Ultimate strength of composite ships’ hull girders in the presence of composite superstructures

Thin-Walled Structures 102 (2016) 122–138

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Ultimate strength of composite ships’ hull girders in the presence of composite superstructures Fattaneh Morshedsoluk, Mohammad Reza Khedmati n Department of Marine Technology, Amirkabir University of Technology, Tehran 15916-34311, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 26 May 2015 Received in revised form 29 October 2015 Accepted 24 January 2016

An extended formulation of the Coupled Beam Theory (CBT) developed by the authors is employed in order to calculate the ultimate strength of composite ships taking into account of the effect of the superstructure. A nonlinear finite element method is applied for solving the equilibrium equations. Behaviour of the stiffened composite panels in tension and compression is modelled by using progressive failure method. Both hull and superstructure of the ship are modelled using beam elements. Connection between beam elements representing hull and superstructure is made using specially developed spring box elements. Accuracy of the extended method is demonstrated using an available experimental result and also the results of finite element analysis. Also, a set of composite ships having different lengths of superstructure is generated and analysed. Efficiency of the composite superstructure in contribution to the ultimate bending strength of the composite ships is finally evaluated. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Composite ship hull Composite superstructure Progressive failure Coupled Beam Theory (CBT) Ultimate strength

1. Introduction Laminated composites can be used in different structures in the fields of aerospace, marine and civil engineering. These sorts of composites are generally assemblies of some layers of fibrous composite materials, which can be joined together with the aid of adhesives or resins, in order to provide required engineering properties, including in-plane stiffness, bending stiffness, strength, and coefficient of thermal expansion. It should be emphasised that different structural arrangements of laminated composites including single-skin, stiffened skin and sandwich panels are used in the engineering structures. Application of such materials in ship structures dates back to the late 1970s. Initially, small boats and ships’ topsides were being built of such composite materials. Over the time, usage of the composite materials in ship construction continued to grow and in recent years, some longer vessels like frigates and passenger ships are made of laminated composite materials. Having larger composite ships in length necessitates assessment of their ultimate bending strength in the early stages of structural design. In most of the steel ships, there is no superstructure in the amidships region and thus, the effect of superstructure on the ultimate strength of the ship is negligible. However, composite superstructures are often fitted in the amidships of the composite ships. This leads to the significant contribution of n

Corresponding author. E-mail address: [email protected] (M.R. Khedmati).

http://dx.doi.org/10.1016/j.tws.2016.01.024 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

the composite superstructure in the bending strength of the composite ship. The ultimate strength of steel ships has been widely investigated by many researchers around the world. Caldwell [1] was the first who estimated the ultimate strength of steel ships employing the fully plastic bending theory of the beams. However, he did not consider the reduction in the load-carrying capacity of structural members after they attain their corresponding ultimate strengths. Smith [2] proposed an approach for calculation of the ultimate strength of the ships. He first divided the ship's cross section into different unstiffened/stiffened plate panels, and then performed a progressive collapse analysis under bending on it assuming that the cross section remains plane after bending and each of the panels behaves according to its corresponding average stress–average strain relationship. Finite element method was applied by Smith [2] in order to obtain the average stress–average strain relationships for unstiffened/stiffened panels. Other researchers made some attempts for derivation of the average stress-average strain relationships for ships’ unstiffened /stiffened plate panels subject to in-plane compression alone or in combination with other loads using analytical approaches. Among them, reference may be made to the work of Khedmati [3]. In all of above-mentioned research studies, the ultimate strength of the ships is calculated by ignoring the effect of the superstructure. On the other hand, the available theoretical methods to estimate the ultimate strength of the ship's hull with a superstructure are mostly based on the simple beam theory or two-beam theory. Mackney and Rose [4] have studied the effect of the superstructure on the longitudinal strength of a ship both

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

Nomenclature

Ai A Β B Cij

dik D eij

Eit

EAi EAit EAij EIi EIit EIij EXi

EXit EXij

Hi Ii i k mj kEAi kEIi kEXi K k ij(x ) L Ls Mi Mx M100

Cross-sectional area of the i-th beam Constant Matrix Diagonal matrix containing the singular values Constant Matrix Bending moment lever on the i-th beam due to the dij j >force i shearing between the i-th and j-th beams ={ 0 j = i Distance the upper fibre of the beam to the −eij j
experimentally and numerically. The simple beam theory and finite element method were applied in their study, while they ignored the effect of the connection between the hull and superstructure. Naar et al. [5] proposed a new approach called Coupled Beams Method (CBM) to evaluate hull girder response of passenger ships. This method is based on the assumption that the global bending response of a modern passenger ship can be estimated with the help of beams coupled to each other by distributed longitudinal and vertical springs. To solve the governed equations, Naar et al. [5] proposed an analytical method that was only applicable when the superstructure is as long as the ship's hull. Very few publications can be found in the literature addressing the issue of ultimate strength of composite ships. Chen et al. [6] were the first who tried to estimate the ultimate strength of composite ships. They proposed a simple analytical method for calculating the ultimate strength of composite vessels. In their method, the behaviour of composite panels was formulated with a simple formula. Chen and Soares [7] extended the above-mentioned method for calculating the ultimate strength of composite ships under bending moment. Two types of the failure modes

M0

Ni Ni[x] pij Qi qi R1 R2 sij R12 Tij U ui ui* uij V viM

vi viM *

vij Xi XT XC YT YC XS zi δiju δijv γs σ1 σ2 τ12 θji

123

Ultimate bending strength of the ship without any superstructure Axial force of the i-th beam Shape functions Transverse (vertical) distributed forces between the ith and j-th beams Shear force of the i-th beam External force of the i-th beam Residual Residual Longitudinal distributed shear forces between the i-th and j-th beams Shear strength in plane 12 Shear stiffness between the i-th and j-th beams Square and orthogonal matrix Axial displacement of the i-th beam Approximate axial displacement of the i-th beam Normal degree of freedom of j-th node from i-th beam Square and orthogonal matrix Transverse displacement of the i-th beam due to the bending Total transverse displacement for the i-th beam Approximate transverse displacement of the i-th beam due to the bending Vertical degree of freedom of j-th node from i-th beam First sectional moment of area of the i-th beam Tension strength in direction 1 Compression strength in direction 1 Tension strength in direction 2 Compression strength in direction 2 Nodal displacement vector of system Distance of the i-th panel to the reference line Relative axial displacement Relative transverse displacement Coefficient of efficiency of the superstructure or superstructure effectiveness coefficient Normal stress in direction 1 Normal stress in direction 2 Shear stress in plane 12 Tangential degree of freedom of j-th node from i-th beam

were considered in their study; the panel buckling as well as fracture of the composite materials. Later, Chen and Soares [8] estimated the reliability of composite ships under bending moment, using the first method proposed by Chen et al. [6]. To calculate the reliability of the reinforced plate buckling failure, the failure of the first layer of the reinforcing plate and the ultimate failure of the reinforced plate were considered. Finally, Chen and Soares [9] used Smith's method, which is a conventional approach and capable of calculating bending moment-curvature curves, in order to calculate the ultimate strength of composite vessels. Application of the composite materials to the construction of long ships is still a relatively new and growing subject, which needs more research to be performed on assessment of the ultimate strength of these types of the ships. Besides, most of the previous studies do not take into account of the effect of the superstructure on the ultimate strength of the ships. As it is understood from the above-mentioned review, ultimate strength of the ships taking the superstructure's effect into account have been studied in many aspects, the most of which are only devoted to the linear elastic material behaviours.

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Furthermore, there has been no investigation to study the ultimate strength of the ships possessing superstructures with a length smaller than the length of the ship. Therefore, diagnosing this shortcoming, the aim of the present study is to develop an approach to investigate effect of the superstructure on the ultimate strength of the composite ship. A finite element approach based on the method described by Naar et al. [5] is proposed. In this method, the composite ship structure is divided into the hull beam and superstructure beam, which are connected together by means of some springs. The beams are constructed of composite stiffened plate panels. The behaviour of each of the panels composing the hull and superstructure beams is represented by the so-called “average load–average strain curve”. This curve can portray the behaviour of the composite stiffened panel under either compression or tension loads, including local failure modes. The average load–average strain curves are derived using a progressive failure nonlinear finite element analysis, the details of which are described by Morshedsoluk and Khedmati [10]. Finally, the ultimate longitudinal strength of the composite hulls with composite superstructures of any length is obtained from the moment–displacement relationship, which is established by imposing progressively increasing load on the hull-superstructure girder.

2. Methodology of ultimate strength calculations The ultimate strength of a composite hull equipped with a composite superstructure is estimated based on the nonlinear form of the Coupled Beam Theory (CBT), which is aimed to evaluate the hull girder response of the whole ship under global bending loading condition [10]. According to the CBT, the ship structure is divided into several separate beams; superstructure beams and main hull beam. The beams are coupled to each other by means of a number of vertical springs and shear springs. The joining springs transfer shear forces and vertical forces among the adjacent beams. Modelling scheme and couplings among the beams are shown in Figs. 1 and 2, respectively. The Coupled Beam Theory is not readily applicable when assessment of the ultimate

Fig. 2. Couplings among the beams in the ship structure model.

strength of the hull girders. This fact is due to the nonlinearity of the stress distribution in the cross section of the hull girder during bending, which leads to ultimate failure of the ship hull. To overcome above-mentioned shortcoming of the Coupled Beam Theory, hull and superstructure beams are assumed to be consisting of several stiffened panels, the stiffness of which will reduce due to collapse during bending. Thus, the non-prismatic beam theory will be applied and the stiffness of the stiffened panels is gained from their corresponding mean stress- mean strain curves obtained by using progressive failure method. Finally, the governed nonlinear equations are solved by means of the finite element method. Linear and nonlinear forms of the governing equations of the Coupled Beam Theory are presented in this section. 2.1. Linear form of the governing equations of the Coupled beam theory To study the effect of superstructure on the ultimate strength of a ship hull girder, the whole ship structure is modelled using Coupled Beam Theory (CBT). A differential segment of the i-th beam with the internal/external/coupling forces acting on it is shown in Fig. 3. The internal forces that are known from beam theory include axial force Ni , shear force Q i and bending moment Mi . In addition, the coupling forces consist of vertical distributed forces pij and longitudinally distributed shear forces sij . The sub-

Fig. 1. Basic concept of discretisation of a multi-deck ship into a set of coupled beams according to the Coupled Beam Theory (CBT).

script ij represents the effect of the j-th beam on the i-th beam. The only external force acting on the segment is qi , which is the result of the difference between weight and buoyancy forces. All these loads change by their corresponding differential values towards the other section of the segment under consideration in the

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125

where the elements of the matrix of shear forces sij and also elements of the matrix of vertical forces pij are as follows

sij j > i sij = { 0 j = i −sij j < i

(3)

and

pij j > i pij = { 0 j = i −pij j < i

(4)

The equilibrium equation for the moments about z-axis gives n

2

∂ Mi ∂x

2

+

∑ pij +

∂(∑ Cijsij )

j=1

∂x

= qi

(5)

where the elements of the matrix C are

dij j > i Cij = { 0 j = i −eij j < i

Fig. 3. A differential segment of the i-th beam with internal/external/coupling forces acting on it.

i-th beam as shown in Fig. 3. Linear forms of the equilibrium equations and the equations of vertical coupling and shear coupling for the segment under study are all presented in Morshedsolouk and Khedmati [10]. In this study, the reference line is fixed to the deck position, which may differ from the centroid position of the cross-section. dik and eij are respectively representing the distance between the upper and lower fibres of the beam to the reference line. The equations of equilibrium for the forces acting on the i-th beam in both longitudinal and transverse directions are

∂Ni + ∂x

∂Q i + ∂x

n

∑ sij = 0 j=1

(1)

∑ pij = qi j=1

In addition to above equations, the coupling equations are used to define the interaction between beams. They include both shear coupling and force coupling. Shear coupling between two neighbouring beams is shown schematically in Fig. 4. Due to existence of the shear element with the shear stiffness Tij , displacement discontinuity δiju causes shear forces sij between beams. It is assumed that this shear force is constant over length dx. Thus, it may be considered as the response of distributed horizontal spring between the two neighbouring beams. Shear stiffness depends on the effective height Hij of the shear element and also its effective area. In this case, as shown in Fig. 4, the effective height is equal to the deck spacing. Therefore, the approximate shear force in the side shell or in the longitudinal bulkhead is equal to

sij(x) = Tij(x)δiju(x)

(2)

(7)

The relative displacement can now be formulated with the help of axial displacement u and deflection v M of beams as follows

δiju = uj + eji

n

(6)

where

viM

∂v jM ∂x

− ui + dij

∂viM ∂x

(8)

is the deflection of i-th beam caused by bending.

Fig. 4. Shear coupling between two neighbouring beams.

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Fig. 5. Vertical coupling between two neighbouring beams.

Substituting Eq. (8) into Eq. (7) results in the following equation for the shear force

sij = Tij(uj − Cji

∂v jM ∂x

− ui + Cij

∂viM ∂x

)

(9)

And the longitudinal shear stiffness matrix is

Tij = {

Tij j ≠ i 0

j=i

(10)

The next type of coupling is the vertical one. This type of coupling is of great importance when the superstructure is weakly supported, as well described by Bleich [11]. The interaction between i-th and j-th beams is described with distributed vertical springs, see Fig. 5. Vertical coupling force pij that depends on

Ni = − EXi

∂ 2viM ∂x2

+ EAi

∂ui ∂x

(13)

In which the parameters EAi and EIi are respectively the axial stiffness and the bending stiffness of i-th beam with respect to the reference axis, EXi is also the value that modifies the internal forces when the reference line differs from the line passing through the centroid of the cross-section. It should be noted that the matrices EAi , EIi and EXi are diagonal. The unknown parameters on the reference line would be ui and viM . Eliminating Mi and Ni from the above equations, results in the following equations n

− ∑ sij = j=1

∂u ∂ 2v M ∂ (EAi i − EXi i2 ) ∂x ∂x ∂x

n

∂(∑ cijsij )

∂ 2v M ∂u ∂2 (−EIi i2 + EXi i ) 2 ∂x ∂x ∂x

(14)

vertical coupling stiffness k ij and relative deflection δijv , is calculated as the difference between beam deflections vi and vj . Hence

qi −

Pij = kij(x)δijv(x) = kij(x)(vj(x) − vi(x))

(11)

The unknowns in Eqs. (14) and (15) are axial deflection ui and transverse deflection induced by bending viM .

Using the beam theory, the relations between the internal forces and displacements are defined assuming that the material follows the Hooke's law. If the values of axial displacement ui and

2.2. Nonlinear form of the governing equations of the Coupled beam theory

deflection viM are known for i-th beam, then the amounts of bending moment Mi and axial forces Ni can be determined as follows (see Crisfield [12])

Mi = − EIi

∂ 2viM ∂x2

+ EXi

∂ui ∂x

(12)

∑ pij − i=1

∂x

=

(15)

To calculate the ultimate strength of the ships considering the effect of the superstructure, nonlinear forms of the above-mentioned equations are to be first derived and then solved. The linear forms of these equations are again summarised in the third column of the Table 1. The equations in the third column of the Table 1 are transformed to their nonlinear forms as presented in the

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127

Table 1 Nonlinear and linear equations of the Coupled Beam Theory (CBT). Type of equations

Equation no.

Linear form

Nonlinear form

Equilibrium equations

1

∂Ni

∂ΔNi

2

∂x ∂Q i

3 Coupling equations

+ ∑nj = 1 sij = 0 +

∂x ∂ 2Mi ∂x2

∑nj = 1 pij

sij = Tij(uj − 2 Beam deformation equations

= qi

+ ∑nj = 1 pij +

1

∂x ∂ΔQ i

∂(∑ Cijsij )

∂v M j Cji ∂x

∂x

− ui + Cij

Mi = −

∂ 2v M EIi 2i ∂x

+ EXi

∂ 2v M i ∂x2

+ EAi

2

Ni = − EXi

∂v M i ) ∂x

+ ∑nj = 1 Δpij = Δqi

∂x ∂ 2ΔMi ∂x2

+ ∑nj = 1 Δpij +

Δsij = Tij(Δuj −

∂(∑ CijΔsij )

∂Δv M j Cji ∂x

∂x

= Δqi

− Δui + Cij

∂Δv M i ) ∂x

Δpij = k ij(Δvj − Δvi )

pij = k ij(vj − vi )

1

= qi

+ ∑nj = 1 Δsij = 0

∂ui ∂x

∂ui ∂x

ΔMi = − EIi

∂ 2Δv M i ∂x2

+ EXi

∂ 2Δv M i ∂x2

+ EAi

ΔNi = − EXi

∂Δui ∂x

∂Δui ∂x

fourth column of that table. For the nonlinear forms of the equations, the coordinate axis is fixed and independent of the beams’ neutral axes. Therefore the C matrix is remained unchanged during the calculations.

3. Average stress-average strain curves As it was mentioned earlier, the average stress- average strain curves of the stiffened panels determine the behaviour of these structural elements under compression or tension and consequently, they could affect behaviour of the global ship structure. Different analytical, numerical and experimental methods are available in order to obtain or reach these curves for unstiffened/ stiffened metallic panels. However, only numerical or experimental approaches have been developed or used in order to derive the average stress- average strain curves for unstiffened/stiffened composite panels. In this study, the average stress- average strain curves of the composite panels are derived using nonlinear finite element method in combination with the progressive failure analysis. 3.1. Progressive failure method Details of the progressive failure method that has been applied herein, are fully presented and validated by Morshedsolouk and Khedmati [13]. They utilised the ANSYS software [14] in their modelling attempts, owing to its capabilities for analysing most of the structural problems regardless of their diversity in structural components. Note that all pre- and post-processing steps of the modelling were carried out by a macro code written in the ANSYS Programming Design Language (APDL). The adopted algorithm for the nonlinear finite element method including progressive failure analysis is outlined in a flowchart shown in Fig. 6. After geometric modelling and descretisation of the panel and furthermore, applying boundary and loading conditions, nonlinear finite element method is employed in order to calculate stresses and nodal displacements at each of the gauge points of the elements. In the next step, the failure criterion index is computed. If occurrence of the failure is detected inside a layer of a typical element, values of the mechanical properties are reduced to zero for that specific layer. The above-mentioned procedure is followed again until the final failure of the whole panel is attained, when it becomes unstable. Tsai-Wu failure criterion is employed in the progressive failure analysis of the composite panels. The use of a precise mathematical equation for estimating the failure is strictly constrained to the number of the possible states of failure. It is also noted that the

Fig. 6. Applied algorithm for progressive failure analysis.

required parameters for two-dimensional failure criteria include longitudinal, transverse and tensile as well as shear strengths. As mentioned above, this study uses Tsai-Wu failure criterion that is in good harmony with experimental results. This criterion is reflected as Eq. (16). σ1, σ2 and τ12 are the normal stress in direction 1, normal stress in direction 2 and shear stress in direction 12, respectively.

(

σ2 σ2 τ 1 1 1 1 + )σ1 + ( + )σ2 + 1 + 2 + ( 12 ) + 2f12 σ1σ2 XT XC YT YC XT XC YT YC S12 ≥1

(16)

If the stacking sequences of the plate and stiffeners are not the same as each other or if there exist several types of stacking sequences in the structure under consideration, then the states of all layers should be investigated for all of the applied finite elements.

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Table 3 Mechanical properties of the material used in this study (Direction 1 is the laminate main direction and direction 2 is the direction normal to direction 1). Mechanical properties

Symbols Magnitude

Module of Elasticity in main direction of the material Module of Elasticity in direction normal to main direction of the material Shear Modulus in direction 12 and 13

E1 E2

15 GPa 13.5 GPa 3.45 GPa

Tensile strength in direction 1 Compression strength in direction 1

G12, G13 G23 XT XC

Tensile strength in direction 2 Compression strength in direction 2

YT YC

204 MPa 224 MPa

Shear strength in direction 12

S12

84 MPa

Shear strength in direction 13

S13 S23

84 MPa

Shear Modulus in direction 23

Fig. 7. Characteristics of the plate models with their stiffening members.

Later, the inferred stresses in each layer have to be transferred to the relevant coordinate system and finally, the magnitudes of stresses are put in the failure criterion. 3.2. Finite element model of composite panels The extent of plate models with their stiffeners is presented in Fig. 7. The extent of the models is so chosen that each model includes two transverse stiffeners and two longitudinal stiffeners. The geometric and dimensional characteristics of the selected stiffened plates are given in Table 2. Table 3 tabulates the mechanical characteristics of the applied composite. Fig. 8 shows a finite element model of a stiffened plate. Lines AB, BH, GH and AG represent the boundaries of the model and lines CD and EF demonstrate locations of the web frames or transverse stiffeners. Given that the transverse stiffeners are stiffer than the longitudinal ones; lateral deformation of the panels is assumed to be zero in the locations of the transverse stiffeners. The boundary conditions imposed on the stiffened plates are as follows:

 Symmetrical conditions are applied along the lines AG and BH.  In the case of odd and even dimensional ratios, periodic con 

tinuous or symmetry conditions are applied on transverse lines AB and GH in Fig. 8, respectively. Since transverse stiffeners are not modelled, the deformation of the plate points along z axis is constrained at the locations of the transverse web frames. Furthermore, longitudinal deformation of the model is coupled at the edge loaded by compressive force.

Owing to the fact that the stiffened plates are thin-walled structures, their out-of-plane stresses are negligible, while their in-plane stresses are determinant. Hence, modelling of these components would be precise enough when employing the shell elements. Therefore, the Shell-181 element, that is appropriate for Table 2 Dimensions of the studied stiffened panel models. Parameter

Magnitude

Unit

a bf d b2 b3 h1 h2 h3 h5

300 1000 18 62 60 50 10 10 12

mm mm mm mm mm mm mm mm mm

Shear strength in direction 23

3.45 GPa 238 MPa 210 MPa

84 MPa

modelling thin and relatively thick plates and is constructed based on Classic Plate Strain and Mindlin theories, was used in the analyses utilising ANSYS software. These elements are composed of 4 nodes, where each node has 6 degrees of freedom. The element is appropriate for both linear and nonlinear solutions with large deformations and great angle variations. Also, this element can be used for modelling composite materials. One of the important factors in the finite element analysis is mesh density that should be so selected to give acceptably precise solutions. Excessive increase in the density of meshing will dramatically increase the time required for the solution. Therefore, the density should be determined accurately. Based on the results of numerous analyses performed by the authors, the density with 40 longitudinal meshes and 15 transverse meshes is finally chosen for the plates. Also, the density with 2 meshes in the height of web, 2 ones in the width of flange and 40 meshes in the length is considered for stiffeners.

4. Application of the finite element method Nonlinear equations presented in Table 1, are solved using Finite Element Method (FEM). Detailed of the stiffness matrices for linear elements and also for spring box elements are derived and given in Morshedsolouk and Khedmati [10]. Using the incremental nonlinear form of the equations given in Table 1, the nonlinear stiffness matrix can be derived in the same way. The adopted concept for discretisation of ship structure into different beam and spring elements is shown schematically in the Fig. 9. As can be seen in Fig. 9, both hull and superstructure are modelled as beams consisting of a number of beam elements. In the connecting region between hull and superstructure, the nodes are so located to have the same abscissa. The beam elements are of three-node type, having a total number of six degrees of freedom, so that the variations in the axial force can be easily considered. The reference line is considered at the deck level. The beams representing hull and superstructure are connected to each other using the so-called ‘spring box elements’. The stiffness matrix of these spring box elements is derived using equilibrium conditions. Any spring box element consists of 9 transverse springs and also 9 shear springs, Fig. 10. The transverse springs and shear springs are simulating respectively vertical forces and shear forces acting between the two beam elements; one inside the hull and the other one inside the superstructure. 4.1. Derivation of stiffness matrices The governing equations to be solved are all summarised in Table 1. Galerkin method is adopted in order to solve the set of

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129

Fig. 8. Dimensional parameters of the finite element model.

equations. Using the Galerkin method, finite element equations are formulated. Length of the ship is divided into m intervals. The first node is chosen at the after perpendicular position, while the last node is placed at the forward perpendicular. Each of the intervals includes three nodes. ui* and viM * are approximate solutions for the ui and viM functions in i-th interval. These approximate solutions are considered as linear combination of the corresponding nodal deflections in the following manner

ui* =

9

∑i = 7 xijNj(x)

viM * =

(17)

6

∑i = 1 xijNj(x)

(18)

Degrees of freedom in above equations are as follows

X i = [ x1i x2i x3i x4i x5i x6i x7i x8i x 9i ] = [ v1i v2i v3i θ1i θ2i θ3i u1i u2i u3i ] where θji , vij and uij are nodal angular displacement and nodal displacements of the i-th beam in normal directions, respectively. The shape functions Nj(x ) are obtained as follows

N1[x] = 1 −

Fig. 9. Adopted concept for discretisation of ship structure into different beam and spring elements.

23x2 66x 3 68x 4 24x5 + − + L2 L3 L4 L5

N2[x] =

16x2 32x 3 16x 4 + + 2 3 L L L4

N3[x] =

7x 2 34x 3 52x 4 24x5 − − − 2 3 4 L L L L5

N4[x] = x −

6x 2 13x 3 12x 4 4x 5 + − + 4 2 3 L L L L

2 3 4 5 { N5[x] = − 8x + 32x − 40x + 16x 2 3 4 L L L L

N6[x] = −

5x 3 8x 4 4x 5 x2 + 2 − 3 + 4 L L L L

N7[x] = 1 − N8[x] =

4x 4x 2 − 2 L L

N9[x] = − Fig. 10. Different components within any of spring box elements.

3x 2x 2 + 2 L L

x 2x 2 + 2 L L

(19)

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Fig. 11. Adopted procedure for the ultimate strength calculations.

Table 4 Dimensions of the model tested by Mackney and Rose. Parameter

Magnitude

Unit

Hull length Hull breadth Hull depth Superstructure depth Superstructure length

2 0.25 0.167 0.116 0.5

m m m m m

Fig. 12. A schematic view of the Mackney and Rose test specimen.

∫x

The residuals would be

x2

Nm(x)(

1

R1 =

∂u * ∂ 2v M * ∂ (EAit i − EXit i 2 ) + ∂x ∂x ∂x ∂ 2v M * (−EIit i 2 2

2

R2 =

∂ ∂x

∂x

+ EXit

n

∑ j = 1 sij*)dx = 0

m (22)

= 7, 8, 9

n

∑ j = 1 sij*

∂ui* )+ ∂x

∂u * ∂ 2v M * ∂ (EAit i − EXit i 2 ) + ∂x ∂x ∂x

∂(∑ cijsij*) ∂x

(20)

=

∫x

1

n

∑ j = 1 pij* − qi*

x2

(21)

Based on the Galerkin method, in order to have the least amount of error, the functions ui* and viM * have to satisfy the following equations

Nm(x)(

∂(∑ cijsij*) ∂ 2v M * ∂u * ∂2 (−EIit i 2 + EXit i ) + + 2 ∂x ∂x ∂x ∂x

− qi*)dx = 0

n

∑ j = 1 pij*

m = 1, …, 6

Integrating Eqs. (22) and (23) by parts results in

(23)

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131

Fig. 13. Model built in Hull_Superstructure_Interaction software for Mackney and Rose test specimen.

Fig. 14. Comparison of the results for Mackney and Rose test specimen.

Fig. 16. Comparison of the bending moment-curvature diagrams for the composite ship hull with the section as assumed by Chen and Soares in the sagging condition.

d * d Mi (x))0L − ( Nm(x)Mi*(x))0L dx dx L d2 ∂ 2v M * ∂u * + Nm(x)(EIij i 2 + EXij i )dx= 2 0 dx ∂x ∂x

(Nm(x)

∫

∫0

L

Nm(x)qi*(x)dx −

∫0

L

Nm(x)Kij(x)(Δv*j (x) − Δvi*(x))dx

− (Nm(x) ∑ cijsij*)0L + {

M

M* dv * d2 *j − Cji j − ui* − Cij dvi ))dx ( ) ( )( N x T x u m ij dx dx dx2 L * ∂u ∂ 2v M * d d (Nm(x) Ni*(x))0L − Nm(x)(EAii i + EXii i 2 )d 0 dx ∂x dx ∂x

∫0

L

(

∫

x= −

∫0

L

(Nm(x) ∑ Tij(u*j − Cji

dv jM * dx

dv M * − ui* − Cij i ))dx dx

(24)

Where, the stiffness matrices for the beam elements and spring box elements can be easily obtained using Eq. (24). 4.2. Assembling stiffness matrix After derivation of stiffness matrices for hull beam elements, superstructure beam elements and also spring box elements, global stiffness matrix of the whole ship structure is to be assembled. Since the shear force and bending moment are zero at both ends of the ship, in order to solve finite element equations, singular points should be eliminated so that rigid body motion of the ship is prevented. The resulting set of the finite element equations is then solved. 4.3. Stiffness matrix of the beam elements Based on the Eq. (24), the stiffness matrix elements for the i-th beam element having nodes 1, 2 and 3 with degrees of freedom u,

Fig. 17. Comparison of the bending moment-curvature diagrams for the composite ship hull with the section as assumed by Chen and Soares in the hogging condition.

v M and θ per each node would take the following form

∫0

L

2 d2Ni d Nj

dx 1 ≤ i, j ≤ 6 dx2 dx2 2 L d2N d Nj Kij = Kji = − EX (x) 2i 2 dx 1 ≤ i ≤ 6, 6 ≤ j ≤ 7 0 dx dx 2 L d2Ni d Nj Kij = Kji = − EA(x) 2 dx 7 ≤ i, j ≤ 9 0 dx dx2 Kij = Kji = −

EI (x)

∫ ∫

(25)

Since the sectional properties of a ship are generally variable along its length, it would be more accurate if the sectional properties of the beam elements can also vary along their length. Therefore, the quantities EAit , EXit and EIit are defined as follows

Fig. 15. Model created in the Hull_Superstructure_Interaction software for analysing the composite ship hull with the section as assumed by Chen and Soares.

132

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

−Cij2TijA

CijTijB

CijCjiTijA −CijTijB

CijTijB

−TijD

−CjiTijB

TijD

2

CijCjiTijA −CjiTijB −Cji TijA

CjiTijB

−CijTijB

CjiTijB

−TijD

KShear = [

TijD

1≤k≤6

∫0

1 ≤ m ≤ 6 [−

∫0

7≤m≤9 [

L

∫0

∫0

∫0

[ Ultimate bending moment [MN m]

Error [%]

Chen and Soares Chen and Soares FEM Present method

2008 2007 2013 2013

126 125 140 133

– 0.8 11.6 5.7

′ CijTijNm Nkdx]

L

L

kijNkNmdx]

L

0

=[

0 kijE 0

Year

Ultimate bending moment [MN m]

Error [%]

Chen and Soares Chen and Soares FEM Present method

2008 2007 2013 2013

185 178 200 197

– 7.8% 8.1% 5.9%

∫0

L

L

[

L

[

TijNmNkdx]

[−

′ CjiTijNm Nkdx] [−

∫0 ∫0

∫0

TijNmNkdx]

[

′ Nk′d CijCjiTijNm L

L

L

′ CjiTijNm Nkd ′ Cji2TijNm Nk′

′ CjiTijNm Nkd (28 )

0

∫0

L

[

kijNkNmdx]

0 0

kijE

0 [−

∫0

L

kijNkNmdx] 0

0 0

]

0 0

0

0 0 0 ] 0 −kijE 0 0

0

0

(29)

4.5. Elimination of singular points in the solution of finite element equations Global stiffness matrix of the whole ship is a singular matrix. This is due to the fact that there are not enough support constraints to prevent rigid body motion of the whole ship structure. In such cases, Singular Value Decomposition (SVD) procedure may be proposed to solve the problem. Any square matrix like global stiffness matrix of system (whole ship structure), K, can be written as below as a product of three matrices:

EI (x) = EI1N7(x) + EI2N8(x) + EI3N9(x) EX (x) = EX1N7(x) + EX2N8(x) + EX3N9(x) (26)

K = KEA1EA1 + KEA2EA2 + KEA3EA3 + KEX1EX1 + KEX2EX2 + KEX3EX3 + KEI1EI1 + KEI2EI2 + KEI3EI3

∫0

∫0

∫0

L

Details of A, B, D and E matrices are given in the Appendix A.

Method

EA(x) = EA1N7(x) + EA2 N8(x) + EA3N9(x)

[−

1≤k≤6

′ Nkdx] CijTijNm

0

kijNkNmdx]

−kijE 0

Table 6 Comparison of the ultimate strength results for the hogging condition.

∫0

[

′ CijTijNm Nkdx]

0

KTrans = [

Year

′ Nk′dx] Cij TijNm

L

and

[−

Method

7≤k≤9

2

′ CijCjiTijNm Nk′ dx] [−

∫0

7 ≤ m ≤ 9 [−

Table 5 Comparison of the ultimate strength results for the sagging condition.

L

[

1≤m≤6 [

Fig. 18. Finite element model of the composite ship hull with the section as assumed by Chen and Soares.

L

]=

(27)

The vector of degrees of freedom for the i-th beam element is [ v1i v2i v3i θ1i θ2i θ3i u1i u2i u3i ]T and the relevant matrices KEAj , KEXj and KEIj are constants, given in the Appendix A. 4.4. Stiffness matrix of the spring box elements Spring box elements are used in order to simulate the connection between beam elements of the hull and beam elements of the superstructure. Forces acting on the beam elements coming from the spring box elements are Σjn= 1sij , Σjn= 1pij and Σjn= 1cijsij , which are respectively representing axial force per unit length, transverse force per unit length and bending moment per unit length. Any of the spring box elements have 6 nodes, a total of 18 degrees of freedom, shear stiffness and transverse vertical stiffness. The shear stiffness matrix and transverse vertical matrix for such elements would be

K = U . Β. V T

(30)

In which U and V are square and orthogonal matrices. Also, Β is a diagonal matrix that contains the singular values. Symmetry of global stiffness matrix, K, imposes equality of the two matrices U and V. Also, since U is orthogonal, then its transpose, UT, will be equal to its inverse, U 1. Besides, inverse of the diagonal matrix, Β , would form another diagonal matrix called Β−1, in which the diagonal elements are just the reciprocals of the diagonal elements of the matrix Β . The problem exists when the system consists of one or more singular values. This means that the value of one or more of the diagonal elements is very small compared to the largest value of other diagonal elements of the matrix Β . As a result of that, the reciprocal of such diagonal elements having very small values would have a very large value, which tends to distort the numerical solution, sending it off to infinity along a direction which is spurious. A good approximation is to throw these spurious directions away completely by setting 1/Βii for the offending singular values to zero. Therefore, K 1 can be written as follows:

K −1 = V . Β−1. UT

(31)

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

133

Fig. 19. Typical cross section of a composite ship hull with a composite superstructure.

Now, the final solution of the finite element equations takes the following form: −1

T

Xs = V . Β . U . F

(32)

where F and Xs are vector of external forces and vector of nodal displacements of the system, respectively.

4.6. Solution procedure The whole process of estimating the ultimate strength consists of 2 different stages; development of the average stress- average strain curves for the unstiffened/stiffened panels composing the ship's hull and its superstructure and then, estimation of the ultimate strength of the whole ship structure. As it was already stated, the average stress- average strain curves are herein

134

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

Vertical Deflection [mm]

120 100 80 60 40 20 0 Fig. 20. Loading condition.

developed by using progressive failure analysis, as documented by Morshedsolouk and Khedmati [13]. As a summary, it can be simply stated that the ship structure is first divided into separate beams. The beams represent ship's hull and its superstructure. In the next step, cross-section of each beam is divided into several unstiffened/stiffened panels. Then, the geometry and information related to the panels are extracted and based on them; the input files for performing progressive failure analysis are created. The input files are now imported to a code capable of doing nonlinear finite element analysis implementing the progressive failure modelling. In the final step, a MATLAB code called ‘Hull_Superstructure_Interaction’ is programmed in order to calculate the ultimate strength of the whole ship based on the Coupled Beam Theory (CBT). Flow of the whole process is shown in Fig. 11.

Ls=0 m Ls=10 m L=16 m Ls=20 m Ls=24 m Ls=30 m

0

-50

-100

-150

-200

-250

Bending Moment [MN.m] Fig. 22. Curves of vertical deflection versus bending moment for different lengths of the superstructure.

5. Verification To verify the developed method, the code Hull_Superstructure_Interaction is tested against the experiment made by Mackney and Rose [4], studies of Chen and Soares [9] and the Finite Element

Fig. 23. Values of the superstructure effectiveness coefficient for different relative lengths of the superstructure.

Fig. 21. Vertical deflections of the hull and superstructure (left) and positions of the failed sections for different superstructure lengths (right).

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

Method. Mackney and Rose performed a set of experiments on a scaled model of a ship possessing a superstructure. The scale factor of the model was 1:60 and it was made of Lucite “L” Acrylic cast. A schematic picture of the model is shown in Fig. 12. Dimensions of the model are also given in Table 4. It should be noted that the whole model was tested under pure bending. Mackney and Rose test specimen was modelled in the software Hull_Superstructure_Interaction as two independent beams, which are connected by spring boxes, Fig. 13. According to description of the experiments made by Mackney and Rose [4], the applied force at the both ends of the structure created a pure bending. Therefore, herein, bending moment is also applied in the same way at the both ends of the structure. Considering the fact that Mackney and Rose performed their test in the elastic range and no failure occurred in the model, mechanical properties of the whole structure, including EI, EX and EA are considered constant throughout the loading. In this study, the hull deflection is calculated and compared with the values provided by Mackney and Rose [4]. Results of this comparison are presented in Fig. 14. In that figure, the solid line with square marks represents the amount of deflection calculated by the Hull_Superstructure_Interaction code and the line with a hollow square marks represent the amount of deflection experienced during Mackney and Rose test. It can be realised that the results of both models are similar in the central areas and the amount of error in the calculations compared with the experimental results is around 1.57 percent. The calculation error is slightly higher in the support areas, and it is about 8.62 percent. This is mainly because the loading procedure was not clearly specified by Mackney and Rose [4] in these areas. As it was mentioned earlier, Chen and Soares [9] have made some study on the evaluation of ultimate strength for the composite vessels without any superstructure. In their research, the ultimate strength of a composite ship hull was calculated using the Smith's method. The composite ship analysed by Chen and Soares [9] is also modelled in Hull_Superstructure_Interaction code as shown in Fig. 15. A comparison between the bending momentcurvature diagrams for both sagging and hogging load cases is presented in Figs. 16 and 17, respectively. In these two figures, solid lines with circular marks represent the solution of Chen and Soares [9], while the line with hollow circular marks are obtained from implementing Hull_Superstructure_Interaction code. In order to further verify the Hull_Superstructure_Interaction code, the structure of the composite ship hull studied by Chen and Soares [9] is also modelled in the commercial ANSYS software [14] and its ultimate strengths in both hogging and sagging conditions are determined. A perspective view on the finite element model built in the environment of ANSYS software can be seen in Fig. 18. Results of the ultimate strength calculations with the aid of ANSYS software are also compared with those obtained from Hull_Superstructure_Interaction code. Boundary conditions applied to the finite element model in Fig. 18 are such that the middle part of the ship's structure is under pure bending. Therefore, the degrees of freedom of the nodes located at the cross-section No. 2 and the cross-section No. 3 are vertically bound, while equivalent vertical forces are applied on the nodes located at cross-section No. 1 and cross-section No. 4. In the sagging mode, the upward vertical forces applied on each of the cross sections No. 1 or No. 4 are equivalent to 37,800 kN. However, these forces in the hogging condition are 55,500 kN in downward direction. It should be noted that the vertical displacement of the nodes located at the cross-sections No. 1 or No. 4 are coupled. Thus, in both cases, the middle region of the model is under pure bending. The results are compared with the results of previous works in Tables 5 and 6. It can be seen that

135

in all cases, the results suggest compliance between the present method and previous research studies. It may be worthy to note that the CPU time required on a PC with an AMD Athlon™ IIX4 620 2.6 GHz Processor with 2 GB RAM in order to calculate the ultimate strength of the composite ship hull studied by Chen and Soares [9] using the finite element method (here with the aid of commercial ANSYS software) was about 160 hours. Of course, this amount of time is not including the time devoted to creating the composite ship model in the finite element code. This is while; calculation of the ultimate strength of the same composite ship using developed Hull_Superstructure_Interaction code takes a CPU time of about 20 minutes only.

6. Case studies In order to perform case studies, typical composite ships with prismatic hull and superstructure are modelled. Length, breadth, width and height of the superstructure are assumed to be equal to 40 m, 9 m, 6 m and 4 m, respectively. The span of the longitudinal stiffeners is 1 m. A sample cross-section of the ships studied herein is shown in Fig. 19. It should also be noted that the composite hull is the same as that studied by Chen and Soares [9], while a composite superstructure has been produced and fitted on it. As it was already described, in order to estimate the ultimate strength of the ships having a superstructure fitted on their hull using the Coupled Beam Theory, the ship structure is to be divided into two separate beams so that each of the beams has to be descretised into 3-node beam elements. Parametric study is now carried out for five different superstructure lengths equal to 10 m, 16 m, 20 m, 24 m and 30 m. In all of these situations, superstructure is located in the middle of the ship. The loading condition used in this study is shown in Fig. 20. It can be simply realised that the loading condition is such that the resultant maximum bending moment lies in the middle of the ship. λ and L are load factor and length of the ship, respectively. The first case study is a ship with no superstructure. This case is used to calculate the superstructure efficiency and also it is regarded as a reference case for comparisons. It is clear that the maximum bending moment is attained at the midship section. Consequently, failure will occur in the midship section. Trend of the vertical deflection and also locations of the failed sections are shown in Fig. 21. The next set of the cases studied herein include superstructures with lengths equal to 10 m, 16 m and 20 m. These superstructures are short compared with the ship's length. As the curvature of the deflected superstructure in such cases is small compared to the ship's curvature, the amount of the bending moment sustained by the superstructure would not be so considerable. However, superstructure bears enough bending moment to shift the failed section to the adjacent sections near the ends of the superstructure and improve the ultimate strength of the ship by 6 percent. The last two case studies are devoted to the composite ships in which the length of the superstructure is relatively large with respect to the ship's length. In such cases, superstructure is either 24 m or 30 m long. The deflection curve and also locations of the failed sections are shown in Fig. 21. The curvature of the superstructure is near to the curvature of the ship's hull, so both of them contribute in the carriage of the bending moments. As can be found, the failed sections in these ships are concentrated at their midship regions. In order to summarise the effect of the superstructure's length on the ultimate strength of the composite ships, the curves of vertical deflection-bending moment for the above-analysed ships are also presented in Fig. 22. The ultimate strength of the ship with no superstructure is the lowest amongst all other case studies. In

136

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

other cases, the ultimate bending moment carried by the ship increases as the length of the superstructure is increased. The level of contribution or participation of the superstructure in the longitudinal strength of the ships is measured in terms of a quantity called the ‘Coefficient of Efficiency of the Superstructure’ or ‘Superstructure Effectiveness Coefficient’, which is defined as follows:

γs =

M100 − Mx × 100 M100 − M0

(33)

Mx , M100 and M0 are the ultimate bending strength of the ship under consideration, the ultimate bending strength of the same ship with a superstructure of 100 percent efficiency and the ultimate bending strength of the same ship without any superstructure, respectively. The trend of the superstructure effectiveness coefficient is shown in Fig. 23. Ls in Fig. 23 is representing the length of superstructure. As can be realised from the results depicted in the Fig. 23, for relative superstructure lengths below 0.4, effectiveness of the superstructure is almost equal to its relative length. As the relative length of the superstructure is increased beyond 0.4, its effectiveness coefficient is increased with a faster trend to a level of nearly 90 percent for the Ls/L equal to almost 0.7. It can be stated that the superstructure fully contributes to the ultimate bending strength of the whole composite ship, when its Ls/L has values higher than 0.7.

Appendix A

0 0 0 0 0 KEA1 = [0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

37 30L

− 15L

7 30L

8 5L

− 15L

22

0 0 0 0 0 0 − 15L 7 30L

0 0 0 0 0 0

0 0 0 0 0 KEA2 = [0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

6 5L

22

2

0 0 0 0 0 0

32 15L

2

16

6 5L

0 0 0 0 0 0

0 0 0 0 0 0 ]

0 0 0 0 0 0 1

2

2

8 5L

0 0 0 0 0 0 − 15L 7 30L

0 0 0 0 0 0

KEX1 = [

22

16

− 15L

7 30L 22

− 15L

− 15L

37 30L

237

988

277

105L2

105L2

0

0

0

0

0

0

0

0

0

0

0

0

−

0

0

0

0

0

0

−

0

0

0

0

0

0

92 35L

− 105L

0

0

0

0

0

0

32 35L

0 35L2

− The research reported in this paper are some part of the results obtained in the PhD studies of the first author under supervision of the second author in the Department of Marine Technology of Amirkabir University of Technology in Tehran, Iran. The research had not any funding support.

0 0 0 0 0 0

0 0 0 0 0 0 − 10L − 15L

237

Acknowledgements

0 0 0 0 0 0

2

16

0 0 0 0 0 0 − 15L − 15L

0 0 0 0 0 0

0 0 0 0 0 0 ]

16

A method based on the Coupled Beam Theory (CBT) is presented in order to predict the ultimate longitudinal strength of composite ship hulls considering superstructure effect. The behaviour of the composite panels in the ship structure is deduced from their mean stress-mean strain curves. The progressive failure method in conjunction with the nonlinear finite element method is used to calculate mean stress- mean strain curves. Verification study shows that the method is accurate enough and it is also much faster than other methods. The CPU time on a PC with an AMD Athlon™ IIX4 620 2.6 GHz Processor with 2 GB RAM for calculating the ultimate strength of a typical composite hull without/with its composite superstructure using the finite element method is something about 160 to 180 h. Of course, this amount of time is not including the time devoted to creating the entire hull-superstructure model in the finite element codes. This is while, calculation of the ultimate strength of the same composite ship using the software developed based on the presented method takes a CPU time of about 20 min only. The results of some case studies have shown that the length of superstructure has significant effect on the ultimate strength of the composite ship and the location of the failed sections. In ships with a short superstructure (having a length roughly less than or equal to half of the ship's length) located at the middle portion of the ship's hull, the sections situating at the superstructure ends and their neighbouring regions are generally failed. Whilst, for the case of longer superstructures (with a length roughly more than half of the ship's length), the failed sections are often located near the midship section of the ship structure.

0 0 0 0 0 0

1

− 15L − 15L

0 0 0 0 0 0 − 15L

0 0 0 0 0 0

2

− 15L − 10L

7. Conclusions

0 0 0 0 0 KEA3 = [0

0 0 0 0 0 0 ]

0 −

32 5L2

92 105L2

−

32 15L2 53

105L2

356

16 21L

− 35L

64

32 35L

0

1 35L

8 105L

92 35L

92 35L

32 35L

1 35L

0

0

0

64

8 105L

0

0

0

11 − 105L

0

0

0

92 105L2

32

53

−

13 35L2

−

0

128

15L2

128 15L2

0

15L2

−

32 5L2

0

988

105L2

−

11 − 105L

105L2 277

35L2

105L2

356

− 105L − 35L 16 21L

32 35L

]

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

0

0

0

0

0

0

0

1024

15L2

21L3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

10 21L

15L2

−

104 105L2

−

21L3

− 35L

24

22 105L

128 35L

64 − 35L

0

0

0

0

0

0

0

22 105L

− 35L

10 21L

16

0

7L2 104

−

105L2

−

7L2

64

344

15L2

105L2

24

− 35L

64

22 105L

0

0

0

24 − 35L

128 35L

24 − 35L

0

0

0

22 105L

− 35L

10 21L

0

0

0

64

796

]

0

0

0

0

0

0

0

0

0

0

0

=[

0

0

0

2

105L 32

15L2

−

92

13 2

105L

−

2

35L

128

32

15L2

5L2

277

988

105L2

105L2

−

8 105L

1 35L

−

0

0

0

0

0

0

0

0

0

0

0

32 35L

− 35L

64

32 35L

0

0

0

0

0

0

16 21L

− 105L

356

92 35L

− 105L

32 35L

16 21L

0

0

0

8 105L

64 − 35L

356 − 105L

0

0

0

1 35L

32 35L

92 35L

0

0

0

53

32 15L2

92

−

277

128

988

15L2

105L2

13

32

35L2

5L2

−

105L2

−

11

105L2

237 35L2

1984

−

105L3

22

−

5L2

1984

2042

4

105L3

21L2

5632

1088

64

105L3

15L3

278 105L

−

2042

105L3

15L3

105L3

1088 3

15L

2042 105L3 2638

− −

5632 3

105L

1984 105L3

− −

768

2

2

105L

105L

35L

58

22

608

105L3

5L2

105L2

22

768

608

15L

35L2

105L2

20

64

7L2

21L2

0 0 0

15L2

2176

5L2

0 0 0

−

3

2176

−

105L2

416

1984

105L2

− 2 KEI1 = [ 105L 416 − 2

−

2638

−

4 21L2

0 0 0

−

20 7L2

64 2

21L

−

4 21L2

0 0 0

13 210

0 0 0

− 70

2419

2032

199

2032

2944

304

199 − 315L

304 − 315L

83 315L

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0 0 0 0 0 0

− 315L − 315L − 315L 0 0 0

]

101

199

2032

− 315L 0 0 0

0 0 0

0 0 0

8 15 11

− 210 B=[ 13L 420

L

− 315 − 126

8

L

L

− 35 − 420 8L

− 105 13L 420

− 420 − 35

L

2L 15

L 15

− 30

L 15

8L 15

L 15

L − 30

L 15

2L 15

D=[

4L

2L 45

101 210

L

13

1 70

4 7

L

8

− 21 − 210

− 15

16L 105

8 105

1 70

0

8L

− 105

0

278 105L

11 210

]

]

2419

1

− 70

4L

128L 315

L

4L

− 315

4L

− 315 2L 45

0 0 0

2032

0 0 0

13 210

− 210 − 126 − 315

4

0 0 0

199

− 315L − 315L

8

− 210 − 7

105L2

0 0 0

− 315L 0 0 0

− 105L − 105

13

2176

2944

0 0 0

8

−

35L2

− 315L − 315L

2

0 0 0

− 315L − 315L − 315L 0 0 0

−

768

22 5L2

304

416

− 21

−

− 315L 0 0 0

105L2

8 105

21L2

608 105L2

304

256

1

−

0 0 0 0 0 0 0 0 0

− 315L

105L2

0

0 0 0

83 315L

15L

210

1088

0 0 0

2638

2638

256

652

105L2

13 210

22

544

]

416

35L

512 105L

0 0 0

15L2

− 105L − 105L

256

0 0 0

544 − 315L

20

0 0 0

− 105L

8 45L

7L2

20

22

0 0 0

9658

7L2

256

0 0 0

105L3

2176

− 105L − 105L A=[ 8 13 − 105 9658

−

5632 − 315L

0 0 0

− 315L − 315L 0 0 0

105L3

−

544

544 − 315L

0 0 0

0 0 0

]

796 105L2

0 0 0

105L

−

64

8 45L

15L3

105L2

3L2

105L2

1088

−

832

12

105L2

2042

12 35L2

0

652

3L

105L3

64 3L2

35L2

796

64

35L2

0

−

−

− 315L − 315L

2

4 − 2 2 21L KEI3 = [ 21L 608 768 − − 2 2

237

0

58

12 35L2 64

0 0 0

105L3

−

21L3

832

0 0 0

11 − 105L

105L2

−

0

53

−

1112

−

0

35L2

− 0

832

3L

KEX3

−

105L2

12

−

0

105L2

1024

0

16

21L3

88

0

10 21L

21L3

21L3

0

104

832

344

− 2 KEI2 = [ 105L 64 − 2

796 105L2

1024

105L2

0

105L2

−

−

2048

16

KEX2 = [ 0

−

−

7L2

−

64 − 35L

64

21L3

64

64

0

15L2

88

21L3

104 105L2

0

344

1024

21L3

16 7L2

0

105L2

1112

344 105L2

137

]

0 0 0 0 0 0 0 0 0

]

138

F. Morshedsoluk, M.R. Khedmati / Thin-Walled Structures 102 (2016) 122–138

[4] 523L

4L

131L

19L2

− 3465

− 63

− 6930

− 2310

4L − 63

128L − 315

4L − 63

2L2 − 315

131L

4L

523L

29L

2

8L2 693

29L2 13860

0

2L2 315

8L

2

− 6930 − 63 − 3465 − 13860 − 693 E=[ 19L2 2L2 29L2 2L3 L3 − 2310 − 315 − 13860 − 3465 1155

2

8L

2

3

32L

3

[5] [6]

2

19L 2310 L3 4620

[7]

] [8]

3

8L 693

0

− 693

L 1155

− 3465

L 1155

[9]

29L2 13860

2L2 315

19L2 2310

L3 4620

L3 1155

2L3 − 3465

[10]

[11]

References [1] J.B. Caldwell, Ultimate longitudinal strength, Trans RINA 207 (1965) 411–430. [2] C.S. Smith, Influence of local compressive failure on ultimate longitudinal strength of a ship's hull, in: Proceedings of the International Symposium on Practical Design in Shipbuilding, Tokyo, Japan, 1977. pp. 73–79. [3] M.R. Khedmati, Simulation of average stress average strain relationship of ship

[12] [13]

[14]

unstiffened /stiffened plates subject to in plane compression, Sci. Iran. 12 (4) (2005) 359–367. M.D.A. Mackney, C.T.F. Rose, Preliminary ship design using one-and two-dimensional models, Mar. Technol. 36 (2) (1999) 102–111. H. Naar, P. Varsta, P. Kujala, A theory of coupled beams for strength assessment of passenger ships, Mar. Struct. 17 (8) (2004) 590–611. N.Z. Chen, H.H. Sun, C. Guedes Soares, Reliability analysis of a ship hull in composite material, Compos. Struct. 62 (1) (2003) 59–66 [4]. N.Z. Chen, C. Guedes Soares, Longitudinal strength analysis of ship hulls of composite materials under sagging moments, Compos. Struct. 77 (1) (2007) 36–44 [5]. N.Z. Chen, C. Guedes Soares, Reliability analysis of ship hulls made of composite materials under sagging moments, J. Mar. Sci. Technol. 12 (4) (2007) 263–271 [6]. N.Z. Chen, C. Guedes Soares, Ultimate longitudinal strength of ship hulls of composite materials, J. Ship Res. 52 (3) (2008) 184–193 [7]. F. Morshedsolouk, M.R. Khedmati, An extension of coupled beam method and its application to study ship's hull-superstructure interaction problems, Lat. Am. J. Solids Struct. 8 (3) (2011) 265–290. H.H. Bleich, Non-linear distribution of bending stresses due to distortion of cross-section, J. Appl. Mech. 29 (1952) 95–104. M.A. Crisfield, Non-linear Finite Element Analysis of Solids and Structures, Volume 1, John Willey & Sons, West Sussex, England, 1991. F. Morshedsolouk, M.R. Khedmati, Parametric study on average stress–average strain curve of composite stiffened plates using progressive failure method, Lat. Am. J. Solids Struct. 11 (12) (2014) 2203–2226. Swanson Analysis Systems Inc., ANSYS User's Manual (Version 7.1), Houston, 2003.