Analysis and Optimization of Prismatic and Axisymmetric Shell Structures Theory, Practice and Software

Shell-type structures can be found almost everywhere. They appear in natural forms but also as man-made, load-bearing components in diverse engineering systems. Mankind has struggled to replicate nature’s optimization of such structures but using modern computational tools it is now possible to analyse, design and optimise them systematically. Analysis and Optimization of Prismatic and Axisymmetric Shell Structures features: comprehensive coverage of the background theory of shell structures; development and implementation of reliable, creative and efficient computational tools for static and free-vibration analysis and structural optimization of variable-thickness shells and folded-plate structures; integrated computer-aided curve and surface modelling tools and automatic mesh generation, structural analysis sensitivity analysis and mathematical programming methods; well-documented, downloadable Fortran software for these techniques using finite element and finite strip simulations which can be readily adapted by the reader for the solution of practical problems or for use within a teaching or research environment. Written by leading experts in finite element and finite strip methods, Analysis and Optimization of Prismatic and Axisymmetric Shell Structures will be of great interest to researchers in structural mechanics and in automotive, aerospace and civil engineering as well as to designers from all fields using shell structures for their strength-per-unit-mass advantages.

I: Introduction.- 1 Introduction.- 1.1 Background.- 1.2 Previous work.- 1.2.1 Structural shape optimization of shells and folded plates under static conditions.- 1.2.2 Vibrating shells of revolution.- 1.2.3 Vibrating prismatic shells and folded plates.- 1.3 Classification of structural optimization problems.- 1.3.1 Classification based on mode of behaviour.- 1.3.2 Classification according to type of design variable.- 1.4 Classification of shells.- 1.4.1 Indian Standard classification of shells and folded plates.- 1.4.2 Simplified classification of shells and folded plates.- 1.4.3 Summary of structures considered.- 1.5 Computer-aided shape definition.- 1.5.1 Shape definition of shells of revolution and prismatic shells.- 1.6 Element technology.- 1.6.1 Elements for shells of revolution.- 1.6.2 Strips for prismatic shells.- 1.7 Automatic mesh generation.- 1.7.1 Automatic mesh generation for shells of revolution and prismatic shells.- 1.8 Structural shape optimization.- 1.8.1 The basic algorithm.- 1.8.2 Sensitivity analysis.- 1.8.3 Sensitivity analysis of static response.- 1.8.4 Sensitivity analysis of dynamic problems.- 1.8.5 Selection and linking of design variables.- 1.8.6 Selection of constraint points.- 1.8.7 Optimization algorithm.- 1.9 Typical shell problems treated.- 1.10 Software developed in this book.- 1.11 Layout of the book.- References.- 2 Structural Shape Definition and Automatic Mesh Generation with contributions from NVR Rao.- 2.1 General perspective.- 2.2 Structural shape definition.- 2.2.1 Three equivalent representations of a parametric cubic spline.- 2.2.2 The cubic B-spline representation.- 2.2.3 Terminology.- 2.2.4 Computer implementation.- 2.2.5 Specification of end condition of splines.- 2.3 Structural thickness definition.- 2.4 Automatic mesh generation.- 2.4.1 General requirements.- 2.4.2 Algorithm for mesh generation.- 2.5 Shape definition and mesh generation in structural analysis.- 2.6 Shape definition and mesh generation in structural optimization.- 2.6.1 Shape design variables.- 2.6.2 Selection of thickness design variables.- 2.6.3 Linking of design variables.- 2.6.4 Perturbation of design variables.- 2.6.5 Prescribed move directions.- 2.7 Other applications of the present tools.- References.- 3 Structural Optimization Methods and Algorithms.- 3.1 General perspective.- 3.1.1 Problem classification.- 3.1.2 Problem definition and formulation.- 3.1.3 Basic algorithm and three-columns concept.- 3.1.4 Other important aspects.- 3.2 Optimization algorithms.- 3.2.1 Overview.- 3.2.2 Mathematical programming.- 3.2.3 Genetic algorithms.- 3.2.4 Approximation concepts.- 3.3 Sensitivity analysis.- 3.3.1 Overview.- 3.3.2 Global finite differences.- 3.3.3 Semi-analytical method.- 3.3.4 Analytical or direct sensitivity method.- 3.3.5 Adjoint variable method.- 3.3.6 Accuracy assessment.- 3.4 Concluding remarks.- References.- II: Static Analysis and Optimization.- 4 Basic Finite Element Formulation for Shells of Revolution.- 4.1 General perspective.- 4.2 Basic formulation.- 4.3 Finite element idealization.- 4.4 Strain energy evaluation.- 4.5 Benchmark examples.- 4.5.1 Cylindrical tank with non-uniform wall thickness.- 4.5.2 Clamped circular plate.- 4.5.3 Spherical dome under uniform pressure.- 4.5.4 Toroidal shell under internal pressure.- 4.6 Closing remarks.- References.- 5 Basic Finite Strip Formulation for Prismatic Shells with contributions from NVR Rao.- 5.1 General perspective.- 5.1.1 Preamble.- 5.1.2 Simply supported Euler-Bernoulli beam.- 5.1.3 Simply supported Timoshenko beam.- 5.2 Right prismatic shells.- 5.2.1 Basic formulation.- 5.2.2 Finite strip idealization.- 5.2.3 Branched strips.- 5.3 Benchmark examples.- 5.3.1 Plates.- 5.3.2 Box-girder bridges.- 5.3.3 Cylindrical shells.- 5.4 Prismatic structures with curved planform.- 5.4.1 Basic formulation.- 5.4.2 Branched strips.- 5.5 Benchmark examples.- 5.5.1 Comparisons with known solutions for right structures analyzed as structures with curved planform.- 5.5.2 Comparison with known solutions for structures with curved planform.- 5.5.3 New solutions for structures with curved planform.- References.- 6 Structural Optimization of Shells of Revolution and Prismatic Shells with contributions from NVR Rao.- 6.1 General perspective.- 6.2 Problem definition.- 6.2.1 Selection of objective function for the problem.- 6.2.2 Selection of constraints for the problem.- 6.3 Sensitivity analysis.- 6.3.1 Analytical method.- 6.3.2 Semi-analytical method.- 6.3.3 Stress resultant gradients.- 6.3.4 Equivalent stress derivative.- 6.3.5 Global finite difference method.- 6.3.6 Volume gradient.- 6.3.7 Strain energy gradient.- 6.4 Shells of revolution examples.- 6.4.1 Clamped circular plate subjected to uniformly distributed load.- 6.4.2 Cylindrical tank under hydrostatic pressure.- 6.4.3 Spherical shell under ring load.- 6.5 Right prismatic shells and folded plates examples.- 6.5.1 Square plates subjected to uniformly distributed load.- 6.5.2 Plates on elastic foundations.- 6.5.3 Single-cell right box-girder bridge.- 6.5.4 Cylindrical shell roof.- 6.5.5 Pinched cylindrical shell.- 6.6 Prismatic shells with curved planform examples.- 6.6.1 Plates with curved planform subjected to uniformly distributed load.- 6.6.2 Single-cell curved box-girder bridge.- 6.6.3 Pinched cylindrical shell with curved planform.- 6.7 Closing remarks.- References.- III: Free Vibration Analysis and Optimization.- 7 Basic Finite Element Formulation for Vibrating Axisymmetric Shells.- 7.1 General perspective.- 7.1.1 Analysis methods.- 7.1.2 Applications.- 7.2 Structural analysis and finite element formulation.- 7.2.1 Finite element formulation.- 7.2.2 Finite element idealization.- 7.2.3 Branched elements.- 7.3 Examples.- 7.3.1 Thin circular plate.- 7.3.2 Annular plates with linearly varying thickness.- 7.3.3 Hemispherical dome.- 7.3.4 Conical shell with variable thickness.- 7.3.5 Cone-cylinder combination.- 7.3.6 Hyperboloidal shell.- 7.3.7 Hermetic capsule.- 7.3.8 Hermetic can.- 7.3.9 Bells.- References.- 8.1 Introduction.- 8.2 Prismatic shells with rectangular planform.- 8.2.1 Basic finite strip formulation.- 8.2.2 Finite strip idealization.- 8.2.3 Branched strips.- 8.3 Examples.- 8.3.1 Square plates.- 8.3.2 Variable-thickness plates.- 8.3.3 Stiffened panel.- 8.3.4 Cylindrical shell.- 8.3.5 Cylinders with interior partitions.- 8.3.6 Two-cell right box-girder bridge.- 8.4 Prismatic structures with curved planform.- 8.4.1 Basic finite strip formulation.- 8.4.2 Finite strip idealization.- 8.4.3 Branched strips.- 8.5 Examples.- 8.5.1 Annular sector plates.- 8.5.2 Two-cell box-girder bridge with a curved planform.- 8.5.3 Right cylinders with interior partitions.- 8.5.4 Cylinders with interior partitions and curved planforms.- References.- 9 Structural Shape Optimization of Vibrating Axisymmetric and Prismatic Shells.- 9.1 General perspective.- 9.2 Problem definition.- 9.2.1 Selection of objective function.- 9.2.2 Selection of design variables.- 9.2.3 Selection of constraints.- 9.2.4 Bounds on design variables.- 9.3 Sensitivity analysis.- 9.3.1 Derivative evaluation.- 9.3.2 Analytical method.- 9.3.3 Semi-analytical method.- 9.3.4 Finite difference method.- 9.3.5 Derivative of volume.- 9.4 Axisymmetric shells.- 9.5 Axisymmetric shell examples.- 9.5.1 Circular plates.- 9.5.2 Conical shell.- 9.5.3 Spherical shells.- 9.5.4 Branched shell.- 9.5.5 Bells.- 9.6 Prismatic folded plates and shells.- 9.7 Prismatic folded plates and shells: examples.- 9.7.1 Square plates.- 9.7.2 Stiffened panel.- 9.7.3 Cylindrical shell.- 9.7.4 Box-girder bridge.- 9.8 Prismatic shells with curved planform: examples.- 9.8.1 Annular sector plates.- 9.8.2 Cylindrical shell segment with curved planform.- 9.8.3 Box-girder bridge.- References.- IV: Dynamic and Buckling Analysis and Optimization.- 10 Buckling Analysis and Optimization of Plates and Shells.- 10.1 Prismatic plates.- 10.2 Strip formulation for prismatic plates with rectangular planform.- 10.2.1 Strain energy