Cycle compression of imperfect steel plates, Part III: Preparations for experimental investigations

Deift University of Technology

Io, STRUCTURES LABORATORY

Cyclic compression

of

imperfect stee1pIates

Part III

PREPARATIONS

FOR

EXPERIMENTAL INVESTIGATIONS

Ir. M.L. Kamin ski

Report No. SSL 323

Second edition, September 1990

Library data

Keywords:

stability, slow buckling, plates, initial deflections, _{extreme loads,}

cyclic loading, repeated loading, viscoplasticity1 _{in-plane compression,}

imperfections tolerances

SSL and the author would appreciate receiving a copy of any work in which the material contained in this work is used or referred to in any context.

SSL and the author assume no liability with respect to any use whatsoever

made of the material and information contained _{in this report.}

Copyright.

(CR) 1990 byM.L. Kaminski, Ship Structures _{Laboratory (SSL).}

All rights reserved. No part of this report may be reproduced without the

prior written permission of SSL or the author.

M.L. Kaminski

Ship Structures Laboratory Delft University of Technology Mekelweg 2 NL 2628 CD Delft The Netherlands Tel 31-(0)].5-786868/6866 Fax 31-(0)15-785602 j-ii

This report is a:part of the work _{on .he'cycIc. cómprssion of imperfect}

plates and describes preparations made for experimental investigations

A series of plate spec imens representing typical plating elements of

primary ship hull struçturé and 'test equipment

are

dêscr thed.

The specimens differ in slenderness; aspect ratio, length, initial deflections amplitudes and patterns.

The different parameters of specimens are determined and regard to their usefulness in further parametric analysis

strength. '

-thickness an4

The spectmen'.s initial deflections

_{are, diScúed in relation to the}

tolerances for maximum allowable plate deformations which are currently

used and which are proposed by the author

A characteristic of the mild steel is given with particular reference to tth:e _{Vis!cO-.pciastió propertIes.}

Test equipment with, siple- and effective technical _{,iútiñs providing}

proper specImen boundary conditions is descr-bed..

discussed,., in

NOMENCLATURE

unit of

Roman symbols: _measure

A - cross-section area _ni2

A - dimensionlEss constant

-a - plate length _ni

- coefficient of the DTFS (the sine-sine part) _m

b - plate breadth _m

b - dimensionless viscoplastic constant B _{- dimensionless constant}

C _{- dimensionless constant in the author's formulae} c _{- dimensionless constant in LR's formulae}

D _{- viscoplastic constant}

D - plate flexural rigidity

E - Young's modulus

e - base of the natural logarithm

f _{- the greatest distance between gauge and specimen m}

I _{- number of the IPD measurement along plate length}

i _{- index of the IPD measurement along plate length} J _{- number of the IPD measurement along plate breadth} j - index of the IPD measurement along plate breadth

K _{- support rotational stiffness per unit length}

N/rad

k - elastic stiffness _N/rn

k - plate buckling coefficient

k - steel parameter in LR's formulae

--1 _{- distance between two points on a plate}

ni

L - gauge length

m

M - moment per unit length _N

M - number of the DTFS coefficients along plate length

m

-mass

_kg

ni

-

index of the DTFS along plate length

n - index of the DTFS along plate breadth

N _{- number of the DTFS coefficients along plate breadth}

-P

-force

_N

p _{- exponent in a viscoplastic law} p _{- exponent in the author's formulae}

p - probability

q - allowable plate deflection

'n-ROH - the standard upper yield stress _Pa

ReL - the standard lower yield stress _Pa

r _{- radius of gyration of stiffeners acting with assumed}

effective breadth of plating m4

T - load period s t - plate thickness m t

-time

s u - displacement in X-direction _m y - displacement in Y-direction _m w - plate deflection _'n rn/rn/s Nm Pa

p E E p p * p o IC À

p

L' a w w

Subscripts and superscripts:

A

B box cr ci.im el eq h i

j

lat lo long rn max n min mod nom -o -P panel -pl plate -st -up -y -E -- plate slenderness - strain - strain rate - density - radious of curvature

- maximum elastic radious of curvature - rotation angle

- coefficient of elastic rotational restraint - reference slenderness of plate buckling - dimensionless coefficient of viscoplasticity - Poisson's ratio

- stress

- circular frequency

- maximum initial amplitude of initial deflections

- refers to the A-type amplitude of plate deflections - refers to the B-type amplitude of plate deflections - refers to square box girder,

- critical, - cumulative,

elastic,

- equivalent, - harmful,

- index of the lFD measurement along plate length, - index of the lFD measurement along plate breadth, - lateral,

- refers to lower yield point, - longitudinal,

- index of the DTFS along plate length, - maximum,

- index of the DTFS along plate breadth,

- minimum, - refers to a model nominal, refers to refers to refers to plastic, refers to stiffener refers to yield, refers to

zero strain rate,

load,

panel plate

upper yield point, strain m/rn m/m/s kg/rn3 m m rad rad Pa rad/s mm v-i unit of

r Others: condition Abbtéviat Ions.: .MTS NEN., SL7 s.sL

- value of x fora given condition

- absolute: vàiue. of4 x

- the mean value of x - tiinederivatj've 'of

uncttonal relation of x and y

- unit of measure of x

- number bibIógraphi'c reference

DTFS - the Double Trigonometric Fourier Series

FEM - Finite Element Method

IPD - Initial Plate Deflections - w0(x,y)

IPC - Initial Plate Curvature

LHF - Large Horizontal Frame

- Lloyd's Register of Shipping - Metal Test System

- Dutch Norm (Nederlandse Norm) - Sea Land 7 - containers ship type - Ship Structures Laboratory

CONTENÏS.

Page

S,,O,'Y

iv

NOMENCLATURE H,

..

DEFINITIONS

REFACE

₁

:.

.

INTRODUCTION

....;..o;...

_i

11

FIELD OF I

STIGATIONS

_i

12

_{CHARACTERISTIC OF WELDED STEEL SmP}

GRILLAGES

.:.

_10.

13

ThE AIMS

₁₃

14

CHOICE OF SPECIMEN CONFIGURATION

₁₄

15

CHOICE OF SPECIMEN SLENDERNESS

₁₆

16

CHOICE OF SPECIMEN BOUNDARY CONDITIONS

₁₆

.1,7.

CHOICE OF REPRESENTATiVE IMPERFECTIONS

.

₁₈

SPEd

I

NS

_..

2.1.

GENERAL

₁₉

2.2.

_{STEELCHARACTERISTICS}

_.20

23.

PREPARATION

...

₂₇

2.4.

DIMENSIONS

..

_{. 27..}

INITIAL DEELECTION$ OF SPECEs'UNS

29

3.!.

GENERAL.

_..

...

...,.,..

,..

.. ....

..

32

IN11tODUCTION OF INITIAL DEFLECTIONS INTO

SPE(II'IENS

..,... ...

o...,...

₃₀

33

DFTERMINATION OF INITIAL DEFLECTION

PARAMETERS

₃₁

34

ALLOWABLE DEFLECTION S

₃₆

:3.5.

CONCLUSIONS

..

_...

₄₂

TEST EQUII'MENT ...

_...

.

44

41

SUPPORTING STRUCTURE

₄₄

4.2.

LOADING SYSTEM

..

. .. .,. 45.

.5.

... ...4

ACKNOWLEDGE I NTS

.. ..

. .,... ..

sé

xli

viii

DEFINITIONS

FIgure. 1 shows an example, of welded steel ship grillage and defines the

terms for its components which are used in the present work _{Figure 2}

shows an example of the initial plate deflections and defines the A- and

B-type amplitudes

The other ternis are defined as follows

initial - refers to an unstraightened structural element which

is already a part of a completed section, block or

whole

_{structure before putting the structure into}

service (before the launching, in the case of marine

strücturés).

- plating

- _{longitudinal s.tiffñer}

3 - transverse stiffener

t- -\ n,,;

J' t,y'

J.' -. J t'. - S-t' .3

-J-g 2 Example of lFD with A- and iB-type amplitudes indicated

'L

¿J

-"J

r;/ '

PREFACE

There is a continuous need for simple design formulas which define, for

instance, the mean strength, the lower or the upper bound of the strength,

or, _{in general, the strength with a given cumulative probability of}

exceedance.

Design formulas which incorporate within wide spectra all the important factors influencing a strength are particularly needed, because such

formulas may be used in the reliability analysis, providing the factors' distributions are known.

Simply design formulas with imperfection factors incorporated are very useful in assessing the tolerance limits for maximum allowable

imperfections according the deterministic criterion (or the reliability

criterion, providing the load spectrum is known).

Buckling is, after brittle fracture, the most crucial consideration in

the strength design of ship structures. In spite of substantial research

and our present understanding of collapse behaviour in welded steel grillages, fatal damage to ships caused by buckling still occurs. The damages usually result from dynamic ship response in rough seas, and

occur in the deck plating of fast containers ships or in slender warship

hulls. An example of damages reported in [1] may be here given. In many

cases, with the exception of warships, the damages might have been avoided by proper operation. The reason for damages is the occurrence of extreme

compressive loads within a finite period of time. The errors lie in the

bad design of properly operating ships - underestimation of design loads

or in the improper operation of properly designed ships - exceedance of

design loads. However, irrespective of where the error lies, the problem

of the estimation of reserve strength which is due to the instantaneous character of extreme loads still exists. The problem may be also put in

reverse: the estimation of a safe time duration for a given extreme load amplitude.

Beside damages caused by a single incident there is also the possibility

of cumulative damages. Ship structure is subjected to random service loads of which a certain number are extreme. Each of these loads can

cause the local yielding of steel and produce finite permanent

deformations and residual stresses. Such repetitive and alternate

phenomena might reduce the plate strength and cause cumulative damage.

A simple design formula may be derived from theoretical, numerical or

experimental studies.

The very powerful Finite Element Method still has its shortcomings. On the one hand, regarding nonlinear problems of complex structures, there

is still research being done on reliable strategies to provide realistic

solution independent of numerical instabilities. On the other hand, current FEM programmes are more check- than design-oriented tools.

Furthermore, good practice recommends several calculations with increasing

structure complexity and increasing allowance for different

nonlinearities. If we add to the above problems those connected with: a

- I.

magnitudes and patterns for imperfections, we conclude that a performance

of the nonlinear FEM calculations is a time-consuming art in itself It

follows that usefulness of FEM in a preliminary design of complex

structúres is, limited'.', .. ''

'''

.;' :" r .. The same may be .stated"n regard. tò 'añalytical and experimentaL methóds.

There are differences in the limitations, advantages and disadvantages of these method, but they should be treated as equally difficult For

instance, theoretical methods are limitad by, 'the existence. o- analytical

- 'solutions and strûcture.. cornplexity.

Summarizing, in order to develop simple design formulas there is a need for apart theoretical, numerical or experimental research on the strength

of parametrically different structures

1.

INTRODUCTION

1.1.

FIELD OF INVESTIGATIONS

This work is concerned with the behaviour of structural elements _which are compressed by prescribed loads. Thus, problems of the determination of loads, and the many elaborations available in the literature are not

addressed in the work. Hereafter, for the sake of further _{consideration}

of the response prediction, only a short characteristic of loads _{is given.} A review of many forms of ship loads and an extensive list of literature

is available in the ISSC reports [2].

Discussions of the preliminary results with other researchers have clearly indicated a necessity for the a priori definition of the terminology _used.

The main reason for this lies in fact that the work refers _{to relative}

terms as: static, quasi-static and dynamic.

The danger of misunderstanding caused by the use of a relative term can

be spectacularly illustrated by an example which is taken from the physics of superconductors,

where the term high temperature refers

to the

temperatures above minus 160 degrees Celsius!

The notion load requires some comments, before the terms: static,

quasi-static and dynamic are specified in the context of the _{present work.}

Generally, this notion is related to the notion of the system, and stands

for the action of the environment on the object. Hence, the meaning of

the notion load is also relative. It depends on a definition of the system and its division into the environment and the object.

In order to illustrate the above, let us give two examples connected with

the subject of the present work. For the sake of simplicity, _{aspects of}

the

system modelling and aspects

related to the

reaction of the

environment on the object response will be omitted.

First, consider the system consisting of a ship hull structure as the object, and the field of gravity together with the still sea as the environment. In this case, under the notion load will be understood: hydrostatic forces and weights.

Then, consider the system consisting of a stiffened plate being a part of

ship hull structure as the object, and the rest of the _{structure as the}

environment. Now, the in-plane forces or even the in-plane displacements

may be understood (defined) under the notion load.

Now, the terms: static, quasi-static and dynamic will _{be specified with} reference to the response of the object with _{one degree of freedom. Let} this object be characterized by the following constant parameters: m -mass, p - damping and k - stiffness. Further, if _{necessary, assume that} the object is cyclicly loaded with a force of the amplitude P and the frequency wi,. Note, that the choice of discrete object with one degree

of freedom implies that the effects related to the stress-wave propagation will here be not illustrated, because they are obviously beyond the scope of the present work.

The response of such an object is described by a solution of the equation of motion. However, under certain conditions, _{depending on the values of}

the object and the load parameters, the response may be described with sufficient accuracy for practical applications

_{by a solution of the}

reduced equation of motion. The terms: static, _{quasi-static and dynamic} are associated with the level of this reduction:

d2u{t) du(t) m dt1 + dt + k u{t) P sin Wp t (1.1.1.) static response kinematic response dynamic response

Consider now conditions for both reductions. _{Regarding a system without}

damping, the behaviour of such a system can be described by the static

response, providing that the load frequency is much lower than the natural frequency:

C', -

_(1.1.2.)

This

simple condition is also valid in the case of the continuous objects, if it refers to the lowest natural frequency.

Figure 3 illustrates how the dynamic response shifts in the frequency domain, depending on the considered problem.

_This

_{shift is, of course,} mainly caused by different values of stiffness _{parameter - from the} heaving stiffness of a floating object _{to the stiffness of an elongated}

steel plate.

The choice between static and quasi-static _{response, or the neglect of}

the damping effect in dynamic response, depends on values of the damping parameter and the load frequency (load rate).

The same loads may cause different responses in different objects and the

same object may respond differently because of the action of _different

loads. Hence, only on the basis of the comparison of the object parameters

with the load parameters, is it possible to predict the type of object

response.

oblect _{w, - load frequency [rad/s]}

.1 1. 10. 100.

pitch of rigid ship _{dynamic response}

vertical bending of ship hull structure

in-plane loaded static response

ship plates

100. 10. 1. .1 .01

T - load period [s]

Fig. 3. Shift of the dynamic response in the frequency _{domain for}

Furthermore, an object response induces internal forces and deformations, which form the loading for. its sub-objects. In general, the type of object

response will differ from the type of sub-object response, because their parameters differ. Such situation is inherent in the object of the present

work. The response of a ship, which is going through the rough sea is

certainly dynamic. But as will be shown, the internal forces induced in

the primary ship structure are not able to cause dynamic plate response.

Dynamic or non-dynamic response?

As was stated, in order to predict the response type of the plate, it is

necessary to compare the

load frequency spectrum with the natural

frequencies of plate vibrations.

First, the plate vibrations will be discussed. Cyclically in-plane loaded

plates may vibrate longitudinally or laterally.

In so far as longitudinal vibrations are obvious, the lateral vibrations

require some comments. Initial imperfections in the plate _{give rise to}

bending moments that excite lateral motion. The plate vibrates laterally at high amplitudes when the loading frequency is twice the natural bending

frequency of the plate: each time the plate bows out to one side or the

other, the axial loading force reaches its maximum and produces bending

moments. More exact analysis shows that this phenomenon may also take

place when the loading frequency is only half the natural

bending frequency. However, in such a case the load amplitude should be close to

the critical load. An extensive treatment of this subject is _{given in a}

book by W.W. Bolotin [3].

The calculations are illustrated in an example of a simply supported plate, because this plate has the lowest first natural frequency. The

other plate parameters are as follows: density _p 7850 [kg/m3], Young's modulus E - 206 [CPa], Poisson's ratio _{0.3, length a = 2.4 [m], breadth}

b 0.8 Em] and thickness t 16 [mm].

The first angular frequency of-the. longitudinal plate _{v-ibratlons is-:}

1ong

(1.1.3.)

"1ong 6700 [rad/s]

and the half of the

_{first angular frequency of the lateral plate}

vibrations is: 01at/2 = a,J p

2(1

1 J I

Et1

Z J

co18 /2 - 210 [rad/s]

_Iexp1e

It is clear to see that for the case considered here, the axially induced

lateral plate vibrations have lover first natural frequency _{than the}

longitudinal plate vibrations. This is also true in the majority of other

cases, because comparison of equations 1.1.3

and 1.1.4

gives the

condition:

b <1fb

a

1ab

for: _{Wjong <} /2 (1.1.5.)

which required an extremely short or an extremely long plate ( a/b > loo or a/b < 0.01 for b/t=50 ).

Now, the frequency of in-plane loads has to be considered. A good recent

source of information about such loads are the reports of the SL-7

research programme [4].

Figure 5 shows a spectrum of the frequency of the _{longitudinal stress.}

The spectrum was evaluated from the records of longitudinal strain which

was continuously registered during extreme operational conditions of an

SL-7 ship: speed 31.5 Kt, Beaufort 9, bow seas. An example of such a

record is shown on Figure 4.

As can be seen the stress frequencies are grouped around two values. The first group, with values in the order of 1 [rad/s], _{represents stresses}

which are mainly produced by the overall hull _{vertical bending response}

to wave excitation.

Whipping - the two-noded vertical hull vibration caused by slamming is responsible for the second group of stresses with frequencies _{in the}

order of 5 [rad/s].

Hence, the natural frequencies of plate vibrations _{are two orders of}

magnitude higher than the frequencies of the in-plane loading. _{This leads} to the following conclusion:

the compression of ships' plates is non-dynamic

Coing outside the scope of the present wörk, note that this conclusion

is also valid in the case of the plates which are directly and laterally loaded with hydrodynamic impact, since the loading rates, as measured by

pressure rise time, may be only as much as 10 times those for whipping

[8].

Quasi-static or static response ?

In order to answer the question _{as to whether the non-dynamic plate}

response should be treated as static or quasi-static, it is necessary to compare the frequencies (strain rates) of in-plane loading already

identified with values of damping (visosic) coefficients.

2.

H

INTERVAL 13

441e:

IWrÈRVAI 16

-APE 143 INIERVAL - 12 RUN 349lic LCAN IAJ'L I-Il INÍLRVAL 48

236T' 12'- r

APE 143 INTERVAL O' 7693

RUN '9l tc:L.EAN - irAPE1:4J IÑTERVAL 48

- 12:

RUN 359 MCLEAN

e

-12-- ,

:

361 MC LEAN TAI& 1.11 INTLRVAi. 68

222e:

ø1 lic LEAN- T 195 .INTERVAL.1

'' RUN 405rFic LE ÎAPE'i4.1rffERVAL 6:

(The vertical scale is stress which is positive in deck tension corresponding to a hogging moment Zero stress corresponds to the sample mean i (kpsil 7 LMPaI The horizontal scala ia time In al].

caseß the duration of the plotted time history Is about 76 [sJ)

Fig,. 4 _{Example shøwing wave- and' siam-indùced longitudinal vertical}

midship bending strains.

circular frequency [rad/s] Fig. 5. Example öf Ioad-frequency spectrum.

First, the rates of the in-plane loading will be considered. The whipping strain is of a significantly higher frequency than the wave bending strain (Figure 4), therefore the rate of their superposition is mainly effected by the rate of the more frequent strain component - the whipping strain.

Approximation of this strain by one harmonic (t) - ¿ sin wet, gives the strain rate ¿(t) w cos wet, with the maximum of _max cw. However,

because the strain rate is equal to zero at the maximum strain, it is more

representative to use the mean strain rate between the minimum and the maximum whipping strain: 7 2/ir which is equivalent to the

saw-formed approximation of the whipping strain.

For observed whipping strain of about 660

[pstrain] (Figure 4) the

corresponding mean strain rate is about 2100 [pstrain/s]. Strain rates due to slamming per se, as measured by pressure rise time, may be as much as

10 times those for whipping. Table 1 shows approximate mean strain-rate values in the steel hull structures of ships, according on their source

[5,6 and 7].

TABLE i

Approximate strain rates in the steel hull structures of ships.

2 000 000 steel against steel: dropped objects

50 000 ice against steel: ice impact

10 000 fluid against steel: slamming, sloshing, bow impact

2 000 whipping

300 wave bending

0.1 loading and unloading of a ship, thermal expansion

Now, the damping will be considered and compared with determined loading rates in order to check whether it may have a considerable effect on the non-dynamic plate response. From the strength point of view, the damping is beneficial, because it produces an additional resistance of the object to the loading. There are different forms of damping. The main distinction

is between internal and external damping.

The external damping is

associated with a resistance of the environment to a deformation and movement of the object. In the case considered here of in-plane loaded plates, this kind of damping may be certainly neglected.

The internal damping of steels depends strongly on steel type, stress

level and temperature. The case of mild steel at room temperatures is here

considered. The reason for this is that in this case which is realistic

for ship structures, strain rate effects are the most evident.

In the elastic range the internal damping which is associated with small local plastic deformations at the grain borders affects slightly the plate

response (max. 2%).

6

Approximate

mean strain rate Source of strains

values

The situation changes in the plastic _{range where internal damping is}

associated with the viscous properties of yielding steel.

There are different empirical laws which model the viscous property of metals and, before applying one of them, it is _{necessary to carefully}

study its validity range i.e.: type of material; temperature; strain range

and strain-rate range: very low (creep), low or high.

The distinction between the terms .] _{and high strain rate has to do with} the transition from the isothermal to the adiabatic condition of plastic deformations [8]. For steels, this transition takes place _{at a strain rate} of about 10 000 000 [zstrain/s]. Thus, it may be stated that strain rates in ship hull structures are low, because the above estimated strain rates

are significantly lower. But, of course, it does not _{mean that their}

effect on the yield stress is small.

The relations based on over-stress formulation are those most frequently

used in literature. Based on Manjoine's results (1944!) _{[9] Symonds and}

Bodner (1962) (see [10]) have suggested the following empirical relation between yield stress and strain rate, for mild steel at room temperature:

ay/co i + (1.1.6.)

where:

= 207 [MPa] _{- static lower yield stress,}

D -

40.4 [strain/s] _{- coefficient of viscoplasticity}

p =

5 - exponent of viscoplasticity,

Regarding the static lower yield stress, the relation above predicts a 30%

enhancement of yield stress for the strain rate of 0.1 [strain/s].

However, Dow et ai (1981) [11], when considering the ultimate ship _hull

strength came to the conclusion that the enhancement of yield due to the

strain rate is not significant. Their conclusion is based on Campbell's

work (1966, see [il]), who shows that the increase in the _{yield stress}

is proportional to the log of the strain rate between 0.1

_{and 1000}

[strain/s], but the increase in yield stress below 0.1 [strain/s] _is

small.

Thus, there is some discrepancy between these models. _{Furthermore, both}

models were

developed

_{for high-carbon mild steels}

_{and mainly for}

high-strain-rate applications.

Therefore, the author decided to carry out additional _{experimental work} in order to determinate the viscous properties of the _{modern low-carbon}

mild steel in the strain rate range

up to 0.1 [strain/sI at room

temperature [12]. Results and conclusions in addition to those presented below are given in Chapter 2.2.

It has been found that the steel is more strain _{rate sensitive than the} existing models predict and than is generally recognized.

The following empirical relation between yield stress and strain rate for mild steel at room temperature has been found:

(1.1.7.)

where:

205 [MPa] static lower yield stress,

(by zero strain rate, see: [30])

b 14 viscoplastic constant,

p -

o.o36 _{- coefficient of viscoplasticity,}

o.000l

[strain/si - the standard strain rate.

Regarding the static lower yield stress, the new relation predicts a 35%

enhancement of the lower yield stress for the strain rate of 0.1 [strath/s].

More spectacular is a 95% enhancement of the upper yield stress, regarding the static lower yield stress, for this strain rate.

Such enhancement of yield stress may give significant strength reserve and therefore it should be concluded that:

the compression of ships' plates is kinematic

In order to more precisely define the area of present investigations, the

author has made an attempt to classify the buckling phenomenon into several types. Table 2 shows the result. For each type of buckling the table gives a load-time diagram and a buckling mode in _{an example of an}

axially loaded strut. Two important values which are decisive factors in a given buckling type are indicated in the load-time diagrams: a value of

the load amplitude in relation to the static critical Euler load and a

duration of the load in relation to the period of the lowest natural

vibrations of a strut. The thin lines and the arrows on the buckling modes

indicate an initial strut deflection and a buckling tendency,

respectively.

The table also

gives maritime examples

of loads and

structures. Further the table is self-explanatory.

The classification made may be extended by addition

_{of the type of}

material response involved during the buckling _{process, e.g.: elastic,}

plastic or visco-plastic.

Thus, according to such a classification the present investigations are

focused on:

,P 4._

DYNAMIC BUCKLING

Pise Ixicklir Step bicklirg

Hit -

Laiir

strucb.ire deck

Classification of the icJdir ithenczinon

Vibìation hicklir Cyclic k1i

òthùi 1oade

t Ióads

friu

loadir 'ciii

waer

of a ship pressure

.]ation

SLOW BUCKLING

dk

Higher-order

ide

L

o w e r - o r d e r

in

o d e

:pLp.

P.

_

deater.

tank arks P - load,

cr - Euler' s load, T -

period of the lowest

natural frequency of a strut, t - time

t

Pulsé iÖd

Step loàd

Vration lad

Qscil1atory1oa.

frOEn mi ile

fiu

1aithz

fran unbalarxd

frczn wave

1.2.

_{CHARACTERISTIC OF WELDED STEEL SmP GRILLAGES}

Figure 1 illustrates a sample of welded steel ship grillage and defines

terms which are used in the work, among others:

Dane]. - longitudinally stiffened plate, between transverse stiffeners;

elate - single plate between longitudinal and transverse stiffeners. The geometrical parameters most strongly influencing compressive strength of longitudinally stiffened panels are the plate slenderness ratio b/t and

the slenderness ratio a/r of longitudinal stiffeners, (r - radius of

gyration of longitudinals acting with assumed effective breadth of

plating).

Numerical investigations carried out by Webb and Dowling [13] have shown

that, depending on these parameters, statically compressed panels may

collapse in three different modes, which are indicated in Figure 6: - squashing (zone 1)

Failure occurs by the yielding of the panel cross-section before extensive plastic buckling occurs. For mild steel, this zone

correspond to:

b/t < 60 and a/r < 65 - overall flexural buckling (zone 2)

Failure occurs

in an overall mode

following failure

of the

relatively slender stiffeners (column-like buckling of stiffeners). For mild steel, this zone correspond to:

a/r > 65 for any b/t

however, certain combinations of higher panel and plate slenderness greater than 60 fall In zone 3.

- plate buckling (zone 3)

The yield stress of the steel exceeds the critical plate buckling

stress, so

that the behaviour is mainly controlled by plate

buckling. This zone can be divided into sub-areas according to the way buckling develops, but it is interesting to note that, despite the different final buckled modes which occur, there appears to be

no significant influence of mode on ultimate strength. For mild

steel, this zone corresponds to: b/t > 60 for any a/r

However, certain combinations of higher panel and plate slenderness greater than 60 fall in zone 2.

o.

ob

0.7

0.6

0.5

-p _{4 I 4}

-Fig. 6. Ultimate loads and collapse modes of longitudinally stiffened

panels.

In order to predict the most probable failure mode for ship panels, the

data from the figure above will be compared with practical values of bt/t and a/r.

A survey of midship deck and bottom designs in existing ships indicated

practical values of:

- the slenderness ratio a/r of longitudinal stiffeners in the range 10 to 120, with a mean at about 30. (96 panel designs; 15 ships of different types, with lengths between 83 and 236 [ni], [14]);

- and the plate slenderness b/t in the range 20 to 90, with a mean at about 45. (as above [14] and 130 tankers built between 1973 and 1986 with lengths between 66 and 390 Em], [15]).

Both distributions are shown in Figure 7. As can be seen, designs of stocky panels dominate. This is clear, because panels in the strength parts of modern long ships have to withstand large in-plane loads which

are caused by vertical bending of the ship's hull. In such cases, panels are usually over-dimensioned from the point of view of local strength.

Comparing the information included in Figure 6 with that of Figure 7 it may be concluded that for panels which are here considered:

all three collapse modes can be expected in practice,

squashing is the most probable mode, about 90% of all designs, in about 90% of all cases, when a/r < 65, plate behaviour may be used as a determinant of panel behaviour.

11

1.0-ZONE

i

próbablllty' t)

30-Fig 7 _{Distributions of the plate ande the panel slenderness in existing}

deck and double-bottom structures

1.3.

THE AIMS

The main aim of the experimental investigations is to provide an empirical expression defining the plate compressive strength as a function of cyclic load, material and geometrical parameters. Having such a relation, called

buckling life, it will be possible to determine a cumulative damage and

to asses

the tolerance limits

_{for plate distortions based on the}

deterministic criterion.

The aims of the part of the work here discussed are as follows:

to design a series of plate specimens representing typical plating elements of primary ship hull structure,

to design test equipment with simple and effective technical

solutions providing proper specimen boundary conditions,

to determine and to discuss different specimen parameters in respect

to their usefulness in further parametric analysis of the plate

strength,

to give a characteristic of the material with particular reference to the visco-plastic properties,

to relate the specimens' initial deflections to the tolerances for maximum allowable plate deformations.

From the beginning, the present report has presented the plate as the

subject of

the investigations. However, at

an early stage of the

investigations different specimen configurations _{were considered. The} choice of the plate as the specimen form will be discussed _hereafter.

1.4.

CHOICE OF SPECIMEN CONFIGURATION

Because the work refers to imperfect structures, it was decided that the the specimens should be full scale. The reason for this was the fact that it is difficult to introduce such residual stresses and initial deflection

in in-scale specimens as those which would have the _{same effect in real}

structures.

Further, the selection procedure was as follows. Two testing machines were

considered: the i _{[MN] Metal Test System (MTS) testing machine and the} 6 [MM] Large Horizontal Frame (LHF) testing machine. _{Both machines are} electro-hydraulic and are mainly used in the Laboratory for fatigue and

fracture testing. Three forms of the _{cross section of specimens were}

considered: plate, square box girder and panel (or grillage). Thus there

were six possible tests to consider. For each case the maximum possible plate dimensions were calculated, based on requirements which derive directly from the scope of the present investigation: _{the behaviour of}

shiD grillages under cyclic extreme in-plane loading. They were:

- the geometry of specimens should correspond to the geometry of

grillages which are a part of ship strength hulls (Figure _7), i.e. the possibility to test specimens with low plate _slenderness ratio b/t 40;

- the testing machine should be able to generate extreme loading, i.e. should have the capability to produce the full plastification of the specimen cross-section.

Table 3 shows how these two requirements effect the maximum possible plate

dimensions,

depending on the

testing machine and the

form of the

specimens. Case "e" is rejected because case "a", which is _{easier in} realization, gives acceptable plate dimensions. Cases "b" and "f" are rejected because the plate dimensions _{are similar to those of cases "c"} and "g" which are more realistic for ships' _{structures. Cases "e" and}

"d" are rejected because the plate dimensions are too small.

From the two remaining cases, "a" and "g", the first was selected. There were two reasons for this.

First of all, the avoidance of the introduction of too many parameters in

a novel

experiment is good practice. The subject of the present

experimental work is certainly novel: there is a lack of knowledge about the behaviour of imperfect plates which are repeatably compressed by loads with extreme amplitude. To date, there has been no data in literature on similar experimental work.

The second reason for selecting "a" was the fact that the _{testing of}

stiffened plates is much more labour consuming and because of a reduction in the Laboratory staff it would be not accepted.

TABLE 3

Maximum possible plate dimensions

Testing machine Maximum possible plate breadth

Panel

Capacity Plate Square box

minimum optimum Name

rnax bpiate bbOX

bpei

Remarks:

a,b,c,d,e,f,g and h denote possible tests.

The resulting nominal plate thickness is given in brackets.

bpiate J

!miìx I

Imin bbOX bpiate

p

i

(n±l)a+n

(b/t) Imin 40 ; a, 235 [MPa] ;

a = A/bt

0.8

n ± 1 - number of stiffeners, "-" and "+" means "without" and "with" stiffeners at both panel sides, respectively;

- n - i = 2 is the minimum acceptable configuration of thepanel,

n + 1 6 is the optimum configuration of the panel.

To summarize:

the plate as a specimen form and the 1 F MN] MTS testing machine were

chosen.

The choice of plate as the form of the specimens has effected _{the whole} of the experimental work in a two-fold manner. First, it has forced the

elimination of residual stress effects from the study, and second, it has introduced a new parameter: plate boundary conditions.

15 MN mm mm mm mm a b c d MTS 1 412 206 192 132 (10) (5) (5) (3) e f g h LHV 6 1010 505 470 322 (25) (12) (12) (8)

1.5.

CHOICE OF SPECIMEN SLENDERNESS

Test plates were designed with various slenderness b/t falling within the

practical range (see Figure 7). The change of plate slenderness was achieved by change in plate thickness, keeping the breadth b - 412 [mm]

constant. The use of three nominal plate thicknesses: lO, 8 and 6 [mm] delivered three groups of specimens with the nominal plate slenderness b/t equal to 40, 50 and 70, respectively.

Besides the plate slenderness parameter b/t, there are also two other

plate slenderness parameters in use:

in which the plate buckling coefficient k is a function of the aspect ratio and the boundary conditions and will be discussed in Appendix I.

The slenderness parameter A defined above

is called the reference

slenderness of plate buckling. It is a means to facilitate the comparison of results obtained from tests on plates with different geometry, material properties and boundary conditions.

Therefore, in the further discussion, the reference slenderness is mainly used.

1.6.

CHOICE OF SPECIMEN BOUNDARY CONDITIONS

A summary of plate boundary conditions, which

_{were chosen for test}

specimens, is given in Table 4. Two of the boundary conditions, both on

long (unloaded) edges, need some comments: in-plane edge deformation normal to the direction of applied load and out-of-plane _{edge rotation}

parallel to the direction of applied load.

The unrestrained (free) in-plane edge deformation was chosen, because technical realization of two other possibilities, _{restrained and}

co-strained edge deformation, is difficult. The author is aware of the facts

that both rejected conditions are more realistic in the case of plates being isolated from a panel, and that the condition has a significant

effect on plate strength.

16 and (1.5.1.) t.J E

l.05/f

ß (1.5.2.) vO.3 where:

- elastic critical stresses:

ir2E

k

t2

I E J

(1.5.3.)

For example, the numerical analysis of square plates (b/t-67, a11/t...O.3)

made by Jazukiewicz has indicated that the change in the boundary

condition of the plates from unrestrained to restrained gives a 15%

increase in the compressive strength.

The condition of out-of-plane edge rotation has also a significant effect

on the plate strength. For example, numerical analysis of rectangular plates (a/b-.3, bIt-67, a11/t-.O.3)

made by the

same researcher has

indicated that change in the boundary condition of the plates from

unrestrained (free) to restrained (fixed) gives a 15% increase in the compressive strength. Either condition is realistic. The unrestrained

(free) condition was chosen, because it allows for larger plate deflections, and thus it was intended to enhance the accumulation of

permanent deflections.

The membrane strains parallel to the loaded edges were restrained by friction between the traverses and short edges. This could induce a

transverse membrane stress in the vicinity of the loaded edge as large as wc. The consequent biaxiality in the region of the loaded edge may affect

the plate strength. However, this effect also occurs in real structures where transverse frames or bulkheads are welded to the plate. In some

measure, therefore, this test condition simulated the action of plates in a ship [16].

Of course, the best manner of modelling plate boundary conditions would

be to test panels in place of testing plates.

TABLE 4

Specimen boundary conditions

In-plane displacements Out-of-plane displacements

Specimen

parallel normal deflection rotations

edge

u V

w

9y

lower fixed

short fixed fixed fixed fixed

free,

upper parallel co-strained

long free free fixed free fixed

1.7.

CHOICE OF REPRESENTATWE IMPERFECTIONS

Initial deflections

A review of IPD and their tolerances is given by the author in [17 and 18]. The modal analysis of initial plate deflection has shown that the

one-half-wave mode is always significantly represented in both directions.

The effect of this mode on the strength of plates under uniaxial

compression depends on the aspect ratio a/b. In short (square) plates

( a/b < 1.4 ) the mode has a harmful effect on plate strength, whereas in

longer rectangular plates

it has a beneficial effect.

In order to

incorporate this aspect in the present work, plate specimens were designed with two nominal lengths: 400 and 1200 [mm], and one-half-wave pattern of

initial deflect ions.

However, three rectangular plates (one for each plate thickness) were designed with the three-half-wave pattern. The reason for this was to ascertain whether this pattern, which conforms to the elastic buckling

mode of the rectangular plates, has a similar harmful effect on the plate strength as the one-half-wave mode in the case of square plates.

Different values for the A- and B-type maximum amplitudes of initial

deflections have been chosen in order to study their effects on the

load-carrying capacity. It was intended to introduce initial deflections with

the amplitudes close to and higher than the mean value for newly built

ships [18]:

ItLBI/t kB (b/t)25, where kB = 7.7.10-6 (1.7.1.)

IWAI/t

kA (b/t)2 , where kA - 4.2l0 (1.7.2.)

One l0[mm]-thick square specimen was designed flat in order to find the

load-carrying capacity in pure squashing.

Residual stresses

The best way to include residual stresses is to test stiffened plates i.e. panels. Since, in this case, introduction of residual stresses and compatible to them initial deflections occur in the same way as they do

in the real structure. In the case of plates there is no simple way (known to the author) to do this. Some sophisticated methods may be invented but it seems that "the game would be not worth the candle", because it would

be probably cheaper and simpler to test panels.

To summarize:

The residual stresses, however. although important have been

excluded from the scone of the present work.

Effects of residual stress on the plate strength have been reviewed in

several papers[l9,20 and 2-1]-.

SPEC

2.Ï. ;GENERAL

- number;

test ctype:

This chapter, first, gives the steel properties then describes the way

the specimens are prepared and finally gives detailed data

on the dimensions of the specimens

For the sake of convenience, specimens are identified by a reference code such as 40R3C denoting nominal b/t='4_Q, ectangular, -half-wave pattrn of initial deflections and cyclically tested More details are given in Table 5

TABLE 5

Re renco code for ècimëns

The elements of the code denote

geometry data:

- noiniral plate slendèrness b/t:

- aspect ratio;

- square specimen, a/b - rectangular specimen, - initial deflect-ion-data

áimóst flat specimen one -half-wave pattern three-half-wave patt1eri.

slow moñötônic

fast moflötönic

cyclic.

specific dáta (optional):

40.... 50.... 70.

-Hi-Lo transfer of cyclic1oad'.aiip1itude - specimen equipped with strain gauges

- specimen covered with stress-coat lacquer L

2.2.

STEEL CHARACTERISTICS

Manufacturer

The specimens were designed in mild steel - the material which is most often used in .sh1pbui1ding.

The actual yield stress of a mild steel is usually higher than the

required one which is 235 [MPa] Its mean value and the standard deviation are 277 and 22 [MPa], respectively [22] Thus, in order to ensure that the

testing machine vili be able to generate average stresses in specimens

equal to yield stresses,

rolled steel plates were ordered with an

additional clause regarding a maximum acceptable value of yield stress

equal to 280 [MPa]. . . . .

Roiled steel plates were delivered by the Hoogovens 'Croep EV, 'Ijrnuiden,

the Netherlands [23]. .

Chemical composition

The chemical composition of the delivered steel plates, as. reported by the

manufacturer [23], and as required in Euronorm 156-80 [.24.]for Fè E 235A

steel, are listed in Table 6. As. can be seen from the table the steel

meets the requirements of the Euronorm.

TABLE 6

Summary of steel chemical analysis

Micro.Structure

As expected, a microscopic study of the te1 gave evidence of. its

ferrite/pearlite structure.., with a grain, size of 8 ASTM.

An additiOnal test was performed in order to check whether the steel structure was affected by the two processes the cold forming and the

annealing (500°C), which were used respectively to introduce the initial

defiections and to relIeve "residual stresses. Three specimens made of: delivered steel, annealed delivered steel, and cold formed and subsequently annealed steel have been analyzed.' As expècted aIi.specimens hadhe same hardness and an unaffactdfèrri-tejpearli-te---s-t-ructurc

023

017

004

004

max min max max

Euronorin 156-80 requirements

007

0.40

0.Qi0;:0i4-

0.00E .0-3---0T06 ---0.028-0.00l-0..i5.7

20

Mechanical properties

The mechanical properties of the steel used,

_{in the direction of a}

rolling, were determined several times: - by the manufacturer;

- in the Laboratory, before annealing, standard tensile tests; - in the Laboratory, after annealing, standard tensile tests; - in the Laboratory, after annealing, standard compressive tests;

- in the Laboratory, after annealing, compressive tests using

different testing velocities.

The tests above differed in: _{test (compressive or tensile), testing}

machine and type of fixing device; specimen geometry and surface quality; annealing and testing velocity. Hence, it is obvious that the mechanical properties determined on a basis of the results of these diverse tests may

differ in their values [25].

A summary of the mechanical properties determined is given in Table 7. The

table also shows

the requirements of the Euronorm 156-80

[24] for

Fe E 235 A steel and the testing strain rates used and recommended in the

Iso

6892-87 standard [26].

The following conclusions have been derived from the table:

- apart from the tensile strength, which is 15% lower than required, the steel fulfills the requirements of the Euronorm for Fe E 235 A steel;

- the values of the tensile strength and the elongation determined

from all tensile tests are practically identical. This suggests that these quantities are not affected by factors discussed above; - there is good agreement between values reported by the producer and

those determined in the Laboratory;

- the elongation obtained is twice the required one. From this and

also the low tensile strength it may be expected that, in general, the steel is ductile and strain-rate sensitive;

- there is evidence of a significant difference between the lower

and the upper yield stresses. Annealing significantly reduces this

difference. The difference is greater in the thinner plate, which

may be attributed to the more intensive rolling process in the case of thinner plates;

- the difference in the results of the two different compressive

tests, "d" and "e", may be attributed to the fact that in _{case "d",}

no alignment device

or bearing block was used which led to

unpredicted bending and thus to local premature yielding of the

specimens.

TABLE 7

Summary of steel mechanical properties

Yield stress

Strain rate

Tensile

tensile

compressive

Elon-Code

_{strength gation}

elastic plastic

upper

lower

upper

lower

Remarks: code "Nx" stands for: N

nominal plate thickness, and x:

-a

- as specified by the producer,

b

- tested before annealing,

c,d - tested after annealing,

e

_{- calculated from the viscoplastic low for the same strain rates as}

in cases "e" and "d",

f

- minimum requirements of EN 156-80 and Iso 6892-87 (strain rates).

- Means ± standard deviations from three tests are given.

- All data refer to the rolling direction, with the exception of the

hardness which was measured on the clean plate surface.

- Tests "b","c" and "d" were carried out in the electro-mechanical

Instron machine using the constant crosshead speed.

- The tensile test specimens "b" and "e" were proportional, 200 mm

long with the length of the reduced section equal to 70 mm. The

surfaces of specimens which were used in case luCut were, contrary to

those used in case "b", machined in order to remove the rolling

scale.

- The compressive test specimens ttdut were in the form of short solid

circular cylinders with a length/diameter ratio of 2. No alignment

device or bearing block was used.

22

6a

255

328

45

6b

17

475

312 ±19

238 ±5

329 ±1

44 ±0

6e

20

475

252 ± 9

221 ±5

332 ±2

43 ±1

6d

5 83

214 ±1

209 ±3

6e

248 ± 5

236 ±1

240 ±4

224 ±1

8a

240

342

47

8b

8

475

303 ± 4

242 ±3

344 ±0

44 ±0

8c

18

475

245 ±15

226 ±6

343 ±1

43 ±0

8d

3 60

215 ±2

212 ±1

8e

248 ± 4

236 ±1

238 ±4

221 ±1

lOa

236

348

40

lOb

11

475

224 ± 1

215 ±0

355 ±0

36 ±0

10e

26

475

232 ± 4

213 ±3

342 ±3

39 ±0

lOd

2

46

201 ±4

196 ±3

10e

250 ± 4

236 ±1

236 ±4

220 ±1

f

15-150 250-2500

235

400

22

all e

Young's modulus

-

206 ±1 [CPa],

hardness

-

56 ±2 [HRb]

Further, a comparison of the results of the tensile tests "c" and the

I compressive tests "e" indicates that:

- for plates of 6 and 8 [mm] nominal thickness:

- the tensile and the compressive upper yield stresses are

practically equal,

- the compressive lower yield stresses are on average 5% higher than corresponding tensile values,

- for a plate of 10 [mm] nominal thickness:

- the above conclusions are also valid providing the lower and

the upper static yield stresses are 5% lower. This may be attributed to the less intensive rolling process applied in

the case of 10 [mm] thick plate.

Now, the viscoplastic properties of the steels used will be separately

discussed.

The viscoplastic properties

The visco-plastic properties of the steel, important from the point of

view of the present investigation, were discussed in Chapter 1.1. It was concluded there that the existing viscoplastic models may not be directly

used for low carbon mild steels. Therefore, the author decided to carry

out an additional

experiment

in order

to determinate the viscous

properties of the modern mild steel in the strain-rate range up to 0.1

[strain/s] at room temperature.

Details of this work are given in [12]. The properties were determined only for a plate material of 8 [mm] nominal thickness. The results are summarized in Table 8 and are presented in Figure 9. An example of the

-stress-strain diagram, obtained during one of many uniaxial compressive

tests made, is shown in Figure 8.

It has been found that the steel is more strain-rate sensitive than the

existing models predict and than is generally recognized.

Regarding the static lower yield stress, the new relation predicts a 20% and 35% enhancement of the lower yield stress for the whipping strain rate of 0.002 [strain/sI and the strain rate of 0.1 [strain/s], respectively.

More spectacular is a 45% and 95% enhancement of the upper yield stress (in other words, delay of an initiation of the plastic deformation), in respect to the static lower yield stress, for these two strain rates,

respectively.

In summary, the steel used is significantly strain-rate sensitive.

Therefore, the plates may have a considerable strength reserve. The more

so in that, usually, during an extreme condition, plates are compressed

a _{rtaJiivelocity, which is far from zero.}

TABLE 8

Summary of the viscoplastic properties

Yield stresses:

l+b- I

in ( cyJ b 14. p10 0.036

The static lower yield stress:

alO _{205 [MPa]}

The standard lower yield stress, known as the yield stress _ROL:

230 Ra [MPa] 248

when:

250 [pstrain/s] 2500

The standard deviation of the lower yield stress: 1%

The upper yield stress:

(stress at the initiation of plastic deformation)

cup

-u-I

;p=l+i4up

_ey J = 0.214 P

=5

The static upper yield stress: - 215 [MPa]

The standard upper yield stress, known as the yield stress R011:

246 R011 [l'iPa] 265

when:

15 e1 [pstrain/s] 150

The standard deviation of the upper yield stress: 6%

The standard strain rate: = 0.0001 [strain/s J

The validity range of both laws: 0.000 000 1 < ¿ [strain/s] < 1.

nominal strain rate

i E4

[fstrain/s]

1 [cui] vertical i O [kNi] compressive force

i [cmi] horizontai 00625 (mmi] endshortening

Fig 8 _{Example of a stress-strain diagram of the steel after an}

annealing, obtained in "static" uniaxial compressftest .

i4

L...

ti

d

î

_Il. ifluIi1'

''

i ii:. I.. .

'H...1w+

ii.-:1;;.

I ff ktftLt 114 i T1 ,ti -I I -I' I i-I 111 .ili.L '

4,

:-I» :'

,t-±'

ffl(q}ft

ill

I '1 -' ,': Tfl1T I i . .

-'-trr

t.i

't

' ./

:

,ti:

j

L -.

lriHH

,

; i t IT f' . iT

hi,

'1 i _11 .41 I'

liiTrlt"l

''

. i i' I ,

J

-Î:rtpT1T;1

.i

,-'

., -:, ¡ r

-H if

L,L h hil tjfI II ij I j

.L-'H

i

:[I!

'i

' j

Ii

' _{1ti Ii! dj:} ti j f HO

iIiI

l JrIIH I

Ililifil

'1I1Ohi1ItiiOiIIii1.

.11OiOIllIOtttlIM.

4 _"i11 n

stress [MPa]

__A

-upper yield stress

-lower yield stress

7

7

I I I I 1.11 I I I I 1111 I I I Il II I I I I 1111 I I _111.41 I I I 141111

6

5

4

3

2

1

logarithm strain rafe [l/s]

Fig. 9. Strain-rate sensitivity of the upper and the lower yield stresses.

Finally, the mechanical properties, which will hereafter be used in a

parametric analysis of the cyclic compressive strength of plates are given in Table 9. It is assumed that tensile and compressive yield stresses are equal.

TABLE 9

Steel mechanical properties used in parametric analysis

O

Nominal Static lower Young's Poisson's

specimen thickness yield stress modulus ratio

mm MPa CPa

6and8

205 10 195 206 0.3 26

400

200

0

2.3.

PREPARATION

The preparation of specimens involved the following steps:

guillotining (6 and 8 sim thick) or band-sawing ( 10 msi thick) to

the following size: 412 [mm] breadth and 1200 or 406 [mm] long,

from delivered rolled plates ( 2 [si] breadth and 3 [m] long); introduction of initial deflections;

annealing, in order to relieve residual stresses introduced in the previous step;

wire-brush cleaning;

grinding of the whole of one plate. surface in order to remove the rolling scale and then covering the plate with stress-coat lacquer

(optional);

machining of loaded sides,

local grinding and polishing and then the mounting of strain gauges (optional);

gluing of smooth steel strips (1 [mm] thick and 12 [mm] breadth)

along both unloaded edges on both plate sides;

local polishing of plate

surface

at places

where the plate

deflection was continuously measured.

The next chapter gives detaiied.data on final specimens dimensions.

2.4.

DIMENSIONS

The real specimen dimensions a,b and t and the slenderness parameters

b/t, ß and A, all with an accuracy of 1%, are given in Table 10. The initial defiections,

being a part of the

specimens' geometry, are

separately discussed in Chapter 3.

---_The_specimens correspond tothe stocky plates, which

are apart of a

primary ship structure (see: Figure 7). Their collapse due to the loss of stability is certainly elastoplastic because the elastic critical stresses

are higher, or close to the yield stress A 1. Therefore, during the

tests, elastic buckling before any plastic deformation is not expected.

The most important parameter from those listed in the table is

the

reference slenderness of the plate buckling A, because it facilitates

comparisons between plates of different geometry, material properties and boundary conditions.

Therefore, the reference slenderness A is used as one of the parameters in the further parametric analysis of the qyclic comDressive strength of elates.

Three nominal plate thicknesses and two aspect ratios gave fOur groups of

specimens with the nominal A = 0.52, 0.67, 0.85 and 1.06. This is, of

course, because of different values of the plate buckling coefficients for square, and rectangular plates, i.e.: k - 6.8 for a/b 1 and k 4.4 for

a/b - 3, which are calculated in Appendix I.

TABLE 10

Summary of specimen dimensions and slenderness parameters

Remarks:

- all specimens were of the same breadth b - 412 [mm];

- ß and A were calculated using the lower static yield stress;

- plate thickness was measured at 5 and 11 positions for square and rectangular plates respectively;

- the standard deviation of plate thickness measurements for one plate was lower than 0.1 [mm].

28

Specimen Thickness Length Slenderness parameter

number code t a b/t _ß A mm mm i 7OS1M 6.0 400 68 2.15 .87 2 7OS1C 6.1 400 67 2.13 .86 3 70R3C 6.2 1200 67 2.10 1.05 4 7OR1M 6.1 1200 67 2.12 1.06 5 7OR1C1 6.0 1200 68 2.16 1.08 6 70R1C2 6.1 1200 67 2.13 1.07 7 70R1C3 6.1 1200 67 2.12 1.06 8 70R1C4 6.1 1200 68 2.14 1.07 9 7OR1F 6.1 1200 67 2.12 1.06 10 5OS1M 7.7 400 53 1.69 .68 11 5OS1C1 7.8 400 53 1.67 .67 12 50S1C2 7.8 400 53 1.68 .68 13 50R3M 7.8 1200 53 1.67 .84 14 5OR1M 7.8 1200 53 1.67 .84 15 5OR1C1 7.7 1200 53 1.69 .85 16 50R1C2 7.8 1200 53 1.67 .84 17 50R1C3 7.9 1200 53 1.66 .83 18 50R1C4 7.8 1200 53 1.67 .84 19 50R105 7.7 1200 54 1.69 .85 20 5OR1F 7.8 1200 53 1.66 .83 21 4OSOM 9.9 400 42 1.28 52 22 4OS1M 9.9 400 42 1.28 .52 23 4OS1C1 9.9 400 42 1.28 .52 24 40S1C2 9.8 400 42 1.30 .52 25 40S1C3 9.7 400 43 1.31 .53 26 40R3C 9.9 1200 42 1.28 .64 27 4OR1M 9.8 1200 42 1.29 .65 28 40R1C 9.2 1200 45 1.38 .69

3.

INITIAL DEFLECTIONS OF SPECIMENS

3.1.

GENERAL

In general,

the compressive strength of plates depends more on the

geometry than on the maximum amplitude of the IPD. Therefore, in practice

estimation of IPD harmfulness implies some problems, which have been reviewed and discussed by the author in [17]. A summary of the author's

conclusions is given hereafter.

The estimation may be done by making the following measurements: maximum Btype deflection

-A use of the maximum B-type deflection WB as a measure of the

harmfulness of the shapes of IPD is recommended when the compressive strength has not to be taken into account.

maximum A-type deflection - WA

A use of the maximum A-type deflection WA as a measure of the

harmfulness of the shapes of lFD is recommended when the compressive strength has to be taken into account. This because the measurement is over a short gauge length, equal to, for instance, plate breadth b, at any point along the length of a plate provides a satisfactory representation of harmful wavelength distortions and localized dents

with lengths in the range of O.5b-l.2b, without including the

amplitude of less significant, longer wavelength distortions.

harmful DTFS coefficient _{- ahi}

The concept of using the ahi coefficients as a measure of the

harmfulness of the shapes of IPD, however, important as it is from the point of view of the explanation of the effect of the shape of

IPD on the compressive strength, has several disadvantages. There

are situations for which the DTFS coefficients may give a misleading

and non-conservative representation of the IPD. The localized IPD (dent) is an example of such an IPD. Numerical and experimental research shows that not

only the

ahi coefficients but also

coefficients representing wave distortions with lengths in the range

O.5b-l.2b have a harmful effect on the compressive strength of rectangular plates. Furthermore,

a check of IPD in regard to

possible tolerances based on DTFS coefficients, which have low

statistical correlation, would be time consuming and would required extensive measurements of IPD using special equipment.

maximum curvature - l/p

In the case of stocky plates, a deflection pattern does not change

with an increase of the compressive load until plastic hinges and

then a local plastic buckling occurs at the positions of the highest

curvature. Therefore, the strength and the failure pattern are determined by thevaluesand.

the

positions of the maximum initial

curvatures, respectively. The problem is how to estimate the value of maximum IPC. One possibility is to make use of the coefficients of DTFS, as it is described further on. However, a check of IPD in regard to possible tolerances based on maximum curvature determined

in that way would be time consuming, and would require extensive measurements of IPD using special equipment. This is difficult to

accept from the point of view of shipyard practice. A second

possibility is, as will be shown, the measurement of maximum A-type

deflection WA over a short gauge length at any point along the

length of a plate, which provides a satisfactory estimation of the initial maximum curvature.

Before further discussion, first the introduction will be given and the

determination of the parameters of the initial deflections will be

described, and then the allowable deflections according to some

representative standards and the author's proposal will be given.

3.2.

LNTRODUCHON OF INITIAL DEFLECTIONS INTO SPECIMENS

Initial deflections were artificially introduced by cold forming. Flat specimens were placed between two rigid frames on a welding bench. The

upper frame had a shape which permitted allowance for plate edge rotation (simply supported). The specimens were bent by means of a hydraulic jack which was placed in the space between the specimen and the welding bench.

Care was taken to ensure that all edges of the free specimen remained straight and lay in one plane. This was obtained by the initial bending

of specimen edges in the directton opposite to the direction of eventual

permanent edge deflections. The initial bending of specimen edges was realized by placing aluminium strips (up to 3 [nun] thick):

- under the specimen, at the corners;

- on the specimen, at the middle of each edge.

Table 11 gives cold forming data for one square specimen from each of

-three groups with the same nominal plate thickness.

TABLE 11

Cold forming of initial plate deflections. Indicatory data

Nominal Initial Maximum Central Permanent

plate deflection central deflection central

Specimen thickness amplitude load at maximum deflection

number of all edges load of free plate

mm mm kN mm mm

30

2 6 -3 30 8.0 3.0

12 8 -2 40 7.5 2.5