plastic design of laterally patch loaded plates for ships

7/29/2019 Plastic Design of Laterally Patch Loaded Plates for Ships

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Marine Structures 20 (2007) 124142

Plastic design of laterally patch loaded

plates for ships

Lin Hong, Jrgen Amdahl

Department of Marine Technology, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

Received 12 March 2007; received in revised form 29 May 2007; accepted 30 May 2007

Abstract

Laterally loaded rectangular plates are used extensively in various marine structures, and they are

often subjected to patch loading during ice action or accidental actions, such as collision and

grounding. Therefore, focus is placed on investigating the resistance of laterally patch loaded plates.

Plastic yield line theory has been adopted in this paper, since considerable plastic behavior is

exhibited. The beneficial influence of the membrane effect during finite deformations is taken intoaccount. The derivation of the roof-top-type patch loading mechanism using work energy

principles is described in some detail. An alternative collapse model, as named double-diamond

pattern herein, is proposed which could reduce the resistance and agrees better with the results from

nonlinear finite element analysis (NLFEA) in plastic bending phase compared to the conventional

roof-top model. Moreover, a plate length restriction factor is introduced to enhance the

applicability of the present formulation when free formation of the collapse mechanism is restricted

by the finite length of the plate. The developed formulae show reasonable agreement with the results

from NLFEA of the plate resistancedeformation relationships. The resistance according to the

proposed formulation is also compared with the recently developed International Association of

Classification Societies (IACS) unified requirements for plating design for polar ships.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Plastic design; IACS; Patch loading mechanism; NLFEA; Accidental loads; Yield line theory;

Load-carrying capacity; Membrane effect; Ship collision

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www.elsevier.com/locate/marstruc

0951-8339/$ - see front matterr 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.marstruc.2007.05.003

Corresponding author. Tel.: +47 73595301; fax: +47 73595697.E-mail address: [email protected] (L. Hong).
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1. Introduction

The stiffened plate is one of the basic structural components for ships and other marine

structures. Usually, fairly uniform lateral load, in the form of hydrostatic or dynamic

pressure dominates in many parts of these structures, so that the performance of the plateunder such a load condition is of significant importance. The lateral load from accidental

actions, such as collision, grounding, is likely to take place over a limited length of the

plate. This is similar to the so-called patch loading, which has been widely studied in ice

design. For rare loads like ship collision and abnormal ice action, significant permanent

deformations should be considered acceptable, provided that no actual collapse takes place

or no fracture occurs. Consequently, the beneficial membrane stretching effect may be

included in the assessment of the resistance of the plate beyond the pure plastic bending

response. For this purpose it is natural to use the plastic design method, as described and

summarized, e.g. by Jones [1,2], which has gained its popularity due to its simplicity for

design purpose. The analysis is simplified considerably when elastic effects are disregarded

and the material is assumed to be rigid-perfectly plastic. The collapse load is determined by

postulating a kinematically admissible plastic mechanism on the basis of yield line theory

and then equating internal and external rate of virtual work.

Recently, the International Association of Classification Societies (IACS) has developed

unified requirements for shell plating for polar ships subjected to ice patch loads adopting the

conventional roof-top-type mechanism model, refer to Daley et al. [3] and IACS [4]. Signi-

ficant simplifications have been made through the development of unified requirements, which

are somewhat conservative. It is the object of this work to reassess the plating resistance by

removing the conservative simplifications in the plastic bending mechanism, and further toinclude the membrane effect at large deformations. Besides, an alternative patch loading mecha-

nism, named as double-diamond model, which shows better agreement with the results from

nonlinear finite element analysis (NLFEA) than roof-top-type mechanism in plastic bending,

is proposed. A correction factor for roof-top-type model is thus employed to adjust the plastic

bending capacity of patch loaded plates. Further, a plate length restriction factor is introduced

when the free formation of the entire collapse mechanism is restricted by the finite length of the

plate. Results from NLFEA are used to verify the usefulness of the proposed formulae.

2. Review of plastic analysis for plates

The prediction of plastic capacity for plates subjected to lateral pressure based on yield

line theory was originally developed for the case of uniformly distributed load. Yield line

theory was first introduced by Wood [5] in the design of concrete slabs and plates for the

case of fully clamped condition. The plastic load-carrying capacity in bending for fully

clamped plates with length, L, and width, b, is predicted by

pc 48Mp

L2z20, (1)

in which, plate aspect parameter z0 is defined as

z0 b

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

b2

L2

s

b

L

0@

1A, (2)

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Mp represents the plastic bending moment capacity of a plate strip, with unit width and

height t, i.e. plate thickness

Mp syt

2

4

, (3)

sy is the yield strength of the material.

Sawczuk [6] included the membrane effect for large deflections. This was further

developed by Jones [7] using an approximate method to estimate the response of beams

and plates under finite permanent deflections, with elastic effects disregarded. The energy

dissipation per unit length of a hinge line for a clamped plate, which satisfies the normality

criterion of plasticity, is established as

D Mp 1 3w2

t2

_ym when

w

tp1, (4)

D 4Mpw

t_ym when

w

tX1, (5)

_ym represents the relative angular rotation rate across a hinge line; w is the transverse

deflection of the plate. Integrating the above expressions over all yield lines, the load-

carrying capacity for uniformly loaded, fully clamped plates is expressed by Jones and

Walters [8] in the following form:

p

pc 1

w2

3t2z0 3 2z0

2

3 z0

!when

w

tp1, (6)

p

pc

2w

t1

z02 z0

3 z0

t2

3w2 1

!when

w

tX1. (7)

Recently, lateral load pattern with finite length or finite height is of great concern and

the prediction of load-carrying capacity has been carried out by a number of authors in a

variety of ways: analytically, semi-analytically or empirically. Based on the plastic bending

behavior of the plates, Daley et al. [3] developed the plate thickness requirement into

practical use by IACS [4], which will be discussed further. Nyseth and Holtsmark [9]

proposed an alternative model for patch loading with a pattern of three parallel hinge lines.

Resistance expressions for loads with finite height and finite length are developed.Hayward [10] investigated the capacity of ship plating subject to loads of finite height

taking into account the membrane effect beyond the plastic bending phase by converting

the loads of finite height or length into equivalent uniform loads, as proposed by

Johansson [11].

3. Roof-top patch loading mechanism

The yield line theory and corresponding collapse mechanism have been proved to be

practical and useful for design of ship plating by Kmiecik [12]. It is generally used for

rectangular plates subjected to uniformly distributed lateral load, but the yield line modelcan also be used for plates subjected to lateral patch load as done by IACS.

The patch loading mechanism model with the shape of roof-top is demonstrated in

Fig. 1 based on the yield line concept consisting of two triangular regions and two

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trapezoidal regions. The plate is assumed to be loaded over the entire width, but only a

part of the length, with the center of the patch in the middle of the plate. Dashed lines

represent yield lines where series of plastic hinges form, and the shaded area represents the

load patch acting laterally on the plate. The short yield lines are located outside the patch

area.

As illustrated, the length of the plate, L, the width of the plate, b, and the span of theload patch, s, are known parameters. The span of the patch may be smaller or larger than

the plate width. The hinge location parameter a and the yield line angle j are unknown, as

is the collapse pressure. The challenge for this mechanism is to identify where the plastic

hinge will form in order to give the least work solution, i.e. to find the hinge location

parameter a and the yield line angle j. It is supposed through the overall derivations that

the length of the plate is sufficient to support developing the yield line mechanism Lba.

Daley et al. [3] introduced considerable assumptions in order to simplify the problem.

First, the length of the mechanism, a, is assumed equal to the span of the load patch s. This

is essentially the same as assuming that the horizontal stiffeners are located at the patch

load boundary which is non-conservative. Later, this non-conservatism is removed byintroducing another simplification into the expression for the collapse resistance. The effect

of the two simplifications counteracts to some extent; the accuracy should thus be

investigated. In the following derivations, these simplifications will be removed.

Regarding the boundary conditions as been discussed by Amdahl [13], shell plating

generally deforms between the supporting members such as frames, longitudinals or

stringers. Provided that these supporting members are capable of supporting the plate

boundary, it is considered that the clamped boundary condition is more relevant than

simply supported, because significant plastic rotations will take place, especially during

finite deformations. The edges are assumed fixed against in-plane inward motion. For

symmetry reasons this is considered reasonable when adjacent panels are subjected tosimilar loading. If this is not the case, the fixity may be slightly overestimated, because

adjacent panels will deform in plane to some extent in order to mobilize the required

membrane stresses at the edges.

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L

Patch loading

s

b

a

p

w

p

w

Fig. 1. Roof-top-type mechanism for laterally patch loaded plates.

L. Hong, J. Amdahl / Marine Structures 20 (2007) 124142 127


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Normally, shell plating can deform extensively before rupture takes place. Local finite

deformation is acceptable provided that this does not compromise the overall strength or

watertight integrity of ships. This is evident when ships suffer from collision and grounding

where large deformations occur without the structure suffering from rupture [14].

Permanent deformations of one times plate thickness are typically allowed in the design ofbow shell plating subjected to slamming loads [15]. This is believed to give enough margins

against possible rupture of shell plating. However, for the moment, there are no universal

acceptance criteria for permanent deformation for shell plating. Wang et al. [16] have made

some efforts to summarize the state-of-art criteria of allowable deformation for shell

plating. Frame spacing, plate thickness or fabrication tolerance may be chosen as the

governing parameters for allowable deformation. Notwithstanding this, the plastic

membrane behavior will be employed herein to evaluate the structural response of plates

under patch loading, especially under the situation of extreme or abnormal ice action and

accidental actions such as ship collision.

4. Structural response of plates under patch loading

The plastic design and yield line theory are based on rigid-perfectly plastic material

model; the elastic part of the plate deformation is disregarded. The yield lines are formed

by a series of plastic hinges; all plastic deformations are assumed to be concentrated at

plastic hinge locations.

First, the internal and external rates of virtual work are calculated. Considering pure

plastic bending, the rate of internal virtual work of the mechanism, dwi, is calculated as

dwi 4Mpatgj bdy, (8)

where dy is the virtual rotation angle of the short yield lines. The rate of external virtual

work, dwe, is

dwe

ZF

p dwx;y, (9)

where w(x,y) is the plate deflection in a point with coordinates (x,y) and F is the patch

loading area. The expression for the rate of external virtual work is obtained by integrating

equation (9):

dwe p dy

123ab b2 tan jb tan j a s3 cot j 3a s2b

. (10)

Equating the internal and external rate of virtual work, dwi dwe, there comes out the

following expression for the load-carrying capacity of patch loaded plates:

p p0 Kp, (11)

p0 is the plastic resistance in bending for a plate strip with unit width, p0 16MP=b2. The

coefficient Kp represents the patch loading factor and is derived as

Kp 31 a=b tan j

3a=b tan j tan j a s=b3 cot j 3a s=b2 . (12)

In case of uniform pressure, all quantities are predefined except the oblique yield line

angle j. In case of patch loading, both the yield line angle j and the length of the



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mechanism, a, are unknown. It is natural to determine j and a such that the critical

pressure is minimized:

dp

da

0;dp

dj

0. (13)

Analytical minimization of Eq. (11) yields tremendously complicated expressions for j

and a. To circumvent this problem, Daley et al. [3] first assumed the length of the failure

mechanism a is equal to the length of the patch load s. By restoring a safety factor later, the

resistance of patch loaded plate is obtained:

p 16Mp

b21 0:5

b

s

2. (14)

And the plate thickness requirement which has been put into practice by IACS [4] is

obtained as

t b

2

ffiffiffiffiffip

sy

r1

1 0:5b=s. (15)

Removing the simplifications used in IACS, numerical optimization is adopted to find

the relationships between the unknown and known parameters.

From the optimization of Eq. (13), the relationships for a and tan(j) with respect to

patch loading aspect ratio, s/b, are shown in Fig. 2.

It is observed from Fig. 2 that the length of the plastic mechanism is approximately

equal to two times the length of the patch load for small patches (s/bo0.5) andasymptotically approaches the length of the patch load for very large patches (s/b46). The

hinge location parameter a and the yield line angle j, in the moderate range, can be

approximated as

a

s 1

1

2s=b, (16)

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Fig. 2. Non-dimensional relationships for a and tan(j) with respect to patch aspect ratio s/b.



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tan j 3

2

1

4s=b. (17)

As shown in Fig. 3, Eqs. (16) and (17) give quite accurate results when they are

substituted into Eq. (12). Introducing Eqs. (16) and (17) into Eq. (12), we obtained:

Kp 6b=s2 18b=s 60 72s=b

b=s3 12b=s2 60b=s 208 432s=b. (18)

Further approximation could be made as

Kp 1:0 1:3b

s 0:18

b

s

2. (19)

The expressions for Kp in Eqs. (18) and (19) are plotted in Fig. 3 under different patch

loading aspect ratio, s/b, together with the numerical optimization results for Eq. (12).

The results from Eq. (19) agree quite favorably with the results from Eq. (18) and the

numerical optimization solutions. As Kp is only a function of patch loading aspect ratio

s/b, the comparison will be valid for all cases.

Plates subjected to large deflections will produce large in-plane membrane stresses which

increase the resistance significantly compared to pure bending. As a matter of fact, pure

bending mechanism is impossible provided that the long edges of the plate are not free to

deform, such as in a continuous plate field, e.g. the shell plating of ships side structure.

This is easily recognized by inspection of the roof-top-type collapse mechanism adopted.

In the middle of the plate, large deformations produce a tendency of pull-in of the

boundary. However, towards the short edges the pull-in effect is much smaller andvanishes at the edges.

By introducing the coefficient Km which represents the membrane effect, into Eq. (11),

the expression for the load-carrying capacity of patch loaded plates can be written as

p p0 Kp Km. (20)

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Fig. 3. Comparison of different expressions for Kp under different patch loading aspect ratios.

L. Hong, J. Amdahl / Marine Structures 20 (2007) 124142130


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The original expressions for the membrane effect for uniformly loaded plates are:

Km 1 1

3z2

3a=b tan j 2tan2 j 1

a=b tan j 1where zp1, (21)

Km 2z 1 tan2 j 1

2a=b tan j 2

1

3z2 1

& 'when zX1, (22)

in which z is the plate deflection normalized with respect to plate thickness, i.e. z w/t.

These expressions will remain the same for the case of patch loading because the factor for

patch loading appears merely in calculating the external virtual work.

The challenge is still to determine the hinge location parameter a and the yield line angle

j. As the yield line mechanism has been formed in the stage of plastic bending, it is

assumed that the mechanism will remain the same until ultimate collapse, i.e. a and j will

not change in the membrane stretching phase. Thus, substituting Eqs. (16) and (17) intoEqs. (21) and (22), the membrane coefficients are expressed as

Km 1 1

3z2b=s2 9b=s 16 36s=b

b=s 12 12s=bwhen zp1, (23)

Km 2z 1 b=s2 12b=s 52

4b=s 48 48s=b

1

3z2 1

!when zX1. (24)

Approximate formulation for Km which will be valid over the entire deflection range

may be obtained by an elliptic fit to the asymptotic solutions:

Km 1; when z ! 0, (25)

Km 2zb=s2 8b=s 4 48s=b

4b=s 48 48s=b

!when z !1. (26)

Then there is obtained:

Km

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2z

b=s2 8b=s 4 48s=b

4b=s 48 48s=b

2s. (27)

The expressions for the membrane coefficient for two patch loading aspect ratios arecompared in Fig. 4. It is observed that the approximate formulation Eq. (27) agrees

reasonably well with Eqs. (23) and (24) (within a few %).

From the derivations above, the required thickness for shell plating subjected to patch

load could thus be formulated as

t b

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

sy

1

Kp

1

Km

s. (28)

5. Proposed double-diamond patch loading mechanism

As will be seen in Figs. 7 and 8 in Section 6, the resistance of patch loaded plate is over-

predicted in the plastic bending stage based on the conventional roof-top-type collapse

mechanism adopted, especially for the case of small patch aspect ratios, e.g. s/b 1. From



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the observation of structural behavior of plates by finite element simulation, an alternative

kinematically admissible patch loading mechanism is proposed in Fig. 5 which is based on

a similar yield line model as the conventional roof-top-type model, named as double-

diamond collapse mechanism. The triangular hinge formations of short edges are

replaced by diamond hinge formations. The model increases the rate of external work, andreduces the rate of internal work, resulting in a somewhat lower collapse load in bending.

When the width of the patch, s, approaches plate length, L, the second yield line angle,

j2, is restricted and will diminish gradually. Consequently, for a uniformly loaded plate

the proposed double-diamond mechanism condenses into the conventional roof-top

pattern.

Three unknown parameters, hinge location parameter a, yield line angle j1 and j2, are

employed in this model, which will make the derivation more complicated than before.

Adopted the same analytical procedure as for the roof-top-type collapse model, the

expression for the pressure of the proposed double-diamond model is derived as

p p0

3a

b

tan2 j2 1

tan j1 tan j2

2a s

btan2 j1 a s=b

2

tan j1a s=b= tan j2tan j1 a s=b

2

tan j1 tan j2 tan j1

:

(29)

The minimization of the pressure requires:

dp

da 0;

dp

dj1 0;

dp

dj2 0. (30)

Apparently, it is even more difficult to get analytical solution for the double-diamondcollapse model. Numerical results show that the bending capacity reduces significantly

compared with roof-top-type model. For instance, for a plate with width 0.6 m, length

3.0 m and thickness 35 mm, the collapse load of plastic bending for different patches is

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Fig. 4. Comparison of different expressions for Km under different patch loading aspect ratios.



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calculated in Table 1. The material is assumed rigid-perfectly plastic, with yield strength

sy 300 MPa.

On the basis of the values in Table 1, an approximate correction factor fB is introducedinto Eq. (11), which adjusts the resistance in pure bending for the roof-top mechanism to

the corresponding resistance for the double-diamond mechanism:

fB 1 0:075s

b

0:5. (31)

It is a very challenging task to determine analytically the effect of membrane stresses on

the resistance in the finite deformation range. Comparison with NLFEA shows (refer

Figs. 7 and 8 in Section 6), however, that the expression ofKm, Eqs. (23) and (24), is a good

predictor also for the proposed double-diamond mechanism. Consequently, this factor

is continuously used as an approximation of the membrane effect. The load-carryingcapacity from the double-diamond mechanism becomes:

p p0 Kp Km fB. (32)

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Table 1

The bending resistance calculated from roof-top model and double-diamond model for different patch aspect

ratios

s/b p

Roof-top Double-diamond

1 10.13 9.36

1.5 7.95 7.42

2 6.92 6.49

2.5 6.32 5.95

Unit: MPa.

L

1

Patch loading

s

b

a

p

p

2

w

w

Fig. 5. Double-diamond collapse mechanism for laterally patch loaded plates.



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As a result, the plate thickness requirement can be written in the form:

t b

2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

sy

1

Kp

1

Km

1

fBs . (33)

6. Finite element verification

NLFEAs have been conducted using the progressive collapse analysis software USFOS

[17] to validate the proposed formulae for the load-carrying capacity of rectangular plates

subjected to lateral patch load.

In the following examples, two plates, denoted plates A and B, are studied for generality.

The different geometry particulars are shown in Table 2.

In Fig. 6, the collapse pattern of patch loaded plate obtained from NLFEA is shown.

The dashed lines represent the patch load boundary, and the stapled lines represent yield

lines. It is obvious that the proposed double-diamond mechanism is appropriate as the

roof-top mechanism when the plate is partially loaded.

For plate A, patch loadings with lengths of 0.40 and 0.60 m are studied. Results from

NLFEA are compared with the resistance predicted by the proposed formula Eq. (20)

from roof-top model in Fig. 7. Comparatively, the prediction by using Eq. (32) adjusted

from double-diamond model is also plotted in Fig. 7.

Three patches with different lengths are investigated for plate B, namely 0.60, 0.90 and

1.50 m, respectively. The resistances predicted using formula in Eqs. (20) and (32) are

compared with the results of NLFEA in Fig. 8.It is seen from Figs. 7 and 8 that the resistances predicted by Eq. (20) overestimate the

resistances in the plastic bending phase, notably for small patch aspect ratios, e.g. s/b 1.

When entering the membrane stage, the resistances agree reasonably well with the results

from NLFEAs. The predicted resistance agrees better and satisfactorily with the results

from NLFEA when it is adjusted by the proposed correction factor, i.e. Eq. (32), notably

for small patch aspect ratios. However, adopting Eq. (32) the resistance becomes

somewhat conservative for plate of large patch aspect ratios, e.g. s/b 2.5, when finite

deformation is allowed.

The principal limitation of the plastic yield line method is that it ignores the elasto-

plastic behavior that foregoes edge hinge formation. This becomes particularly importantfor small patch aspect ratios. Hence, the proposed formulation should be used with care, if

small or no permanent deflections are acceptable. On the other hand, the formula adjusted

by the correction factor from double-diamond mechanism model shows great

applicability, and it can be applied for patch loaded plates in most cases.

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Table 2

The geometry particulars of two plates

Plate A (m) Plate B (m)

Width 0.4 0.6Length 2.0 3.0

Thickness 0.02 0.035

sy 300MPa, E 2.1 105, n 0.3.



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The formulation predicts the resistance quite well in the range of finite deformations,

where a favorable increase of the resistance is experienced. Examples where finite

deformations could be accepted are cases when the plate is subject to extreme/abnormal iceactions or accidental actions such as ship collision.

Fig. 9 shows the resistances of plate B obtained from proposed Eq. (32) and IACS rule,

Eq. (14). Only small discrepancy is observed between the collapse bending resistance.

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Fig. 6. Collapse pattern for laterally patch loaded plates from NLFEA.

Fig. 7. Resistance predictions for plate A according to proposed Eq. (20), Eq. (32) and NLFEA.



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Finite deformations and the beneficial effect of membrane forces are not taken into

account in IACS plating design formulation.

It is also of interest to investigate the required plate thickness by using proposed formulaEq. (33) and IACS I2 structural requirement [4], Eq. (15). The design pressure is set to

5 MPa. The results are shown in Table 3 for a plate with width 0.6 m for various patch

aspect ratios and different allowable permanent deformations.

ARTICLE IN PRESS

Fig. 8. Resistance predictions for plate B according to proposed Eq. (20), Eq. (32) and NLFEA.

Fig. 9. Resistance predictions for plate B according to Eq. (32) and IACS rule.



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It is observed that when no permanent deformation is allowed, the required plate

thickness attained from Eq. (33) is virtually identical to the IACS requirement. The

maximum deviation is 1.4%. This shows that the IACS requirementin spite of the two

simplifications introducedis surprisingly good when the length of the plate is sufficient to

support the mechanism.

If permanent deformations are allowed, a noticeable thickness reduction is experienced.

For example, when the allowable permanent deformation is set equal to half of plate

thickness, the thickness reduction from adjusted formulae is up to 8.4%. If the allowable

permanent deformation is one times plate thickness, the required plate thickness may be

reduced up to 23%. The reduction becomes more significant if more deformations areacceptable.

Permanent deformations may often be accepted for abnormal ice loads or accidental

loads like ship collision, but acceptable criterion should be set appropriately due to

considerations of strain levels and proper safety margins with respect to fracture.

It should also be noticed that the present formulation and the IACS requirement are

appropriate as long as the plate length is much longer than the patch length, i.e. stiffeners

on short edges do not play any role. If the patch length approaches the plate length, these

stiffeners will impose restrictions on developing the mechanism and the resistance increases

correspondingly. This effect is particularly significant for plates with small aspect ratios.

Fig. 10 shows the plate resistance in bending versus patch aspect ratio s/b using variousformulations. The resistance is normalized versus that of a plate strip, p0, corresponding to

an infinitely long plate. In all these formulations, it is implicitly assumed that the plate

length, L, is sufficiently larger than the patch length, s, so as to contain the mechanism

yielding the minimum resistance. In the same diagram, the resistance for the plate with a

stiffener at the patch short edge, i.e. L s implying uniform loading, is plotted for

comparative purpose. It appears that the increase in resistance is significant for small plate

aspect ratios, but negligible for large aspect ratios. The IACS formulation fails to give

credit to this increased resistance. The present formulation may take this into account by

imposing restrictions on the mechanism length, a, and the second yield line angle j2 when

the plate length is limited for developing the entire mechanism. For this purpose, anotherfactor containing the plate aspect ratio must be introduced to enhance the applicability of

the proposed formulae. The restriction influence of plates with finite length will be

evaluated in Section 7.

ARTICLE IN PRESS

Table 3

Required plate thickness according to proposed formula Eq. (33) and IACS rule Eq. (15) for various patch aspect

ratios and different allowable permanent deformations

t s/b

1 1.5 2 3

tIACS 25.8 29.0 31.0 33.2

tEq. (33) (z 0) 25.6 28.6 30.5 32.8

tEq. (33) (z 0.5) 24.4 27.0 28.6 30.4

tEq. (33) (z 1) 21.6 23.4 24.4 25.5

sy 300 MPa, unit: mm.



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Fig. 11 shows numerical analysis including unloading of plate B with a patch loading of

s 0.9 m. For the present geometry the unloading stiffness is large, and the totaldisplacement is fairly equal to the permanent (plastic) displacement. This substantiates

direct comparison with results from plastic analysis by neglecting the elastic displacement

component.

In the present investigation, it is assumed that the boundaries are fixed against pull-in.

For illustration purpose, the resistance when the boundaries are free against pull-in, but

constrained to remain straight, is also illustrated in Fig. 11. It is interesting to see that there

is a significant increase of the resistance during finite deformations for straight boundaries,

although less pronounced than that for fixed boundaries. Actually, the roof-top

mechanism will imply development of internally balancing in-plane tensile and

compressive stresses as long as the boundaries remain straight. The pure bendingmechanism is therefore hypothetical when finite deformations are considered.

7. Patch loaded plates with finite length

It is explicitly assumed for all derivations that the plate is sufficiently long that no

restrictions apply for the development of the double-diamond mechanism. When the

plate is not much longer than the patch width, restrictions must be imposed on the

mechanism length, a, and the second yield line angle j2. This occurs when

Loa b tan j2, in which a and j2 are the optimal values for an infinitely long plate.

By applying restrictions to Eq. (29), the minimum resistance increases. The yield lineangle j2 vanishes gradually and the mechanism length a approaches the plate length L as L

approaches the span of the patch loading s. The relative increases in the bending resistance

for patch aspect ratios of s/b 1, 1.5 and 2 are shown in Fig. 12. It is observed that for a

ARTICLE IN PRESS

0

1

2

2.5

3

1 3 5

normalizedbendingresistance

Uniform loading

IACS

Roof-top model

Double-diamond model

3.5

1.5

0.5

2

patch aspect ratio s/b

4

Fig. 10. Normalized plate bending resistance versus patch aspect ratio s/b for an infinitely long plate using various

formulations.



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square patch (s/b 1), the stiffeners/frames have no influence as long as the plate length is

L4(1.7s 1.7b). The relative resistance increases exponentially up to a maximum value of

1.31 when L s, i.e. a square plate under uniform loading. For a longer patch (s/b 2),there is no influence of stiffeners for L4(1.25s 2.5b). The maximum relative increase of

the resistance is 1.1 for L 2s, i.e. a uniformly loaded plate. In this aspect, it is concluded

that stiffeners/frames play virtually no role when patch aspect ratios s/b42.

A reasonable approximation for plate length restriction factor is given by the following

expression:

fL 0:84 0:45L

s

L

b

!1X1. (34)

The length restriction factor should be no less than 1 in all cases, i.e. it will be employedinto Eq. (32) only when fL is larger than 1. The final expression for the load-carrying

capacity of laterally patch loaded plates is

p p0 Kp Km fBfL. (35)

Some NLFEAs have been carried out for plate B subjected to a square patch load to

validate fL when the plate length is insufficiently long to allow free formation of the

double-diamond mechanism. From Fig. 12, the plate length restriction will play roles

when L=so1:7 for a plate with a square patch loading. For illustration purpose, L/s 1.4and 1.2 are considered herein. In Fig. 13, the results from NLFEA are compared with that

from simplified calculations when fL is introduced. It is noted that the resistance increasessignificantly when the plate length is finite and insufficient to develop the entire

mechanism. The resistance compares reasonably well using Eq. (35) with respect to the

results from NLFEA, especially when finite deformation is allowed.

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Fig. 11. NLFEA predictions of resistance versus deflection of patch loaded plates with fully clamped and axially

free, but straight boundary conditions, together with a loadingunloading history curve (b 0.6m, s 0.9 m,

t 0.035 m, sy 300 MPa).



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8. Conclusions

The plastic, large deflection resistance of laterally patch loaded plates has been analyzed,

adopting the approach developed by Jones and Walters [8] for uniformly loaded plates. It

presupposed that the patch loading extends over the entire plate width, but only a part of

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Fig. 13. Resistance predictions when the collapse mechanism is restricted by the finite length of the plate.

Fig. 12. Length restriction factor for plates with insufficient length under various patch aspect ratios.



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the plate length. The approach resembles the one adopted by IACS in their unified

requirements for the design of plates against ice action, but the simplifications introduced

in the IACS formulations are not used and effect of membrane stresses are employed.

The expressions for yield line angle j and patch mechanism length a are obtained in the

present study for the conventional roof-top-type collapse model. The resistancepredicted with the proposed formulations is compared with the results from NLFEAs.

The resistance is somewhat overestimated in the plastic bending phase, notably for small

patches, but becomes increasingly accurate when entering the stage of finite deformations.

Hence the formulation seems to be a versatile tool for predicting the plate resistance when

finite, permanent deformations are accepted. This may, for example, be the case for plates

subjected to extreme/abnormal ice action and accidental actions such as ship collision.

In addition, an alternative and more comprehensive yield line mechanism model for

patch loaded plate is proposed, named as double-diamond collapse model. This model

yields improved prediction of the pure bending resistance, but at the expense of very

complex analytical solutions. Instead, the new model is used to derive an approximate

correction factor, which is implemented into the proposed formulae.

The required plate thickness according to the IACS unified requirement for polar ships

is in very good agreement with that obtained from the proposed formulae when no

permanent deformation is allowed, and shows that the IACS requirement is reasonable for

plates with sufficient length supporting the mechanism.

In cases of plates with small plate aspect ratios and patch length approaching plate

length, a plate length restriction factor is introduced which takes into account of the

influence of the plate with finite length imposing significant restrictions on collapse

mechanism. The formulation of the length restriction factor is verified through NLFEAwhich shows great agreement.

Acknowledgments

The present work is carried out within the scope of the Strategic University Programme

(SUP) ScenaRisC&G in Norwegian University of Science and Technology (NTNU)

which is funded by the Research Council of Norway (NFR). The authors wish to thank the

Research Council of Norway for supporting this project.

References

[1] Jones N. Structural impact. Cambridge: Cambridge University Press; 1989.

[2] Jones N. Review of the plastic behavior of beams and plates. Int Shipbuild Progr 1972;19:31327.

[3] Daley CG, Kendrick A, Appolonov E. Plating and framing design in the unified requirements for polar class

ships. In: Proceedings of the 16th international conference on port and ocean engineering under arctic

conditions, vol, 3, Ottawa, Canada, 2001. p. 77991.

[4] International Association of Classification Societies. I2 Structural requirements for polar class ships, 2006.

[5] Wood RH. Plastic and elastic design of slabs and plates. New York: The Ronald Press; 1961.

[6] Sawczuk A. On initiation of the membrane action in rigid-plastic plates. J Mech 1964;3(1):1523.

[7] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. Int

J Solids Struct 1971;7:100729.[8] Jones N, Walters RM. Large deflections of rectangular plates. J Ship Res 1971;15(2):16471.

[9] Nyseth H, Holtsmark G. Analytical plastic capacity formulation for plates subject to ice loads and similar

types of patch loadings. In: Proceedings of 25th international conference on offshore mechanics and arctic

engineering (OMAE2006), Hamburg, Germany, p. 12.



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[10] Hayward RC. Plastic response of ship shell plating subject to loads of finite height. Master thesis, Faculty of

Engineering and Applied Science, Memorial University of Newfoundland, 2001.

[11] Johansson BM. On the ice-strengthening of ship hulls. Int Shipbuild Progr 1967;14:23145.

[12] Kmiecik M. Usefulness of the yield line theory in design of ship plating. Mar Struct 1995;8(1):6779.

[13] Amdahl J. Plate and stiffener ice action design consideration for the Shtokman platform. Technical notes,

Trondheim, 2006.

[14] Wang G. Some recent studies on plastic behavior of plates subject to large impact loads. J Offshore Mech

Arctic Eng 2002;124(3):12531.

[15] Wang G, Tang S, Shin Y. Direct calculation approach and design criteria for wave slamming of an FPSO

bow. Int J Offshore Polar Eng 2002;12(4):297304.

[16] Wang G, Basu R, Chavda D, Liu S. Rationalizing the design of ice strengthened side structures. In:

Proceedings of maritime transportation and exploitation of ocean and coastal resources, vol. 1. Lisbon,

Portugal, 2005. p. 54957.

[17] /www.usfos.comS.

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plastic design of laterally patch loaded plates for ships

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