lr class direct - shipright...non-linear structural collapse analysis for plates and stiffened...

Non-linear Structural Collapse Analysis forPlates and Stiffened Panels

May 2016

ShipRightDesign and Construction

Additional Design Procedures

© Lloyd's Register Group Limited 2016. All rights reserved. Except as permitted under current legislation no part of this work may be photocopied, stored in a retrieval system, published, performed in public, adapted, broadcast, transmitted, recorded or reproduced in any form or by any means, without the prior permission of the copyright owner. Enquiries should be addressed to Lloyd's Register Group Limited, 71 Fenchurch Street, London, EC3M 4BS.

Non-linear Structural Collapse Analysis for Plates and Stiffened Panels, May 2016

CONTENTS Section 1 Introduction 1.1 Scope 1.2 Application Section 2 Analysis Procedure for Stiffened Panels 2.1 Workflow chart 2.2 Structural model 2.3 FE mesh and coordinate system 2.4 Boundary conditions 2.5 Applied loads 2.6 Geometrical initial imperfections 2.7 Solution on the non-linear FEA Section 3 Analysis Examples for Stiffened Panels 3.1 General 3.2 T stiffened panel Section 4 Appendix: Ultimate Strength of Unstiffened Plates 4.1 Structural model 4.2 FE mesh and coordinate system 4.3 Boundary conditions 4.4 Applied loads 4.5 Geometrical initial imperfection 4.6 Solution on the non-linear FEA 4.7 Analysis examples for unstiffened plates Lloyd’s Register is a trading name of Lloyd’s Register Group Limited and its subsidiaries. For further details please see http://www.lr.org/entities Lloyd's Register Group Limited, its subsidiaries and affiliates and their respective officers, employees or agents are, individually and collectively, referred to in this clause as ‘Lloyd's Register’. Lloyd's Register assumes no responsibility and shall not be liable to any person for any loss, damage or expense caused by reliance on the information or advice in this document or howsoever provided, unless that person has signed a contract with the relevant Lloyd's Register entity for the provision of this information or advice and in that case any responsibility or liability is exclusively on the terms and conditions set out in that contract.

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Non-linear Structural Collapse Analysis for Plates and Stiffened Panels ■ Section 1 Introduction 1.1 Scope

1.1.1 As part of the ShipRight Design and Construction procedures, Lloyd's Register (hereinafter referred to as LR) has developed and introduced a procedure for non-linear Finite Element Analysis (non-linear FEA). 1.1.2 This procedure is for performing structural collapse analysis of plates and stiffened panels. 1.2 Application 1.2.1 The procedure is to be applied on a non-mandatory basis at the request of the ship owner, ship designer, ship builder or LR where it is considered necessary to carry out non-linear FEA to determine the capacity of structural components under considered loads or stresses. 1.2.2 This procedure can also be used to derive load-shortening curves of plates and stiffeners for hull girder ultimate strength analysis. 1.2.3 The procedure is applicable to steel plated structures of all ship types. Other materials will be specially considered.


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■ Section 2 Analysis Procedure for Stiffened Panels 2.1 Workflow chart

2.1.1 The chart of the workflow for performing structural non-linear FEA is presented in Figure 1.2.1 Workflow chart for performing non-linear FEA.

Figure 1.2.1 Workflow chart for performing non-linear FEA

2.2 Structural model

2.2.1 The midship region of a ship can be considered as a prismatic section made up of stiffened panels, which consist of closely spaced secondary stiffeners supported by more widely spaced primary members. The buckling and ultimate strength of such stiffened panels are fundamentally important to the local and global strength of a ship. Therefore, stiffened panels are the basic components of the hull girder to be analysed. 2.2.2 The model extent of a stiffened panel to simulate the buckling behaviours of a stiffener in a non-linear FEA is illustrated in Figure 1.2.2 Illustration of the model extent of a stiffened panel: • In longitudinal direction (parallel to the stiffeners): ½+2+½ frame spacings. • In transverse direction (perpendicular to the stiffeners): 4 stiffeners (5 stiffener spacings).


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Note a = primary members spacing, b = stiffener spacing.

Figure 1.2.2 Illustration of the model extent of a stiffened panel

2.3 FE mesh and coordinate system

2.3.1 Element type: plate, stiffener web and flange should all be modelled using shell elements. The analysis may be carried out using 4-node shell elements. 2.3.2 Element size: The mesh should be fine enough to reflect the local deformations and stresses developed during buckling and collapse. The following criteria should be used: • Plate: eight elements between stiffeners. The number of elements in the longitudinal direction should be selected so that the

element aspect ratio is as close to the unit as possible. • Stiffener web: minimum three elements across the web depth and the element size should not be larger than the element in the

plate. • Stiffener flange: one element across the stiffener flange for angle- and bulb profiles and two elements for T profiles and

asymmetrical profiles with overhang. • It should be noted that the primary supporting members are not to be explicitly modelled, but their functions will be reflected by the

boundary conditions applied in the FE model. 2.3.3 The coordinate system: the system is in a right hand Cartesian coordinate system with x-axis (in stiffener direction) and y-axis (perpendicular to the stiffener) at the mid-surface of the plate thickness and z-axis is in the direction of the stiffener webs. A model and the system with FE mesh is shown in Figure 1.2.3 Finite element model, the coordinate system and edges and locations where boundary conditions to be applied for illustration.

Figure 1.2.3 Finite element model, the coordinate system and edges and locations where boundary conditions to be applied


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2.4 Boundary conditions

2.4.1 The definition of all edges and locations where boundary conditions need to be applied is shown in Figure 1.2.3 Finite element model, the coordinate system and edges and locations where boundary conditions. Four edges: B1, B2, B3 and B4; four corners: C1, C2, C3 and C4; and three primary supporting members: Fr1, Fr2 and Fr3, are to be considered. 2.4.2 All Four edges (B1, B2, B3 and B4) shall remain straight during the analysis. 2.4.3 Edge B1, except for the corner nodes C1 and C4, is restrained from rotations about the y- and z-axes. Translational displacement in x-direction is fixed. 2.4.4 Edge B2, except for the corner nodes C2 and C3, is restrained from rotations about the y- and z-axes. Translational displacement in x-direction is to follow the displacement of corner node C2. 2.4.5 Edges B3 and B4, translational displacement in z-direction is fixed. Rotation about the x-axis is fixed. A ’slider’-function is applied to all nodes on edges B3 and B4 constraining them to move on a straight line between the end nodes of each edge. Additionally the edges are constrained to stay parallel. This is obtained by forcing the displacements in y-direction of the corner nodes C2 and C3 to follow the y-displacement and z-rotation of the corner node C4 as illustrated in Figure 1.2.4 Constraints to keep the plate model edges parallel. This gives the following set of equations:

42 CzCy LU −− = θ

443 CyCzCy ULU −−− += θ

where corner node C1 is fixed in y-direction and L is the model length.

Figure 1.2.4 Constraints to keep the plate model edges parallel

2.4.6 Frames Fr1, Fr2 and Fr3, correspond to the positions of primary supporting members. At these locations the plate nodes are fixed in the z-direction. The y-displacements of the nodes in the stiffeners webs are to follow the nodes at the intersections between the stiffeners webs and the plate. The constraints in the model are shown in Figure 1.2.5 Multi-point constraints to represent primary supporting members.

Figure 1.2.5: Multi-point constraints to represent primary supporting members


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2.4.7 A summary of boundary conditions is presented in Table 1.2.1 Summary of boundary conditions. Table 1.2.1 Summary of boundary conditions Area Boundary condition B1 except corner nodes Translational displacements in x-direction are fixed

Rotations about y- and z-axes are fixed B2 except corner nodes Translational displacements in x-direction are constrained to follow C2

Rotations about y- and z-axes fixed B3 except corner nodes Translational displacements are constrained to move on a straight line between C1 and C2

Translational displacements in z-direction are fixed Rotations about x-axis are fixed

B4 except corner nodes Translational displacements are constrained to move on a straight line between C3 and C4 Translational displacements in z-direction are fixed Rotations about x-axis fixed

C1 Translational displacements in x-, y- and z-directions are fixed C2 Translational displacements in z-direction are fixed

Translational displacements in y-direction are constrained to ensure edges remain parallel C3 Translational displacements in z-direction are fixed

Translational displacements in y-direction are constrained to ensure edges remain parallel C4 Translational displacements in x- and z-directions are fixed Fr1, Fr2, Fr3 Translational displacements in z-direction are fixed

The y-displacements of the stiffener webs are to follow the nodes at the intersections between webs and plate

Additional boundary conditions in eigenvalue analyses for generating local imperfections are presented below. Please note these boundary conditions should be used only in eigenvalue analyses, not in the non-linear collapse analysis. Intersection between plate and stiffener web

Translational displacements in z-direction are fixed

Intersection between stiffener web and stiffener flange

Translational displacements in y-direction are to follow the y-displacements of the intersection between plate and stiffener web

2.5 Applied loads

2.5.1 Applied loads, in a general case, are the in-plane stresses (Sx, Sy and Sxy) on edges as shown in Figure 1.2.6 Applied edge loads (Sx, Sy and Sxy) and lateral pressure on the plate (normal to the plate).

Figure 1.2.6 Applied edge loads (Sx, Sy and Sxy)

2.5.2 The applied longitudinal stress Sx is parallel to the stiffeners. A point force is applied at corner node C2, which is the master node for the displacement of the edge B2. The constraint equations will ensure that the loads are distributed along the edge and make the edge remain straight. The magnitude of the point force is determined by:

22

BxC

x ASF =

where 2BA is the cross-section area of the edge B2 (total area of the plate and stiffeners).


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2.5.3 The applied transverse stress Sy is perpendicular to the stiffeners. Point forces are applied at all nodes of edges B3 and B4. The magnitude of each point force is determined by:

3

3

B

plateByy N

ASF −=

where plateBA −3 is the plate cross-section area of the edge B3 (plate only) and 3BN is the number of plate elements along the edge

B3 (or B4). However, at each of the corner nodes C1, C2, C3 and C4, the point force to be applied shall be half the above magnitude. 2.5.4 The applied shear stress Sxy is along the edges B1, B2, B3 and B4. It is assumed to be carried only by the plate. For edges B1 and B2, point forces are applied at all nodes of the plate. The magnitude of each point force is determined by:

1

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B

plateBxyBxy N

ASF −=

where plateBA −1 is the plate cross-section area of the edge B1 (plate only) and is the number of plate elements along the edge B1 (or

B2). However, at the corner nodes C1, C2, C3 and C4, the point force to be applied shall be half the above magnitude. For edges B3 and B4, point forces are applied at all nodes of the plate. The magnitude of each point force is determined by:

3

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B

plateBxyBxy N

ASF −=

where plateBA −3 is the plate cross-section area of the edge B3 (plate only) and 3BN is the number of plate elements along the edge

B3 (or B4). However, at the corner nodes C1, C2, C3 and C4, the point force to be applied shall be half the above magnitude. 2.5.5 The applied lateral pressure is acting on the plate surface either on the stiffened side or the unstiffened side. 2.5.6 The load application steps are as follows: in cases where the panel is loaded in a combination of in-plane loads and lateral pressure, the loads should be applied in two load steps. The first load step is to apply the lateral pressure of the specified magnitude. The second step is to apply the in-plane loads proportionally (Sx : Sy : Sxy), while keeping the lateral pressure constant.

2.6 Geometrical initial imperfections

2.6.1 One of the important steps of a non-linear FEA is to define the shape and magnitude of geometrical initial imperfections. This is to take into account initial deformations from fabrication. Additionally, geometrical imperfection will be needed to trigger the non-linear FEA response. 2.6.2 The shapes of initial imperfections can be divided into two parts: the local mode and the global mode. 2.6.3 The local mode: three local modes are to be considered on the first elastic buckling mode of the model: (1) under considered in-plane loads; (2) under longitudinal loads only and (3) under transverse loads only. The ultimate strength takes the minimum value obtained from the three analyses using the three local modes (combined with the global mode) respectively. A magnitude of b/200 is to be used for the plate, where b is the stiffener spacing. The additional boundary conditions in eigenvalue analyses for generating local imperfections, as specified in Table 1.2.1 Global mode: overall deflection with tilted stiffeners, should be used in this case. 2.6.4 The global mode has an overall deflection with tilted stiffeners (See Figure 1.2.7 Global mode: overall deflection with tilted stiffeners). The magnitude for overall deflection and stiffener sideway displacement are a/1000, where a is the primary supporting member spacing. The overall deflection is to be determined by:

)sin()cos(1000 B

yaxa

zππδ =

where B is the overall width of the model. The sideway displacement of the stiffener is to be determined by:

)cos(sinaxyy

πθδ =

where )1000

(sin 1

wha−=θ , and wh is the stiffener web depth.


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Figure 1.2.7 Global mode: overall deflection with tilted stiffeners 2.6.5 A visual inspection on the created initial imperfections should be carried out before performing non-linear FEA. This is to make sure that appropriate shapes are applied in the model. An example of the combined local and global modes of a stiffened panel under longitudinal loads is presented in Figure 1.2.8 An example of initial imperfection under longitudinal loads. 2.6.6 Combined local and global initial imperfections should be used in the non-linear FEA. 2.7 Solution on the non-linear FEA 2.7.1 Analyses should be performed using the arc length solution algorithm or its variations, such as the modified displacement control and the orthogonal trajectory methods. This is necessary in order to be able to trace the equilibrium curves past limit (or peak) points. 2.7.2 It should be noted that the increment size of each load step is to be sufficiently small to capture the ultimate capacity of the model.


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Local mode (x100)

Global mode (x50)

Combined mode (x50)

Figure 1.2.8: An example of initial imperfection under longitudinal loads


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■ Section 3 Analysis Examples for Stiffened Panels 3.1 General

3.1.1 Experience has shown that the quality and accuracy of non-linear FEA results are highly dependent on the skills of the structural analysts. The difference between FE results assessed by different analysts can be up to 15%~20% for the same model with same specified procedures. This is considered unacceptable in view of using such sophisticated approaches. The reason for such big differences may be due to the lack of experience by some analysts in implementing the specified conditions into FE software packages. Therefore, validations and calibrations of using the procedure are important and necessary before performing production runs. 3.1.2 This Section sets out a number of examples for validation and calibration purposes. 3.1.3 It is recommended that the seven load cases for a T stiffened panel, as in Section 3.2, should be performed, and acceptable results should be obtained before performing other tasks for either design purpose or other assessment purposes. 3.2 T stiffened panel 3.2.1 The geometric dimensions of the T model, in mm, are a=4300, b=815, t=17,8 for the plate; hw =463, tw=8 for the T-stiffener web; bf=172, tf=17 for the T-stiffener flange; Materials Young's modulus, E=205800 N/mm2; Poisson ratio=0,3; Yield stress=315 N/mm2; and Strain hardening parameter, ET=1000 N/mm2. 3.2.2 The mesh density for T model: eight elements in the plate between stiffeners; five elements in the stiffener web and two elements in the stiffener flange. The element length (in x-direction) is 98mm. 3.2.3 Load Case 1: the result of ultimate strength, under longitudinal load (Sx), is 269,9 N/mm2. The load-shortening curve and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.1 Load-shortening curve of the T panel in longitudinal load case and the mid-surface von-Mises stresses at the ultimate state.


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Figure 1.3.1 Load-shortening curve of the T panel in longitudinal load case and the mid-surface von-Mises stresses at the ultimate state


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3.2.4 Load Case 2: the result of ultimate strength, under transverse load (Sy), is 113,2 N/mm2. The load-shortening curve and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.2 Load-shortening curve of the T panel in transverse load case and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.2 Load-shortening curve of the T panel in transverse load case and the

mid-surface von-Mises stresses at the ultimate state


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3.2.5 Load Case 3: the result of ultimate strength, under shear load (Sxy), is 181,9 N/mm2. The load-shortening curve and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.3 Load-shortening curve of the T panel in shear load case and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.3 Load-shortening curve of the T panel in shear load case and the mid-surface von-Mises stresses at the ultimate state


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3.2.6 Load Case 4: the result of ultimate strength, under combined in-plane loads, is (Sx : Sy)=(0,7 : 0,3)=(213,9 : 91,7) N/mm2. The load-shortening curve in the longitudinal direction and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.4 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy)=(0,7 : 0,3) and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.4 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy)=(0,7 : 0,3) and the mid-surface von-Mises stresses at the ultimate state


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3.2.7 Load Case 5: the result of ultimate strength, under combined in-plane loads with shear, is (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3)=(203,1 : 87,1 : 87,1) N/mm2. The load-shortening curve in the longitudinal direction and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.5 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3) and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.5 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0.7 : 0.3 : 0.3) and the mid-surface von-Mises stresses at the ultimate state


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3.2.8 Load Case 6: the result of ultimate strength, under combined in-plane loads, with a lateral pressure of 0,2 N/mm2 applied at the unstiffened side, is (Sx : Sy : Sxy)=(189,1 : 81,0 : 81,0) N/mm2. The load-shortening curve in longitudinal direction and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.6 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3) with pressure of 0,2 N/mm2 applied at the unstiffened side and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.6 Load-shortening curve in the longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3) with pressure of 0,2 N/mm2 applied at the unstiffened

side and the mid-surface von-Mises stresses at the ultimate state


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3.2.9 Load Case 7: the result of ultimate strength, under combined in-plane loads, with a lateral pressure of 0.2 N/mm2 applied at the stiffened side, is (Sx : Sy : Sxy) = (188,1 : 80,6 : 80,6) N/mm2. The load-shortening curve in longitudinal direction and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.3.7 Load-shortening curve in longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3) with pressure of 0,2 N/mm2 applied at the stiffened side and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.3.7 Load-shortening curve in longitudinal direction of the T panel in the combined load case (Sx : Sy : Sxy)=(0,7 : 0,3 : 0,3) with pressure of 0,2 N/mm2 applied at the stiffened side and the

mid-surface von-Mises stresses at the ultimate state


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3.2.10 The summary of the results of the ultimate strengths for the seven load cases are presented in Table 1.3.1 Summary of ultimate strength for the T stiffened panel. Table 1.3.1 Summary of ultimate strength for the T stiffened panel

Load Case

Ultimate Strength (MPa) Pressure (MPa) Sx Sy Sxy

LC1 269,9 0,0 0,0 0,0 LC2 0,0 113,2 0,0 0,0 LC3 0,0 0,0 181,9 0,0 LC4 213,9 91,7 0,0 0,0 LC5 203,1 87,1 87,1 0,0 LC6 189,1 81,0 81,0 0,2 (applied at the unstiffened side) LC7 188,1 80,6 80,6 0,2 (applied at the stiffened side)

■ Section 4 Appendix: Ultimate Strength of Unstiffened Plates 4.1 Structural model

4.1.1 The extent of unstiffened plate model contains one frame spacing in the longitudinal direction and one stiffener spacing in the transverse direction. 4.2 FE mesh and coordinate system 4.2.1 Element type: The plate should be modelled using shell elements, such as the 4-node quadrilateral elements. 4.2.2 Element size: The mesh should be fine enough to capture the local deformations and stresses developed during buckling and collapse. For plates that collapse under longitudinal and transverse stresses, eight elements over one stiffener spacing are normally sufficient. The number of elements in the longitudinal direction should be selected so that the elements aspect ratio is as close to the unit as possible. 4.2.3 The coordinate system: The coordinate system used for the unstiffened plate model is illustrated in Figure 1.4.1 Constraints to keep model edges straight and parallel. 4.3 Boundary conditions 4.3.1 The definition of all edges and locations where the boundary conditions need to be applied is shown in Figure 1.4.1 Constraints to keep model edges straight and parallel. Four edges, B1, B2, B3 and B4 and four corners, C1, C2, C3 and C4 are to be considered. 4.3.2 All Edges (B1, B2, B3 and B4) are simply supported and remain straight during the analysis. 4.3.3 Edge B1 is restrained from translation in z-direction and rotation about the x-axis. Translational displacement in x-direction is fixed at all nodal points. 4.3.4 Edge B2 is also restrained from translation in z-direction and rotations about the x-axis. However, the translational displacement in x-direction is to follow the displacement of corner node C2. 4.3.5 Edges B3 and B4 are restrained from translation in z-direction and rotation about the y-axis. A ‘slider’-function is applied to all nodes on Edges B3 and B4 constraining them to move on a straight line between the end nodes of each edge. Additionally the edges are constrained to stay parallel. This is obtained by forcing the in-plane displacements of the corners nodes C2 and C3 to follow the y-displacement and z-rotation of the corner node C4 as illustrated in Figure 1.4.1 Constraints to keep model edges straight and parallel. This results in the same constraint equations as for the stiffened panel models discussed before.


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Figure 1.4.1 Constraints to keep model edges straight and parallel 4.3.6 A summary of boundary conditions for unstiffened plate is presented in Table 1.4.1 Summary of boundary conditions for unstiffened plate. Table 1.4.1 Summary of boundary conditions for unstiffened plate Area Boundary condition B1 except corner nodes Translational displacements in x- and z-directions are fixed.

Rotations about the x- and z-axes are fixed. B2 except corner nodes Translational displacements in z-direction are fixed, but translations in x-direction are to

follow the displacement of corner node C2. Rotation about the x-axis is fixed.

B3 except corner nodes Translational displacements are constrained to move on a straight line between C1 and C2 Rotation about the y-axis is fixed.

B4 except corner nodes Translational displacements are constrained to move on a straight line between C3 and C4 and Edges B3 and B4 remain parallel after deformation. Rotations about the y-axis fixed.

C1 Translational displacements in x-, y- and z-directions are fixed. C2 Translational displacements in z-direction are fixed. Translation displacements in x- and y-

directions used as master nodes in MPCs to ensure edges in parallel. C3 Translational displacements in z-direction are fixed. Translational displacements in x- and y-

directions used as master nodes in MPCs to ensure edges in parallel. C4 Translational displacements in x- and z-directions are fixed. Translational displacements in x-

and y-directions used as master nodes in MPCs to ensure edges in parallel. 4.4 Applied loads 4.4.1 Applied loads, in a general case, are the in-plane stresses (Sx, Sy and Sxy) on edges, and the lateral pressure on the plate (normal to the plate). 4.4.2 The in-plane stresses and the lateral pressure are applied to the unstiffened plate models in the same way as they are applied to the stiffened panel models, as described before. 4.4.3 The load application steps: in cases where the panel is loaded in a combination of in-plane loads and lateral pressure, the loads should be applied in two load steps. The first load step is to apply the lateral pressure to the specified magnitude. The second step is to apply the in-plane loads proportionally (Sx : Sy : Sxy), while keeping the lateral pressure constant. 4.5 Geometrical initial imperfection 4.5.1 One of the important steps of a non-linear FEA is to define the shape and magnitude of geometrical initial imperfections. This is to take into account initial deformations from fabrication. Additionally, geometrical imperfections will be needed to trigger the non-linear FEA response. 4.5.2 For non-linear FEA of unstiffened plates, the local mode of initial imperfections is included. 4.5.3 The local mode: three local modes are to be considered: the first elastic buckling mode of the model (1) under considered in-plane loads; (2) under longitudinal loads only and (3) under transverse loads only. The ultimate strength takes the minimum value obtained from the three analyses using the three local modes respectively. A magnitude of b/200 is to be used for the plate, where b is the length of the shorter edge.


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4.6 Solution on the non-linear FEA 4.6.1 Analyses should be performed using the arc length solution algorithm or its variations, such as the modified displacement control and the orthogonal trajectory methods. This is necessary in order to be able to trace the equilibrium curves past limit (or peak) points. 4.6.2 It should be noted that the increment size of each load step shall be sufficiently small to capture the ultimate capacity of the model. 4.7 Analysis examples for unstiffened plates 4.7.1 This Section sets out examples of non-linear FEA of unstiffened plate models for validation and calibration purposes. It is recommended that the three load cases for the specified unstiffened plate, as in this Section, should be performed and acceptable results obtained before performing other tasks for either design purpose or other assessment purpose. 4.7.2 The geometric dimensions of the plate model, in mm, are a=4205, b=841, t=16.4. Materials Young's modulus, E=207000 N/mm2, Poisson ratio=0.3, Yield stress=315 N/mm2, and Strain hardening parameter, ET=1000 N/mm2. Load Case 1: the result of ultimate strength, under longitudinal load (Sx), is 241,5 N/mm2. The load-shortening curve and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.4.2 Load-shortening curve of the plate in longitudinal load case and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.4.2 Load-shortening curve of the plate in longitudinal load case and the mid-surface von-Mises stresses at the ultimate state


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4.7.3 Load Case 2: the result of ultimate strength, under transverse load (Sy), is 95,47 N/mm2. The load-shortening curve and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.4.3 Load-shortening curve of the plate in transverse load case and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.4.3 Load-shortening curve of the plate in transverse load case and the mid-surface von-Mises stresses at the ultimate state


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4.7.4 Load Case 3: the result of ultimate strength, under combined in-plane loads, is (Sx : Sy)=(0,7 : 0,3)=(175,98 : 75,42) N/mm2. The load-shortening curve in the longitudinal direction and the mid-surface von-Mises stresses at the ultimate state are shown in Figure 1.4.4 Load-shortening curve in the longitudinal direction of the plate in the combined load case (Sx : Sy)=(0,7 : 0,3) and the mid-surface von-Mises stresses at the ultimate state.

Figure 1.4.4 Load-shortening curve in the longitudinal direction of the plate in the combined load case (Sx : Sy)=(0,7 : 0,3) and the mid-surface von-Mises stresses at the ultimate state

© Lloyd’s Register Group Limited 2016 Published by Lloyd’s Register Group Limited

Registered office (Reg. no. 08126909) 71 Fenchurch Street, London, EC3M 4BS

United Kingdom

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