Ship vibration analysis by finite element technique

Issued by the Council

REPORT NO. 107 S December 1967

(S 3/85-85a)

NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

NETHERLANDS SHIP RESEARCH CENTRE TNO

SHIPBUILDING DEPARTMENT LEEGHWATERSTRAAT 5, DELFT

*

SHIP VIBRATION ANALYSIS BY FINITE

ELEMENT TECHNIQUE

PARt I

GENERAL REVIEW AND APPLICATION TO SIMPLE STRUCTURES,

STATICALLY LOADED

(ANALYSE VAN SCHEEPSÏRILLINGEÑ DOOR MIDDEL VAN DE ELEMENTENMETHODE

DEEL I. ALGEMEEN OVERZICHT EN TOEPASSING OP EENVOUDIGE CONSTRUCTIES,

STATISCH BELAST)

by

Ir. S. HYLARIDES

VOORWOORD

Dit rapport is zeker niet het eerste over het onderwerp scheeps-trillingen dat door het Scheepsstudiecentrum gepubliceerd wordt, maar het is wel het eerste waarin geen gebruik wordt gemaakt van de kiassieke balktheorie.

Het is reeds lang bekerìddat de gebruikelijkè aanpak, waaÈbij het schip wordt beschouwd äls een elemeñtaire balk met Over de lengte variërende massa, buiging- en afschùifstijtheid niet ge-heel voldoet voor de gecompliceerde langwerpjge doosconstructie die de scheepsromp eigeñlijk is. Dit blijkt vooral het geval te zijn voor de trihingen vañ hogêre orde waarbij de afstand tussen de knopen van de trillingsvorm vän dezelfde ordé van grootte wordt als de dwarsafmetingeñ van de balk.

Een methode die o.a. iñ de civiele en vliegtuigbouwkundige techniek wordt toegepast voor het analyseren van gecompli céerde constructies Iijkt ook grote mogeijkheden te bieden voor de analyse van zowel hetstatische als bet dynamische gedragvan de scheepscoñsthictie.

Deze rekenmethode wordt aangeduid als de ,,elementenrne-thodà". Een constructie wordt hierbij beschouwd als te zn opge-bouwd uit een aantal elementen van eenvoudige vorm die onder-ling verbonden zijn in knooppunten. Uit de elastische

eigen-schappen van deze elementen kunnen dan vergelijkingen ver-kregen worden die het verband tussen vervorming eñ belasting van dà tòtale constructià beschrijvàn. Hierbij wordt gebruik ge-maakt van matrix notatie dià zich zeer goed leent voor verwer-king met elektronische rekenapparatuur.

In dit rapport geeft de auteur, die werkzaam is bij de Research groep Sterkte en Trillingen van het Nederlandsch Scheepsbouw-kundig Proefstation, een algemeen overzicht van de methode, ge-illustreerd met een tweetal statische toepassingen op eenvoudige constructies.

Een vervoig, waarin speciaal de analyse van trillingen be-handeld wordt, is in bewerking.

MET NEDERLANDS SCHEEPSSTIJDIECENTRUM TNO

PREFACE

This report is not the first one treating ship vibrations that is published by the Ship Research -Centre, but it certainly is the first oñe in whith no use is made of the classical beam theory.

It has since long been recognized that the usual way of con-sidering a ship as an elementary beam with mass, bending- and shear stiffness varying along its length, does not yield completely satisfying results fòr the complex oblong box-like construction that a ships' hUll really is. This especially appears to be the case for higher Order vibrations where the distance between the nodes of the vibration profilà becomes of the same- order as the trans-verse dimensions of the beam.

A method which is among others applied in civil and aircraft-engrneering for analysing complex structures also seems to offer great possibilities for analysing both the statical and the dynam-¡cal behaviour of the ship structure.

This computational procedure is referred to as the "finite-element method". For this a structure is considered as an assem-blage of a number of siinple elements, mutually connected in nodes. From the elastic properties of these elements equations can be derived that describe the relatiOn between deflection and loading of the whole structure. For this use is made of matrix notation which is very well suited to the use of electronic com-puting machinçs.

In this- report the author, who is employed by the Research-group Strength and Vibrations of the Netherlands Ship Model Basin, gives a general review of the method, illustrated by two statical applications to simple strutures.

A continuation in which particularly the vibrational analysis

is treated is-being prepared.

CONTENTS

page

Summary.

. 7

I Infrodúction . 7

2 Variation of the strain energy 8

2.1 The stiffness matrix of a rectangular plate - 8

2.2 Transformation to the standard set of axes 10

3 The derivatives of the virtual work of the loadings . . 11

3.1

The loading of rectangular elements ...

Il 3.2

Transformation of the co-ordinate system ... .

12

4 The solution of the problem 12

5 Calculation of the stresses . 13

6 Vibratory loading 14

7 Application tÓ some simple structures 14

7.1 The clathped p1te 14

7.2 A square plate with a äirulàr hole - 16

8

Conclusions and outlines offuture research ...

. 18

References ...

19

LIST OF SYMBOLS

a, b Length and breadth of a rectangular plate a, b. COefficients

dv, df irfinitesimal volume and suìface elements respectively

E Modulus of elasticity

E Matrix expressing the stress displacement relations

K Stifffness matrix

k Acertain number of nodal points, k < 3n

L Load matrix

M Mass matrix

n Total number of nodal points

P Potentiâl energy

p, q r

Surface forces in the X-, Y and Z-direction respectively s Parameter along the boundary of a plate element

T Used as a superscription of a matrix expressing the transposed of this matrix T Transformation matrix of the axes

t Thickness of a plate

U Unit matrix

u Displacement matrix

u1, y1, w Displacements of node i in cartesian co-ordinates

V Strain energy

W Vfrtual work

X Y Z Standard set of axes, related to the whole construction

X*Z*

Private setof axes

X, Y Volume forces

y Angle used in stiffness matrices of triangular plate and bar elemebt (fig 2 and 3)

ô Differential operator

i'

Pson's ratio

Patheter aloñg an edge of an elemeht, O i

°, °'

° _{Stress components in cartesian co-ordinates}

t,

r _J

i

Introduction

For the strength and vibration calculations of ship structures,, the construction should be considered in detail. Due to the complexity of a ship this cannot be realized by doing this analytically but discrete methods should be developed. For this purpose the construction is divided into a number of simple elements, joined to each other in certain points, the nodal points or nodes. The simplicity of these elements must be of such a degree, that the relations between the loading forces and the corresponding deformations can be deter-mined Ship structures consist for the greater part of plates, so- it is acceptable to consider only forces and no moments in the nodal points.

By means of various principles, i.e. the equilibrium of the forces in the nodal points [1], the principle of virtual work [2], the variational principle of Reissner [3] and others, the equations governing the problem can be obtained. In this report the principle of virtual work or that of the minimum of the total potential energy of the system has been applied.

Expressing the deformation of an element j as a function of the displacements of the joined nodes, the strain energy V, can be obtained as a function of these displacements. It is assumed that Hooke's law holds. The principle of virtual work for the whole system can be written as

ÔP = {

VffJ(Xu± Fv+Zw)dv+

_Jf(pu+qv+rw)df j = O

(1.1) which expresses that the variation ò of the total poten-tial energy P of the system should be zero [2] and in which.

*) Publication no. 300 of the Neth. Ship Model Basin.

SHiP VIBRATION ANALYSIS BY FINITE ELEMENT TECHNIQUE

PARTI

GENERAL REVIEW AND APPLICATION TO SIMPLE STRUCTURES, STATICALLY *

by

Ir. S. HYLARIDES

Summary

Forthe analysis of the stiffñess against defoñnations ship structures can be divided into elements whose stiffñess can be deterúiiñed

with a high degree faccuracy. These elements are connected to each other in nodal points. Expressing the vfrtual work of the

con-struction in the unknown displacements of these nodal points and differeniiàting this expression with respect to the displacements, leadsto the same number of linear equations for these displacements as the number of the yet unknown displacements. This set of equations represents the-stiffness of the construction. The finer the division into elemeñts, the higher the accuracy will be.

Dynamical problems can also be solved. For this purpose the masses of the elemeñts are coicentrted in the nodal points and so inertia forces are introduced. Determination of the eigenvalues Of the set ofquatioñs leads to the natural frequencies of thestructure. Finally this calculation method is appliód to two simple structures statically loaded. The results are compared with an accurate solutiôn. It appears that the reliability of the method is good.

V is the sum of the strain energies of all the m elements into which the structure has been divided;

X, Fand Z are the forces per unit volume in the co-ordinate directions;

u, ú and w are the displacements of the points of application of the forces;

dv is an- infinitesimal volume element; p, q, r are the surface forces per unit area;

df is an infinitesimal area element on the surface. If y lume forces are important it is convenient to con-centrate them in the nodal points, so we omit in equa-tion (1.1) the volume integral.

The deformation of the elements is a function of the unknown displacements of the nodal points. Accord-ingly equation (1.1) has been varied with respect to these displacements and can be written as

a. a

6F =

6u1 + -- 6v1 + - 6w1 = 0

(1.2)

i=i au, av1 aw,

where P represents the total poteñtial energy and u1, y1 and w1 the displacements in the three co-ordinate directions of the- nodal points. This equation is for ar-bitrary 6u,, 6v1 and 6w, only satisfied by

au1 öv, aw, (1.3)

which results in a set of 3n equations. The unknown quantities are the 3n displacements of the nodal points. The strain energy is a quadratic function of the dis-placements, whereas the virtual work of the loading is a linear function, so equations (1.-3) are- linear -functions of the unknwon displacements. From this set Of -equations the unknown displacements can be solved if the loading forces are given. The considerable

number of equations necessitates, however, the appli-cation of a high speed computer.

The displacements of the nqdal points being known, the deformations of and the stresses in the elements can be determined.

This report deals exclusively with problems for which the deformations are small in such a way, that the geometry of the structure is sensibly unaffected by deformation. This implies that the conditiOns of equi-librium of compatibility or other conditions can be considered for the. undistorted structure.

2 Variation of the strain energy

The elements are joined in the nodal points. The defor-mation of an element is therefore a function of the displacements of the connected nodal points. To deter-mine these functions and from these the stiffness relations with a sufficient deee of accuracy, we have to choose these elements of a simple geometry, i.e. rectangular or triangular platesand bars, their corners or ends coinciding with the nodal points.

It is acceptable in ship structures to neglect the lateral stiffness of the elements as here plate elements are dóminant and their lateral stiffness is very small. Só för the determination of an element the in-plane loadings are only considered whereas the lateral forces must be absorbed by elements, perpendicular to that under consideration.

We introduce a standard co-ordinate system XYZ, related to the whole construction. In general, however, the elements are situated at random and, in order to obtain the stiffness relations in a simple way, we choose a private frame of co-ördinates, XY'Z to each ele-ment and determine their stiffness relations first in this private set of co-ordinates. The transposition of these relations to the standard axes is indicated in section 2 2

As an illustration of the proposed method the stiff-ness relations of a rectangular plate are derived in detail (sectioñ 2.1). In a similar way the stiffness relations of the other elements are determined (appendix 1).

2.1 The st(ffness matrix of a rectangular plate

Ship structures consist mainly of rectangular plates and in stiffness analysis the attention has to- be con-centrated on these elements. As everything m this section is referred to the private set of axes X'' the, asterisks will be omitted. Let the XY-plane coincide with the middle plane of the plate (figure l)

Since the thickness i is-very small compared with the length a and the breàdth b we may assume. that the displacements and strèsses are constant over the height and equal their mean value. Therefore two-dimensional considerations may be applied.

Fig. 1. Rectangulär plate elèmeñt. Numbering the corners is

counter-clockwise, corner i cóincides with the origin of the co-ordinate system related tO the element. On the element dxdydz the nomenclature of the stresses has been indicated.

Further it is assumed that the plate elements are only loaded along their boundaries which is acceptable as an element is in general only loaded by forces applied by the other elements and out-of-plane loadings can only be resjsted by oût-of-plane elements. Hence the stresse o, and r, are negligible in comparison with

o, i and r,.,.

The stress-strain relations for a state of plane stress are [2 and 4] E ¡au av

=

-

_{\aX ay} E

ía,

au\

=

(,+v)

(2.1)

lv ay

ax E (au av 7(l+v) \ay

ai

in which Eis the modulus of elasticity and vis Poisson's ratio. The equations of equilibrium are

ax ay

ax ay

In agreement with the remarks in section 1 the volume forces are put zerO.

In order to express the strain energy, V, as a function of the displacements of, the nodal points, we need a

relation between the translations u andy of each point

and of the corners of the plate for each element. There-fore we take

u =

a0+a1x+a2y+a3x2-Fa4y2+a5xy

y = b0+b1x+b2y+b3x2+b4y2+b5xy } (2.3)

Now the equations (2.1) are written

-lv2

{a1 +vb2+(2a3+vb5)x+(a5+2vb4)y} E F 2(1+v) {a2+b1+(a5 +2b3)x+(2a4+b5)y} (2.4)

and the equations of equilibrium lead to

4a3+2(1v)a4+(1+v)b5 = O (1+i')a5+2(1v)b+4b4 = o

The displacements of the corners of the plate are equal to the displacements of the joined nodal points. To-gether with the two conditions of equilibrium we have the disposal of ten relations for the twelve coefficients

a and b. (i = 0,1,2, 3,4, 5). Two additional equations

can be freely chosen

b5

=

2a4

a5

=

2b3

and, hence, equations (2.5) are reduced to a3 = va4

b4 vb3

Substituting the equations (2.6) and (2.7) in (2.3) and writinga3 instead ofa4 (for the sake of regularity) we find

u = a0 +a1x+a2y±a3(vx2 ±y2)-2b3xy y = b0+b1x+b2y+b3(x2+vy2)-2a3xy

Numbering the corners of the plate by ito4(figure 1), two relations between the coefficientsa1andb. and the

displacements of the corner in question are found for each corner. Evaluation of these relations leads to

a0 = u1 u2u1 _v1v2+v3v4 +1) a 2b u4u1

+

v1v2+v3v4 a2= b 2a viv2 +v3v4 a3 = -2ab } (2.6) J (2.8) { a b

+ (1_v)_(4_v2) U2+

{ a (2.9) + (iv)--*(2+v2)- u3+ b a 9 b0 =V1 b1

+

(2.9) a v4v1 b2= 2b u1u2-l-u3u4 b 2a b3 = 2ab

With the equations (2.6) and (2.7) and changing also the suffix ofa4 into 3,the stresses become

7a1+vb2 2 2b3y \ 1v = E

(1±2

2a3x) (2.10) a2 + b1 = E 2(1-Ev)

The strain energy per unit volume is given by [4]

y0 1+v (a+a)2 2ixcry+2Txy2j (2.11)

2E t 1+v

Integration over the volume of a rectangular plate with length a, breadth b, and thickness t, using for the stresses the equations (2.10), leads to the strain energy V of the whole plate

v=

_tf!V0dXdY

=

= Eabt a+b+2va1b2

+

21)2

+

L 2(1vi) 4(1+v) _a3b2a_aib3b+aa2+bb2_va3b3abJ (2.12)

The derivative of the strain energy with respect to one of the displacements of the plate corners and, thereby, with respect to the displacement of the related nodal point, can be written as

3

av aVaa. avab.

au,

(2.13)

where u is a generalization of one of the displacements u ory of the corners. The same holds fora3andb. As

an example the strain energy yhas been differentiated with respect to u1, the derivativesaa/au1 and ab!a1

have been determined from equations (2.9).

av Et i a b

I (1_v)_+*(4_v2)_

u1+

.10

a bi

+ (1v)-+(2+v2)-ju4+

±

(l±)vi(l_3v)v2_(1+v)v3±(l3v)v4]

(2.14)

The other derivatives are given in appendix i in matrix notation These equations can be contracted

to 6V = K4u.

Similar equations hold for the thangularplate (figure 2) and for a bar-like element (figure 3), these are also given in appendix 1.. The order of the square matrix K and of the column matnces 6V and u have been

deter-Fig. 2. Triangular, plate element; Numbering the corners is counter-clockwise, corner I coincides with the òrigin of the co-ordinate system. "The X*axis is, orientated along the side

1t22.

X u

mined by thè configuration of the elemeñts. Bearing this in mind the derivatives f the strain energy with respect to the displäcements of the surrounding nodal points of each element can be written m'generai as

6V =

Referring to the omission of-the asterisks in 'the be-ginning of this section, the correct way of writing this equation is

6V = K* .u* . ' (2.15)

2.2 Transformation to the standard set of axes In generaÏ an element is not situated parallel to the XY-plane of the standard co-ordinate system of the construction. Therefore the in-plane displacements u and v of a certäii element should be expressed as a function òf u, y and w belonging to the standard set of

w

Fig. 4. XYZ-system related to the construótion, X*Y*Z*.system to the elements

X*axjs has the corners Çi, q',, q'3 with X-, Y- and Z-axis

respectively;

-Y*axis hasthe corners ip,, ,,with X', Y- ând Z-axis

respect-ively.

axes. With the direction cosines of the private set of axes (figure 4) we write for the displacement u'

u = u1 cos q' ± v cos q'2 + w cos q'3 (2:16) in which cos q', COS q'2 and cos q' are the direction cosines giving the position of X*axis with respect to the standard set of axes. The Y*axis is given by cos cos and cos As we consider only right-handed co-ordinate systems, the Z*_axis is given by' the X4-and Y4-axes.

AS an example and because of its dominant ppli-cation in ship structures we examine a rectangular plate in detail again. Similar considerations hold for the other elements, the results are given in appendix 2.

Tie in-Ìane displacemeñts u and v' of the rectan-gular plate corners, which plate is arbitrarily situated in regard to the standard co-ordinate system, can be

of _{given by}

u. = T4u

(2.17)

The column matrices u and u and the rectangular transfo rmatiòn matrix T4 are based on equation (2.16). They are' presented in appendix 2.

The derivatives of the strain energy are given by

av

av au.*

av av1

=

* au1

+

av* au1

These equations can be presented in matrix notation 'by

ÔV = T6V

(2.19)

(2.18)

Fig. 3. Bar-like element. The origin coincides with end i

where the column matricesÖV and .â*V are identical to

the column matrix ö*V given iri appendix 1, the

rectangular matrix T is the transposed of T4. It is obvious that similar equations hold for the other elements. So in the generalized case we can write

= T.0

6V = T'.öV

In section 2.1 it is already found that in general holds K* . u' (the asterisks indicate that this equation is only valid in the plane of the element, viz, the X*Y* plane) Substitution of equations (2 20) in this latter equation leads to

ÖV = TKT.0

The square matrix [TTK*T] çan be replaced by the

matrix K and, hence, for an arbitrarily situated

element holds

ÔV=K.0

(2.21)

3 The drivatives of the virtual work of the loadings The variátion of the virtual work W of the surface forces p, q and r is given by

ÔW= (5ff (puH-qv+rw)df (3.1)

where the integration refers to the total surface of the construction [4]. The volume forces are neglectèd as explained in section 1. In the preceding section we have concentrated the applied fôrces on the edges of the elements The in-plane displacements of the points on these edgès are linear functions of the displacements of the jóined nodal points (see equations (2.2) and (2 9)) So the virtual work W is a linear function of displacements of the nodal points and in the variation

n

aw

aw

Iâu+--6v+

6w1 \aU1 ay, } (2.20)

the expressions aW/au1, aW/av1 and aW/aw do not yet contain the unknown displacements

In general there will be a. force applying in a certain nodal point. Only the variations with respect to the displacements of that special nodal point, will contain the components of this force in the direction of the displacement in consideration So the variation of the virtual work with respect to the displacements of the othernodes will not contain this force.

In analogy with the derivation of the equation of the variation of the strain energy in section3.1,the dériva-tion of the variadériva-tion of the virtual work for a, rectan-gular plate is given in its private set of axes. In sectiori

3.2 this is generalized for an arbitrarily situated

ele-ment.

3.1 Loading of rectangular elements

For the sake of simplicity the asterisks, distinguishing the private set of axes from the standard axes, are omitted again in this section.

The in-plane loading on the edges of a rectangular element has been indicated ¡n figure 5. Since the plate is of uniform thickness t and the forces can be assumed to be constant over the. thickness of the plates, it follows' fòr the virtual work

w

_{= ff (pu+qv)df.= t}

_(pu+qv)ds. (3.3)

where s. is a parameter along the boundary (figure 5). In detail equation (3.3) becomes

w -

t]

₊

(3.2) + tf (p2ux=a+q2vxp4ux=0 ±q4v=0)dy (3.4)

Fig.. 5. The in-plane loading on a rec-tangularelement. In accordance with' the numbering of the corners the numbering of the sides is also côunter-clockwise, be ginning at the side along the X*axis. For the triangular plate and the bar. a

similar way of numbering holds.

is the generalized co-ordiñate along the edges of the plate.

12

and for the derivative of the virtual work, say with respect to u1, we can write

av

aw au0

aw aux_a

+

aU1

a0 au,

aUxa au,

aW aU a 6u =b

a=0

±

au1 _{aUY_b '1}

x0

au1

aW aV

aw av =

aw av =b

+

y y

aVxa au1

av0 au1

aVYb au,

With the relations in equations (2.8) for the displace-ments and the relations in equations (2.9) for the coefficients a and b. (i = 0, 1,2,3) we find

a

l'i

xl

x\

=

tJ

(i

-

+

(i

_-

-) q1+Op3 +

au, a 2b a o b

x/

x

vyl

y\

+ - (i

_{-) q3j}

dx+tJ j0P2 +

\l

_-

-) q1 +

2b a 2a b o

/

y\

vyf

y\

1

+

Similar equations hold for the other variations, so in contracted. matrix notation holds, referring to the private set df axes,

o*w=fL*.p*.dE

(3.7)

The column matrices ô' Wand p are given in appendix 3, just as the square load matrix L*; ¿ i the dimension-less, generalized co-ordinate along the edges of the plate. It is obvious that for the other elements a similar equation can be derived. These equations are also de-fined in appendix 3.

3.2 Transformation of the co-ordinate system

In the preceding section the elements have been loaded only by in-pläne forces p and q* (The asterisks rèfer to the co-ordinate system X*Y*Z*). To relate these in-plane loadings to the loadings in the standard set of axes with components p, q and r, we coäsider again a rectangular plate element. For the other . elemeñts the same considerations hold.

Referring to figures 4 and :5 it is easy to see that equations similar to (2.16) and (2.17), must hold for the stresses p and q*

= T4.p (3.8)

T4 is defined in appendix 2, p and p are in agreement with the column matrix p given iá appendix 3.

The variation of the virtual work with respectto the displacements is

aw

aw au,*

aw av

= a

±

aw

(3.5) (3.6) (3.9)

This equation is similar to the equations (2.18) des-cribing the variation of the strain energy of an element. In matrix ñotation it becomes

ÔW = T.ò'W

(3.10)

It is clear that omission of suffixe 4 in equations (3.8) and (3.10) leads to the generalized relations. In the preceding section it has been found that for any element in its plane for the virtual work the relation (3.7) holds

where ¿ is a parameter along the edges. Combination of this relation with the generalized equations (3.8) and (3.10) leads to

ÖW=JL.P.4E

(3.11)

where L = TTL*T. The several matrices are deter-mined by the configuration of the element to which they refer.

4 The solution of the problem

In section 1 it has been outlined that the solutiön is given by equations (1.3).:.

a

_{= - = - = O}

ap ai

for i = I

n

where P = V W represents the total potential energy of the construction änd, hence, equations (1.3) can be Written in matrix notation as

ÖVÖW

(4.1)

Nòw the total strain energy V is the sum of the strain energies V of all the elements and for the whole con-strúction can be written

ÔV=Ku

(4.2)

where K is composed of the stiffness matrices E of the elements in accordance with the numbering of the elements in the displacement column matrix u (This

can be verified by writing out all the equations

öV = K.

and summing up all these expressions. It appears that the resulting set of equations is of the form of equation (4.2)).

The virtual work of each element is formed by the internal forces of the construction and by the external forces. By compösing the virtual Work of the whole construction the internal forces cancel each other so that only the external forces are. of importance for the détermiñation of the virtual work. In the contracted

matrix notation we write for the variation of the

ôW=Lp3d ={L.p.d

X

(4.3)

where L and p have been composed of thé thá±rices L and p of all the m elements.

Sübstitution of the equations (4.2) and (4.3) in equation (4.1) leads to the final equation that represents the solution of the unknown displacements of the nodal points as functions of the external surface forces, viz.

K.u=JL.p.d

(4.4)

or

u = K1 fL.p.d

(4.5)

o

where K' is the inverse of K, so K'. K = U and U represents the square unit matrix

l00 ....O

ol0...0

001

000...1

If all the displacements of the n nodal points are

un-known, it appears that the matrix K is singular, that means, its determinant vanishes and its inverse does not exist. This corresponds to the case that the system is free to move as a rigid body under the action of the loading and, hence, we can overcome this by supplying boundary conditions sufficient to prevent these rigid body degrees of freedom.

Suppose that from the 3n displacements the quan-tities u1 to Uk(k < 3ñ)are not prescribed whereas Uk+ to u3, are. (This can be enabled in equation (4.4) by re-arränging the terms u and in agreement the terms in the matrices K, L and p). So equation (4.4) can be written as

LK21K22] k2] - -' LL21L22] _Liii]

[K11K121

ÍuiI

-

¿IL11L121 iPil _{dE (4;6)}

where u1 is a column matrix consisting of the unknown quantities u1 to uk and u2 comprises the prescribed displacements Uk+I to u3. In agreement with this partition of u into u1 and U2 the matrices K, L and p have been divided. For example K11 is a square matrix of order k by k.

In the supporting nodes the forces in the direction of the prescribed displacements are unknown as they are the reactions to the loading of the system. If the boundary conditions are restricted to some ñodal points, these unknown reactions are presented by _P2 and of the load matrix L the submatrix L12 is a null matrix and L22 is a unit matrix. From equation (4.6) the unknown displacements u1 to. Uk can be calculated

U ==JL11pdEK12u2

Iri.gènèal the prescribed quantities Ük+ i to u3 will be zero Imposing these support conditions results in striking out columns and rows in the stiffñess matrix K of equation (4.4) corresponding to the zerO dis' placements. This leads to the stiffness matrix K1 of equation (4.7) of reduced order and non-singular.

From equation (4.6) also follows an expression for the reactions

K21u1+K22u = [L21p1±L22p2]dE

o

Substitution of u1 given by equation (4.7),u2 = O and

L22 U results in

P2 =J [K21K11L11L21Jp1dE (4.8)

where use is made of

j

Up2 .dE =

5 Calculation of the stresses

In section 2.1 the equations (2.9) and (2.10) relate the stresses to the displacements of the corners of a rec-tangular plate. From equations (2.10) it follows that a and o, are linear functions of y and- x respectively and, hence, they have to be calculated oñ each of the edges to obtain the maximum values. In appendix-4

the matrix equation is giveñ in detail, in contracted form it reads

= E4'.0

(5.1)

where the asterisks refer to the co-ordinate system X*Y*Z* related to the element.

The relation between the displacements uK and the displacements u in the standard set of axes has al-ready been given in section 2.2 equation (2.17)

u*rT4.0

For a rectangular- plate element we may write

-= E4.T4.0

(-5.2)

Similàr relations can be obtained for the other ele-ments (appendix 4), sothe omission of thé suffix4 in equation (5.2) gives the generalized equation for the stresses.

It is not required to express the strésses of an elément in quantities that are oriéntated on the standard axes, as we are only interested in the stresses in the elements. (4.7)

14

6 Vibratory loading

For the calculations of the vibratiOns of a structure loaded by periodical forces we have to take into ac-count the masses of the several elements. In accordance with the principle of d'Alembert we introduce inertia forces. For that purpose the masses are concentrated in the corners of the elements. This is done in such a way that the centre of gravity of a lumped element coincides with that of the original element. Therefore the mass concentrated in the corners of an element has to equal the total mass of that element divided by the number of corners. The total mass rn in nodal point i is thus formed by a number of lumped masses from the surrounding elements.

At vibrations with circular frequency w the inertia forces mw2u1, mco2v and »iw2w1 in the three co-ori-nate directions have to be introduced in that point i. The derivative a W/au of the virtual work W of a force concentrated in node i equals this force itself. So the inertia forces can be introduced by adding to the column matrix L11p1d of equation (4.7) a column matrix co2Mu1, where

m100...O

-O m2 -O ... -O

M=

O

O O ...tn

J

is the mass matrix. It is acceptable to suppose that in dynamical cases the supports are fixed, so u2 = O. Then equation 4.7) becomes

K11.u1 =fL11.p1dE+w2.M.ui or

[1Á11_w2M]u1J L11 p1dE Hence

U1 [K11 M]1.LL11 .p1d (6.1)

Here we have tacitly made use of the fact that the structure is loaded by harmonic forces p, sin cot and damping can be neglected, so the displacements can be written as u sin cot. Then all the terms of thesé latter equations can be divided by sin cot.

In general the inverse of the square matrix [Ku--w2M] of order k will exist. For certain values, of co it becomes, however, singular and this means that the vibration is at resonance These values for co are the natüral frequencies of the construction and they are found by puttingthe determinant [K1 1co2M] equal to zero. To each of these solutions belongs a set of values u/u1 (i = 1. . . k) for example, which 'determines a configuration of the vibration, the natural mode

7 Application to sorne simple structures

To check the outlined method we consider some cases for which the statical deformations are given with a high degree of accùracy. First we take a flat plate, the length equalling four times the 'breadth and of thick-ness one. This plate is clamped at One short side and loaded at the opposite side by.. a constant shearing force (figure 6). Next we consider a flat square plate with sides a and thickness one. A circular hole with a diameter a is centrally situated. This plate is fixed in such a way that its centre cannot move and the plate cannot rotate around its centre. The' loading consists of shearing forces at the stràight edges (figure 7).

7.1 The clamped plate

A plate of length 4b, of breadth b añd of thickness 1 has been loaded at one small side by a uniformly applied shear such that f rdy = F (figure 6). At the other small side the plate has been clamped. The KY-plane of the standard system coincides with the KY-plane Fig. 6. Clamped plate divided intó ,1.mutual equal elements. The encircled figures indicate the number-ing of the elements.

of the plate. First the elementary beam theory, in-cluding shear effect, has been applied for the calcula-tion of the displacements ol the points 2, 3, 4 and 5 (figure 6). From symmetry considerations the dis-placements of the points 6 to 9 aré then known. Beam theory is used as a reference for the matrix calculations and, by using a low number of elements, the values of the beam theory will have agreater reliability.

Fôr the finite element method the platé has been consideréd to be composed of 1, 2 or 4 elements. To illustrate the described calculation technique the com-putation of the variation of the total potential energy with respect to the displacement u of node 4 (figure 6) is given here in detail.

In section 1 we have stated

(1.3) According to section 4 this statement leads to'

av

aw

The strain energy.Í/ is the sum of the strain energies V1 of all the elements and, hence,

4

_ çaV

au4 -

a,4

Because of the fact that the private set of axes of each element is parallel to that of the whole system the transformation equations (2.20) and (3.10) of the derivatives of the strain energies V1 and of the virtual works W1 related to the standard set ofaxesare reduced to av1

av

a-4 - auk* and (7.2) ah'1 au4 auk*

in which the corner k of the element j coincides with node 4.

Referring to the stiffness matrix in appendix i we can find fot.: element 1, with length .a and breadth b (figure 6 and figure 1)

ai',

:Et I a bl

au'

8(1_v2) L

(lv)

+ * (4-va)

uj' +

+1(l_v)(4_r2)!u2*+I_(l_)+

- *

(2+v} U3*

+

{_(1_v)

± (2+)

ju*

±

± (1+v)vi*_(1_3v)v2*_(l+v)v3*+(l_3v)Ü4*] Substituting a = b = 1, y = 0.28

ui* =

vl* = V4 u2* = u5 v2* = V5 u3* = u6 V3* = V6 u4* = u7, V

-V7

and taking equations (7.2) into accouñt we find

ay,

Et

=

2 [3.33u4-i.89u5-2.10u6+0.66u7 +

au4

8(1-v)

+ l.28v4-0.16v5-i.28v6±0.16v71

(7.1)

Table I. The dëflections of a clamped, shear loaded plate, calculated in various ways.

Similarly we find for element 2

For the elements 3 and 4 it is easy to see that their-cohtribiitibn in the väriation of the total strain energy V with respectto the displacement u4 is zero, because they are not connected to node 4. So equation (7.1) leads to

av

av, av2

=--+

au4 au4 au4

Et

=

_8(1-v)

2 l.89u3-j-6.66u4-l.89u5-2'.lOu6± + 1;36u7 -2.10u8+0. 16v3+O . v4-0.16v5+

-1.28v6+0.v7-l.28v8]

By adding the variations of the virtual work of each element the variatiOn of the virtual work 'of the whole system has been obtained. In our example only element 1 and 4 are loaded by external forces As remarked äbove the virtuâl works of the internal loadings cancel each other (section 4).

Referring to the end of section 4 we concentrate the reaction forces in the nodes i and 10. The displace-ments of these points are zero so the load matrix is formed by the load matrix on element i only and we find 6W = 6* W4.

_{= j}

.L4*p4*d (see section 3 and appendix 3) From this element only the edge 2 (see figure 5) is loaded. The load is a cons'taiit shear strèss

-r, so the only non-zero term in 4* is q2*

= -ir.

The variation with respect to the displacement u4 is

awaw4

/vb2t

-au4

au1*J 2a

Taking into account that a = b we get a w4

_{- -T}

vbt

au, * ₁₂

(73) 15

E

.U2 EV2

E-

.U3 E E.U4

- E

Beam theory

21

24.28

36

84.57

45

168.84

48

'

265i2

. I element , . .

48

20224

(-23.8%)'-2elements -36.24 -77.4 . 48 -25o.8 i -. (+0.39%) (&47%)'

"',

4 elements 21 24

23 7

3624

83 54 45 24 -- 167 38

48

263 08 (+067°/) (-2 4%) (+039V) (-1 2/e) (+031V)

(l l7/)

(-077/e) av2 Et 1.89u3+3.33u4+O.66u7-2.10u8 -1.28v4-0. 16v7+ 1.28v8]

+

au4

_{8(1-v )}

2 -l-

'0.16u3-16

In the example r has been chosen such that r = F/b; so that finally we find foi- equation (4.1)

- l.89u3+6.66u4l.89u5-2.10u6 +

+ l.36u7-2.10u8-l-O.'16v3-0.16v5 + 2(1 v2)vF

- l.28v6l.28v8 =

3E

The other equations are found in the same way. So we find a set of 16 lineär equations in the unknown dis-placements u and y of the nodes 2 to 9. The nodes i and 10 are constrained and, hence, their displacements are zero. Then only the variations of the potential energy with respect tq the displacements of the nodes 2 to 9 need to be considered (see the end of' section 4). The results of' these calculations are gwen in table I. The differences between these results and those' based on the beam theory are given in the brackets below the numerical values.

As could be expected the accuracy becomes higher with an increasing' number of elements. Älrea4y at a division into 4 elements the accuracy is surprisingly high and this suggests large reliability of the calculation

method. It is reasonable to state that the use of

square or nearly square elements is preferable.

7.2 A square-plate with a circular hole

A square flat plate of thickness one and side length a is loaded by shearing forces at its edges'. A circulàr hole with diameter a is located centrally. This model has been chosen, because plates with manholês are of

w

HIV

w

(7.4)

importance in ship structures and because there are accurate solutions of similar problems.

These accurate solutions are given by MEYERS [5] and by BAILEY and HiCKS [6]. These authors consider an infinite end-loaded plate completely perforated with closely spaced circular holes, forming a square pattern. Several pitch to diameter ratios were studied (figure 7) For our purpose it is sufficient to consider oniy the ratio of two. The infinite plate is loaded by a uniform shear at its straight edges.

We take a square el6ment of this plate as indicated in figure 7 and let the XV-plane of the standard system coincide with the plate. The origin has'been placed in the centre of the hole. From symmetry it follows that the normal stresses on the edges of these elements are zero and that the elongation of the edges is also zero

Consequently the displacements y of all points on the side x = a are the same.

In the finite element technique we replace the circu-lar hole by an octagon of equal area and divide the plate into elements in accordance with this octagon (figure 8A). In the proposed calculàtioñ method the shear stresses in a rectangular element 'are constant as is given by equation (2.10), so at each side element we apply a shear stress uniformly divided over the element. The stresses 'are chosen in such a way that' the con-dition of equal displacements of the nodal points along the edges is satisfied.

Then the ratio between the stiffness of the calculated plate and that f a similar plate without a hole is given by GO/G = 0.525. For the infinite plate, completely perforated, has been found [5 and 6] that G0/G =

Fig. 7. Square element of an infinite plate completely perforated, with a pitch to

A

E

B

D

Fig. 8. Several ways of dividing a square

plate with circular

hole into elements.

= 0.4787, so the matrix solution is 9.7% too stiff. Considering the rough división into elements the deviation is acceptable indeed.

In general a plate with a hole will be considered as one element, so the shear stresses will be constant over the whole side. With the same division into elements as above, such a loading gives G0/G = 0.45 1, taking into account the mean displacement along the edges. This is 16.4% lower than the first and this difference is mainly caUsed by the fact that the uniform stress distribution differs largely from the more accurate distribution represented in figure 9 [6]. The distribution of constant stresses per element, satisfying the requirement that the elongation of the edges is zero, shows a good re-semblánce with this accurate stress distribution (figure

10).

The. influence of replacing the circular hole by an octagon has been tested by leaving the triangular ele-ments out in the division of figure 8A Again it is loaded by a constant shear along úíe edges of the plate. The result, G0/G = 0.236, referring to the mean displace-ments again, has to be compared with a plate with circular hole of the same area as the enlarged square hole. For this configuration it appears [6] that GO/G = 0.403, so the solution of the described method is

0.5 0.3 0.1 0.5 0.3 0.1

t'

_turn £ 0.1 0.3 0.5

Fig. 9. Accurate shear stress distribution along the edges of a square elemeñt of a completely perforated infiñite plate with a pitch to diämeter ratio of two.

0.1 0.3 0.5

Fig. 10. The total shearforce divided into constant stresses per boundary element, satisfying the conditión that the nodes on the edges of the plate have the same displacements along the edge on which theyaresituated.

41.4% too low. Apparently the triangular elements largely affect the stiffness of the plate.

The accurate distribution of figure 9 [6] has also been applied. The plate was first divided into elements as above (figure 8A), later the division was refined in several ways (figure SB to D) At last the circular hole was replaced by a regular sixteen-side polygon (figure 8E) The results are given in table II From these

re-Table II. Deformations of a square plate with circular hole,

loaded by the accurate shear distribution given in figure 9 [6] with

several division systems (figure 8); u1 represents the displacements

of the corners of the square plate and Um represents the mean displacements along the sides. GO/G refers to the mean dis-placement. 17 Division system u1 Um G0/G A 4.66882 4.77275 0.536 B 5.56107 5.5859 0.458 C 6.03753 5.99323 0.427 D 6.52006 5.7963 0.442 E 5.16397 5.19424 0.493 accurate method [6] 5.347817 5.347817 0.4787

18

suits it may be concluded that the division of the circular hole into a polygon and the division of the plate into elements must be of the same order. If this is done the accuracy of the method is high in case of a sufficient division into elements. How many elements have to be taken, cannot be prescribed, but, taking the elements as quare as possible and keeping the divi-sions of the several parts in the same order, will be a reliable gúiding principle.

The accurate stress distribution has to be accom-pamed by uniform displacements of the nodes parallel to the edge on which they are situated. Discrepancies occur, however, and are likely caused by a singularity in the loadings of and the stresses in these boundary elements. The loading consists of a pure shear stress, divided along the edges of the plate as indicated in figure 9, whereas the shear stresses in the elements are constant, equation (2.10). Further it is clear that, generally, the normal stresses in these elements will not equal zero, so neither will those along the edges!

An important structural member in a ship is a rec-tangular plate with a man-hole. It is recommendable to require that tbe displacements along its edges are con-stant, wherth.s this will nearly be the case in a loaded construction. The possibilities to realize this require-ment have not been revealed yet.

8 Conclusions and outlines of future research

Applications of the described calculation method on simple structures, for which accurate solution exist, show a high degree of accuracy if the division into elements of the structures is deliberately made. As the method has been developed especially for complex three-dimensional structures, for which an accurate solution does not exist, it is obvious that experimental data must be collected tò check the method. Since the loading of an existing structure is hardly known, it is necessary to examine an appropriate model to get an insight into the accuracy and the reliability of the proposed method.

Due to the fineness of the division of the structure the number of unknown displacements, three times the number of nodal points, will always be very lárge and so the capacity of the electronic computers will soon be too small. To meet this difficulty we can take fewer nodal points or group some simple elements to a composite element. These composite elements are connected to each other at so-called outer nodes, numbered I to k. The other nodal points k+ ito 3n are

called the inner nodes. For the procedure to be given the supports have to cóincide with the outer nodes.

Referring to equations (4.6) we can write for a composite element

[:]

E:]

f[uL12]

[]

d (8.1)

where u1 is the column matrix of the outer nodes and u2 of the inner nodes. The partition of the matrices K, L and pis of course in agreement with that of u

LU2

Elimination of the displacements of the inner nodes gives

rit -1 _i.

Lit

L2 22 21ju1

-=

J

[L1 1K12K'L21][L12K12KL22]

['j d

which can be contracted to Ku

₌

j Lp

d

This is a similar equation as found in section 4 for the original elements. So the stiffness matrix of a structure, made up of composite elements, can be derived in the same way and for the solutiQn the same remarks hold. In this case it is surely necessary to compare the results with experimental data, as it is to be expected that the influence of round-off errors, introduced by repeatedly employing the equation (8.2), would be noticeable. The detailed performance of the outlined theory will be the next step in the scope of this research.

It is also possible to diminish the unknowns by considering only the displacements in one direction. This is acceptable if it can be shown that the

displace-ments in the other two directions can be neglected. Also in this case data from an appropriate model are

absolutely needed.

On account of the nature of the underlying calcula-tion method, the matrices found have been composed for a great deal of zeros. The non-zero terms have been situated on the principal diagonal of the matrix or in lines parallel to it. it is, however, to be expected that a numbering of the nodal points with delibration will lead to a better conditioned matrix than a numbering

at random. This investigation has also to be

per-formed in the near future.

At last also damping has to be taken into account for the determination of the response of structures to periodical loadings. Thi is possible by introducing complex numbers and solving the real and imaginary parts of the equation obtained.

References

I. TURNER, M. J., R. W. CLOUGH, H. C MARTIN and L. J. Topp:

Stiffness and deflection analysis of complex structures. Journal of the Aeronautical Sciences, VoL 23 no. 9, Sept.

1956.

TIMOSHENKO, S. and J. N. GOODIER: Theory of elasticity. McGraw-Hill, 1951, 2nd edition.

Jor.as, R. E.: A generalization of the direct-stiffness method of structural analysis. A.I.A.A. Journal, Vol. 2, no. 5,

May 1964.

BIEZENO, C. B., and R. GRAMMEL: Technische Dynamik. Springer-Verlag, Berlin, 1953, Vol. I.

MEIJERS, P.: Doubly-periodic stress distributions in

per-forated plates. Thesis Technological University Delft

1967.

BAUEY, R. and R. HIcKS: Behaviour of perforated plates under plane stress. Journal Mechanical Engineering Science, Vol.. 2, No. 2, 1960.

20

ApeDdix i

The derivatives, of the strain energy of a rectangular plate with respect to the displäcemehts of the cOrners in matrix notation The displacements have been defined in figure 1 y is Poisson s ratio, a and b are the length and breadth dimensions of the plate and t is the thickness

av

(1v)

b

(1v)

(1v)

b b au1 *

+«4_v2)

_(4_2)

4(2+v2) _+I(2+v2)

av

(lv)

(1v)

(1v)

(1v)

_(4_2)

_+1(2+v2)

+v)

av

(lv)

_.(2+v2)

(lv)

+(2+v2)

1 . (1=v)

+(4_v2)

(1v)

_(4_v2)"

au3

av

(1v)

(1v)

(1v)

(1v)

au4* _{b b b b} Et

±2+v2)

a

+v)

a

_42)

-

a

+(4_v2)

a

av

= 81_v2)

(1+v) (1-3v)

(1±v)

(1-3v)

av *

av

_(1-3v)

(l+v)

(l-3v)

(L±v) av2 *

av

(1+v)

(l-3v)

(1+v)

(1-3v)

av3 *

av

(1-3v)

(Í+'v)

(l-3v)

(1+v)

av4*

(l+v)

(l-3v)

(l-Fv)

(l-3v)

(l-3v)

(l+v)

(l-3v)

(l+v)

21 +«4_v2)

+«2±v2)

_4(2l_v2).

_«4_v2)

a b b b b

(lv) -

(lv) -

(lv) -

(lv)

-a a a a _v2* 41(2+v2)

+«4_v2)

_.(4_v2)

_(2+v2)

b b b b

(lv)--

_a

(lv)

_a

(lv)

_a

(1v)

_a _*(2+v2)

_.(4_v2)

+(4_v2)

+(2+v2)

b b b b

(lv)

(lv)

(Iv)

(1v)

a a a

a

_(4_v2)

_(2Iv2)

±(2+v2)

+(4_v2)

b b b b

(lv)

(lv)

(lv)-

(1v

a a a v1*

(1+v)

(l-3v)

(l±v)

(l-3v)

U3 *

(l-3v)

(1+v)

(l-3v)

(1+v)

U * UI U2*

22

For a triangular plate element (figure 2) the functions for the in-plane displacements are accepted for the sake of simplicity to be represented by ui" = ao±aix*±a2y*

v = bo±bix*+b2y*

Then the derivatives of the strain energy are given by

Contracted ô*V _ K3*.u*

The derivations of the strain energy of a bar (figure 3) are based on the relations u* = a0 + a1x* and v = b0+ b 1x* and can be written as

where A is the area of the cross section of the bar having a length i, y is the angle between the bar and the positive X*,axis and is in counter-clockwise direction positive (figure 3).

In contracted matrix notation these equations become Ô*V

av

cos2 y

cos2 y

siny cosy

sin y cosy

*

au1 *

av

EA

cos2 y

cos2 y

sin y cosy

sin y cosy 2

au2*

av

/

sin y cosy

sin y cosy

sin2 y

sin2 y

*

dv,

av

sin y cosy

siny cosy

sin2 y

sin2 y v2*

aú2*

av

bsiny

lv

(b cosya)2

bsiny

lv(bcosya)cosy

au1 * _a

+=--

2 absiny a 2

asiny

av

bsiny

l-1'

(b cosya) cosy

b siny

+

lv b cos2y

a 2

asiny

a 2

asiny

av

lv b cosya

1v cosy

Et 2

bsiny

2

siny

av

2(1v2)

I±v b cosya

bcosya

lvbcosy

av1 * 2 a

y

a 2 a

av

bcosy

l---vbcosy--a

l+i b cosy

av2* V a

+

2 a 2 a

av

av3*

lv b

cosy-2

bsiny

lv cosy

2

siny

lv

a 2

bsiny

lv

2

lv

2 o

-1+vbcosya

bcosy

1vbcosya

+

2 a a 2 a

bcosya

1vbcosy

1+vbcosy

7)

+

a 2 a 2 a

2 2

(bcosya)2

1vbsiny

(bcosya)cosy

1vbsiny

bcosya

absiny 2 a

asiny

2 a

bsiny

(bcosya)cosy

1vb siny

bcos2y

1vbsiny

cosy

asiny

2 a

asiny

2 a

sin'

b cosya

cosy a

bsiny

siny

bsiny

UI * U2 * U3 * v1* V2 * * V3 23

1v

1v

-7)

V o

Appendit 2

Transformations

Reötängular plate (figure 4)

Contracted u* = T4 u Triangular plate

[

v2* v3* Symbolically Bar-like element

or U*=T2.ú

COS 971 o o cos ip1 o o

cos 97 O cosq2 O cosq O

o cosç O cos O cosç'3

COS ip O COS 972 0 CQS₉₇ O

o cos O cos 972 0 cos3

I

u' u2 vi W2 UI u2 cosq2. O O O cos O O O H O COS O O O cosç3 O O u4 O O cosq2 O O O COS 973 0 vi 0 0 0 cos O O O cos3

cos2

O O O cos'çv3 O O O u3 O Cosp2 O O O

cos3

O O V4 0 0 cos 972 0 0 0

cos3

O w1 O O O COSI/J2 O O O COS W2 w3 w4 u1 0 0 COS 92 O O COS 973 0 0 u3

COsq71 O O COS 92 O O COS 92 O

vi o

cös1

O O C0S972 O O cosç,3 V2 O O cos 972 0 0 cos3 O O V3 sp1 O O COS1p2 O O CoS973 O o cös tp1 O O COSt/)2 O O cos973 u cosq1 Q O O O C0S911 O O O O cosq1 O O O O cos 971

-

cos O O O V2 O COS1/1 O O v3 O O cos 971 0 V4* O O O cQstp1

Appendix 3 The:virtual work of the inplane loading on a rectangular plate. The symbols of the loading are defined in figure 5 with ,p,*(C) and q*(), where

.E (O

C,

I') is the dimensionless generalized co-ordinate along, the edges of the plate (figure 5).

Contracted ö*W_JL*.p*.dC

ow a2 i'b2 . a2 vb2 a(1 -C) o b(1.E) 2b a2

C(1C)

2a vb2

E(lC)

2b a2

E(lC)

2a vb2. Ou, * ow o va2 b(lC). bC o b2 o a(lC) va2 o bC b2

--C(lC)

2b a2 2b a2 --C(1'C)-2b

--C(lC)

2a vb2 ¿(1e) 2a vb2 , ---C(1C) 2a ---C(lC) 2b a2

C(lC)

2b û2

--E(lC)

2b E('l--E) 2a vb2

C(lOE)

2a vb2 -C(l--C) 2a * p3 p4.-Ou2 * 'a

w

Ou3 * ow Ou*

Ow

--C(lC) 2b . va2 C(l-C). 2a b2

C(.lC)

2b .va'

C(lC)

2a b2 a(1C) O O' b(lC)

*,

q1 vi*

_Ow

---C(lC)

2b va2

--E(lC)

2a b2 ----C(1C)' 2b va2 ---E(lE) 2a b2 b(1E) o o * 2 0v2* ow

C(lC)

2b va2'

-¿(1C) 2b E(.lC) 2a b2

-¿(lE) 2a 4('l---C) 2b va2 - - ¿(1e) 2b

C(1C)

2a b2

-¿(1E) 2a 0 O bC o . a O bE q3 q4 0v3* ow * V4

26

The virtual work of the in-plane loading on a triangular element.

or

where c =V'a2+bl_2ab cosy p and q* are functions of the parameter along the edge in consideration (O. E 1) (figure 2).

The virtual work of the loading along a bar-like element

aw

-au1 *

aw

au

aw

av1 *

aw

av2 o O O O

lE

O E

shoitly ÔW

_JL2'' p* . dE

where pK äñd q* ate fUnctions fE = x*f1 cosy (figure 3), the parameter along the bar (O E 1).

aw O

b(lE)

O O _Pi* aui*

aw

O O O O _P2*

aw

O

c(1E)

bE Ö O _P3* dE

aw

O O

a(lE)

b( 1E)

q1* avi*

aw

O cE O q2* v2 *

aw

O O O. O

c(lE)

bE q3* av3*

Appendiì 4

The stresses in the rectangular plate element (figure 5)

A triangular plate (figure 6)

a

y a

cosy 1

_\

A bar (figure. 7

r

i I cosy cosy siny siny

iai=Ei

--

-Li

L'

t.

¡ ¡ Contraóted

û = E .*

Contracted * = E3*.u*

2 v2

y b b

2v2

b b b b a . a

a_

where for eamplë O3 represents o àlôñg tue side 3 (figuÈe :5) in the X*, Y, Z co-ordinate, system. Contracted

a = E4* .

u1 * u2* vI.* v2i * V3

2\asiny

bsinyl

27 i

-

O

¡cosy

i

\

cosy r

a v \a siny b sin y!i.

-

a sin y b slny

V cosy i COSy u3*

à äsiny

bsny

asiny bsiny vi *

1v cOsy

lv

1

1vi

1v!

V2* 2 asiny 2 bsiny a v3*

2v2

2v2

y2 y2 a a a a

2-v2

2v2

a a y y y y a a a a V V V V a a a a

1-v

lv

lv

lv

b b

.b

b

2-v2

2v2

y2 b

bb

lv

lv

.1 - V E i Xt X3 E .Jy4 2(1_v2

PUBLICATIONS OF THE NETHERLANDS SHIP RESEARCH CENTRE TNO (FORMERLY THE NETHERLANDS RESEARCH CENTRE TNO FOR SHIPBUILDING AND NAVIGATION)

M = engineering department S = shipbuilding department

C = corrosion and antifouling department PRICE PER COPY DFL.

IO.-Reports

I S The determination of the natural frequencies of ship vibrations

(Dutch). H. E. Jaeger, 1950.

3 S Practical possibilities ofconstructional applications of aluminium

alloys to ship construction.. H. E. Jaeger, 1951.

4 S Corrugation of bottom shell plating in ships with all-welded or

partially welded bottoms (Dutch). H. E. Jaeger and H. A.

Ver-beek, 1951.

5 S Standard-recommendatiòns for measured mile and endurance

trials ofsea-goingships (Dutch). J. W. Bonebakker, W. J. Muller and E. J. Diehl, 1952.

6 S Some tests on stayed and unstayed masts and a comparison of

experimental results and calculated stresses (Dutch). A. Verduin and B. Burghgraef, 1952.

7 M Cylinder war in marine diesel engines (Dutch).. H. Visser, 1952. 8 M Analysis and testing of lubricating oils (Dutch). R. N. M. A.

Malotaux andJ. G. Smit, 1953.

9 S Stability experiments on models of Dutch and French

standard-ized lifeboats. H. E. Jaeger, J. W. Bonebakkerand J. Pereboom, in collaboration with A. Audigé, 1952.

I O S On collecting ship service performance data and their analysis.

j. W. Bonebakkër, 1953.

I I M The use of three-phase current for auxiliary purposes (Dutch). J. C. G. van Wijk, 1953.

12 M Noise and noise abatement in marine engine rooms (Dutch). Technisch-Physische Dienst TNO-TH, 1953.

I 3 M Investigation of cylinder wear in diesel engines by means of labo-i:aory machines (Dutch). H. Visser, I 954.

14 M The purification of heavy fuel oil for diesel engines (Dutch). A. Bremer, 1953.

I 5 S Investigations of the stress distribution in corrugated bulkheads with vertical troughs. H. E. Jaeger, B. Burghgraef and I. van der Ham, 1954.

16 M Analysis and testing of lubricating oils Il (Dutch). R. N. M. A. Malotaux and J. B. Zabel, 1956.

17 M The application of new physical methods in the examination of

lubricating oils. R. N. M. A. Malotaux and F. van Zeggeren, 1957.

18 M Considerations on the application of three phase current on board ships for auxiliary purposes espacially with regard to fault pro-tection, with a survey of winch drives recently applied on board of these ships and their influence on the generating capacity (Dutch). J. C. G. van Wijk, 1957.

19 M Crankcase explosions (Dutch). J. H. Minkhorst, 1957.

20 S An analysis of the application of aluminium alloys in ships' structures. Suggestions about the riveting between steel and aluminium alloy ships' structures. H. E. Jaeger, 1955.

21 S On stress calculations in helicoidal shells and propeller blades. J. W. Cohen, 1955.

22 S Some notes on the calculation of pitching and heaving in

longi-tudinal waves. J. Gerritsma, 1955.

23 S Second series of stability experiments on models of lifeboats. B.

Burghgraef, 1956.

24 M Outside corrosion of and slagformation on tubes in oil-fired boilers (Dutch). W. J. Taat, 1957.

25 S Experimental determination of damping, added mass and added

mass moment of inertia of a shipmodel. J. Gerritsma, 1957. 26 M Noise measurements and noise redUction in ships. G. J. van Os

and B. van Steenbrugge, 1957.

27 S Iñitial metacentric height of small seagoing ships and the in-accuracy and unreliability of calculated curves of righting levers. J. W. Bonebakker, 1957.

28 M Influence of piston temperature on piston fouling and pistonring wear in diesel engines using residual fuels. H. Visser, 1959. 29 M The influence of hysteresis on the value of the modulus of

rigid-ity of steel. A. Hoppe and A. M. Hens, 1959.

30 S An experimental analysis of shipmotions in longitudinal regular

waves. J. Gerritsma, 1958.

31 M Model tests concerning damping coefficiènt and the increase in the moment of inertia due to entrained water of ship's propellers. N. J. Visser, 1960.

32 S The effect of a keel on the rolling characteristics of a ship. J. Gerritsma, 1959.

33 M The application of new physical methods in the examination of lubricating oils (Contin. of report 17 M). R. N. M. A. Malotaux and F. van Zeggeren, 1960.

34 S Acoustical principles in ship design. J. H. Janssen, 1959.

35 S Shipmotiôns in longitudinal waves. J. Gerritsma, 1960.

36 S Experimental determination of bending moments for three

mod-eIs of different fullness in regular wavès. J. Ch. de Does, 1960.

37 M Propeller excited vibratory forces in the shaft of a single screw tanker. J. D. van Manen and R. Wereldsma, 1960.

38 S Beamknees and other bracketed connections. H. E. Jaeger and

J J. W. Nibbering, 1961.

39 M Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system. D. van Dort and N. J. Visser, 1963.

40 S On the longitudinal reduction factor for the added mass of vi-brating ships with rectangular cross-section. W. P. A. Joosen and J. A. Sparenberg, 1961.

41 S Stresses in flat propeller blade models determined by the

moiré-method. F. K. Ligtenberg, I 962.

42 S Application of modern digital computers in naval-architecture.

H. J. Zunderdorp, 1962.

43 C Raft trials and ships' trials with some underwater paint systems.

P. de Wolf and A. M. van Londen, 1962.

44 5 Some acoustical properties of ships with respect to noise control.

Part. I. J. H. Janssen, 1962.

45 5 Some acoustical properties of ships with respect to noise control.

Part II. J. H. Janssen, 1962.

46 C An investigation into the influence of the method of application on the behaviour of anti-corrosive paint systems in seawater. A. M. van Londen, 1962.

47 C Results ofan inquiry into the condition ofships' hulls in relation

to fouling and corrosion. H. C. Ekama, A. M. van Londen and P. de Wolf, 1962.

48 C Investigations into the use of the wheel-abrator for removing rust and millscale from shipbuilding steel (Dutch). Interim report. J. Remmelts and L. D. B. van den Burg, 1962.

49 S Distribution of damping and added mass along the length of a

shipmodel. J. Gerritsrna and W. Beukelman, 1963.

50 S The iniluence of a bulbous bow on the motions and the

propul-sion in longitudinal waves. J. Gerritsma and W. Beukelman, 1963.

51 M Stress measurements on a propeller blade of a 42,000 ton tanker on full scale. R. Wereldsma, 1964.

52 C Comparative investigations on the surface preparation of ship-building steel by usingwheel-abrators and the applicatiòn of shop-coats. H. C. Ekama, A. M. van Londen andJ. Remmelts, 1963.

53 S The braking of large vessels. H. E. Jaeger, 1963

54 C A study of ship bottom paints in particular pertaining to the behaviour and action of anti-fouling paints A. M. van Londen,

1963.

55 s Fatigue of ship structures. J. J. W Nibbering, 1963.

56 C The possibilities of exposure of anti-fouling paints in Curaçao, Dutch Lesser Antilles, P. de Wolf and M. Meuter-Schriel, 1963. 57 M Determination of the dynamic properties and propeller excited

vibrations of a special ship stern arrangement. R. Wereldsma,

1964.

58 S Numerical calculation of vertical hull vibrations of ships by

discretizing the vibration system. J. de Vries, 1964.

59 M Controllable pitch propellers, their suitability and economy for large sea-going ships propelled by conventional, directly coupled engines. C. Kapsenberg, 1964.

60 S Natural frequencies of free vertical ship vibrations. C. B.

Vreug-denhil, 1964.

61 S The distribution of the hydrodynamic forces on a heaving and

pitching shipmodel in still water. J. Gerritsmaand W. Beukelman,

1964.

62 C The mode of action ofanti-fouling paints: Interaction between anti-fouling paints and sea water. A. M. van Londen, 1964. 63 M Corrosion in exhaust driven turbochargers on marine diesel

engines using heavy fuels. R. W. Stuart Michell and V. A. Ogale,

1965.

64 C Barnacle fouling on aged anti-fouling paints; a survey of pertinent literature and some recent observations. P. de Wolf, 1964.

65 S The lateral damping and added mass of a horizontally oscillating

shipmodel. G. van Leeuwen, 1964.

66 S Investigations into the strenght of ships' derricks. Part I. F. X. P

Soejadi, 1965.

67 S Heat-transfer in cargotanks of a 50,000 DWT tanker. D. J. van

der Heeden and L. L. Mulder, 1965.

68 M Guide to the application of Method for calculation of cylinder liner temperatures in diesel engines. H. W. van Tijen, 1965. 69 M Stress measurements on a propeller model for a 42,000 DWT

tanker. R. Wereldsma, 1965.

70 M Experiments on vibrating propeller models. R. Wereldsma, 1965.

71 S Research on bulbous bow ships. Part Il. A. Still water perfor-mance of a 24,000 DWT bulkcarrier with a large bulbous bow. W. P. A. van Lanimeren and J. J. Muntjewerf, 1965.