Ultimate Longitudinal Strength of Composite Ship Hulls

Curved and Layer. Struct. 2017; 4:158–166

Research Article Open Access

Xiangming Zhang*, Lingkai Huang, Libao Zhu, Yuhang Tang, and Anwen Wang Ultimate Longitudinal Strength of Composite Ship Hulls

DOI 10.1515/cls-2017-0012 posite materials. There are now all-composite naval ships Received Dec 15, 2016; accepted Mar 12, 2017 in service. Longitudinal bending moment carried by ship hull Abstract: A simple analytical model to estimate the lon- girder increasing with the increasing length of composite gitudinal strength of ship hulls in composite materials ship hulls, the ship safety related to longitudinal ultimate under buckling, material failure and ultimate collapse is strength becomes more important at design. According to presented in this paper. Ship hulls are regarded as as- traditional ship design rules, the longitudinal strength of semblies of stiffened panels which idealized as group of the ship hull built of steel with length exceeding 60 m plate-stiffener combinations. Ultimate strain of the plate- must be assessed. The stiffness of composite materials is stiffener combination is predicted under buckling or ma- generally low. Longitudinal deformation of ship hulls in terial failure with composite beam-column theory. The ef- composite materials is usually larger than that built of fects of initial imperfection of ship hull and eccentricity of steel. Recent years, with the rapid development trend and load are included. Corresponding longitudinal strengths momentum of construction of large-scale and super large of ship hull are derived in a straightforward method. A composite ship hulls, the length of ship hulls in composite longitudinally framed ship hull made of symmetrically materials are increasing so that the study of longitudinal stacked unidirectional plies under sagging is analyzed. ultimate strength of composite ship becomes increasingly The results indicate that present analytical results have a important and urgent. good agreement with FEM method. The initial deflection of The ultimate strength of steel ships has been widely ship hull and eccentricity of load can dramatically reduce investigated by many researchers. Caldwell [1] was the the bending capacity of ship hull. The proposed formula- first who estimated the ultimate strength of steel ships tions provide a simple but useful tool for the longitudinal employing the fully plastic bending theory of beams. In strength estimation in practical design. his method, the hull girder was divided into a number Keywords: Composites; Ship hull; stiffened panel; Ulti- of panels and the collapse load of each panel was cal- mate longitudinal strength; analytical model culated through strength reduction factor due to buckling for compressive loading. However his model did not take into account the post-buckling strength of the panels. 1 Introduction Smith [2] proposed an approach for calculation of the ultimate strength of ship hulls. He divided the ship’s cross section into a series of stiffened panels and then performed Laminated fiber reinforced composites have been widely a progressive collapse analysis under bending where the used in marine and ship structures. The application of post-buckling behavior of the stiffened plates was consid- such materials in ship structures dates back to the late ered. The collapse of a girder section is assumed to occur 1970s. Over the time, the usage of composite materials in between two adjacent frames, being induced either by the ship construction continues to grow due to the improve- inter-frame flexural beam-column collapse of panels un- ment of design, fabrication and mechanical performance der compression or by the inter-frame yielding of panels of advanced composites. Currently, some large vessels like under tension. The behavior of the stiffened plates under frigates and passenger ships are made of laminated com- compressive load can be characterized through so-called load-shortening curves. The resulting strains and stresses in the stiffened plates making up the cross section were *Corresponding Author: Xiangming Zhang: College of Science, calculated using the predefined load-shortening curves. Naval University of Engineering, Wuhan, China, 430033; Email: Based on the general approach of Smith’s method, Dow et [email protected]; Tel.: +86 02765460796 al. [3] and Ueda and Yao [4] developed a method for deter- Lingkai Huang, Libao Zhu, Yuhang Tang, Anwen Wang: College of Science, Naval University of Engineering, Wuhan, China, 430033 mination of load-shortening curves of beam-column ele-

© 2017 X. Zhang et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Ultimate Longitudinal Strength of Composite Ship Hulls Ë 159 ments, which similar to the Smith’s beam-column method cal method is needed for practical engineering at prelim- with the difference that a large portion of the hull girder inary stage of ship design. For this purpose, an analyti- plating with multiple stiffeners was considered. Viner [5] cal approach to estimate the longitudinal strengths of ship proposed an expression for calculating the ultimate bend- hull in composite materials was developed in this paper. ing moment assuming that the elastic behavior is main- Ship hulls are modeled as assemblies of stiffened com- tained up to the point where the compression flange reach posite panels. Stiffened panels are idealized as a series of the collapse state resulting in immediate hull collapse. plate-stiffener combinations. Ultimate strain is predicted Khedmati [6] made attempts for derivation of the average under buckling or material failure with beam-column the- stress-average strain relationships for stiffened plates sub- ory, considering the effects of initial imperfection of ship ject to in-plane compression alone or in combination with hull and eccentricity of load. The corresponding longitu- other loads using analytical approaches. Frieze and Lin [7] dinal strengths of ship hulls are derived with a straightfor- expressed ultimate bending moment capacity of the ship ward analytical method. hull as a function of a normalized ultimate strength of the The primary modes of failure for ship hull girder is col- compression flange. Paik and Mansour [8] presented anex- umn or beam-column type collapse of plate-stiffener compression for predicting the ultimate strength of single and bination as a representative of stiffened panel. Two failure double-hull ships under vertical bending moments. Gar- modes of beam-column in composite materials are consid- batov [9] performed hull girder ultimate strength verifica- ered in this paper. The first failure mode is buckling. The tion according to the Class Society rules based on experi- tensile and compressive strengths of FRP laminates com- mental results and the dimensional theory. Some other ul- monly employed in hull design are approximately equal timate strength approaches can be found in [10–12]. to the yield strength of mild steel, and the Young’s modu- Very few publications can be found in the literature lus of such laminates are only 5-10% of that of steel. Thus on evaluating the ultimate strength of composite ships. buckling may be an important failure mode of ship hull Chen et al. [13] firstly tried to estimate the ultimate strength in composite materials. The second failure mode is mate- of composite ships. They proposed a simple analytical rial failure. The material failure of a local area in ship hull method for calculating the ultimate strength of compos- may occur with the increase of longitudinal bending mo- ite vessels. But his model is too simple to handle the stak- ment, which will influence structural performances. Con- ing sequence of real composite materials and fail to give sequently, it is necessary to take material failure as an im- procedures to handle the composite hat-stiffened panels. portant failure mode of ship hull in composite materials. Chen and Soares [14] conducted a calculation of ultimate strength of composite ships under bending moment with nonlinear finite element method. Two types of the failure 2 Bending stiffness of composite modes were considered in their study; the panel buckling as well as fracture of the composite materials. Later, Chen ship hulls and Soares [15] used Smith’s method to calculate the ultimate strength of composite vessels. Morshedsoluk [16] Longitudinal framed structure system is usually used for applied a Coupled Beam Theory to calculate the ultimate large ships. The deck plate, bilge plate and side plate are strength of composite ships taking into account the ef- strengthened with a series of longitudinal stiffeners (in- fect of the superstructure. The behaviour of the compos- cluding Stringer). They are all hat-stiffened laminates. Hull ite panels in the ship structure is deduced from their mean girder consists of a number of structural elements of stiff- stress-mean strain curves. The progressive failure method ened plates (Figure 1). According to the Smith analysis in conjunction with the nonlinear finite element method method [17], each stiffener of the hull girder (including is used to calculate mean stress- mean strain curves. The attached flange and plating) works almost independently efficiency of the composite superstructure in contribution from other stiffeners. Consequently, each plate-stiffener to the ultimate bending strength of the composite ships is combination can be analyzed alone. The load capacity of evaluated. Case studies showed that the length of super- structural element of stiffened panel and ship hull girder structure has significant effect on the ultimate strength of can be calculated with composite beam-column theories. the composite ship. To determine the bending stiffness of ship hull girder, Application of composite materials to the construc- the section is divided into a set of representative plate- tion of large ships is a relatively new and growing subject, stiffener combination (Figure 2). The plate-stiffener com- which needs correct assessment of the ultimate strength bination is then divided into a set of slats [18]. Each strip of these types of the ships. Especially, a simple analyti- can be calculated as a laminate. 160 Ë X. Zhang et al.

Figure 2: Section of a plate-stiffener combination

1 G = A l t 66

Where, El1 is effective tensile/compressive elastic modulus at x direction; El2 is effective tensile/compressive elastic modulus at y direction; Gl is effective shear modulus.

2.2 Stiffness of laminate plate-stiffener combination

The stiffened panel is idealized as a number of plate- stiffener combination (stiffener including attached flange) or a beam-column. Typical section of a beam-column is Figure 1: Hull girder section shown in Figure 2. Beam-column is divided into a set of strips. Assuming 2.1 Stiffness of symmetrical balanced these strips to possess the same longitudinal strain, which laminate are equal to that of stiffened laminate, we have P P Symmetrical balanced laminate presents orthotropic ε = = i (4) (EA) E Ai properties. The planar stiffness coefficients are ps l1i

2 2 Where, P is axial force of beam-column; Pi is axial force of A12 A12 (Et)l1 = A11 − ;(Et)l2 = A22 − ; (1) A22 A11 each slat, (EA)ps is tensile/compressive stiffness of beam- column; El1i is the effective elastic modulus of each slat; (Gt)l = A66 Ai is the sectional area of each strip. Where, (Et)l1 is effective tensile/compressive stiffness coefficient at x direction; (Et) is effective ten- ∑︁ P ∑︁ P ∑︁ l2 P = Pi = El1i Ai = (El1i ti)bi (5) (EA)ps (EA)ps sile/compressive stiffness coefficient at y direction, (Gt)l is effective shear stiffness coefficient. Where ti is thickness of strip; bi is width of strip. The ten- Bending stiffness coefficients of symmetrical balanced sile/compressive stiffness of beam-column is laminate are (︂ 2 )︂ D2 D2 ∑︁ ∑︁ A12i 12 12 (EA)ps = (E t )b = A − b (6) Dl1 = D11 − ; Dl2 = D22 − ; Dl12 = D66 (2) l1i i i 11i A i D22 D11 22i ∑︁ Where, Dl1 is effective bending stiffness at x direction; Dl2 = E0 λi bi is effective bending stiffness at y direction; Dl12 is torsional stiffness. Where, E0 is the elastic modulus of an arbitrarily selected Effective elastic modulus of symmetrical balanced standard reference material and laminate is (︂ 2 )︂ A12i (︂ 2 )︂ (︂ 2 )︂ λi = A11i − /E0 (7) 1 A12 1 A12 A22i El1 = A11 − , El2 = A22 − , (3) t A22 t A11 Ultimate Longitudinal Strength of Composite Ship Hulls Ë 161

Similarly, Where, α is the angle between the slat width and the neu- ∑︁ tral axis. (GA)ps = A66i bi (8) Bending stiffness of shifting neutral axis of slat to that The sectional area of beam-column is of beam-column is ′′ 2 2 ∑︁ ∑︁ (EIy)li = El1i ti bi(zi − zcps) = E0λi bi(zi − zcps) (17) Aps = Ai = bi ti (9)

The effective elastic modulus of beam-column is Bending stiffness of beam-column is expressed as ∑︀ ∑︁ ′ ′′ λi bi (EIy)ps = [(EI) li + (EI)li] (18) Eps = ∑︀ E0 (10) bi ti Similarly, bending stiffness of beam-column about its Assuming the bending deformation of each strip keeps symmetric axis is expressed as consistent with that of beam-column ∑︁ ′ ′′ Ml M l (EIz)ps = [(EIz) li + (EIz) ] (19) φ = = i (11) li (EI)ps (EI)li where, Where, M is the bending moment of beam-column, (EI) ps 1 1 (EI )′ = (Et) (b cos α)3 = E λ (b cos α)3 (20) is the bending stiffness of beam-column; Mi is the bending z li 12 li i 12 0 i i moment of slat; (EI)li is the bending stiffness of slat; l is the ′′ 2 2 (EIz)li = (EAi)li(yi) = E0λi bi(yi) length of beam-column; φ is the relative rotation of beam- column end sections. ∑︁ M ∑︁ 2.3 Bending stiffness of composite ship hull M = Mi = (EI)li (12) (EI)ps girder The bending stiffness of beam-column is obtained from The sectional area of ship hull gird is sum of that of all beam-column ∑︁ (EI) = (EI) (13) ps li ∑︁ A = Aps (21) At local reference coordinates system shown in Figure 2, the neutral axis of beam-column is determined from The composite beams are mainly distinguished from traditional beams in structure. Different types and spec- ∑︀ E A z z = l1i i i (14) ifications of reinforcement as well as different layup are cps ∑︀ E A l1i i utilized in different parts of a composite beam. Compos- (︁ A2 )︁ ∑︀ A − 12i b z ∑︀ 11i A22i i i λ b z ite beam theory should be applied. Area conversion meth- = = i i i (︁ A2 )︁ ∑︀ ∑︀ 12i λi bi ods should be used to dealt with composite beams [19, 20], A11i − bi A22i namely, an elastic modulus of an arbitrarily selected ma-

Where, the zcps is the coordinates of neutral axis of beam- terials as standard reference modulus, denoted E0, and cross-sectional area of other materials are transformed in column; zi is the centroidal coordinate of each slat. When the width of slat is parallel to the neutral axis of the coefficient beam-column section, bending stiffness of the slat about Epsi 훾i = (22) its own neutral axis is in form of E0 (︂ D2 )︂ (EI )′ = D − 12 b (15) The axial stiffness of ship hull gird is y li 11 D i 22 i ∑︁ ∑︁ (EA)gir = (EA)psi = E0 (훾i Apsi) (23) When the width of slat is at an inclined direction which is not parallel to the neutral axis of beam-column Setting a global reference coordinates at the bottom of section, bending stiffness of the slat about its own neutral ship hull, the neutral axis is determined as axis is expressed as ∑︀ ∑︀ (EA)psi zpsi (훾i Apsi)zpsi 1 zcgir = ∑︀ = ∑︀ (24) (EI )′ = E t (b sin α)3 (16) (EA)psi (훾i Apsi) y li 12 l1i i i (︂ 2 )︂ Z -centroidal coordinates of ship hull girder sec- 1 A12 3 1 3 cgir = A11 − (hi) = E0λi(hi) tion. 12 A22 i 12 162 Ë X. Zhang et al.

Bending stiffness of ship hull girder is Or simply

∑︁ [︁ 2]︁ 2 (EI)gir = Dgir = (EIy)psi + (EA)psi(zpsi − zcgir) (25) ϕ1σmf + ϕ2σmf − 1 = 0 (31)

The moment of inertia of ship hull girder section is Where

[︃ ]︃ 4 2 2 4 2 2 ∑︁ (EIy)psi 2 ϕ1 = F11m + 2F12m n + F22n + F66m n (32) (︀훾 )︀ (︀ )︀ Igir = + i Apsi zpsi − zcgir (26) 3 3 E0 − 2F16m n − 2F26mn ϕ = F m2 + F n2 − F mn When the neutral axis of beam-column is not parallel 2 1 2 6 to that of ship hull girder, the moment of inertia should be The ultimate strength of composite stiffened panel un- transformed. der material failure is given by

(EIy)ps + (EIz)ps √︁ (EIy)ps = (27) 2 2 −ϕ2 + ϕ2 + 4ϕ1 σmf = (33) (EIy) − (EIz)ps 2 + ps cos 2θ 2 The material failure strain is

Where, θ-orientation of the neutral axis of beam-column σmf εmf = (34) with respect to that of ship hull girder, which clockwise is Eps positive.

4 Column buckling 3 Material failure The column buckling load of plate-stiffener combination The lamina stresses in principal material direction under with uniform cross-section is given by uniaxial stress are π2(EI) ⎧ ⎫ ⎧ ⎫ P = ps (35) σ σ cr 2 ⎨⎪ 1 ⎬⎪ ⎨⎪ x⎬⎪ l σ2 = [Tσ] 0 (28) ⎪ ⎪ ⎪ ⎪ The column buckling stress is ⎩τ12⎭ ⎩ 0 ⎭ 2 ⎡ 2 2 ⎤ ⎧ ⎫ ⎧ 2 ⎫ π (EI)ps m n 2mn σx m ⎨⎪ ⎬⎪ ⎨⎪ ⎬⎪ σcr = 2 (36) ⎢ 2 2 ⎥ 2 Aps l = ⎣ n m −2mn ⎦ 0 = n σx 2 2 ⎪ ⎪ ⎪ ⎪ −mn mn m − n ⎩ 0 ⎭ ⎩−mn⎭ The critical buckling stress including shear deformation is in form of [22] Where, Tσ is transform matrix; m = cos θ, n = sin θ; θ 2 is the orientation angle of lamina. π (EI)ps 2 To identify the material failure of stiffened composite Aps(µl) σcr = 2 (37) π (EI)ps panels, the Tsai-Wu criterion [21] is adopted in this paper. 1 + 2 (GA)ps(µl) Material failure is considered to have occurred as the following equation is satisfied at any lamina The ultimate buckling strain of stiffened composite panels is F = F σ + F σ σ = 1 i, j = 1, 2, 6 (29) i i ij i j σcr εbf = (38) Eps where σi denotes the stress components referred to the principal material coordinates; Fi and Fij are material strength coefficients. Introducing Eq. (28) into Eq. (29), we have 5 Initial imperfection and

(︁ 4 2 2 4 2 2 eccentricity of load F11m + 2F12m n + F22n + F66m n (30)

3 3)︁ 2 −2F16m n − 2F26mn σmf The initial defection of beam-column is assumed to be 2 2 πx + (F1m + F2n − F6mn)σmf − 1 = 0 w = w sin (39) 1 0 l Ultimate Longitudinal Strength of Composite Ship Hulls Ë 163

Where, w0 is the amplitude of initial deflection. l is The maximum stress is length of beam-column. x is longitudinal coordinates. P M P (︂ M )︂ σ = + max = 1 + max (44) The total deflection is given by wt = w1 + w, where w max A W A ρ is bending deflection. P (︂ P w )︂ e π2 Pe The differential equation of beam-column is given by = 1 + E 0 + + A ρ(PE − P) ρ 8ρ Pcr d2w (︁ πx )︁ (EI)ps + Pw = −P w sin + e (40) with dx2 0 l (EI) Where, P is axial compressive load; (EI)ps = Dps is the W = ps ; ρ = W/A flexural stiffness of plate-stiffener combination; e is the ec- El1zl centricity of load. where zl is the maximum vertical coordinates measured Particular solution of Eq. (40) is given by from neutral axis, A is the cross-sectional area. πx πx Adopting the approximation: w = C sin + D cos − e (41) p l l P P cr ≈ 1 + Substituting into Eq. (40), we obtain Pcr − P Pcr [︂ (︂ π2 )︂ ]︂ πx (︂ π2 )︂ πx C k2 − + k2w sin + D k2 − cos = 0 Substituting into Eq. (44) yields: l2 0 l l2 l [︂ (︂ 2 )︂ ]︂ P e + w0 π e P 2 σmax = 1 + + w0 + (45) where, k = P/(EI)ps. Solving for C and D, we have A ρ 8 ρPcr (︂ 2 )︂ (︂ )︂ π Pcr Ultimate strength is reached when material failure oc- C = w0/ − 1 = w0/ − 1 k2l2 P curs:

= ηw0/(1 − η); D = 0 [︂ (︂ 2 )︂ ]︂ e + w0 π e σu σu 1 + + w0 + = σmf (46) ρ 8 ρσcr Where, η = P/Pcr. General solution of Eq. (40) is in form of or simply πx w = A sin kx + B cos kx + ηw0/(1 − η) sin − e (42) 2 2 l σu + µσcr σu − ω(σcr) = 0 (47) Boundary conditions are where, x = 0, l w = 0 (43) ρ + e + w ρσ /σcr µ = 0 ; ω = mf (48) π2 e π2 e Substituting into Eq. (42), we get w0 + 8 w0 + 8 e A = − e cot kl; B = e The ultimate strength of composite stiffened panel sin kl with initial imperfection and eccentricity of load is given Solution of Eq. (40) is written as by

πx kl −µ + √︀µ2 + 4ω w = ηw0/(1 − η) sin − e cot sin kx + e cos kx − e σ = σ (49) l 2 u 2 cr πx e (︂ kl )︂ = ηw /(1 − η) sin + cos kx − − e 0 l kl 2 The ultimate strain is cos 2 σu Bending moment is determined as εime = (50) Eps

M = P(w + w1 + e) πx Pe (︂ kl )︂ = Pw /(1 − η) sin + cos kx − 0 l kl 2 cos 2 Maximum bending moment is 6 Longitudinal strength of

k2l2 composite ship hulls M = Pw /(1 − η) + Pe + Pe max 0 8 π2 P To simplify the problem, the following hypotheses are = Pw0/(1 − η) + Pe + Pe 8 Pcr made: 164 Ë X. Zhang et al.

Table 1: Dimensions of stiffened composite panels at deck and side

Name a b b1 b2 b3 h tp tb tc Size(mm) 600 300 18 62 50 60 4.1 3.5 2.1

Table 2: Dimensions of stiffened composite panels at bilge

Name a b b1 b2 b3 h tp tb tc Size(mm) 600 400 38 124 100 90 7.2 6 3.1

1. The hull girder is assumed as an Euler-Navier beam. material, and the material properties are shown in Table 3. The transverse cross-sections of ship remain plane when subjected to bending moments and the influ- The strength coefficients for Tsai-Wu failure criterion ence of transverse restraint on longitudinal stress is can be computed from the failure strengths in Table 3. negligible. When the ship is under sagging, the stiffened panels in 2. Longitudinal failure only occurs between two adja- the deck are in compression and those in the bilge are in cent transverse frames. tension. Tables 1 and 2 show the dimensions of stiffened 3. The failure of each stiffened composite panel is as- panels at bilge are larger than those at deck. It is certain sumed to occur individually and independently. that the buckling and material failure of the stiffened panels at deck will be prior to that at bilge. So the first step is The normal strain ε varies linearly across the perpen- to calculate the buckling and material failure strain of the dicular direction of cross-section. stiffened panels at deck. M ε = z κ = z (51) The sectional properties of plate-stiffener combina- i i i D gir tion and ship hull are computed and shown in table 4. where z is the coordinates measured from neutral axis; κ The plate-stiffener combination is assumed to be sim- is the curvature; M is bending moment. ply supported at the ends. Initial deflection w0 is taken The ultimate strength of ship hull is defined as as 10 mm and eccentricity of load e as 5 mm, The buckling failure strain εbf and material failure strain εmf at Dgir εu Mu = (52) deck and ultimate strain with initial imperfection and load zps max eccentricity εime are evaluated. Then the buckling failure strength Mbf , material failure strength Mmf and ultimate strength Mime of ship hull under sagging are obtained from 7 Numerical example Eq. (52). The results from present analytical solution are showed in table 5. It indicates that the initial deflection of A longitudinally framed structure ship hull built of fiber- ship hull and the eccentricity of load can remarkably de- reinforced plastic was considered. The cross-section of crease the bending capacity of ship hull. The present ana- mid-ship is shown in Fig. 1. Breadth moulded is 9 m; Depth lytical method has been correlated with simulations using moulded is 6 m. The thicknesses of the plate are 4.1 mm the nonlinear finite element method described in [14]. The at deck and side, and 7.2 mm at the bilge, respectively. results indicate that it gives very similar results to that of The geometry of the stiffened composite panels at mid- the FEM. ship section is shown in Fig. 2. The dimensions of the stiffened panels at deck and side are shown in Table 1. The dimensions at bilge are shown in Table 2. The skin of 8 Conclusions deck is made of 32 symmetrically stacked unidirectional plies [90/45/90/-45/0/45/90/90/-45/0/45/90/90/-45/0/-45]s. An analytical model accounting for the effects of initial im- The measured ply thickness is 0.13 mm. The stacking se- perfection of ship hull and eccentricity of load is presented quence of stringer is [90/45/90/-45/0/45/90/90]s and the in this paper to estimate the longitudinal strength of ship ply thickness is 0.13 mm. The head of the stiffener is re- hull in composite materials under buckling failure, ma- inforced and made of 17 plies [90/45/90/-45/0/45/90/90/- terial failure and ultimate collapse of the deck, based on 45/0/45/90/90]s. The ship is made of one kind of composite Ultimate Longitudinal Strength of Composite Ship Hulls Ë 165

Table 3: Material properties of composite material

Property Value

Elastic modulus in 1 principal material direction E1 15.7 GPa Elastic modulus in 2 principal material direction E2 14.8 GPa Poisson’s ratios 12 ν12 0.127 Shear modulus in 2-3 principal material plane G23 0.34 GPa Shear modulus in 1-3 principal material plane G13 0.34 GPa Shear modulus in 1-1 principal material plane G12 0.34 GPa Tensile strength in 1 principal material direction XT 238 MPa Compressive strength in 1 principal material direction XC 204 MPa Tensile strength in 2 principal material direction YT 210 MPa Compressive strength in 2 principal material direction YC 224 MPa Shear strength in 2-3 principal material plane R 23.5MPa Shear strength in 1-3 principal material plane S 23.5MPa Shear strength in 1-2 principal material plane T 104MPa

Table 4: Sectional properties of plate-stiffener combination and ship hull

Property Value −3 2 Sectional area of plate-stiffener Aps 1.73×10 m 4 2 Bending stiffness of plate-stiffener (EI)ps 1.31×10 Nm 5 Shear stiffness of plate-stiffener (GA)ps 6.5×10 N Effective elastic modulus of plate-stiffener Eps 15.5GPa Neutral axis Coordinates of mid-ship hull zcgir 2.917m 10 2 Flexural stiffness of hull girder Dgir 1.736×10 Nm

Table 5: Results of failure strain and ultimate strength

failure strain (×10−3) ultimate strength (×107Nm) buckling material Imperfection buckling material Imperfection failure failure Present analytical 8.63 13.8 4.3 4.88 7.8 2.39 Finite element 9.4 14.2 4.6 5.31 8.03 2.43 accurate strain computation of stiffened composite pan- mate strength was investigated. The results show that the els with a simple analytical method. In the present model, initial deflection of ship hull and the eccentricity ofload detailed procedures to evaluate the stiffness of symmetri- can dramatically decrease the bending capacity of the ship cal balanced laminate, composite beam-columns and ship hull. Owing to lack of information on ultimate strength of hull girder is described. Because of the special characteris- composite ship hulls, the present method was compared to tics of composite material, the buckling and material fail- the finite element simulation. The results indicate present ure may be the important failure modes of ship hull in analytical results have a good agreement with FEM meth- composite materials. The Tsai-Wu criterion is adopted to ods and the present approach provides a simple but useful identify the material failure of stiffened composite pan- tool for the longitudinal strength estimation of ship hull in els. Because of the lower stiffness of composite materials composite materials. at transverse direction, the critical buckling stress including shear deformation is adopted. A longitudinally framed Acknowledgement: This work is supported by Grant no. structure ship hull in composite materials made of sym- 51479206 of National Natural Science Foundation of China. metrically stacked unidirectional plies under sagging is analyzed as an example application. The longitudinal ulti- 166 Ë X. Zhang et al.

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