19
* Corresponding author. Tel.: 0086-21-6384-2238. E-mail address: yurenhuc@online.sh.cn (Y. Hu). Marine Structures 12 (1999) 585}603 An approximate method to generate average stress}strain curve with the e!ect of residual stresses for rectangular plates under uniaxial compression in ship structures Yuren Hu!,*, Jiulong Sun" !School of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University, 1954 Huashan Road, Shanghai 200030, People's Republic of China "Shanghai Rules and Research Institute, China Classixcation Society, 1234 Pudong Ave., Shanghai 200135, People's Republic of China Received 6 April 1999; received in revised form 7 December 1999; accepted 6 January 2000 Abstract The average stress}strain curve for rectangular plates under uniaxial compression in ship hull structures is generated on the basis of the existing design formulae. The dual-term equation for ultimate strength of uniaxially compressed plates is employed to generate the average stress}strain curve for long plates. Two alternatives proposed by Faulkner and Frankland, respectively, are considered. An approximate method to take into account the e!ect of residual stresses is proposed. By investigating the variation of the stress distribution at di!erent stages of compression, the average stress}strain relationship of the rectangular long plates with the e!ect of residual stresses is derived from equilibrium. The behaviors of the wide plates are also discussed. The assumption proposed by Valsgard for the idealized stress distribution of wide plates under uniaxial compression is adopted. The average stress}strain relationship for wide plates is derived from the existing design formulae. Two equations proposed by Hughes and recommended by API for critical stress of the central portion of the wide plate are considered. The e!ect of residual stresses is also taken into account for wide plates. ( 2000 Elsevier Science Ltd. All rights reserved. Keywords: Rectangular plates; Axial compression; Residual stresses; Ultimate strength; Buckling; Post- buckling; Post-collapse 0951-8339/99/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 1 - 8 3 3 9 ( 0 0 ) 0 0 0 0 2 - 2

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  • *Corresponding author. Tel.: 0086-21-6384-2238.E-mail address: [email protected] (Y. Hu).

    Marine Structures 12 (1999) 585}603

    An approximate method to generate averagestress}strain curve with the e!ect of residualstresses for rectangular plates under uniaxial

    compression in ship structures

    Yuren Hu!,*, Jiulong Sun"!School of Naval Architecture and Ocean Engineering, Shanghai Jiaotong University, 1954 Huashan Road,

    Shanghai 200030, People's Republic of China"Shanghai Rules and Research Institute, China Classixcation Society, 1234 Pudong Ave., Shanghai 200135,

    People's Republic of China

    Received 6 April 1999; received in revised form 7 December 1999; accepted 6 January 2000

    Abstract

    The average stress}strain curve for rectangular plates under uniaxial compression in ship hullstructures is generated on the basis of the existing design formulae. The dual-term equation forultimate strength of uniaxially compressed plates is employed to generate the averagestress}strain curve for long plates. Two alternatives proposed by Faulkner and Frankland,respectively, are considered. An approximate method to take into account the e!ect of residualstresses is proposed. By investigating the variation of the stress distribution at di!erent stages ofcompression, the average stress}strain relationship of the rectangular long plates with the e!ectof residual stresses is derived from equilibrium. The behaviors of the wide plates are alsodiscussed. The assumption proposed by Valsgard for the idealized stress distribution of wideplates under uniaxial compression is adopted. The average stress}strain relationship for wideplates is derived from the existing design formulae. Two equations proposed by Hughes andrecommended by API for critical stress of the central portion of the wide plate are considered.The e!ect of residual stresses is also taken into account for wide plates. ( 2000 ElsevierScience Ltd. All rights reserved.

    Keywords: Rectangular plates; Axial compression; Residual stresses; Ultimate strength; Buckling; Post-buckling; Post-collapse

    0951-8339/99/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 9 5 1 - 8 3 3 9 ( 0 0 ) 0 0 0 0 2 - 2

  • Nomenclature

    a length of the plateb actual width of the platebe

    e!ective width of the plate at post-buckling stageb%.

    e!ective width of the plate at collapseE Young's modulus of the materialRd in#uence factor of the initial de#ectionR

    rd interaction factor between the initial de#ection and the residual stresst thickness of the plateb slenderness of the platebe

    e!ective slenderness of the platebL

    longitudinal slenderness of the wide plated initial de#ection of the platee average strain of the plateey

    yield strain of the materiale6 dimensionless straine6 H dimensionless strain at which the e!ect of the residual stresses vanishesg factor of residual tension stress block widthpa

    average apply stress of the platepa,1#

    average stress of the central portion of the wide platepa,1%

    average stress of the edge portion of the wide platepra

    average apply stress of the plate with residual stressesprda

    average apply stress of the plate with residual stresses and initial de#ectionpe

    edge stress of the platepm

    ultimate average stress of the platepm,pc

    ultimate stress of the central portion of the wide platepm,1%

    ultimate average stress of the edge portion of the wide platepr

    compressive residual stress in the central portion of the platepy

    yield stress of the materialp6 dimensionless stress*p stress reduction in the central portion of the plate

    1. Introduction

    Rectangular plates with four edges supported by longitudinal and transversesti!eners are the basic structural components in ship structures. The averagestress}strain relationship of the rectangular plates under uniaxial compression, in-cluding the nonlinear post-buckling and post-collapse behaviors, is essential inestimating the ultimate longitudinal strength of a ship hull [1}3]. Generally, anelasto-plastic large de#ection bending analysis by using classical theory or by using

    586 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • the nonlinear "nite element method is required to establish the average stress}straincurve for a plate. It is a very complicated task.

    For the purpose of practical application, it is bene"cial to develop a simpli"edmethod to generate the average stress}strain curve of plates under uniaxial compres-sion. In 1990, Billingsley [4] "rst proposed a method to generate the averagestress}strain curve on the basis of the widely used design formulae for ultimatestrength of plates. He generated the average stress}strain curves for both long andwide rectangular plates based on a design formula adapted by US Navy [5]. However,the load shedding at the post-collapse stage was not taken into account, so theaverage stress was assumed to keep constant after the ultimate strength was reached.The e!ect of residual stresses and initial distortions were not considered either.

    In 1993, the method on the basis of the design formulae was further developed byGordo and Guedes Soares [6] to take into account the post-collapse load sheddingbehavior. The average stress}strain curve they generated was based on the designformula proposed by Faulkner [7]. The e!ects of residual stresses and initial distor-tions were also considered.

    Since the design formulae have been found to prove excellent agreement with testdata, the average stress}strain curves generated by the above-mentioned method areexpected to have enough accuracy. The method is superior to those using classicaltheory and nonlinear FEM in that it is simple and convenient in practical use.

    In this paper, a similar method to that developed by Gordo and Guedes Soares isemployed to generate the average stress}strain curve of rectangular plates underuniaxial compression. The e!ect of residual stresses is discussed. An approximatemethod to treat the e!ect of residual stresses, which is di!erent from that of Gordoand Guedes Soares, is proposed. The average stress}strain curve of wide plates is alsogenerated by adopting the assumption proposed by Valsgard [8] and by using theexisting design formulae. The e!ect of residual stresses is also considered for wideplates.

    2. Average stress}strain curve generated from design formulae

    In predicting the ultimate strength of rectangular long plates under uniaxialcompression, the e!ective-width approach has been widely adopted for years. A plateis considered long if the length a of its unloaded edges is greater than the width b of theloaded edges. For long plates, it is assumed that at collapse the load is entirely takenby two strips adjacent to the supported edges, while the remaining central portion isunstressed. The width of the load-carrying strips, b

    %., is called the e!ective width of

    the plate. von Karman suggested that the critical stress of the loaded portion of theplate should be calculated by using the equation for critical stress of simply supportedlong plate of the width b

    %., and at collapse the critical stress should equal the yield

    stress of the material, thus derived the well-known von Karman's equation as follows:

    b%.b

    "pm

    py

    "1.9

    b, (1)

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 587

  • where b is the actual width of the plate, pm

    is the maximum average platestress, p

    yis the yield stress of the material, and b is the slenderness of the plate

    de"ned by

    b"b

    tSpy

    E,

    (2)

    where t is the thickness of the plate, and E is the Young's modulus of thematerial.

    However, experimental results have shown that von Karman's equation over-predicts the ultimate strength of the plate at low b's. To overcome this shortcoming,a second term has been introduced and the following dual-term formula has beenproposed by many researchers [7]:

    b%.b

    "pm

    py

    "G1, 0)b(1,

    C1

    b!

    C2

    b2, b*1.

    (3)

    Faulkner has proposed C1"2 and C

    2"1 in the above dual-term equation

    for plates with simply supported edges. Faulkner's equation has shown excellentagreement with test data and has been widely used in practice. Another well-known equation is the US Navy plate strength equation attributed to Frankland,which takes C

    1"2.25 and C

    2"1.25 for plates with simply supported edges.

    Frankland's equation has also found wide applications [9]. There are manyother empirical formulae applied in aeronautical, civil engineering and navalarchitecture. A historical review and detailed listing of the formulae can be foundin [7].

    For plates with clamped edges, the dual-term formula of Eq. (3) can also be appliedwith the values of the coe$cients adjusted to C

    1"2.5 and C

    2"1.5625 according to

    Faulkner [7]. However, in estimating the ultimate longitudinal strength of ship hulls,it is a general practice to assume that the plate has simply supported edges, since thiswill give a result on the conservative side. Therefore, only simply supported plates arediscussed in this paper. Nevertheless, the method is also valid for plates with clampededges.

    von Karman also suggested generalizing the idealized stress distribution of the plateat collapse to the post-buckling stage from buckling to collapse. He assumed that atany moment after buckling the load was entirely taken by the e!ective width of theplate. Under this assumption, the above-mentioned empirical formulae are alsoapplicable at the post-buckling stage, provided that an e!ective slenderness b

    eis

    de"ned as follows:

    be"

    b

    tSpe

    E"bS

    pe

    py

    ,(4)

    588 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • where pe

    is the edge stress of the plate. Thus, Eq. (3) can be generalized into

    beb"

    pa

    pe

    "G1, 0)b

    e(1,

    C1

    be

    !C

    2b2e

    , be*1,

    (5)

    where pa

    is the average apply stress, and be

    is the e!ective width of the plate at thepost-buckling stage. Eq. (5) forms the basis of generating the average stress}straincurve of the long plates.

    To generate the average stress}strain curve from Eq. (5), the "rst step is to rede"nethe e!ective slenderness of the plate in terms of the average strain as follows [6]:

    be"bJeN , (6)

    where e6"e/ey

    is the dimensionless average strain and ey"p

    y/E is the yield strain. In

    addition, Eq. (6) is extended to the post-collapse unloading stage. That is to say, thee!ective width of the plate is assumed to be governed by the average strain even aftercollapse. Substituting Eq. (6) into Eq. (5) yields the following equation of the e!ectivewidth in terms of the average strain:

    beb"

    pa

    pe

    "G1, 0)e6( 1

    b2,

    C1

    bJe6!

    C2

    b2e6, e6*

    1

    b2.

    (7)

    Further assume the stress}strain relationship of the plate edge is the same as that ofthe material. For steels used in naval architecture, it is reasonable to assume anelastic}perfectly plastic relationship, which can be expressed by the following equa-tion:

    p6e"

    pe

    py

    "G!1, e6(!1,e6 , !1)e6(1,1, e6*1.

    (8)

    Notice that for convenience the stress and strain are positive in compression in theabove equation.

    From Eqs. (7) and (8), a dimensionless average stress}strain relationship can bederived for rectangular long plates under uniaxial compression.

    p6a"

    pa

    py

    "pe

    py

    pa

    pe

    "Ge6 , 0)e6(

    1

    b2,

    e6 AC

    1bJe6

    !C

    2b2e6 B,

    1

    b2)e6(1,

    C1

    bJe6!

    C2

    b2e6e6*1.

    (9)

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 589

  • Fig. 1. Average stress}strain curves of compressive rectangular plates.

    Fig. 1 shows typical average stress}strain curves generated from Faulkner's equationand from Frankland's equation. It can be seen that at a given average strain, theaverage stress obtained from Faulkner's equation is a bit lower than that fromFrankland's equation. That is to say, the average stress}strain curve generated fromFaulkner's equation is slightly more conservative.

    3. Approximate method to take into account the e4ect of residual stresses

    With the contraction arising from the welding process, tension residual stresses atabout p

    ycan be found in that portion of the plate in the immediate vicinity of the

    supporting frames. The residual tension stress block typically extends several thick-ness (gt) out from the weld each side. In the central portion of the plate far from thewelds, compressive residual stresses of value p

    rexist to maintain equilibrium in the

    longitudinal direction. An idealized distribution of the residual stresses is shown inFig. 2 [10]. The following relationship can be obtained from equilibrium,

    pNr"

    pr

    py

    "2gt

    b!2gt . (10)

    A method to take into account the e!ect of the residual stress in generating theaverage stress}strain curve is proposed by Gordo and Guedes Soares [6] by adoptingthe assumption made by Cris"eld [11]. In their method, the central portion of theplate initially under compression is assumed to lose the load-carrying capacity at

    590 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • Fig. 2. Idealized distribution of residual stresses.

    a strain of eN"1!eNr"1!pN

    r, while the tension blocks at plate edges can be loaded

    elastically until the strain reaches eN"2. The tangent modulus of the material in thetension block varies at eN"1!eN

    rdue to the e!ect of the loss of the load-carrying

    capacity of the central portion. A detailed description of this method can be foundin [6].

    A di!erent method is proposed in this paper by investigating the variation of thestress distribution at di!erent stages of compression. The basic assumption adopted isthat the e!ective width of the plate is governed by the average strain (Eq. (7)) in thewhole process of compression and is not a!ected by the residual stresses. The averagestress}strain relationship is derived from equilibrium.

    The following four stages of compression are considered.

    Stage (1): 0)eN(1/b2. At this stage, the plate is fully e!ective. The stress in the plateis the sum of the applied stress and the residual stress, as shown in Fig. 3(a). Obviously,the average stress and the average strain have the following linear relationship.

    pN ra"eN (11)

    Stage (2): 1/b2)e6(1!p6r. At this stage, the applied stress is fully taken by the

    e!ective width of the plate, while the distribution of the residual stress keeps un-changed. The resultant stress distribution is shown in Fig. 3(b). Since inside the rangeof the e!ective width the relationship between the applied stress and the average strainis linear, the following relationship can be derived.

    p6 ra"p6 r

    eAC

    1bJe6

    !C

    2b2e6 B"e6 A

    C1

    bJe6!

    C2

    b2e6 B. (12)Stage (3): 1!p6

    r)e6(e6 H. Except for the residual tension stress block of width 2gt,

    the remaining portion of the e!ective width at plate edges (of the width be!2gt)

    yields at e6"1!p6rand starts unloading with an increase of the strain. The residual

    tension stress block can carry the load until the average stress}strain curve intersects

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 591

  • Fig. 3. Stress distribution of long plates under compression with residual stresses.

    592 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • the unloading curve without the e!ect of the residual stresses at a strain e6 H. Mean-while, the central portion of the plate also starts unloading and transfers the load tothe residual tension stress block. If the stress reduction in the central portion of theplate is *p, then the stress in the residual tension stress block becomes pr

    e!p

    y#

    ((b!be)/2gt)*p, while the stress in the remaining portion of the e!ective width keeps

    the value of the yield stress py. The resultant stress distribution at this stage is shown

    in Fig. 3(c). Equilibrium yields

    prab"pr

    e2gt#(p

    y!p

    r)(b

    e!2gt)"Ee2gt#(p

    y!p

    r)(b

    e!2gt). (13)

    From Eqs. (7), (10) and (13), the average stress}strain relationship for this stage can beobtained as follows.

    p6 ra"(1!p6

    r)A

    C1

    bJe6!

    C2

    b2e6 B#p6r(e6!1#p6

    r)

    1#p6r

    . (14)

    Stage (4): e6*e6 H. The stress in the central portion of the plate decreases to zero ate6"e6 H. At the same time, the stress in the residual tension stress block reaches p

    y, i.e.

    pre!p

    y#

    b!be

    2gtpr"p

    y. (15)

    After that, the e!ect of the residual stresses vanishes and the average stress}straincurve is the same as the unloading curve without the e!ect of the residual stresses.Therefore, the average stress}strain relationship at this stage is

    p6 ra"

    C1

    bJe6!

    C2

    b2e6. (16)

    The intersection point of the curves for stages (3) and (4), e6 H, can be solved from thefollowing equation:

    e6 H#(1#p6r)A1!

    C1

    bJe6 H#

    C2

    b2e6 HB"2. (17)However, it is di$cult to solve this equation explicitly. For the convenience in

    computer programming, the average stress}strain relationship for both stages (3) and(4) can be written as

    p6 ra"minC(1!p6 r )A

    C1

    bJe6!

    C2

    b2e6 B#p6r(e6!1#p6

    r)

    1#p6r

    ,C

    1bJe6

    !C

    2b2e6 D, e6*1!p6 r .

    (18)

    In summary, the average stress}strain relationship for rectangular long plates underuniaxial compression with the e!ect of residual stresses derived in this paper can be

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 593

  • Fig. 4. Average stress}strain curve of compressive rectangular plates with residual stresses.

    expressed as follows:

    p6 ra"

    Ge6 0)e6( 1

    b2,

    e6 AC

    1bJe6

    !C2b2e6 B,

    1

    b2)e6(1!p6

    r,

    minC(1!p6 r )AC

    1bJe6

    !C2b2e6 B#

    p6r(e6!1#p6

    r)

    1#p6r

    ,C

    1bJe6

    !C2b2e6 D, e6*1!p6 r .

    (19)

    Fig. 4 shows typical average stress}strain curves for di!erent levels of residual stresses.The curves are for a plate with slenderness b"2 and the Faulkner's equation is used(C

    1"2 and C

    2"1).

    The present method di!ers from the method proposed by Gordo and GuedesSoares mainly in the treatment of stage (3). In Gordo and Guedes Soares'method, theintersection point between the average stress}strain curve of stage (3) and the unload-ing curve without the e!ect of the residual stresses is pre-assumed to be e6 H"2, andfrom this assumption derived the average stress}strain relationship for this stage. Thecondition of equilibrium is not satis"ed. In contrast to this, the present methodderived the average stress}strain relationship directly from the equilibrium condition.Then the intersection point is obtained, which does not necessarily equal 2. This seemsmore reasonable. Fig. 5 shows the di!erence of the average stress}strain curvesderived from the two methods for a plate with b"1.99 and with residual stress

    594 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • Fig. 5. Comparison of average stress}strain curves with e!ect of residual stresses.

    p6r"0.22. The curve obtained from experiment data by Brad"eld for a plate specimen

    of the same parameters, specimen No. 45S1W in [12], is also plotted in the "gure.A fairly good agreement can be found between the experimental results and theapproximate curve generated by using the present method.

    For the purpose of further veri"cation, average stress}strain curves generated byusing the present method for three plates are compared with those obtained from the"nite element method by Cris"eld [11] in Fig. 6.

    It should be pointed out that the e!ect of the initial de#ection is not addressed inthis paper. Gordo and Guedes Soares have proposed a method to account for thee!ect of the initial de#ection in [6], which can be used in generating the averagestress}strain curve for long plates. Following this method, an in#uence factor of theinitial de#ection, Rd(e6 ), and an interaction factor between the initial de#ection and theresidual stress, R

    rd(e6 ), can be introduced and the average stress}strain relationship ofthe rectangular long plates can "nally be expressed as

    p6 rda(e6 )"p6 r

    a(e6 )Rd (e6 )Rrd(e6 ). (20)

    4. Average stress}strain curve for wide plates

    The discussion in the previous sections is only valid for long plates. Although longplates are dominant in modern longitudinally framed hull structures, there are stillwide plates especially in side shell and longitudinal bulkhead structures. For trans-versely framed hull structures, wide plates are dominant. Therefore, it is of practicalsigni"cance to study the behaviors of wide plates.

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 595

  • Fig. 6. Comparison of average stress}strain curves with FEM results.

    It has been found [13] that the ultimate strength of wide plates under uniaxialcompression is much lower than that of long plates, since the central portion of thewide plate is less supported by the frames at plate sides. The typical stress distributionof a wide plate under uniaxial compression is shown in Fig. 7(a). When the stress in thecentral portion reaches its critical value, it collapses and loses the load-carryingcapacity. The plate edge can take further load until its stress reaches the yield stress.The whole wide plate collapses at this moment.

    Valsgard [8] assumed that at collapse the behavior of the edge portion of the wideplate at each side is the same as that of a long plate of width a

    2, and the behavior of the

    central portion of the wide plate is approximately the same as an in"nitely wide plate.Therefore, an idealized stress distribution of the wide plate at collapse can be obtainedas shown in Fig. 7(b). Notice that the stress of the edge portion plotted in the "gure isthe average stress of that portion. Finally, the ultimate average stress of the wide platecan be expressed from equilibrium as follows.

    p6m"a

    bp6m,1%

    #A1!a

    bBp6 m,1# (21)where the ultimate average stress of the edge portion, p6

    m,1%, is calculated from the

    equation for long plates as described in the previous sections of the paper.To generate the average stress}strain curve for wide plates, a further assumption is

    made that at any given strain the stress distribution in a wide plate can be idealized in

    596 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • Fig. 7. Stress distribution in wide plates.

    the same manner as that at collapse. Therefore, the following relationship can beobtained:

    p6a"

    a

    bp6a,1%

    #A1!a

    bBp6 a,1# , (22)where p6

    ais the dimensionless average stress of the whole wide plate. p6

    a,1#is the

    average stress of the central portion of the wide plate. p6a,1%

    is the average stress of theedge portion calculated from Eq. (9). Notice that the slenderness b in Eq. (9) shouldbe changed to the longitudinal slenderness b

    Lof the wide plate de"ned by

    bL"

    a

    tSpy

    E.

    (23)

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 597

  • Now, we derive the average stress}strain relationship for wide plates on the basis ofthe following two equations.

    (1) Hughes+ equation: The following equation is derived by Hughes [13] fromanalyzing the numerical results obtained by Smith [14]:

    p6m,1#

    "G1, 0)b

    L(1,

    0.63

    b2L#3.27dM

    , bL*1,

    (24)

    where dM "d/t is the dimensionless amplitude of the initial de#ection.To derive the average stress}strain relationship for wide plates, it is assumed that

    the stress}strain relationship of the central portion of the wide plate is linear beforecollapse and after collapse the stress remains unchanged. Thus the following equationis obtained.

    p6a"

    a

    bp6a,1%

    #A1!a

    bB p6 a,1#

    "Ge6 , 0)e6(

    0.63

    b2L#3.27dM

    ,

    a

    be6#A1!

    a

    bB0.63

    b2L#3.27dM

    ,0.63

    b2L#3.27dM

    )e6(1

    b2L

    ,

    a

    be6 A

    C1

    bLJe6

    !C

    2b2Le6 B#A1!

    a

    bB0.63

    b2L#3.27dM

    ,1

    b2L

    )e6(1,

    a

    bAC

    1bLJe6

    !C

    2b2Le6 B#A1!

    a

    bB0.63

    b2L#3.27dM

    , e6*1.

    (25)

    Fig. 8 shows typical average stress}strain curves for a wide plate ofa/b"0.25, b

    L"2 having an initial de#ection of d"0 and d"0.12b2

    Lt, with Faulk-

    ner's equation (C1"2 and C

    2"1) and Hughes' equation used for the edge portion

    and central portion of the wide plate, respectively. An average stress}strain curve fora corresponding long plate of b"2 is also plotted in the "gure for comparison.

    (2) Equation recommended by API: The following design equation is recommendedby the American Petroleum Institute for the ultimate strength of wide plates [15]:

    b%.b

    "pm

    py

    "a

    bAC

    1bL

    !C

    2b2LB#0.08A1!

    a

    bBA1#1

    b2LB

    2)1. (26)

    where the coe$cients are taken as C1"2 and C

    2"1 (Faulkner's equation for long

    plates) by API. A similar equation is adopted with C1"2.25 and C

    2"1.25 (Frank-

    land's equation for long plates) in [9] to estimate the ultimate strength of wide platesin reliability assessment of ship structures.

    598 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • Fig. 8. Average stress}strain curve of wide plate.

    It can be found that this equation has the same form as Eq. (21). Now p6m,1%

    is againcalculated by Faulkner's equation or Frankland's equation, while the equation forp6m,1#

    is

    p6m,1#

    "0.08A1#1

    b2LB2. (27)

    By manipulating in a way similar to that of Hughes' equation, the following averagestress}strain relationship for wide plates can be obtained:

    p6a"

    a

    bp6a,1%

    #A1!a

    bBp6 a,1#

    "Ge6 , 0)e6(0.08A1#

    1

    b2LB

    2,

    a

    be6#0.08A1!

    a

    bBA1#1

    b2LB

    2, 0.08A1#

    1

    b2LB

    2)e6(

    1

    b2L

    ,

    a

    be6 A

    C1

    bLJe6

    !C

    2b2Le6 B#0.08A1!

    a

    bBA1#1

    b2LB

    2,

    1

    b2L

    )e6(1,

    a

    bAC

    1bLJe6

    !C

    2b2Le6 B#0.08A1!

    a

    bBA1#1

    b2LB

    2, e6*1.

    (28)

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 599

  • Fig. 9. Comparison of average stress}strain curve for wide plates.

    A typical average stress}strain curve for a wide plate of bL"2 generated from

    Eq. (28) with C1"2 and C

    2"1 is also plotted in Fig. 8 together with the curves

    generated from Eq. (25) by using Hughes' equation with d"0 and 0.12b2Lt. It can be

    found from the "gure that the two sets of curves are fairly close. However, Eq. (28)cannot account for the e!ect of the initial de#ection. Since wide plates are moresensitive to the initial de#ection than long plates, Eq. (25) is more suitable for use ingenerating the average stress}strain curve for wide plates.

    For comparison, average stress}strain curves for wide plates ofa/b"0.33, b

    L"1.57 and 2.63 generated from Eq. (25) and from numerical procedure

    by Valsgard [8] are plotted in Fig. 9. It can be found that the curves generated by thepresent method are in fairly good agreement with those obtained by numericalprocedure. The average stress}strain curve for a wide plate of a/b"0.33, b

    L"1.57

    generated by Billingsley's method is also plotted in the "gure. It can be seen that fora given average strain the average stress obtained from Billingsley's curve is muchhigher than that from the curves given by this paper. The reason might be that inBillingsley's method the critical stress of the central portion of the wide plate iscalculated by using Euler's equation for elastic buckling of in"nitely wide plate and noinelastic e!ect is concerned. Another di!erence is that Billingsley determines theload-carrying e!ective width according to the full width of the wide plate, while in thepresent paper it is determined according to the width of the edge portion of the totalwidth a as proposed by Valsgard.

    600 Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603

  • Fig. 10. Average stress}strain curve of wide plate with residual stresses.

    To take into account the e!ect of residual stresses, it is reasonable to assume thatthe residual stresses only a!ect the behaviors of the edge portion of the wide plate.Hence, Eq. (19) can be used in calculating the average stress of the edge portion of thewide plate (p6

    a,1%in Eq. (22)). Fig. 10 shows typical average stress}strain curves for

    a wide plate (bL"2 and d"0.12b2

    Lt) with di!erent levels of residual stresses.

    5. Conclusions

    The average stress}strain curve of the rectangular plates under uniaxial compres-sion, including the nonlinear post-buckling and post-collapse behaviors, is essential inestimating the ultimate longitudinal strength of a ship hull. To generate the averagestress}strain curve on the basis of the existing empirical design formulae is a simpleand e!ective method in practical application. This method is adopted in the presentpaper. The average stress}strain curve for rectangular long plates is generated fromthe dual-term design formulae. Two alternatives proposed by Faulkner and byFrankland are discussed and compared. The results show that the curve generatedfrom Faulkner's equation is slightly more conservative than that from Frankland'sequation.

    A new method to approximately take into account the e!ect of the residual stressesis proposed in the paper. Variations of the stress distribution in the plate at four stagesof compression with the involvement of residual stresses are investigated. It is foundthat stage (3) is the one a!ected by the existence of the residual stresses. The average

    Y. Hu, J. Sun / Marine Structures 12 (1999) 585}603 601

  • stress}strain relationship of this stage is derived from equilibrium. The e!ect of initialde#ection on the behaviors of long plates is not discussed in the paper. Gorgo andGuedes Soares' work [6] can be referred to for this matter.

    Wide plates are also discussed in the paper. Under the assumption made byValsgard for idealized stress distribution in wide plates, the edge portion of the wideplate is treated as a long plate, while the critical stress of the central portion iscalculated from the empirical design formulae for in"nitely wide plates. Two cases ofusing the Hughes' equation and the equation recommended by API are considered.Since Hughes' equation can take into account the e!ect of the initial de#ection, it ismore suitable in practical use.

    The method described in this paper has the advantages over the classical elasto-plastic analysis and the nonlinear FEM method in that it is simple and convenient inpractical use and it has enough accuracy since the empirical design formulae havefound quite good agreement with the experimental data. The resulted averagestress}strain curves for both long and wide plates can be further used to generate theload-end shortening curves for sti!ened panels in predicting the ultimate longitudinalstrength of ship hulls by using the simpli"ed method.

    Acknowledgements

    The authors wish to express their appreciation to the China Classi"cation Societyfor supporting this research. However, any views in this paper are those of the authorsand do not necessarily re#ect the o$cial views of the CCS.

    References

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    [2] Adamchak JC. An approximate method for estimating the collapse of a ship's hull in preliminarydesign. Proceedings of Ship Structure Symp '84, SNAME, Arlington, VA, 1984. p. 37}61.

    [3] Gordo JM, Guedes Soares C, Faulkner D. Approximate assessment of the ultimate longitudinalstrength of the hull girder. J Ship Res 1996;40:60}9.

    [4] Billingsley DW. Hull girder response to extreme bending moments. Proceedings of Fifth STARSymposium SNAME, Coronado, CA, 1980. p. 51}63.

    [5] Frankland JM. The strength of ship plating under edge compression. US EMB Report 469, May 1940.[6] Gordo JM, Guedes Soares C. Approximate load shortening curves for sti!ened plates under uniaxial

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    [8] Valsgaard S. Ultimate capacity of plates in transverse compression, Det norske Veritas Report No.79-0104, 1979.

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  • [11] Cris"eld MA. Full range analysis of steel plates and sti!ened plating under uniaxial compression,Proc Inst Civil Eng Part 2, 1975;59:595}624.

    [12] Brad"eld CD. Tests on plates loaded in in-plane compression. J Construct Steel Res 1980;1:27}37.[13] Hughes OF. Ship structural design. New York: Wiley, 1983.[14] Smith CS. Imperfection e!ects and design tolerances in ships and o!shore structures. Trans Inst Eng

    Shipbuilders Scotland 1981;124:39}46.[15] American Petroleum Institute. Bulletin on design of #at plate structures. API Bulletin 2V, 1987.

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