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Appendix A Illustrative Examples

A.1

Appendix A

Illustrative Examples A.1 Axisymmetric Spherical Shell Subject to Step Pressure (VPSX003) A.1.1 Description

A segment of a spherical shell of radius 22.27 inches is clamped and subjected to step external pressure of 600 lb/in2 as shown in Figure A.1.1. The pressure is applied using a step function i.e. instantaneously and then left constant and the material is modelled using a von Mises elasto-plastic yield function with linear hardening. The solution algorithm utilised is explicit dynamic, which is most suitable to capture the high frequency response that results from the step loading.

A

0.41in

26.67°

p

22.27in

B

C

D

Figure A.1.1 Problem Description

A.1.2 Finite Element Model Finite Element Mesh The shell is modelled as an axisymmetric problem, using a regular mesh of quadrilateral elements as shown in Figure A.1.2. Four elements are utilised through the thickness of the shell. This is sufficient to reproduce the correct bending response exhibited by this type of application.

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Illustrative Examples Appendix A

A.2

Figure A.1.2 Finite Element Mesh

Material Properties The material is modelled using a von Mises elasto-plastic constitutive model with linear hardening.

Property Value Young's Modulus 10.5 x 106 lb / in2

Poisson's Ratio 0.3

Initial Yield Stress 0.024 x 106 lb / in2

Density 2.45 x 106 lb / in2

Hardening Modulus 0.2143 x 106 lb / in2

Constraints The spherical shell is clamped along AB and CD. Loading The spherical is subjected to an external pressure load of 600 lb/in2 applied to the external surface BD. The instantaneous step load is modelled using the following load function;

Pseudo-Time Load Factor 0.0 1.0

1.0 1.0

Solution Algorithm The application is solved using a explicit central difference solution algorithm. Time Step The initial time step is 0.2E-6s which is less than the critical time step for the initial mesh.

This is kept constant unless it exceeds the automatically computed critical time step, in which case 0.9∆tcr is used.

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Appendix A Illustrative Examples

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Termination The solution is terminated when either:- • 5000 time steps have been performed • the solution time is 1.0E-3s.

No minimum time step for termination was used as structural failure is not anticipated. Output The output frequencies are Results 100 time steps Plotting data 500 time steps

A.1.3 Neutral File The neutral file has the following form

Element_group_data { IGROUP { 1 } Element_type_number { 9 } Material_number { 1 } Element_options { NELOPS { 50 } F F F F F F F F F F ... F F F F F F F F F F } }

SAEQ_4 4-noded axisymmetric element Material number 1 No element options

Element_data { IGROUP { 1 } Element_topology { NNODE { 4 } NELGP { 40 } 4 22 29 26 .. 55 25 2 13 } Element_numbers { NELGP { 40 } 1 2 3 4 5 6 7 8 9 10 .. 31 32 33 34 35 36 37 38 39 40 } Assignment_surfaces { 1 1 } }

40 elements in the group

Geometric_property_data { IGROUP { 1 } Material_angle_flag { 0 } }

Geometric properties for element group with isotropic material

Node_data { 0 Node_numbers { NPOIN { 55 } 1 2 3 4 5 6 7 8 9 10 .. 51 52 53 54 55 } Coordinates { NDIMN { 2 } NPOIN { 55 } 0 22.68 ... 9.171063 20.628847 } }

55 Nodes Nodal coordinates

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Material_data { 1 Material_name { "vpsx003" } Elastic_material_flags { NFGELA { 4 } 0 1 0 0 } Elastic_properties { NMPRP { 15 } 1.05e+007 0 0 0.3 0 0 0 0 0 0 0 0 0.000245 0 0 } Plastic_material_flags { NMFPLS { 2 } 0 7 } Plastic_properties { NPRPLS { 1 } 24000 } Hardening_material_flags { 1 0 } Hardening_properties { 2 2 0 2 24000 452600 } Number_state_variables { 12 } }

Material 1 "vpsx003" Elasto-plastic von Mises Elastic and plastic properties present Isotropic elastic constants - no temperature dependence Young's Modulus = 1.05E7 Poisson's Ratio = 0.3 Density = 0.000245 Von mises model - linear hardening Yield stress = 24,000 Hardening curve Default 12 state variables

Operation_assignment_geometry { Assignment_lines { 3 1 3 4 } }

Geometry lines with assignments

Load_curve_data { ILOAD { 1 } Time_curve { NLFUNC { 1 } 0.0 } Load_factor { NLFUNC { 1 } 1.0 } }

Step load function for load case 1

Element_load_data { ILOAD { 1 } Face_load { NDIMN { 2 } MFNOD { 2 } NFASET { 1 } 600 0 600 0 } Face_load_pointers { NFAASM { 10 } 3 4 8 12 16 20 24 28 32 36 40 10*3 10*1 } Face_load_line_pointers { 1 2 1 1 } }

Pressure/Face loading (normal direction) List of pointers Geometry lines and pointers with face loading

Support_data { 0 Displacement_codes { MGDOF { 2 } NPDGRP { 2 } 1 0 0 1 1 0 } Displacement_code_pointers { NDPGRP { 10 } NSETS { 2 } 1 2 3 4 23 24 25 26 27 28 1 2*2 1 3*2 3*1 } Displace_code_line_pointers { 2 2 3 4 2 1 } }

Two fixity conditions U prescribed (symmetry) U and V prescribed List of nodes List of pointers 1 - Symmetry condition 2 - Clamped end Geometry lines with fixitiy conditions

Load_case_control_data { 0 List_loadcases_input { NLODIN { 1 } 1 } List_active_load_flags { NLODIN { 1 } 2 } }

A single active load case

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Appendix A Illustrative Examples

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History { 1 Name { "disp_y" } Output_steps { 10 } Nodal_output_values { 1 "Y_DISP" } Nodes { NHNODE { 1 } 1 } Tracking { 0 } }

High resolution history results of displacements in y every 10 analysis steps for node 1

Control_data { Control_title { "elastic_shell" } Initial_time_step { 2e-007 } Maximum_time_step_change { 101.0 } Minimum_time_step { 0 } Maximum_time_step { 1 } Factor_critical_time_step { 0.9 } Solution_algorithm { 1 } Maximum_number_time_steps { 5000 } Termination_time { 0.001 } Output_frequency_results { -1 } Output_frequency_plotfile { 250 } Output_frequency_restart { 0 } Output_time_results { 0 } Output_file_plotfile { 0 } Output_time_restart { } Screen_message_time { 0 } }

Time step = 2e-007s 1% of time step growth permitted No minimum time step limit If ∆tcr exceeded use 0.9*∆tcr Central difference Terminate after 5000 steps or Terminate after 0.001s Output last result Output plot data every 250 steps No restart file required Not used Not used Not used Not used