Proefschrift van R. Nielsen jr

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ARC BEF

LABORATORIUM VOOR

SCH EEPSCONSTRUCTI ES

TECHNISCHE HOGESCHOOL

- DELFT

RAPPORT Nr.

BETREFFENDE:

Proefschrift van R. Nielsen ir.

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SSL 101

Proefschrift van R. I'Tjelsen jr.

1. Mogelijkheden en beerkingen van de berekensmethode

De methode die de auteur ontwikkeld heeft maakt het mogelijk orn grote, orthogonaal verstijfde plaatconstructies op doorbuiging en krachtopnarne te berekenen. }iulp van een electronische rekenmachine en een groot aantal sub-programmats in procedurevorm is noodzakelijk, maar dan is ook zeer veel mogelijk. Dit kan het beste blijken uit een bespreking van wat voor andere methoden vaak beperkingen zijn.

Belastingen. Hieraan worden geen beperkingen gesteld zolang zij te herleiden zijn tot belastingen op de stijien. Puntlasten op een plaat-paneel zijn dus niet zonder meer te behandelen, maar behoren ook niet

voor te konien.

Randvoorwaarden. Moeten bekend zijn.

Profielafmetingen. Dienen constant te zijn over de lengte van het be-treffende constructiedeel.

Sti lafstanden. Inconstante stijlafstand wordt door de auteur niet behandeld. (pag. 27, S

i).

Torsiestijfheden. KuinériIñ rëkenTig gebrachtworden.

De methode heeft volgens Nielsen weinig bezwaren behalve dat het aantal stijlen voldoende groot moet zijn. Beperkingen worden gesteld doordat de

formules hanteerbaar moeten blijven en doordat het geheugen van de computer niet onbegrensd is.

Verder blijft het een methode waarbij een orthogonaal verstijfd plaatveld als open raster wordt beschouwd. (zie 2). Het is dus altijd een benaderings-methode.

Bij de gewone rasterberekening wordt de statisch onbepaalde constructie, statisch bepaald gemankt door alle staven tussen de knooppunten los te ne-men. De continuteitsvoorwaarden voor elk knooppunt vormen de basis voor een systeem van vergelijkingen waarmee de onbekenden (reacties of verplaat-singen) worden opgelost. Nielsen brengt elke serie knooppunten van

drager onder in n differentlaal vergelijking die opgelost moet worden. i

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De rekenmethode die in Nielsen's dissertatie besproken wordt is gebaseerd op het geval van een belaste balk op een verende onderlaag. Dit is bij vrijwel alle roosterberekeningen zo, en de gedachtengang die tot dit uit-gangspunt geleid heeft wordt bekend verondersteld.

Een bestudering van een voorgaand artikel van Nielsen en Micheisen (lì, dat tevens als inleiding tot de dissertatie kan dienen, werkt verheiderend, zoals biijkt uit het voigende.

Uit de bijgaande copien van de eerste twee pagina's van het artikel (i) zijn de aangestreepte fragmenten van belang. Zij worden hieronder in de tekst opgenomen, de copien zijn volledigheidshalve bijgevoegd. De almea boyen fig. 2 luidt:"In many engineering problems such as grillage ship structures, one encounters beams not supported partially by a continuous elastic medium but rather by a finite number of equally spaced springs. A simple case is the deck girder supported partially by the deck beams." Dit mag dan voor een rasterwerk-analyst normaal zijn, voor een praktische scheepsbouwer is dat niet het geval. De laatste beschouwt een dekdrager als

teunpunt voor de balken en niet omgekeerd. Verderop (pag. 2) wordt alles duideiijk. Ret begin ligt bij de aangestreepte almea onder fig. 3 _{en de} sleutel wordt gevormd door de formules (O.Li) en (1.4), die volgens simpele statica afgeleid zijn. Ret geheel luidt als voigt:

"It is therefore required to obtain a relationship between the deflection of any stiffener and the force exerted by the stiffener onto the girder. To be able to account for the live load q, a uniform load of qa per unit

length must be considered carried by the stifferBs. The freebody diagram of a stiffener, including effective plating, is shown in Figure 1+, where i represents the force between girder and stiffener. The deflection at R due

to the uniform load is given by

4

5qab

y(q) =

384 EI s

whereas the deflection due to the reaction R at the point of application is given by

Rb3 =

-48 EI

(4)

The total deflection is therefore LI-5

q ab

RR

(q) ₊ ₌ y R R b3

38k

EI k8

EL

s

Since R represents the loading on the girder, this is the quantity sought, and, from Equation (1.3), it follows that

5 '+8E1

qab- __s

(1.3)

3

De verdiensten van het werk worden het beste gellustreerd door eens na te gaan

8 b2

If this concentrated force is assumed to be uniformly distributed over a stiffener spacing, one can say that the girder is loaded by a distributed load of magnitude

R ₅

48E1

=

qb-5 y

(i.k)

a 8 ab3

where the subscript has been dropped since Equation (1.1+) is valid over the whole, length of the girder. In addition to distributing the elastic reaction

b

it is noted that Equation (1.4) also distributes the concentrated load uniformly over a stiffener qacing. This fact is not too serious as far as the elastic reaction is concerned if the number of stiffeners is sufficiently large, and in regard to the other part of the loading on the girder, it will be shown that no ayntage is to be gained by distributing this since concentrated forces Can be handled quite readily."

Uit formule (i.4) is nu evident waarom de cirager beschouwd wordt als een belaste en verend-ondersteunde ligger.

In de dissertatie wordt de vergelijking voor de elastische lijn van de verend ondersteinde ligger op een andere manier afgeleid dan in (1) i.v.m. het streven naar een zeer algemene behandeling van het probleem met als consequentie een

streng mathematische algemene formulering van de ranclvoorwaarden. De differenti-aalvergelijking of de gekoppelde differentidifferenti-aalvergelijkingen wordt of worden opgelost met behuip van Laplace transformaties. Dit is een wiskundige methode waarbij een differentiaa1ergelijking wordt getransformeerd tot een lineaire algebrasche vergelijking. Een definitie is te vinden in (2) (hoofdstuk XIII).

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wat er van het lijstje bezwaren overblijft dat Ando

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geeft in een over-zichtje van plaatveldberekeningsmethoden. (Zie bijgaande copien).

Bezwaar I De torsiestijfheid van de stijien kan in rekening worden

gebracht. (Heeft meestal weinig invloed. (Nielsen pag. 55

e.v.)

)

De torsieatijfheid van de stringers kan in rekening worden gebracht middels een i1ratieproces dat snel zal

convergeren. De torsiestijfheid van dubbele bodems mag niet verwaarloosd worden.

Bezwaren II t m IV \Iervallen.

1iteratuur

(i) Michelsen, F.C. and Nielsen, R. "Analysis of Grillage Structures by

means of the Laplace Transform." Schiffstechniek 13d.

9 -

1962 - Heft _+9.

Handboek der Wiskunde red. Kuipers, L en Timman, R.

Ando, N. "On the Strength of orthogonally stifened plate" Transp. Techn. Research Inst. Rep. nr. 'i-8, maart 1962.

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An analysis of grillage structures is 1)rescnted which is based un the solution of girder dtílections as uhtained from Laplace Transforms of givuriiing riiffirentia! equa-tions. The resulting formulation is allIlieahle to Shy type of loading, including 'onetiitrntcd forces, 511d lends itself

readily for computer programming. The method of solution

therefore offers the opportunity to determine optimum distribution of material with a minimum airmunt nf labor and in a short tizno.

The analysis of grillage stni°tures has reoeived a great deal of att4ntiIn in the past. n view if their great impor-tance, this is not surprising. Considering the ship

strue-ture as an example, one observes immediately that imtany

of the primary strength members consist of plating sujm-ported by mutually orthogonalsets of stiffeners

- i. e., a

grillage structure.

lt

is immediately realized that such a structure is

highly redundant and that strength calculations would be cumbersome if they were based on elementar heamn formulae. Early researchers, among them Timoshenko

and Föppl, soon realized that the grillage structure could he

analyzed by the use of deflection equations derived for beams on elastic foundations in much the same manirer

as in the analysis of railroad tracks [5]. I)r. G. Vedeler [2J

and Hetenyi [6] have contributed much to perfect this method of analysis, but it can be said that, since their treatments are based on the classical solutions of the governing differential equations, their method becomes

quite complicated to handle for some types of

loa-ding such as the case of a considerable number of

concentra+,ed forces.

The purpose of this paper is to present an analysis of the grillage structure based on the elastic foundation approach. However, the differential equations of deflec. tions are solved by means of Laplace Transforms Instead of by the use of classical solutions. This approach leads to

s general formulation valid for all types of loading.

Subscript notation has been used extensively to save space and to attempt to make the problem more trac-table. In doing so, the computer programming was kept w mind, but even more important is the ability of this form of notation to reveal important functional relation-.hlp between physical parameter..

1. lesa sa sa elastic hsiidMløs

A beam supported along its entire length by a con-tinuous linear elastic foundation will have a distritiuted

load which at every point is proportional to the deflect w ni

at that point. The load intensity due to tl

eLastic

IowsdatM* M then represented by Ks,, whete K i a

ent

sailed the modulus of fotindatlos If the hie

t&t bausa

M dded by! (z), the toi.J lMsal ld

Analysis ¿

_{i G nuage Strui ures by means}

of the Laplace Transform

f'. C. M irheh,en arid R. N

iclsen, Jr.

on the beam becomes / (z) - Ky. Figure 1 shows the loading in the, case of a simply sujported beam on an

elastic foundation.

Ut)

iiu,,llIIøIIPIIHlllIftpu,p,i

-Jig. 1. SImply iupporLed besmOD_{aa oh,.sU trrndat1os}

For the coordinate system chosen, and with the usual definition of positive bending moment, the following

relationships hold;

d'y

M

dx'

EJ

d' M

t

where p (z) is the total lateral load as a function of z. From equation (1.1), the differentia' rqustiue govarn-ing the beam on an eLastic foundation then bceogsim

d'

EI-±Ky=/(x)

This equation is, of course, only valid within the aump. tions of the Bernoulli-Euler beam theory.

fin many engineering problems such as grillage .hlp structures, one encounters beams not supported partially by a continuous' elastic medium but rather by a fütit. iumber of equally spaces springs. A simple case M tba

deck girder aujpnrted partially by the deck beaa.

Schematically, this situation is shown in Figure t

,,,11itIIIIIIIIlllllItlHIIIIllllttlllHllhIllll nu,..

A R

- - +

Such a problem would indeed be difficult to solve If

exact physical nature of the foundation were to b. tùa* into account. If, however, there are s auf fieleM .mSu

of closely spaced spring., the beam can be assamed t. be partially supported by an equivalent oont&nwss

h

K

foundation of modulus K = -, an smptlsa wWò ii

very good for small deflection, of besa theory. To selve Equation (1.2),

tIse de4srminaon of, the eqs1vs

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can he done uy means of simjile lsam formulae. Asan

example, consider the ca. of Figure 3. A reetngiilar panel of dimensions L and b is stiffened li one girder in

one threetion and several closely sjiacM stiffeners in the other. Wjthot loss of generality, the _{noel is nssiiiiied t4 i} be loaded by a uniform had of _{q lbs. per square inch, and}

- q yr r"T

J'

1Ig. 3

the stilft'ners are assunied to he simply siuppoited at beth

ends. As ment i ned _{)rev o usiv, the girder iS MtIJ)J >rted} elastically by the stiffemiersfli _{is therefore required to} obtain a relationship between the _{deflection of aims'} stiffener and the force exerted by the stiffener onto the

girder. To be able to account for the live_{load q, a uniform}

load of qa per unit length must be considered carried hr

the stiffeners. The freebodv diagram of a stiffener, inelud.

ing effective plating, is sh;wn in Figure _{4, where R} represents the force between girder and stiffener. The

deflection at R due to the uniform _{load is given by}

- 384E!.

whereas the deflection due to the reaction _{R at the point}

of application is given by

!FT

y(R) =

-. 48E!.

The total deflection is therefore

5qab'

Rb

Ya =

_y,dq)+ys(R)

_(1.3)

- 384 EI.

48E!,

Since R represents the leading on the girder, this is the quantity sought, and, from Equation (1.3), it follows that

R_---8-qab

b3 YR

(pli)

5

_48E!,

II this concentrated force is _{assumed to be unifr.rnly} distributed over a etiffener spacing, _{one can say that the} girder ii loaded by s distributed losd of magnitude

R 5

48K!,

(1.4)

a

8

51l.is.1

Ek4115

. 5

where the subscript has been dropped sInce Equation

(1.4) is vaho over the whole length of the girder. _{In addi.}

tion to distributing the elastic _reaction

_{y, it i.}

noted that Equation (l.4 _{also distributes the} _eoneen. trated load uniformly over a stiffener spacing. This ¡set

is nit too serious as fa r as t he elastic reaction is concerned.

if the number of stiffeners is sufficiently large. and, in regard to the other part uf the loading_{ofl the girder, it}

will be shown that uo advantage is to he gained by distri.

but ing this siLJce concentrated forces _{can be handled ((Uit.}

readily.

The different lumI eu 1uation for the girder, fiir t he simple

case considered timemi becomes, by Equation (1.2),

d'

48E!

E1u-L'.I-

'Y =R5

where it is noted that represents the spring eon.

stani K, and f (r) Of Equation (1.2) is equal to R

su ho'hu represents a series of concentrated forces, each

n1agmmitu(le'- _{gab acting at intersections of girders and}

stiffeners. The solution of Equation (l.2) by means nf

LapIner' Transförni techniques for various loadings and gromuuetrv is the stmhjeet. of Reference 1. [Example

_{I of}

Appendix I IJJ shows that Equation (1.2) is_{a special case of}

the noire general problem of coupled equations trestod

here.

If the panel is supjsrted by_{more than one girder, it is} advantageous to generalize Equation (1.3) as follows:

-

R1 ,, (i

= 1,2,'

N) (1.6)

Total deflection of stiffener at the ith girder 1}"flection at the ith girder due to live load

and no girder suppomi.

Influence index - i. e., the deflection of the stiffener at the ith girder due to a unit load at the ith girder.

R1 _{Vorce between ith girder and the stiffener.}

N -= Xumber of girders.

The usual convention of summing over repeated suhiscriptis observed. So!ving Equation (1.6)

_{fr the}

equivalent distributed loading on the ith girdeti, _one

obtains (1.5) where where R, a é = --- (ne su'nmation) a ri j=ri11;

i.j

=0;

i=j.

The differential equation for the ith girder thenbecomes

EI1,

dz4 == é(d5 - y' - (1.8)

where

ô15=I;

=L

=0; j+k

= Moment of inertia of ith girder if j ¡ 0;

j+i.

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Ando, N. "On the strength of orthogonally stiffened plate"

rep. k8 _{of Transportation Technical Research Inst.}

maart

_1962.

i. e.,

(_{1 ')} Met h d nf

calculation by regard ng it as grid structure.'

Ç 2 ' Energy method,'1'

Ç 3 ) Met hod of

ticulation by applying the orthotropic plate theory.

Merits and demerits contained in thse methods may he considered as

folh )W$

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Energy methxl

Merits:

(i)

As the solution may he obtained without difficulty by coiaratively

siml)le calculation, this is the convenient method for estimating the

scantling of these structures, and strength corresponding

to

varia-tion of scantling will be readily obtained.

l)emerits

(i)

Solution obtained will satisfy only one of the basic equation or

boundary condition. Consequently, it will contain considerable error.

2

-Ç 1)

Grid structure

Merits

i)

lt

is very plain to lead the basic equation, and eben if

the end

cunctions are these of built in or more complicated, it can do as

simple as the case of simy supported edges.

('ii)

As the value uf local ele,nen: can he exactly calculated,

it may

be available fur designing details of respective element.

I )emerits

( i ) In this method, torsional rigidity is ignored, because every stiffener

with effective breadth of plate

is cut off and calculated

indepen-dently as a simple beam. The calculated value, therefore, will differ

from actual structure, especia lv in case uf the structure with

corn-I)arativelv large toronal rigidity such as the double bottom of ship.

(ii )

Though it

is usually very easy to

lead

basic eluations, solving

these equations will be considerably troublesome.

As those equations have different

yiì according to the number

of stiffener, loading, end condition, etc., trial calulation wifl be

dif-ficult at the time of designing these structure.

The way of distributing the given load to respective stiffener

is

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(ii

)

There is nu adequate method to check the

_{n'atgnitide of error.}

2)

Ort hotropi plate

rits

(i )

As it holds the continuity of plate, and torsinnal

_{rigidity is}

consi-dered reasonably, it will he the most precise

_{one anong aforesaid}

met hd6.

(il)

\'alti1 corresponding to given end conditions

or loings

be

readily obtained by the formulas or graphs which my be

previou-sly prepared.

(iii)

_{As variation of stress or deflection corresponding to the variation}

of scant lings mai'

be easily observed from the formula derived

from orthotropic plate

theory, it

_{is very convenient for practical}

design.

erits:

(i)

It IS consiØerably troublesome to obtain the solutjoi of

orthotro-pic plate with complicated boundary conditions. And the investigation of these soltition