Door opening

72

I3

6

v

11

10

2

z

5

0

4

350

300mm

.- Stress panel on this face

t__

50

FIG. 5.2. ELEMENT MESH USEO TO MODEL SECTION

85.

Êi^r.Õ.¿ Fabrication

It can be seen from Fig¡ure 5.2. that the section of bus

that v¡as modelled consisted of box tubing strengtheneo with both flat

and curved stress panels. Since the tubes could not be cast directly

nor bent after the Epirez had set, they were fabricated from four fl-at

strips. These strlps, and the stress panels as ulell-r were cast on a

layer of heat resistant oven vrrap that lay on a glass plate. Tlre

thickness of the casting was controlted by the thicl<ness of the ABS

strips that constituted the sides of the moulcls and v¿hich tryere fastened

to the oven wrap with cloubl-e-sideci tape. The Epirez uras carefull.y

levell,-rd of F to the required thickness as it vras poured. (Various

staqes in the construotion are shown in Plates 5.1. anci 5.2.) ' Since

Epírez shrinks as it sets it \{,as found to be advantageous to heat it

to about 3OoC so that the temperature and hence the setting rate would

be unil'ornr throughout. This procedure prevented wrinl<l-es fronl form-

ing on the surfaces.

After the sides of the box tubes had set and had cured for

15 days, they u¡ere fastened together. Two sides were clamped into

a special !:racl<et and Eplrez vlas poured into the joínt to form haff

a tube. When tu¡o such half-tubes had been formed, they were .joined

togetlrer in the same way to complete the tube. Wlren all of; the tubes

were complete they were post cured.

The curved pane'ls were first cast flat, in the same mannËr

as the flat paneÌs, but after the Epirc.:z had set and before jt had

hardened, the panels were placecl in the oven on speci.ally curved

moulds. The panels were then heatec.l and forced into shape by weighted

male mou1ds.

86.

PLATE 5.I. MOULD USED FOB JOTNING THE WALLS OF TUBE MEMBERS

I

PLATE 5.2. THE METHOD USED FOR POSÏTIONING STRIPS DURING

COI'JSTRUCTION OF CURVED TUBES

87.

Eventual.]y the various components urere joined together to

form sections and these sections urere post-cured to strengthen the

joints. It vras flound that the post-cured Epirez was unaffrected by

subsequent heating. Finally these sectj-ons viere joÍned together and

the complete structure was post-cured in a specially made polystyrene

l.oam box (see Plate 5.3.). This box was necessary because the struc-

ture vras too large to fit in the oven '

since it was irnpossible to vier,v the box =t"tion" through a

normal polariscope, tlre inside surfaces of the tubes and the bacl<

surfaces of the panels vúere sprayed with an al-unliniunr-based spray

pai-nt. Light vras refLected by this layer and the reflecting polar-

iscope, shown in Plate 5.4., was used to examine the model.

The end pieces of the structure required consideration since

1t was necessary to load the structure without causing stress concen-

trations, at the surpports, that mighL have indt-lced craclcing. A1so,

it uras necessary that there be no movement betl'¡een the model and the

end pieces at the supports. fn addition it rvas decided that some

of the supports should be free to rotate abou'b one axis, in order to

represent nìore realistically in tl-re test, the stresses that would

occur in the actual- vehicle. A satisfactory end piece t'vas nbtained

by cutting lvooden plugs sliohtly srnall-er than the size of the tubes'

These were covered with rubber pads before being glued lnto the ends

of the tubes. supports that were free to rotate about one axis

werc achieved either by drilling holes in the wooden bl-ocl<s and

pivoting them about stee]. pins, or by means of steel hinges which

lvere bolted to the ends of the bfocks. An end piece is shown in

plate 5.5. and plates 5.6. and 5.?. are phoLugraplrs of the finishcd

mode1.

88.

-..u-i./¿

PLATE 5.3. FOAM Ii/ALLED OVEN USED TO POSTOURE THE FTNISHEDÍ\4ODEL.

THE NVEN VJAS HEATED Ì¡TTH THE BLOWER-HEATER ANDTI]i: TEI"4PERATURE V/AS REGULATED BY ADJUSTING THESIZE OF THE OPENÏNG.

B9

PLATE 5.4. THE REFLECTTNG POLARISCOPE

SCBEW JAGI( AND PROVTNGRTNG

ODEAN

NEDP

ND BLOCKIVOT

¡

PLATE 5.5. WO

90

PLATE 5.6. EPIREZ MODEL VIEWED FROM POSITION CORRESPONDING

TO THE INSIDE OF THE BUS.

PLATE 5.?. OUTSTDE VIEI¡J OF MODEL.

91 .

5.3.3. The testin o of the ohoto-elastic 'model.

The testinçy of the model was carrieC out in the steel frame

shown in Plate 5.8. Horizontal- loads lvere applied by means of small

screw jacks. Vertical loads were applied by suspended weights.

Tlre appJ-ied horizontal loads rvere measured urith proving rings that

vrere placed Lretween the jack and the point of application of the foad.

The supported points were alf free to rotate about one axis and this

axis could be altered by changingl the direction of the steel hlnge or

by altering the cjirection of the steel- pin about vthich the joint

pivoted.

Three different tests M/ere made on the modef:

(f) n horizontal load rlas applied to position B

(see F,eJure 5.4.)

(Z) n vertical foad t^¡as applj-erl to position B (see

Fiçrure 5.5. )

(¡) n horizontal load was appJiecl to positJ-on 5

( see Fiç¡ure 5 .6 . )

For each of these three tests a series of blacl< and white

photographs were taken of the isocl-i.nics and a series of colour

photographs v¡ere tal<en of the isochromatics at varying loads. Dis-

p.ì-aceorents of parts of the loaded structure were recorded ttith dj-aI

gauges. These can be seen in Plate 5'9.

5.4. Dj-scussion of Experimental and Theoretj-cal Results

5.4 . I. Disolacements of rroints on the model

Displacement was measured at various l.ocations with dial

gaugÉs. Tlre measured and predicted displacements have been tabulated

i-n Tab1e 5.1. and fr.om this table lt can be seen that the measured

displacements were considerably g¡reater than the predlced values '

I92.

2.48

2.87

6.1s

1.69

1 .14

DÏSPLACEI.ÍENTLOADING ONE

Horizontal LoadNode I

Measured Predicted

. i'i9

-l-.2? -. 181

LOADING Tì¡/O LOADING ÏHREE

verrical- load Node t"'il:::*äI Load

IMeasured Predicted Measured PredictetNode

No.Direc-tion

TABLE 5.1.

X

X

X

X

X

X

X

2

4

6

?

I

10

07

38 -I.46

-1,30

-2.70

-1. tB

-r.37

-.294

-.0?5

-.292

- .345

+ .u63

+ .09

006

.o47

+ .055

.099

+ .D40

.645

Y

Y

Y

3

B

.19 +.118 + .o2I

+ .25 3{7 -L.r7

DTSPLADEMENTS OF POTNTS ON MODEL.N0DES SHOIIJN 0N FIGURE F¿,2.

L4 5

4

c

6

z

z

z

z

.'t

16.0

2.16

2.16

LOCATION OF

56

10

93.

PLATE 5.8. MODEL IN LOADTNG FRAME.

)f

PI-ATE 5.9. ISOCLINICS IN MODEL AND LAYOUT OF DIAL GAUGES

5

94.

1000 N

100 N

I

v

v

x

z

1

FIG. 5.4. HORIZONTAL LOAD FHOM RIGHT

11 12

5

I

x

z

1

FTG. 5.5. VEHTICAL LOAD ON RIGHT

95

00N

11,12

FIG. 5.6. HOHIZONTAL LOAD FROM REAR

5

v

X

11 12

FIG. 5.7. LOBATION OF ROSETTE STHAIN GAUGES.

Gauge C is on j-nside =r"¡u"r oC panet opposite guage B,

5

I

v

z

X

1

I

z

+ B,(C)

A+

+D

96

Although some of this disparity could possibJ-y be attributed to the

inherent stifl'ness of the fini-te el-ements used in the analysi.s, it

is lilcely that the main cause of this large difference is the movement

of the supposedly rigid supports. Although some improvement was

achieved by bracinçl the supporting structure and by improvì-ng the

model to end piece connection, there u,ras stiLl some movernent of the

supported ends. The l.arqe di1'f erence in tl-re displacernent of point 5

in loadlng 3 for exampJ-e, 1s believed to be due to movement of the

lateral- support at poi nt 8. This matter 1s discussed further in

Section 5.4.3.

5.4,2. Strains at the rosette strai-n oauoe l-ocations

Four electrical resistance rosette strain gauges vrere placed

on the model at the locations shovvn in Figure 5.7. Jhe gauges vJere

all located at positions corresponding to the centroids of elements

used in the Finite ef enlent analysls. Strain readinqs rvere tal<en for

the three loading cases. The stresses at the gauges were cal-culated

and these have been tabul-ated in Table 5.2,, r¡rith the stresses pre-

dlcted by the finite element analysis.

There is ç1ood aglreement between the measured and predicted

planar stresses at tl-¡e location at rvhich there \ryere gauges on either

side of the sheet. This sugçJests tl-rat we could expect better agree-

nlent for the measured and predicted planar stresses at the other

locations l¡here gauges r¡iere attached to one side of the sheet only,

if the component of stress due to bending v¿as el"lminateci . ïl-re pre-

dj-ctions for bending moments in the elements do not match the observed

values very wel1.

5.4.3. Tl-re irrulirratiorr of tlre principal stresses and thedifference in macJnitude betvreen the tv¡o principal

stresses

The difference between the two principal stresses rc I -o'2,vras obtained experimentally from the phobographs of the isochronratics.

97.

STRESSES TN MPa

LOADING ONE LOADING TWO LOADING THREESTRESS

L0cArroN t""i:::tä1 Load u'il:å:.å t'"0 t"i:::*Ëf Load

Location Stress Measured Predicted Measured Predicted Mcasured Predicted

A

A

dx

ay

90

+

-I.84

+ . iJ06 +.01

+.132

+.040

, I05

.015

.016

-.16 -.18 r.32

o3 -1.28

-.I4

1 .82

70

.62

T2

L.22

57

3il-.91

90B

D

D

(rx

oy

-I. 12 -.3? ?6

03

-,u2

+.I2 .o?

la

035ì . Lltrz

-. oiJ 07 . o09

+, .004 +.03 -.O2

and

mean

TABLE

OX

oY

¿xy

Mx

My

+.I?

+.01

.11

.003

.0til

.e4

.?I

.68

.48

.15

+

c

62

43

03

o4

28

5.2. MEASURED AND PHEDICTED STHESSES. POSITIONS OF

GAUGES AND DIRECTIONS OF STRESSES ARE SI-{OWN ON

FIGURE 5.7.

98,

PLATE 5.10. ISOEHROMATIC PATTERN TN JOTNT LOADED I,TTH A3OOON HORIZONTAL LOAD ON THE RIGHT.

TSOCHROMAT]C PATTEBN HESULTING FROM A 1OON LOAD

APPLIED OUTWARD ON THE END OF THE ROOF RTB.PLATE 5.11.

99

During the calj-bration test, when the fringe constant of Epirez was

being determined, a record rves kept of the stresses corresponding to

the chanç1inç¡ colours for the thickness of Epirez used in the caLibra-

tion test. It was therefore possible to match the colours on the

photograph to those in the cal-ibration test and hence calcul-ate the

value of o1 -c2. In addition, the inclination of the principa.l

stresses was obtaincd f rom tlre photoç,Jraphs of the isoclinics. Both

the experinrental and the theoretical values of ol- -o 2 have been

plotted on Fiqures 5.8. to 5.13. Quite reasonable agreement was

obtained betlveen the experiment and the computed values of ol -o2.

It s.hould be note'd that in many locations the stresses were quite

small. Because of this fact it u¡as difficult to determine accurately

the stresses in the experimental model since only a few different

colour I'rÌng¡es were present. AIso because of the small- stresses, the

initial stresses in the mode1, themselves very sma11, tended to affect

the results. An additional hindrance to the accurate determination

of the stresses by photo-elastj-city uJas the darl<ening of some colours

as the polariscope v,ras rotated. This phenomenon, v'rhich prevented

some colours from being correctly recognised, occurred because the

thickness of the euarter-wave plate rvas not equal to a quarter of the

wavelength of all the colours present.

Although there was generally reasonable agreement between

theory and experÍnent for the direction of the prlncipal stresses,

there ì¡/ere sorne instances where large differences were observed.

Some of the differences, in these instances, might have been due to

errors j-n determining the angles from the experiment. The isoclinics

were often difficult to separate and despÍte the fact that photographs

t¡rere tal<en only cvery J.So, some overlapping did occur. Agaln too,

100.

L

Loading IFis . 5.4

qI - o2 "1, "f$,4nA Peal< obs

' Sf,r.

l-oading 2

Fig. 5.5

Oì-Or "1" ofMpa Peal< obs

str.

ÊiÃ

Loading 3

Fis.5.6

I,BPeak obs stress 3.1

Peak predictedstress. Elementcentroid.

Peak obs . str¡lss.Element centroids

TABLE 5.3.

2.? 87/, I ,4 s6?d 5.3 54','o

2.8 1Tft 1.8 ?2/ 8¡B 90/'

RELATIONSHTP BETIVEEN THE PEAK AND OBSERVED VALUE9 OF

oÊ - o; AND THE PEAK VALUES 0F o-; - o; OBSEBVED ANDpAgorcfeo AT posrrroNs coBRESPoNórNG fo rtE¡,tElrltCENTIìOTDS.

+

++

+

x

X

X

I

f

*

f+?P+

?

Ê

*

I

x

x

+

+++++

++ss

t

+

t

Ì I

ltlÊ

ì

ì

+b

ff I + + ++tÊ I

T

-+)

ìt

-t

I

x>{

lF

.lr

+&

+

T

L

Y"

xxx

x

*.Á

"Å

x

&

iF

)F

T+s't+tx

x

x

l(

t

FiGURE 5.8OESE¡VED RI.¡D FREI] ICTEOVflLUEs OF ThE D¡FFERENCEEEThEEN TI-E PRINCIPRLPLHI.¡ßR STRESSES.

FRONT VIE¡T.

IOOO NEhITOI.¡S NPFLIEDHORIIONTRLLY RT THE

END OF THE CRNT RRILON THE RICHÏ.SEE FIGURE 5.',I

+ úB5E8VE0 PRINC¡PRL SfnESSO¡FFERETCE

-+ PRtlllCfEIl PRINC¡PAL SIRESSDIFFEflETCE

rHE s'Iî{BOLS F8E ßLLIB¡EOPNRNLGL TO T}É O8€ER\EORIJO PED ¡CTEI] PRINC¡PH.6TRE6€ES.

scRLE ltxrtz.25lfno

+

à ++++ t

à

-.4 a

.-ù â +++Xx,.ì à+ârroXXX

+ t ++ * + + xx XXa

-t f x

xx I t Ê f + + , ,

+ fi +It

I )r + fx

+

+

++

+

I t ¡ +

+

+

+

+

¡ ì

\

ì

\'t

FICURE 5.9OESEßVED f,I.IO PRED ICIEDVFLIJES OF TI-E DIFFERENCEEET}üEN ThE FRINC¡FRLPLFÀNR SÍRESSES.

FRONT VIE}I.

IM NE}ITOItrì RFFLIET]DOTôIIffiDS ON TTE E¡.ID

OF THE CANT RFTL S.¡

T}E RIGHÎ.8EE FIflNT 5.5

+ OBSEñ/E0 fßnc¡PFL SlßtSSD¡FFER€'CE

+ PRültclfD mUIclPil SIßESSDIFFF¡E}CE

il€ SïrB0LS AE RLL¡ertoPñRnLr¡L 10 n€ @6[f,]\r€OSD PNED¡CTTD PRINCIFH-efRE6€Ê6,

SIRLE lûrÊt.6llfnolu

t ¡ t I , a

¡ + 1. +

+ + + + ++

4 + I

+++ + + + + + + |r I +¿

x >É X X \.*P*x ì $ * * ,t x

+ + + + + + ++ +

+ + + )( +

+

+

f

*

¡ I ¡

I x

+ + +

I + ,FIGURE 5.10OBSE¡VED RT.ID PRED ICTEDVRLUES OI TI.E DIFFERENCEEETh¡EEN THE FRÍNCIFRLPLFMR SIRESSES.

FRONT VIEH.

IOO I{EHIONS' RPPLIEDOUTI.IRRI]S RT TTIE END

OF THE ROOF IIEHBER.SEE FIGURE 5.6

+ 0858t[vED PRINCIPFL SfBESsOTFFER€IGÊ

-r PRtOIcTtll PRINC¡PÂL SIRESSO¡FFEßE¡f,E

ftE S'ITBOLS RNE RLLIEI|€DPÂRNLI€L TO T}€ O8€ER\€ORIID PFEDICTIO PßINCIPFL6TRE5€€6. o(^,

scâLE r(l{tt{.5t{P8

+

+

Y x + + ++fÈ

I fff

I

,

I

++++ Ê* +* XXX)F

)

t

,

I

1

FIGURE 5.1 1

OESENVED ßIiD FRED ICTEDYRLUEs OF TI-f DIFFERENCEEET}.¡EEN T}E PRINCIFRLPI.F¡¡Rß STRESSES.

BßCK VItr{.t000 ilEtJToÀs RPFLTEDHORIZONTßI.LY RT T}EEIü OF THE CRNT RRILOTI THE LEFT.SEE F¡GURE S.q

+ 0BSEßVE0 PRIXcIPft SfRESSDIFF€REI€E

-r PRfllIcTtD PRINC¡PßL SfßtSSD¡FFERETüE

ITIE sIî€OLS Rtr RLL¡6¡€DPÂRfl-l-EL T0 Tì€ t86tfil'€OilD PNEO¡CTTI' PßINCIPFLGTRE66E6. o

ÀscflLE l(¡{lt2.25tfn

t .a + + !

\

I

¡

1

¡

,

t:r.ìl)t lì a KXÉ î

X'*

FIGURE 5.12OÂSERVED 8Iü] FRED ICTEDVRLUES OF TI.E DIFTEREI{CEBETTËEI.J TTË FRII[IPFLPLFI{RR STRESSES.

BSCK V¡EÏ.IOO IIEHTOIIS 8PPLIEDDOhS.II{RRI5 ON TIE END

OF THI D8NT RNIL O,ITI€ LEFT.SEE FICTRE 5.S

+ 08SEEVE0 lßfiC¡PRt SfRt6So¡ÍFEn€t€€

+ PRII)ICIIÐ Pß¡xclPßL SfßtSsD IFFEfiE¡CE

llE s'lltoLs 88€ f,LLIÊlC0PÊRÂLI¡L 16 T}C

'SGÊfiVEDSO PRE¡]¡CTT¡ PR¡I¡CTPfl.€TREê]EG.

scRLE lo{aÊl.slfro(¡

r

++++ \

t

*

f+

I

\

b

!

t

,

a

,l

a

*++T

+tIÌ

IùuI,,

F IGURE 5. 13OESEEIED ñfi¡ FREOICTEDVH.If,S Of TTf DIFFEREI€EEETI.[O{ fT€ PRINC¡FALPLFMR STßESSES.

BRCK VIEI{.

100 NEltIotffi ßFFL!tDOUThNRI]S ON T}E END

OF TTE ROOF IiEJ{BER.s€E FldxE 5.6

+ OBSEf,YE¡ IR¡IG¡Fff. 5TNT6SD¡FFET€IGE

{ FffO¡CTfD tßllclPßL slRtss0¡ÍFEIE}CE

IHE SìlOlLS ffi RLLtEtg¡zmtt.lÁL T0 ΀ og€tfilEDmo rru¡cro PR¡tlclPÈ€f߀€€€-

SCSLE IüTÊ6ll?ß

oOl

IO7.

in areas that were lorv1y stressed, the initial stress v,ras observed to

be affecting the results. Desplte these factors, hovrever, not alÌ

the divergence could be blamed upon the observation of the isoclinics.

In the third loading case (see Figure 5.6.), for instance, the observed

torsion in the left hand section of the cant raj-I, as shown in Fig-¡ure

5.10., ì¡res considerably larger than the predicted torsion. f n

addition to this, however, the observed displacements of the loaded

end (see Table 5.1.) were almost three tj-mes larger than the predicted

values. A possible explanation for these results vuas that the right

hand support, node I, was not as rigid as the left one, node 1, þnd

hence more of the load was carried in torsion in the left hand section

of the cant rail

The peak stresses observed in these tests did not occur at

the centroids of the elements. Since the peak stresses will g¡ener-

ally not be expected to occur at these points, a llmitation is placed

on the ability of finite element analyses, v¿hich only , OUlpUt ' stresses

at element centres, to preclict the peal< stress concentrations. The

relationship between the peak value of o1 - 612 observed in each of the

three tests r and the peal< values observed and predicted at the element

centroids is shown in Table 5.3. The relationshi.p varies vuith the

type of loading and v¡ith the shape of the structure but u¡ilL be

expected to improve as more detailed element meshes are used.

5.11. Investiqation of the Accuracy rnore detailed Ele nnt Meshes

In order to examine how the accuracy of finite element

analyses was affected by the use of more detail.ed element melshesr a

smaller Epirez model. was made and tested. This srnaller model was

constructed from hollow tube sections simil-ar to those used in the

I08.

doorway structure. The dimensions of the new model have been shown

in Figure 5.I4. as has the uray in which the model was tested. 0n1y

one type of loading was applied and several photographs were taken of

the isochromatics as the l-oad was increased. At a loading of 63

Newtons the model failed. The broken model is shourn in Plate 5.12.

Two different finj.te element analyses were made of the test using tvro

different finite element meshes. The First mesh vuas similar to that

used in the analysÍs of the larger dooruray sect.r on. . The second mesh

contained four times aS many elements. The efement meshes are shovln

in Figures 5.15. and 5.16. The experimental and theoretical values of

úl - a2 in the el-ements along the lines A-4, B-B and C-C, shown in

Figures 5.15. and 5.16. have been plotted on pigures 5.17. and 5.18.

The results of the more detaiLed analysis agreed more closely with the

experiment but still underestimated the value of o-I -cr2 on line C-G

by up to 30.¡6. The relatj-onship between the peak observed vafue of

ol -o-2 and the peak predicted value is tabulated in Table 5.4.

Stresses at load of 53 NewtonsaI <-2

MPa"/6 of peak

observed valueofo-1 -o-2peak observed value cl -o-2

peak observed value o-1 -o-2at efement centroid. Coarse mesh.

16.5

13. s e2l"

peak predicted value aL -o-zat element centroid. Coarse mesh.

peak observed value oI -o2at element centroids. Fine mesh.

peak predicted values o-1 -o-2at etrenent centroids. Fine mesh.

9.4

16.5 1oo%

1r.6

5?"1,

?ú1,

TABLE 5.4. Comparison of the peak observed value ofo-f -o-2ffiñTffi the observed and predicted peak values at the elementcentroids j-n the two finite element analyses.

o(.o

$

oN

HJ '-A

'F"r

r09,

A

B,C

IA

t A

B c

FIG. 5.14. DTMENSIONS AND LOADING ARRANGEMENT OF SMALLEPIREZ MODEL.

460mm

50NConstructed from 50mm x 41 x I.Z}mm

ho1low box tubing.

Eeof-r\l

BJ [c

FTG.5.15. GOARSE ELEI,¡ENT MESH

L

FIG. 5.16. FTNE ELEMENT MESHB,C

il0

DIFFERENCE BETWEEN PRINCTPAL STRESSES ALONG SEGTTON

A-A (FrG. s.ls) - GOAHSE ELEMENT MESH -

20

15

FrG. 5.1?.

âr observed stresses on Section A-A (fig. 5.15)b. predicted stressesc. predictions made when in-p1ane rotatÍonal stiffness

of elements was very large (see section 4.2.2,L),

ft)

-lr{(úoûrdo-rúu,0)

=

aú)LrJÍrt-at)

a

1 0

5

b

c

0100 50 0 50

DTSTANCE ALONG SECTTON A-A (plg. 5.ls)100

(vtittimetres)

FIG. 5.19.

111.

DIFFEBENCES BETWEEN PRINCIPAL STRESSES ALONG SECTTONS

B-B and C-c (rig. 5.16) - FINE ELEMENT MESH -

â¡ observed stresses on sectj-on C-C (pig. 5.16)b. predicted stresses on section C-Cc. observed stresses on section B-B [pig. 5.16)d. predicted stresses on section B-8.

20

15

10

a

arlrlrúoa(ú(LofÐ0)

=

aat-rj(rFa

,

I

br'-

c

d

\,

\I/

/\

\I .---- - -:

\

\a

5

II

Ial

J--/

- _tl

0 500DISTANCE ALONG VEHTICAL MEMBEH

50 100

(uittimetres)100

112.

It is interestingl to note that, for the l-oad at which the

model- failed, the predicted stresses in the model vJere several tirnes

smaller than those at which the Epirez would normal.ly be expected to

fail. AIso the planar stresses, as indicated by the photo-elastic

fringes, vrere lllceivlse quite snral.l. Therefore, if the possibility

o1" a r¡real<ness in some section of the Epirez 1s iqnored then it fol-Ior¡¡s

that large bending mornents existed in the wal-ls of the tubes and that

these moments were not accurately preclicted by the finite efement

analysis. This suogestion, although supported by the resul-ts of the

strain gauge readingJs discussed in Section 5.4.2,, has not, however,

been sufficiently investJ-gated for any fÍrm conclusion to be reached.

5.6. SUMMARY

ïn the experiment that has .iust been described, the finite

element method has confirmed that it is a useful- tool for analysing

conrplicated structures. Although the mesh that v¡as used was fairÌy

coarse, the planar stresses predicted at the centroids of the elements

have been shown to be a good approximatj.on to the stresses that have

been observed by photo-elastj-c means. Since bending stresses do not

affect the photo-el-astic results, it has been impossjble, however, to

checl< the predictions for the out of plane bending in this experiment.

Although finite element analyses urhich only calculate stresses

at element centroids can not be relied upon to predict peal< stress

concentrations, they do present a useful guide. Furthermore the

results of the analysis have been shown to be closer to the peak

stresses r¡¡hen more detaÍl-ed element meshes were used.

PLATE 5.12. EPOXY-HESIN MODEL OF T-JOINT AFTER FAILURE

Q

lr4.

6. STRAIN MEASUR EI4ENTS IN BUS BODY

6.I. Introduction

when the civil Engineering Department was approached by

the Tramways Trust to determine the possible causes of the crackinç¡

that was occurring in their buses, it was agreed that a partially

constructed bus should be made available for testing. The bus

that v¡as provided had been fitted with a body frame and a floor but

had no outer shelt, ceiling, tvinclows or seats (see plate 6.I.).

At this stage of the construction it was possible to fasten elec-

trical resistance strain gauges to the frame and to car::y out a

series of static and dynamic tests. The initial aim of these

tests was to determine the stress distributions that arose flrom the

trial loadings. It uras hoped that these results would enable the

in-service stresses in the critical sectj-ons of the bus to be pre-

dicted.

when it was found that the buses were cracking, attempts

were made to strengthen the frames. Shortly before the tests that

are described below were carried out, extra stress paneJ-s were

installed in the rear door pillars of all new buses. Al-1 the

tests lvere carried out tr,vice, both with and without these panelst

in order to j-nvestigate their effect.

6.2, Positioninq of the Strain Gaucres

The layout of the strain gauçies is sholn on Fig¡ure 6.1.

fi4ost of the cracking had been found in the left side-wall of the

busrso all of the strain gauges were placed on this side. The

l-eft side-wa]I is v¿eakened by the presence of the trryo doorurays.

The g'auges were concentrated in those regions 1n whj-clr cracking had

occurred and they were placed at locations that would, it was hopedt

11 5.

F

'ì

ri

b

-

PLATE 6.1. TI¡JO VIEWS OF THE INCOMPLETE BUS THAT WAS USED

FOR TESTING.

-

116.

enable the behaviour of the critÍcal tubes and panels to be deter-

mined. It was hoped that a mechanism that was causing failure might

be discovered and subsequently remedied.

6.3. Comouter analysis of the static tests

A matrix analysis of the left hand sj-de of the bus was

carried out using a relatj-vely simple plane frame analysis. The

layout of the representation used to model the side-vlallr j-s shorvn

in Figure 6.2. The left chassÍs member was included in the analysis

by assuming it to be in the plane of the side-walI and by connecting

it to the side-wall with smaIl members whose axial and bending

stiffness was such that they simulated the bending stiffnesses of

the outriggers which join the chassis to the side-wall. The analy-

sis was done using the ACES programme(Ze). All stress panels were

ignored in the analysis except those in the door pillars which vuere

included with the two tube members to form a single pi11ar member

fsee Figure 6.3.).

Because only half the bus frame was modelled it was not

possible to analyse torsional or non-symmetrical loadings. fn

addition, the analyses were based on the assumption that all loads

r¡rere shared equalLy between the two sides of the bus. This assump-

tion will not generallY be true.

6.4. Static Tests

6.4.I. Jacl<ing the Bus from the Fìear

The bus rvas foaded at the rear ends of the two longitud-

inal chassis members by two large jacks. Vertical loads of 9 KN

urere applied first to the left side¡ dnd then to the right side of

v2(a) h21

6)d37(4o) d3'1,ß4)

1)

LOGATION OF STRAIN GAUGES.GAUGE NU¡ilBERS SHOWN. OUTSIDE GAUGES IN BRAGKETS.ALL GAUGES ON LEFT STDE OF BUS.

)

v(564)h(58A)

V 53)o)h )

((

.pco¡r

lJ.

.{

1618

11

657i,8

141312

fi7-28

027-v30,(35)

u

Rear doorvray

2V

h9d1

v36(39)u

44 t,45

46t,4748û,49

v5 2, h50, d51

v (58)(d) h ( 56)

d(5Ð

(6263æ

k) 59A(e)(59)

FIG. 6.1.

45 35 31 28 25 19

1

1830 27 24

REPRESENTATTON USED FOR Í!,IATRTX ANALYSIS OF LEFT BUSSIDELTALL. CHASSIS I,IEMBER DEFINED BY 5?-50-48-58.

10 5 441

6 44 38 34

FïG.6.2.

MF0 160

SECTION OF DOOR PILLARS. This section wasrepresented in analysis by a single member.

o

2

1

B

I

36

2

14

13

21 15

01523

622

51 50

33

5253

293237

047

42 39 36

FrG.6.3.

119.

the busr €rrìd finally a total load of 18 KN was applied evenly to

both sides. The strains were recorded at the first 41 gauges all

of rvhich v/ere located around the rear door on the left si.de of the

bus. The measured strai-ns are shown on Figures 6.4., 6.5. and

AEU ¡(J ¡

The peak recorded straÍns were measured j-n the door and

vrindow pillars. At these points it was noticed that there were

large straj-ns developed when the left side of the bus was loaded¡

but very smal1 strains r,ryhen the riqht side '",las loaded. The çleak

strain recorded in the pillars was 301 microstrain v¡hich is about

a quarter of the yield strain. Ihe other interesting area is the

stress panel above the door. In this reçrion plate bending was

oh.¡served and it was seen that the largest strain was often developed

v¡hen load was applied to the right hand side of the bus. Also the

sign of the strains at many of the gauges changed vthen opposíte

sides of the bus v¡ere loaded. Of particul.ar interest are the plate

bending moments observed in the vertical gauges in the stress panels

above the rear door. These moments vvere presumably caused by

moments in the adjacent roof ribs.

VJhen the stresses in the pillars were compared with those

predicted by the matrix analysis (see Figure 6.?.), it rvas observed

that there was reasonable agreement. ft seems lilcely that the

omission of the stress panels above the cant rails over the doors

and acljacent windor¡¡s in the analysis was responsible for some of

the observed diff erence betyueen the values. The matrix analysis

predicted that larger stresses would have been developed in other

lvi-ndow pillars where no strain gauges had been placed.

7+150,-52,+112+2 3,-3 0,-4

+23,-30,-4

û -200,0,-204y' On-200,7,-193

' t -68, o,-67\ edge-286, 26,-27 O

t -8,-23,-30

-301 ]4,-2931

+238, -9,+229 t

./,/,

top -13,+3,-20bot -1 1, O,-17in s -61 +28 ,-22ou t(+38,-31,+7)

v -12,10,9,(6,4,11''+25'-3h 1 2, 0,-1 2,(1 5, 1 ,1

.l)

d-29, 1 1,-25 l-5, 0;1 2) S-38-,:3Or77

-92154;154

-63,+41 ,

-12

v -7 8,92,29,( 38 ;53,-2 9)h 9,-51,- 5 4,(48-7 1,-23)

Jd 13, 2 0,28,(93,-71j5)

135,-1 27,15

v -36,-1 5 0 717 5 I 27,120)h 0,74,68,(11,83,92 )

Vd, - ,-,-,( 20,68,81)

9I155173

V ,-177h+ 0,+8,+8d+ 7,-5,+10

FIGS.

Bear

Rear doorway

6.4, 6.5, 6.6. STRATNS 0BffiRVED I{HEN BUS ltAS JAG<ED FFOM

BEAR TN MTGROSTRAIN.Results in brackets indicate external gauges.Order of Results (A<ru left jack, 9KN right jack,IBKN on both)

l'\)o

150

150

Stresses in cant-ratl

FrontRear

t

t

BUS JAGKED FBOM BEAR.OBSERVED AND PREDICTED BENDING STRESSES IN THE WINMWPILI.AHS AND CANT RAIL OF THE LEFT SIDEIVALL.SYMMETRTCAL LOAD. TDTAL LOAD TS 18KN

X Observed stresses.

r\J-

FfG.6.7.

lmm. represents 1O MegaPascaLls

'I22.

6.4.2. Loadinq beh ind rear axle

The static stresses due to the weight of the eng¡ine and

the rear section of the body urere investigated by loading the rear

section of the bus. Load lvas applied by stacking a line of flat

vreiç;hts on the floor. The line of weights was placed directly

over the centre of the engine about 2 metres behind the rear axle'

A maximum load of almost 18 KN was applied '

The measured strains have been shotryn on Figures 6'B' and

6.9. For this loading case the strains were recorded at the ç¡auges

around both the doors. As might be expectecl the measured strains

r'¡ere similar in distribution to those recorded in the loading case

described in section 6.4.1. when the rear of the bus ì/vas iacked on

both sides. For the neur loading, however, the strains tvere

smaller and, of course, were opposì-te in sign to those in the

earlier test. The strain readings around the front door lvere found

to be almost as large as those around the back door and as before

quite reasonabl-e agreement was obtained betvreen the observed

stresses and those predicted by the matrix analysis in the cant

rail and in the door and windor¡r pillars. The calculated and the

rneasured bending stresses in these members have been plotted on

Fi-gure 6.10.

6.4. 3. Loadinq betle en the two axLes.

Inordertoinvestigatethestressesthatarisefrom

se11=vleight ancl passenger l0adings in the section of the bus between

theaxles,thebuswasloadedwithflatweights.TheweÍghts

were placed along the floor of the bus in three lines corresponding

tothepositionsofthetworowsofseatsandthecentreofthe

aisle as is shown in Figure 6'1I' A total load of 4O KN v¡as

-108

+222

+81

top +31

bot +42i ns +50ou t(+29)

ù +'159edge +147

t +53edge +167

-41+18

+40

+2_6

v +34,(+2 0)h +21, (+1 8)

d+43, (+34) ¡l

+118

+65

+32

v +39f+1 'l)

h +44f+26)d + 5,F12)

v +142(-66)h - 18 (-43)

v d -,(-23)-I1

v +107h+ 15d+ 20

Rear dooruray

LOAD BEHIND REAH AXLE - STRATNS IN GAUGES ABOUND REAR

æ08 IN ùIICRO STRAIN. External gauges in brackets '

rear

N)(J

FrG. 6.8.

124.

v -35h +32d +14v

v +49h +16d +31(+24)

- G7)

ins -9ou t(-3'l)bot -34

t+9+32

t +66edge +50

FRONTD OO RWAY

û -164edge - 1 85

t -20e e -78

LOAD BEHIND REAB AXLE - STRAINS IN GAUGES AHOUND

FRONT DOOR IN MIDRO STRAIN. External gauges ínbracketsr '

v (-38)h(-60)d(-63 )

FIG.6.9.

II

,(

50

50

Front

I

I

,

f

r

ßear

FIG. 6.10. LOAD BEHINO REAR AXLE.OBffiRVED AND PREDIDTED BENDINGSTRESSES IN THE WINDOW PILLARS AND CANT RATL OF THELEFT STDEWALL.

X Observed stresses

1\,(¡

lmm. represents 5 MegaPascalls

L26.

distributed evenly between the two axles. When the bus vras unl-oaded

the right hand ror¡¡ of weights was removed first and a new set of

straln readings tvere taken. Next the centre row of ureights rlere

removed and further readings were made. From the three sets of

strain measurements it was possible to examine the effects of both

uniform and unsymmetrical loadings.

The recorded strains are shown on Figures 6.12 and 6.13.

Once again the largest strains v/ere recorded Ín the door and v¡índow

pillars. The strain values in these members astree well with the

values predicted by the matrix analysis vuith the exception of the

strains in the main front door pillar which were the largest

observed strains and which v¡ere underestimated in the analysis by

a factor of tr¡ro. The observed and predicted stresses are plotted

on Figure 6.l4. The matrix analysis again predicted that the

mernbers that were strain gauged were not the most highly stressed.

Reasonably large plate bending strains rÀJere measured in

the stress panel above the rear door. The strains measured at

gauges 30 and 35, (see Figure 6.15), were possibly caused by a

moment in the adjacent roof rib. The strains measured when only

the left l-i-ne of vreights rlas applied v¡ere, in most cases, equa] to

about half the strains recorded when all three lines of weights

v;ere in position. Also in every significant case a load on the

left side was found to produce a strain with the same sJ-gn as a

load on the riqht side.

Almost all the measured strains developed by the loading¡s

beLvreen the axles have opposite signs to those straj-ns devel-oped

rryhen the bus was loaded behi-nd the rear ax1e. Therefore when

all o1' the bus is ]oaded, these strain components will tend to

127,

il ht

front rear

lef t

FIG . 6.1T. LOCATION OF I¡JEIGHTS FOR BETWEEN AXLE LOADING

Gurved stress panel

lL

30,(35)¡l

Rear door

FIG. 6.15. LOGATTON OF GAUGES \IJHICH RECOBD PLATE BENDINB INSTRESS PANEL ABOVE REAR DOOR

Gauge 30 inside' gauge 35 outsidÞBoth gauges vertical.

lt

FDnErn-c:EEqEEE I] EI E

EE]EEE

v - B,- 6, o,(1 o,Blåi5,+eh 11,6,5,(13,9,11)d 0,0,0 , (7 ,6,9 ) J

+159,+124,+81

-'l 89,- 135,-B

57,-41;25

40;29;14

2) -_42,-29,-13

;1-21

¡1,rledse

5

063

I

2

t3

49

2

3

9

1

6

5

1

-232dget

e

top -4¡10,-5bot -]B;15,-3ins-1 4;17, O

out( 0,0,0)

+168+130,+100+45+30,+32

+9,+9,+'16

t -169,-128;82

-85,-7 2,42

-153,-128,-ggf+gh 39,35,37,(82,69,64)jd -16;1 5;11,(1 23jO4,U)

186,145,119

v +23,+19 ¡351+27¡2a*lh -50,-33;23,(-9;5,0)

Vd -,-,-,(0,0,0)

8,-6¡18

v-16, 11+5h+12,+17*13d+5,+6,0

FIG . 6.12. LOADS BETWEEN AXLES.STRAINS IN GAUGES AROUND REAR DOOR TN MICRO STRAIN.

RESULTS OF THREE LOADTNGS GÏVEN:(+Of¡ evenly distributed, 2?(N right hand-side weightsremovedr 13KN left hand side weights only)External gauges in brackets. Í\,

CD

l 29.

ins +15¡lZt7out(+45+3 5,+19)bot +50,+39,+'19

v +6O+42,+25h -68,-48,-23d -38,-25,-11

d(-56,-31,-17)

(-38, -27-14)

v (-88,-49,-20)h (-50,-39,-22)d (-62-47,-27)

7-(0, 0,0) ô -30-21,-11

,-40,¿3t -127,-gB,-49

edge -98i73r42

v(+58,+45,+28)h(+90,+64,+35)d(+90,+68,+39)

I +228,+195,+125edge -98;73,-42

t +20,+.l8,+9edge+106,91,55

LOADS BETWEEN AXLES STRATNS TN GAUGES AROUND FBONTDOOR TN MICRO STRATN. BESULTS OF THREE LOADTNGSGIVEN: (AOfru evenly distributed, 2Z(N right hand sideweights removed, 13KN l-eft hand side weights only).External gauges in brackets.

FIG. 6.13.

75

75

xI

Front

I

I

x

fI

Bear

4OKN R/ENLY DISTRIBUTED BETWEEN AXLES. OBSEFII/ED AND

PREDTCTED BENDTNG STRESSES IN THE ì¡.lÏNDoW PTLLARS AND

CANT RAIL OF THE I-EFT SIDEWALL.

X 0bserved stresses

G,o

FIG. 6.14.

lmm. represents 5 MegaPascalls

l_3r.

cancel- each other.

6,4.4. Loadi- f orlvard of the front axle.

Thi-s test rvas carried out to examine the effects of self-

weight and live Ioads on the front section ol' the bus. The bus

was foacled 1.4 metres in front of the front axle with f1.at weightsI

placed on the floor. The arrangement of the vreights can be seen

in Plate 6.2. A total load o1'9 KN was applied and strain readings

riere tal<en ar.ound botlr tlre front and rear doors. Tlre strajns

developed by thls ]oacling are recorded on Figure 6.L6 and 6.12-

They were generally smal-ler than those measured for the other

loadingJs. The strains in the winclow and door pi 11ars ag;reed

reasonably r,ve]l r,^rith the vafues precJicted by the matrix analysis

although the strain in the top of the front door pi11ar vras twice

as large as the precJicted value. It is interesting to note that

a simíl-ar. difference was observed in this member r,vhen the centre

o1'the bus lvas loaded. The strains observed in the cant rai1s,

hovrever, did not agree r¡¡el-l vlith the predicted values. This was

undoubtedly due i-n part to the absence of the stress panels in the

analysis. The observed stresses in the cant rail above the rear

cioor, horvever, were opposite in direction to those predicted by

the analysis. This last difference may have been due to errors

in determining the bending in the cant rail above the rear door t

for, as can be seen from Fig¡ure 6.17, the only strain gauges in

this section r,vere located in the stress panel. The observed and

predicted stresses have been plotted on Fj'gure 6'18'

PLate bencling was ag¡ain observed in the stress panel

above the rear cloor and the strains in gugges 30 and 35, (see

Figures 6.15 ancl 6.f?) i-ndicated a plate bending moment that could

have been causecl by a moment in the adjacent roof rib.

132.

V

h-53+7

d+4v

v(+107)h(+39)d(+59)

40

(+49)

- (+18 )v

ins 0out (-40)bot (-41 )

t +2336

t +122edge +79

FRONTDOORWAY

t -20edge -36

t+9

gKN LOAD FORIìJARD OF FFONT AXLE. STRAINS IN GAUGES

AROUND FRONT D00R IN MICHO STRAIN. External gaugesin brackets.

ee -9

v (-39)h(t2)d(+65)

FrG. 6.16.

+j

+_4

-1v+1,(-2)h -7 ,(-3 )

d 0,(-3 ) \¡

+71

l+1'l

-31 I

I

top -10Þot -14ins- 7ou t(- 1 2)

t +29

,,/edSe +20

' t+9\ edge+32

7-44-1 5

-9

+6

v + 53,(-34)h- 27,ç37)d+ 7,Ç45) j

v -35,(+13)h +12,(+10 )

v d -,(+10)+29

V

h-9d-l 5

gKN LOAD FORWABD OF FRONT AXLE.around rear door i-n micro strain.in brackets.

Strains in gaugesExternal gauges

(,(,

FrG. 6.17.

3s

35

fxf

t

TtI

Rear

9<N LOAD FOBWARD OF FBONT AXLE. OBSEBVED AND

PREDIGTED BENDING STHESSES TN THE WINDOW PTLI-ARS

AND CANIT RATL OF THE LEFT SIDEWALL.

X Observed stresses

(,À

FrG. 6.18.

lmm. represents 2.5 lvlegaPascalls

135.

6.4.5. Disnlacing the Wheels

The effect of small static displacements of the v¡heeIs

was investigated by d¡iving each wheel in turn on to a 3 inch blocl<

and observing the change in strain. Unfortunately this test vuas

carried out at the start of the series when only 20 strain gauges

hacl been fastened to the bus and these only in the area around the

rear door, In addition only two gauges had at that time been

fastened to the stress panel above the rear door where the greatest

torsional effects v/ere observed in the other tests. Despite

these limitations it vtas possible to mal<e several interesting obser-

vations about the measured strains. Firstlyt as can be seen in

Figures 6.19 and 6.20 the bending strains in the door and window

pillars were Elenerally smal1. The horizontal strains in the stress

panel above the door and the vertical strains at the top of the door

pillars vrere larger by comparison. Also the sign of the strains

produced in the more highly stressed gauges lvhen the different

wheels v,rere raised indicated that the torsional component of the

load on the bus lvas responsible for greater strains at these gauges

than the bending component'

6.5. Effect of the Addition of Extra Stress Pane1s to the FìearorP ars.

As was mentioned in section 6.1., additional stress panels

were added to the rear door pillars of all new buses after the first

cracking was discovered. The rear door pillars had previously

consisted of two 50 mm x 4O mm rectangular tubes 400 mm apart and

connected on their inside face by a 2.5 mm stress panel. The new

stress panel was similar to this and was urelded to the outside face.

top + 4,-8,-2,-9bot + '1, +0,-8,-'l 1

i ns +15,-9 ,+1 O,-17out - 4,+9,-10,-5_

7l-23,+19;30+11I -1 5+1 4,-19,+2

I -13 ,+1 1,-16,+2

l+6,- 3,-3 ,-4

-2,-6,-7,-5 |

,/,

+14,-14,+12,-14

-61,+45,-51,+31t-

+ 5Z-6_8+57,- 45

V- ,+ rl-h- 4 ,-5 ,-28 +3d-4,+6,-9,-4

Rear doorvray

FIGS. 6 .19, 6.20 STBATNS IIEASURED IN GAUGES AROUND BEAR DOOBS

I'/HEN WHEELS UJERE BAISED 75mm. IN TURN .(urcno srnnrru)ORDER 0F RESULTS: fUeft Front, Hight Front,Right Rear, Left Bear). External gauges inbrackets.

(^)O)

13?,

Tvro complete series of tests were macle both with and lvithout the

aclditional panels and the conclusj-on reached v¡as that the addition

ofthepanelsreducedstressesslig¡htlyinthereardoorpillarsbut

had little effect on the rest of the frarne. For this reason a com-

prehensivesetofresultsforthetestswiththepanelsaddedhasnot

been included.

6.6. DYnamic Testinçl

Aknowledgeofthebehaviourofbusesunderstaticloads,

although necessary for their design, is not suflficient for designing

for dynamic conclitions. Because of the difficulties encountered in

extendin¡¡ structural analyses to dynamic problems of this complexityt

a standard practice in the past has been to al-low for dynamic loads

by multiplying the static load by a factor of about 2'

fnordertocheckthesuitabilityofthismethodandto

examinethedynamicstressesintheframe,aseriesofdynarnictests

Werecarrledoutonthestrain-gaugedbus.Twelvegaugeswhichhad

given}argeresponsestothestaticloads'V'/erechosenandthese

guagesWereconnected,fourgaugesatatimetofourbridgecircuits

andfromtherntoafourchannelpenrecorder.Thelayoutofthe

recording equì-pment is shown in Plate 6'3' The ç¡auges were all in

the rear section of the bus, in the door and window pillars and in

the stress panel over the rear door (see Figure 6.21). The strain

recordswerecalibratedbyswitchinga560Kohmresistoracrossthe

gauges.Thisproducedachangeofresistanceinthebridgecircuit

arm equivalent to a strain of 1OB microstrain'

13 B,

rl11

ilT I

I

u.. ¡.-!

I

, ib---

PLATE 6.2. LOADING FORìJlJARD OF FBONT AXLE.WEIGHTS.

ABRANGEMENT OF

I

INC0IvIPLETE BUS PRIOR TO DYNAMIC TESTING.THE RECOHDING ESUTPMENT IS AT THE FRONT AND

THE gKN LOAD OVEB THE BACK AXLE TS VISIBLEAT THE REAR.

t'I

t.

Fl¡

Il

t

I!

ì

,

¡

r/\

I

I

ì.

J

.l,

I

,1 trr _1

PLATE 6.3.

15

16

7 c.1.,8 edge

4

4

INSv36H383

OUT3941L

10

Bear doorwayFìear

(,!o

FIG. 6.21. L0GATI0N oF GAUGES USED IN DYNAI,4IC TESTS

I4A.

Prior to the tests, a 9 KN load was placed in the rear of

the bus over the enç¡ine sc that the worst expected dynamic strains

could be observed. This load can be seen in Plate 6.3. Strain

records were made when the bus was being driven through the streets

and urhen the bus r¡as driven over obstacl,es of known height. The

observed strains were compared with the estimated static strain at

each g¡auoe.

6.6.1. Determination of the estimated static strain

The static strain at any of the gauges prior to the dynanric

testing was tal<en to be the sum of the strains caused by:-

(t) The weight of the engine and the bus body behind the

rear axle.

[Z) ffre weight of the bus between the axles.

(:) ffre weight of the bus forward of the front axle.

(+) ffre weight of the additional load placed over the

engine prior to the dynamic tests.

The strains in each of the gauges due to any of these

forces was estimated by multiplying the force by the strain recorded

in the appropriate static test. The magnitude of the forces men-

tioned above was estimated by assuming the mass of the partially

finished bus to be 6 tonnes of which 2 tonnes was the engine, 2.5

ùonnes was the partly finished body and 1.5 tonnes was the wheels

and axLes. The weight of the body was considered to be uniformly

distributed along the length of the bus and the weight of the wheels

and a Xles was assumecl to have no bearing on the stresses in the

frame. As menti-oned before, tl,ro complete sets of tests were carried

out both with and without the additional stress panels. Because

the difference between the tests was so minor, the results from these

L4T.

two series have been combined and all the quoted values for dynamic

strains and estimated static strains are averages of the tr¡¿o sets of

tests. The estimated static strains and the components from which

they vrere f'ormed are given in Tab1e 6.1.

GaugeLocation

GaugeNo.

27 KNbehind 12.3 KNrear betv¡eenaxle axles

Extra5 KN load

forward behindof front rearaxle axl-e

Estimatedstaticstrain

Rear doorpillar

B316T2

166

+1+2-1+

7L]

I410

+

5??o3553

+

i+

514942

oU

+16+14-20-17

+ 205+ 250- 12s+ 205

Wi-ndowpi11ar

1516

- 2r7+ 316

+42ÊÊ

-16+12

67+ 100

258372+

Horizontalgauges in stresspanel above reardoor

34

3B4\

1833B3974

4+55-16

Ã

+29-32+14+11

5611T2t'7

- 2I4315390

Vertical- gaugesin stress aboverear door

3639

+ 2D4- 114

-27+14

+6?36

+ 244- I23+

0?

TABLE 6.1. ESTIMATED STATIC STRAINS FOR GAUGES USED IN DYNAMÏCTESTS (t'rticrostrain )

6.6.2. D namic strains rneasured when the bus v'ras driven overat blocks

For these tests the bus was driven to a quiet secti-on o1"

road and a record of the strains was kept while the bus was driven

over a series of different arrangements of flat weights. The bus

v'ras driven over the blocks at a speed of 5O l<ilornetres per hour.

Four basic arrangements of blocks were used -

(1) A 50 mm high obstacl-e on the left. P1ate 6.4. shows

the left front wheel- passing over this obstacle.

L42,

(Z) A 100 mm high obstacle on the left'

(¡) A 50 mm high obstacle on both leFt and right'

(4) A 50 mm obstacle on the right. This arrangement was

used for four gauges onIY.

As the bus passed over the blocks two pulses corresponding

to the front and rear lvheels striking the blocks were recorded'

The strains produced by the rear"'vlheels stritcing the obstacles were

generally larger than those caused by the front wheels. The reasons

for this were that more weight was carried by the rear wheels and

also because all of the gauges used in the dynamic tests were sited

near the rear wheels. The effects of the obstacLes were observed

to last about 2 seconds. A copy of one of the strain records i-s

shown in Figure 6.22, To enabl-e a compari-son to be made betlveen

the runs, the difference between the maxj-mum and minimum straÍns has

been tabulated in Table 6.2. for the different runs. The peak

strain oscillations recorded at the glauges vrhen 50 mm obstacles were

placed on both sides were about ?5 to lgtP/o of the sum of the two

peak strains that were observed in the two runs when the 50 mm blocks

were placed on only one side at a time '

The strains developed when the bus tvas driven over a 100

mm bl-ock on the left were, on averager just under 2 times larger

than the strains developed by the 50 mm blocl<. There are consider-

able grounds for qualifying the comparison of the different runs for

these tests since there was probably considerable variation in the

speed and alignment of the bus. Also there rvas some doubt cast on

comparisons involving the 100 mm obstacle, which was formed by plac-

ing tvro 50 mm blocks on top of each other, since the top block was

occasionally displaced by the passage of the bus '

Gauç¡eLocation

GaugeNo.

50 mm

Blockon

Left

50 mm

Blockon

Risht

50 mm

blocks onboth sides

143.

Ratio l00mm100 mm blockbloek on lefton over 50 mm

left on left

Reardoorpil1ar

7B

I410

5?0Ea CJ-LJ

2?5645

600u=u

9809254?O830

75Q890470550

I1I

3??9

WindowpilIar

15I6

5705?0

5?O640

00

8696

9601150

r.')2.O

Rear doorstress panel

horizontal

34

3B4T

430290100140

610510zcJO

370

650550410420

1.s1.94.13.0

Rear door stresspanel vertical"

3639

320250

52555tJ

720620

2.3é-¡J

median value l-.9

TABLE 6.2. DIFFERENCE BETWEEN I4AXIfu'IUM AND MINTÍ\4UM STFAINS WHEN

BUS \¡JAS DRIVEN OVER OBSTACLES (microstrain) .

The maximum and minimum dynamic strains recorded when 50

mm blocl<s were placed in the path of both sets of wheels have been

compared with the estimated static strain and the results have been

tabulated in Table 6.3.

J

GA UGE N0.38 -çå-¿r---?'

GAUGE N0.39

rear whee I

Direction of chart50mm = 4secs"

frontwheel

400 microstrain

420 m icrostrai n

,l

Stra ins caused by obstacle.

n f f nfr"r"A^^^.¡/',,ÂÂ^¡!vu,vral

ItI lr :t L'ir-r[

ilill.rl

II

It-

I

FTG.6.22. STRAIN GAUGE TRACE OF GAUGES IN PANEL ABOVE REAR U]ORDURING DYNAMIC TESTS. Both wheels were driven over50mm. obstacles.

èÞ

,'

DYNAMIC STRAINSBoth wheels hitting

50 rnm bl-ocks

145.

Ratio of Dynamic Strainto est. static strain

GaugeLocation

Reardoorpi1lar

V/indowpi11-ar

Rear cloorstresspanelhori.zontal

+ 205+ 250- 125+ 205

- 258+ 3?2

- 2I43l-5390

GaugeNo.

l516

Est.Stati cStrain

positivepeal<

neg,¡ativepeak

max.positiveratio

max.negativeratio

- r.7- 1.8- 1.3- 1.4

+ 2.7+ 1.5+ 2.5+ 1.8

7B

T410

- I.2- 7.4- 2.D- 1.3

+ I.9+ 9.0+ 3.5+ 2.8

'l

43B4L

95++

1.91.8

I.5

Rear doorstress panelvertical

3639

+ 244- r29

+ 305+ I95

- 220- 360

9J1

+ 1.3+ 2.8

rnedian values + 2.2 - 1,45

TABLE 6.3. COIT4PARfSON BETI'JEEN PEAK STRAINS FEGORDED FOR BOTHU/HEELS STRTKING 50 IVl¡.î BLOTI<S AND THE ESTII/ATED

STATIC STBATN.

The dynamic strains are defined as being¡ the dil=ference

between the observed strains and static strain. The maximum posi-

tive dynamic/static strain ratio, which is the ratio of the peak

dynarnic strain with the same sign as the static strain over the esti-

mated static strain¡ wâs fairl-y constant for al-l gauges and had a

median value of 2.2. The maximum negative ratio v¡as also fairly

constant and had a median value of about 1.5. Some g¡auges hov,ever,

notably gauges 4 and 38, had peak dynarnic/static ratios that v¡ere

mr-rch higJher than the other .cjauges. Possil-¡l-e reasons for tl'ljs

occurrence are that (l) The estj.mated static strains r'úere small

ancl hence vJcre susceptible to smaIl errors in the statictests.

(Z) The gauges uJere sensitive to torsionaf loads.

+ 545+ 365+ 160+ 365

- 340- 460- 310- 290

- 480- 350

+ 390+ 660

+ 260+ 230+ 105+ I20

- 410- 2BO

1c)F

- 2s5

14Éi.

(¡) The strains due to loads on the front and

bacl< of the bus cancelled out when the bus was statj-ca1ly loaded but

l,u,ere larger vlhen the bus v,ras subjected to uneven dynarnic foads.

The estimated static strain was added to the peak dynamic

strain and a val-ue was obtained for the larçJest strain devel-oped

during these clynamic tests. 1070 microstraj.n lvas observed at gaug¡e

16 at the top of the v¡inslow pil1ar just behind the rear door rvhen the

left side of the bus was driverr over the l-00 rnrn obstacle. This

strain is equivalent to a stress of 90iå oJ' the yield stress.

Stresses of this magnitude would cause the bus to fail J-f they vuere

repeated often enough. In order to determine hov,r often large strains

coul-d be expected to occur, strain measurements were taken lvhile the

bus was being driven around the streets.

6.6.3. Dynar,nic strains duringt normal running conditions

An attenrpt v¡as made to deternine the stresses that could

be expectec.l to occur duríng the life of the buses and the frequency

of their occurrence. Strain records were made for the tv¡elve gauges

while the bus was being driven around a street circuit l¡hich included

cornering, bralcinç¡ and parking. Each circuit took almost four

minutes to complete. The strain records were analysed in tr¡io vJays.

Firstly¡ the difference between the maximum and rninimum strains

recorded during the run was determined and then the maximum strain

oscillation that occurred on average once every 15 seconds u/as found.

AssumÍng the buses to have a life of 15 years and assuming that they

will be operating on average 20 hours a weel<, then a frequency of

once every 15 seconds lvill resul-t Ín 5 x 106 cycles Ín the buses

lifetime. The maximum amplitude of strain oscillation for the test

and the rnaximum amplitude o1= strain oscillation with an average

period less than 15 seconds were both compared with the estimated

static strain. These values have been tabulated in Ïable 6.4.

14?.

Ratio Ratio 15Þeak second

stralrr P eal< strain strainEst. Peak versus oscil-lation versus

static strai-n static every staticstrain oscillation strain 15 secs. strain

i'i" . =i". I (t':." . =t". ) (t) (t'i" . str. ) (t)

Locationof

Gauç¡eGauge

No.

Rearuoorpi1lar

205250r25205

+

i+

T4

?6

10

I1

4B40929B

')t.56.73.48

?5':E6373

')a.22.50.36

ü/indowpi I 1ar

15l6

- 258+ 372

16?_

T7B.63.48

8B100

342?

Horizontal 3gaug;es in 4stress panel 38over rear 4Idoor

- 2r431q1

-90

11r913546

^t)cl.66.51

B8552D40

1.4I.B.38.44

verticalgauges instress panel

medi-an val-ues t .56+ .3?

TABLE 6.4. PEAK STRAIN OSCILLATTON IN TEST' AND I4AXIMUM STRAINOSCILLATTON OCOUHBTNG ON AVERAGE EVERY 15 SECONDS.

COI,IPAHTSON OF THESE VALUES WTTH THE ESTT¡,IATED STATICSTRAIN. STREET CTRCUIT ON GOOD ROAD SURFACE.

The ratios of the dynamic strains urith the estimated

static strain were fairly constant for all gauges with the exception

of gJauge 4. The possible reasons for the ratios for this gauge

being high have been discussed in the previous section.

Vlhen g¡auge 4 w¿rs ignored the ratio of tl-re peak dynamic

strain amplitude with the estimated static strain varied between

.33 and .?3 vrith a median.value of .56 and the ratio of maximum

dynamic strain amplitude that occurred every 15 seconds with the

3620

92?B

.38

.60+ 244- I29

2?45

655B

static strain, varied betlveen .22 and.50 with a median val-ue of

..J /.

The road surface of the street circuit used \À/as generally

good as Plate 6.5 shovrs. Strai-n records were taken for four gauges

on a stretch of road wj-th a poorer surface, see Plates 6.6 and 6.7t

and higher dynamic strains v¡ere observed, see Tabl"e 6.5.

149.

Ratio peak Ratio Peal<dynamic Peak 15 sec.strai-n s Lrain strain

Est. peak versus anrplitude versusstatic strain static every staticstrain amolitude strain l-S secs. strain

(':.c . =t". ) (t'i-" , str. ) (1) (t'i. . str. ) (t)Gauge

LocationGauge

No.

fissp doorpil1ar

?B

+ 205+ 250

180200

.BB

.8012s155

.61

.62

l'lindowpillar

GaugeLocation

\1/indorvpl 1lar

lvlean values

Batio ofPeak dynamic

Good road Poor road

1516

GaugeNo.

- 258+ 372

f+

sur ace

.60

225OEE¿.JJ

surf(t

.81

.o /

.68rto195

6652

TABLE 6.5. DYNAT',TC STRATNS AND COI"IPARISONS \'V]TH EST]NIATEDSTATIC STRAIN. POOR ROAD SURFACE.

The dynamic strain/static strain ratios for the good and

bad road surfaces have been tabul-ated for the four gauges in Table

6.6.

strain static strain

Fatio of15 sec.

Perj.od dynamicstrain/stàtic strain

Gffisurface surface

tt) (t)) )

Rear doorpi1lar

oo. LrU

.80.37.22

.61.62

ace(

342?

?256

7ó

1516

.66q-

.613u

CO¡,ÍPARISON OF DYNAMIC/STATIC STRATN RATTOS FOH G00nAND BAD ROAD SURFACES.

.63

.48.B?.68

TABLE 6.6.

149.

ET¡F

---.-

PLATE 6.4. FRONT WtIEEL OF BUS PASSING OVEB 50mm HIGHOBSTACLE.

PLATE 6.5. TYPICAL SECTION OF 'IHE GOOD ROAD SURFACE

INCLUDED TN THE ROAD CIRCUÏT.

150

t \L-:- -

PLATE 6.6. PART OF THE STRETCH OF ROAO ITITH THE POORERSUHFACE.

t .-,..'¡l I .r ;.:.i

PART OF THE STHETCH OF ROAO WITH THE POOHER

SUBFACE.PLATE 6.?.

151.

The nlean value for the peak clynamic/stati-c strain ratio

uras 30i/o higl-rer l=or the poor road surface and the maxj.mum strain

oscillations occurring every 15 seconds were tt''¡ice as high.

Therefore, f or the passage of tl-re bus on sealed rQads and

at normal operating speeds, strain variations of between tAOi', -ncl

Â

lBO/ oF the static straj.n could be expected to occur 5 x l¡-times

in the buses 1i1=e. For this reason it is suggested that buses

should be designed such that tlre static stress j-n the l:ocJy is no

greater than 5û',å of the endurance strength of the metal. It should

l¡e notecl that a1l the gauges used for the dynarnic tests were in a

section of bus near the rear door. ft is possible that there may

be some difference in the size of the dynamic effects in different

sections of the bus. AlfredsonIt), ro" example,establi-shed that'the

peak variations in vertical acceleration varied along the length of

the bus and he obtained useful es'timates of the dynamic stresses by

using these accelerations in conjunction r¡rith the results of hjs

static analyses.

6 . ?. SUt¡t'tARY

The tcsts carried out on the partially completed bus pro-

duced some interesting results. 0f interest lvere the generally lovr

values of strain observed in tlre static tests. Using these static

tests as a guicle it has been predicted that, even for a fu1ly loaded

bus, the static strains in any of the gauges rvould not be more than

435 micros'train which corresponds to a stress v¡hich is 36$ of the

yield stress. The static tests also gave some indication ofl the

amount of load sharecl betr¡reen the two sidewalls for uniform and

for unsymmetrical. loaCs.

Ì52.

Torsional loads were observed to be transl'erred in part

by the roof ribs, and to cause stresses in the connectionÇ betleen

the roof ribs and the cant rails. The suitability of a simple

ntaLrix analysis of the static l-oads in the sidev,rall was examined

for eymmetrical loadings. Although the gauges were not sited rvith

the intention of verifying the analysis, the resuLts that could be

used, sholved that even a simple representation could provide use-

fu1 predictions. The analysis also predicted that the gauges were

not always located on the most highly stressed members. Some 1ocal

discrepanci-es in parts of the cant rail and in the front door pi1lar

are believed to be, to some extent, due to the omission of the stress

panels above the doors and in the sidewall around the vlheel openings.

Allorvance for these panels could be made r¡¡ithout too much difficulty

using finite el-ements. fi4atrix analyses could be extended to pre-

dictinq the effects of torsional loadings and the method appears

suitable for analysing static loads on bus fran¡es.

The dynamic tests revealed that very large strain oscilla-

tions were developed in some of the dynamic tests. A maximum

strain of 1150 microstrain or 95"/" of the elastic strain at yield

lvas recorded r,vhen the left wheels were driven over a 100 mm block

at 50 kilometres per hour. An effort was made to relate the dynamic

strains to the static strains in the frame during the dynamic tests

by using values for the static strains predicted from the results of

the statj-c tests. Generally there lvas good correlation between the

tr,vo although for some gauges the dynami-c strains vuere higher than

expected, The dynamic stresses observed when the buses were driven

around a street circuit viere considerably l-ess than those measured

lvhen the bus was driven over the blocks. The maximum changes of

153

strain vrere generally less than tfOO/" of the static strain and it

j-s estimated that variations of betrv u"n lAeÁ anU tef/" of the static

strain, depending on the quality of the road surfaces, vrould be the

largest variations that would occur 5 x 106 times in the buses

lifetime.

154.

7. CONCLUSIONS

One of the relatively simple conclusions that can be

drawn from the research described in this thesis is that, for normal

finite element analyses, the value of the predictions j-s dependent

upon the amount of effort, measured in data preparation and comput-

ing time, that is usecl in obtaining the solution. In the experi-

ments carried out, the predicted stresses vJere closer to the maximum

observed stresses lvhen finer el-ernent meshes ì,vere used because of the

inherent accuracy of finer finite element meshes and because, lvith

close-packed meshes, the element centroids wiÌI generally be nearer

to the more highly stressed points in the structure. The accuracy

of stresses determined at positions other than the el-ement centroids

lvas investigated but the predictions were in general, l.ess relj-abl-e.

As more complex finite element representations are used, the cost of

the analysis increases because of the necessity to process more

nodes and elements. Eventually the point is reached lvhere a further

increase in accuracy v'ii11 not justify the necessary increase in cost.

Tvro methods of decreasing the amount of computation viith-

out decreasing the accuracy of the analysis were examined. The

first involved rnal<ing allowance for the fact that many sections

along the length of the bus are almost identical and hence the body

can be represented by a large collection of elements, calLed here

a super-element, repeated several times along the bus. A computer

programme uJas viritten to test the effectiveness of reducing the

super-e1ement, by removing internal nodes, prior to the formation

ol. the stiffness matrix of the complete bus. It uras found that the

el'ficiency of this procedur"e vJas largely dependent upon the fraction

of nodes that vlere removed from the super-elernent. For the structure

155.

examined, vrhich cons'isted of B super-elements, it was found that

uJhen a quarter of the nocles riïere removedr no time was saved by reduc-

ing the super-elements. When half the nodes lvere rernovecl the analy-

sis tool.. only 25iå of the time required for the unreduced analysis.

The removal of nocies from the analysis rneans that fe','¡er node dis-

¡-rlacenrents are obtained. It is, hourever, sti11 possible to calcul--

ate the stresses in all the original elements. The fraction of

nocles that can be renrovecl is lirnited by the co¡rclition that afl nodes

in the super-elemcnt that are in contact ivith other elernents must bc

retained in order to rrlaintain compatibility of clisplaccment on the

l¡oundaries.

The other methocl of obtaining greater accuracy lvithout

marlcedly increasing the amount of computation vras to isolate critical

sections, such as the corners of door and rvj.ndorl openings, model thern

in deta1l r¡iith finite elements ancl load these finite element models

vrith foads that were obtained from a less detailed analysis of the

complete structure. This procecjure has advantages over incorporat-

ing the detailecl representation of the critical section directly

into the l-ess detai]ed analysis of the structure in that it is suit-

ablo i'or analysing joints and sections that are repeated in many

parts of the bus and because it does not necessarily require compli-

cated chanç¡e-overs frrom coarse to fine element grids or from beam

type mernbers to other types of elements '

The method vras usecl to investigate the cant rail-door

pi-llar joint lvhere tlre cant rail is stepped dov¡n to accommodate

the door opening mechanj-sm. v/hen the section was model-l-ed in

epoxy resj-n and the stress predictions were checked ag¡ainst the

stresses observed by photo-clastic means, it was observed that the

156.

peal< predictecl stresses for the tests vJere, on averasJer about 65J/o

of the peal< observed stresses. The finite element representation

that v¡as used 1=or this comparison contaj-ned 355 elements. It

appears that this process r¡¡oul-d be useful for contparing different

design details such as the difference between curved and angular corners

at door and windol openings. ft would possibly be less useful for

determining the rnaximum stress concentrations at these points unless

very close element subdivisions, which vuould involve considerabl"e

expenser w€T.e used.

The comparison of the'simple matrix anerÌysis of the bus

si.der,,¡all and the static tests carried out on the partially finished

bus, shorved that useful predictions could be obtained from even the

simplest analyses. The success of the simple analysis supports the

design method proposed by ¡rJarclilt(fa) which utilized a series of

g¡raduated analyses for successive stages of a desigln.

Evidence that forces were being transmitted through the

roof ribs rvas found during the static tests. Thj-s interaction

betlveen the two sides of the bus was observed to cause large strains

in the stress panels around the doors. From the success of the

earl.ier analysis it seems possible that a three dj-r¡ensionaL frame

analysj-s of the r¡¿hole bus'could be used to investigate these forces.

The fÍnite element method of analysis appears suitable for applica-

tion to the design of bus bodies despite the fact that the design

t^roul-d be complicated by such matters as stress concentrations,

dynamic loadings, fatigue in r¡relded joints and initial- stresses.

For many design cases the analysis of the frame, chassis and stress

panels, in which the main members would be represented by beam type

1s7

elements and the stress panel-s by plate elements, ltould provide

sufficient information for a satisfactory design. The extension

of the analysis to investì-gate the effects of the cladding could be

done although it would involve a consj-derably more complicated finite

element representation. Before clad buses could be analysed with

confidence, it would be necessary to investigate such matters as the

connections between the claddlng and the frame and the effects of the

discontinuity between the deflected boundaries of beam and plate

elements if both nrere to be used together.

The dynamic tests that were carried out on the bus enabled

the relationship between the static and the dynamic strains to be

studied. In general a good correlation betr¡reen the two was obtained

although it was observed thai;, at locations which lvere affected by

interaction betu¡een the two sidewalls, the ratios between the maximum

dynarni-c strain oscillations and the estimated static strains were

hi-c¡her than at otlrer points. Since the estimated static strain vras

cal-culated assuming an evenly loaded bus, it seents that a realj-stic

torsional loading t¡¡oul-d have to be incl-uded in any statÍc analysis.

In general, however, it was possible to concl-ude from the

strai-n records taken when the bus vlas driven in conditions si-miIar

to those expected in normal runni.ng, that the dynamic strain varia-

tions rarely exceeded tfOO/. of the static strains. Straln variations

of betrryeen + 4}ofo anA tgO"/. of the static straj-n, depending on the

quali.ty of the road sur'l'ace, were observed to occur at a frequency

corresporrcJlng to 5 x 106 cycles 1n the buses lifetime. Although

all the gauges used in the dynamic tests were located in a flairly

small section of the bus it appears reasonable to conclude from these

results, that a suitable desig¡n criterion rr;ould be to ensure that the

static stresses in the bus r¡;ere less than SOi'/o of the enclurance

strength o1' the metal .

1

-L

158.

B]P,L]OGRAPHY

f.4ICHILBERGER, P. "\/lirl<ung cler Turoffnun-oen auf das l(raftespiel- derOnrnibuskarosseri-eni. Periodica Polytechnica (Budapest). V':t e, l'lo 2,

. I (l¡î-L .]Q C- .

ERZ, l(. ilUber ciie clurch Unbenheten der Fahrbalrrr hervorgerufereVerclr.enung von Strassenl'ahrzeugenrr. A.T.Z. lJo 4: No 6: I'lo 11: lJo 12t

1957.

TIDGURY, G.H. "The structt-rral Design o1' Bus Bodies'r. BodyEngineering Symposium, Cranf ieldr l-970.Proceedings Institution of llìeclranical Engineers 1969-70, Vol 1E'4'

Pt 31"4.

BRZOSI(A, Z. rlBas,ic pr:oblems in the statics of self supportirrgvehicle boclies" . Archv¡im Budoury l,,laozin, vol 2, \lo 4 I 1955 ; Vo1 3 t

No 1, 1956.

Y0SHIf\4It\E, l(. 'rstrength calculertion for cur bodyr', Japan SailrlayEng j.neerirrg , VoI 4, No 2 r 1963 .

YOsllIMrNE, l(.; ITO, H.; and ARAI, H. "speed-up of car Body streng¡thCalculation by the Llse of Electronic Contpu'berr'. BuLletin ofJt=rpanese Society of fr,lr:chanical Engineers, Vol f0, No 3t-J, 196?'

pp. 27O-2?e.

3

4

t:

6

? ALFREDSON, R.J. "The structural Analysis of a stressed ski-n Bus

Bodyrr. The Journaf ofl the Institution oF Engineers, Australia.0ctober-trlovernber , l96f , pp. 18 t-198.

OKUBO, Y.; I<ARATA, T.; ancl NAKAYA, l-1. Journal of society ofAutomobile Engineers Japan, Vol 24, No 4, I9?O'

ful0oRE, G.G. rrstructural Analysis of car Body shells usingComputer Techniques'r. Body Engineering Symposium, Cranfield 1970'Proceeclings of the Institution of tt4echanical Engineers 1969-70'

VoI 184, Pt 3l'Jl.

L2. PETERSEN, r¡/ . S .A .E . Journal Paper No .

l-3. l(IRIOl(4, K.; OKURO, Y.; and H0TTAT Y.

?10263, JanuarY 1371.

VJARDTI--L, G.A. "Sma1l Computer Procetlures as Tool-s for StructuralDeslgners'r. Body Engineering Symposium, Cranfield l'970'Proceeclinqs of the Institution of Meclranical EngineBrs 1969-70'

Vol 1t4 ' Pt 31v1.

B. llicKENlrJA, E.R. "Computer evaluation o1' automobile body structure'¡,4.5.8.E. Annual Technical Conventi-on, 1961.

9. NORVILLE, C.C.; and IJILLS, B. t'An applied finite e]-ement displace-,nent anaiysi.s of a sinrple integral construction vehicle body shel-lrr.International Journaf oF l:lechanical Science, Vol 14r PP.809-821'

I9?2.

10.

1t_ .

S.A.E.?ror3'7.

Journal Paper No.January, I9?I.

].4.

159.

15. VOLVO BODYI^JORK \\JORKSHOP BULLET]N. JULY 1971. Product B.No l-.

Group 80.

16. DTRECTIVES FOR THE Í.,'IANUFACTURE OF BUS BODIES ON MERCEDES'BENZ BUS

CHASSIS - I\,IODEL O 305. No 305 009 Cl 96,

I?. DEVJAR, J. Lutter from Chief Eng¡ineer, Trucl< D Bus Divisiont_Leylan¿ Australj-a to the Chiel' Engineer, Municipal Tramways Trust'Adelaide . 29 N'larch , I9?4.

1g. pA¡M, y.A.; et a]. "Experimental investigations of unified body

construction". Avtom. Prom. NJo 12, Dec., 1965r p 9-I0'

19. ATOYAN,strength19ô0, No

20. ELTZALDE,Trrelfth F.

ANDERSON,Chassis FJ [¡urnal o

23. LOUJE' l/ìl.T

of Smal-l

K.[1. (Lvcv Bus \a/orks). I'The t'orsj-ona]- sti.ff=nessoF a bus lraving no chassis framerr. Avtorn. Prom.,2, pp l4-i.7; M.f .Fì.A. Translation No - I6/bO.

andFeb,

2T

22

ZIENKIEV,IICZ, O.C. "The Finite Element t'4ethod in Engineering5cience", lvlcGraw-Hi1l. .

C.CI.S

D.Tramef lJe

. "Study of Stresses in Chassis-less Busesrr.

.I.T.A. Congress. Paper No. 3-03r May 19-25r 1968.

.i and MILLS, B. "The Dynamic Analysis oF a CarUsing the Flnite Element llethod", International

chanical Science, Vol 14, 1972, pp 799-808.

.; AL-HASSANI, S.T.S. and JOHNSON, W.J. 'rlmpact Behaviourscale iJotor coachesrr. Journal- of Automotive Engineering,

Vol 3, No 1, JanuarY I9?2.

24

25

26

RTJMPH, l/J

School- B

tion lt/ee

.C. fBtue Bird Body Co.). "static Load Test Code forus Body Structure'r. Combined Pourerplant and Transporta-ting, October 18-21' 1965. S.A.E. Paper 6507I?.

RUDNAI, G.; nnd |I4ATOLCSYr t¡.1,4ain Structural l'/iembers of Bus

rr0omparative Investigations of theBoclies'r. A.T.Z. November, I9?Ol

Vot. 12, No 11, p 410-414.

27.

2B

ATOYAN, K.M.; AKOPYAN' R.A.; ANd ROMANIV, O.N. ''ThC StTCNgthProperties of the Profiled, VJelded Tubes Used in the Load-BearingSystems of LAZ Buses'r. Avtom. Prom., Februaryt 1967, No. 2,

PP. 12-15.

lvlELOSH, R.J.; and BAMFORD, R.M. "Efficient sol-ution of Load-lJeflection Equations". Journal of the Structural DivisiontA.S.C.E.' April 1969' ST4' pp 661-676.

ACES, Analysis of civil Engineering structures. A computerprogramme prepared by the south Australian Public service.

tvlEEK, J.L. "l,rlembrane analysis of unframed high rise buil-dinqsr"tne öivi-t Engineering Transacti-ons of the Instj.tution of EngineerstAustralia, 1973, pP 87-93.

29.

160.

30. GREEIJE, B.E.; STROME, D.R.; and WEIKEL' R.C. "Applications oF thestiffness method bo the analysis of shell- structuresr'. Proc.Avia'bion Conference American Society of lJechanicaì- En¡JineerstLos Anç¡eles, l.4arch 1961.

URIìA'I'Ä

(1) Page 32 L'ine 9

"rectangul-ar quadratic displacement elernents"

These elernents are rectangular e.lements of the"serendj-pity" family described by ZIENKIEWICZ(Ref. 2I, section 7.3). They have nodes at the cornersand at the mid poÍnt of each side. The displacementfunction is parabolic along the boundaries.

(2) Page Bf , Section 5"2 Line 6

"triangular and rectangular linear shell elements"

These elements h.ave six degrees of- freedom pernode namely three displacements and ttrree rotations-The in-plane stiffnesses are the same as Lhe linearplane stress elements described by ZIENKIEWICZ (Ref- 2Iõfrap. Z and Section 7.3). The in:plane rotational stiffnessrs zero.

The out of plane stiffnesses are the same as thosedescribed in lìef . 27 (Sections 10.4 and 10.6) fornon-conforrning rectangular and triangular pJ-ate bendlngelements with only corner nodes.