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# Structural Analysis: With Applications to Aerospace Structures

## byO. A. Bauchau, J.i. Craig

### Hardcover | August 17, 2009

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The authors and their colleagues developed this text over many years, teaching undergraduate and graduate courses in structural analysis courses at the Daniel Guggenheim School of Aerospace Engineering of the Georgia Institute of Technology.

The emphasis is on clarity and unity in the presentation of basic structural analysis concepts and methods. The equations of linear elasticity and basic constitutive behaviour of isotropic and composite materials are reviewed. The text focuses on the analysis of practical structural components including bars, beams and plates. Particular attention is devoted to the analysis of thin-walled beams under bending shearing and torsion. Advanced topics such as warping, non-uniform torsion, shear deformations, thermal effect and plastic deformations are addressed. A unified treatment of work and energy principles is provided that naturally leads to an examination of approximate analysis methods including an introduction to matrix and finite element methods.

This teaching tool based on practical situations and thorough methodology should prove valuable to both lecturers and students of structural analysis in engineering worldwide.

This is a textbook for teaching structural analysis of aerospace structures. It can be used for 3rd and 4th year students in aerospace engineering, as well as for 1st and 2nd year graduate students in aerospace and mechanical engineering.

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The following ISBNs are associated with this title:

ISBN - 10:9048125154

ISBN - 13:9789048125159

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Table of Contents

Part I Basic tools and concepts; 1 Basic Equations of Linear Elasticity .1.1 The concept of stress; 1.1.1 The state of stress at a point; 1.1.2 Volume equilibrium equations; 1.1.3 Surface equilibrium equations; 1.2 Analysis of the state of stress at a point; 1.2.1 Stress components acting on an arbitrary face; 1.2.2 Principal stresses; 1.2.3 Rotation of stresses; 1.2.4 Problems; 1.3 The state of plane stress; 1.3.1 Equilibrium equations; 1.3.2 Stresses acting on an arbitrary face within the sheet; 1.3.3 Principal stresses;1.3.4 Rotation of stresses; 1.3.5 Special states of stress; 1.3.6 Mohr's circle for plane stress; 1.3.7 Lamé's ellipse; 1.3.8 Problems; 1.4 The concept of strain; 1.4.1 The state of strain at a point; 1.4.2 The volumetric strain; 1.5 Analysis of the state of strain at a point; 1.5.1 Rotation of strains 1.5.2 Principal strains; 1.6 The state of plane strain; 1.6.1 Strain-displacement relations for plane strain; 1.6.2 Rotation of strains; 1.6.3 Principal strains; 1.6.4 Mohr's circle for plane strain; 1.7 Measurement of strains; 1.7.1 Problems; 1.8 Strain compatibility equations; 2 Constitutive Behavior of Materials; 2.1 Constitutive laws for isotropic materials; 2.1.1 Homogeneous, isotropic, linear elastic materials; 2.1.2 Thermal effects; 2.1.3 Problems; 2.1.4 Ductile materials; 2.1.5 Brittle materials; 2.2 Allowable stress; 2.3 Yielding under combined loading; 2.3.1 Tresca's criterion; 2.3.2 Von Mises' criterion; 2.3.3 Comparing Tresca's and von Mises' criteria;2.3.4 Problems; 2.4 Material selection for structural performance; 2.4.1 Strength design; 2.4.2 Stiffness design 2.4.3 Buckling design; 2.5 Composite materials; 2.5.1 Basic characteristics;2.5.2 Stress diffusion in a composite; 2.6 Constitutive laws for anisotropic materials; 2.6.1 Constitutive laws for a lamina in the fiber aligned triad; 2.6.2 Constitutive laws for a lamina in an arbitrary triad; 2.7 Strength of a transversely isotropic lamina; 2.7.1 Strength of a lamina under simple loading conditions; 2.7.2 The Tsai-Wu failure criterion; 2.7.3 The reserve factor; 3 Linear Elasticity Solutions; 3.1 Solution procedures; 3.1.1 Displacement formulation; 3.1.2 Stress formulation; 3.1.3 Solutions to elasticity problems; 3.2 Plane strain problems; 3.3 Plane stress problems; 3.4 Plane strain and plane stress in polar coordinates; 3.5 Problem featuring cylindrical symmetry; 3.5.1 Problems; 4 Engineering Structural Analysis; 4.1 Solution approaches; 4.2 Bar under constant axial force; 4.3 Hyperstatic systems; 4.3.1 Solution procedures; 4.3.2 The displacement or stiffness method; 4.3.3 The force or flexibility method; 4.3.4 Problems; 4.3.5 Thermal effects in hyperstatic system; 4.3.6 Manufacturing imperfection effects in hyperstatic system; 4.3.7 Problems; 4.4 Pressure vessels; 4.4.1 Rings under internal pressure; 4.4.2 Cylindrical pressure vessels; 4.4.3 Spherical pressure vessels; 4.4.4 Problems; 4.5 Saint-Venant's principle; Part II Beams and thin-wall structures5 Euler-Bernoulli Beam Theory; 5.1 The Euler-Bernoulli Assumptions; 5.2 Implications of the Euler-Bernoulli assumptions; 5.3 Stress resultants; 5.4 Beams subjected to axial loads; 5.4.1 Kinematic description; 5.4.2 Sectional constitutive law; 5.4.3 Equilibrium equations; 5.4.4 Governing equations; 5.4.5 The sectional axial stiffness; 5.4.6 The axial stress distribution; 5.4.7 Problems; 5.5 Beams subjected to transverse loads; 5.5.1 Kinematic description; 5.5.2 Sectional constitutive law; 5.5.3 Equilibrium equations; 5.5.4 Governing equations; 5.5.5 The sectional bending stiffness; 5.5.6 The axial stress distribution; 5.5.7 Rational design of beams under bending; 5.5.8 Problems; 5.6 Beams subjected to axial and transverse loads; 5.6.1 Kinematic description; 5.6.2 Sectional constitutive law; 5.6.3 Equilibrium equations; 5.6.4 Governing equations; 6 Three-Dimensional Beam Theory; 6.1 Kinematic description; 6.2 Sectional constitutive law; 6.3 Sectional equilibrium equations; 6.4 Governing equations; 6.5 Decoupling the three-dimensional problem; 6.5.1 Definition of the principal axes of bending; 6.5.2 Decoupled governing equations; 6.6 The principal centroidal axes of bending; 6.6.1 The bending stiffness ellipse; 6.7 Definition of the neutral axis; 6.8 Evaluation of sectional stiffnesses; 6.8.1 The parallel axis theorem; 6.8.2 Thin-walled sections; 6.8.3 Triangular area equivalence method; 6.8.4 Useful results; 6.8.5 Problems; 6.9 Summary of three-dimensional beam theory; 6.9.1 Examples; 6.9.2 Discussion of the results; 6.10 Problems; 7 Torsion; 7.1 Torsion of circular cylinders; 7.1.1 Kinematic description; 7.1.2 The stress field;7.1.3 Sectional constitutive law; 7.1.4 Equilibrium equations; 7.1.5 Governing equations; 7.1.6 The torsional stiffness; 7.1.7 Measuring the torsional stiffness; 7.1.8 The shear stress distribution; 7.1.9 Rational design of cylinders under torsion; 7.1.10 Problems; 7.2 Torsion combined with axial force or bending; 7.2.1 Problems; 7.3 Torsion of bars with arbitrary cross-sections; 7.3.1 Introduction; 7.3.2 Saint-Venant's solution; 7.3.3 Saint-Venant's solution for a rectangular cross-section; 7.3.4 Problems; 7.4 Torsion of a thin rectangular cross-section; 7.5 Torsion of thin-walled open sections; 7.5.1 Problems; 8 Thin-Walled Beams; 8.1 Basic equations for thin-walled beams; 8.1.1 The thin wall assumption; 8.1.2 Stress flows; 8.1.3 Stress resultants; 8.1.4 Local equilibrium equation; 8.2 Bending of thin-walled beams; 8.2.1 Problems; 8.3 Shearing of thin-walled beams; 8.3.1 Shearing of open sections; 8.3.2 Evaluation of stiffness static moments; 8.3.3 Shear flow distributions in open sections; 8.3.4 Problems; 8.3.5 Shear center for open sections; 8.3.6 Problems; 8.3.7 Shearing of closed sections; 8.3.8 Shearing of multi-cellular sections; 8.3.9 Problems; 8.4 The shear center; 8.4.1 Calculation of the shear center location; 8.4.2 Problems; 8.5 Torsion of thin-walled beams; 8.5.1 Torsion of open sections; 8.5.2 Torsion of closed section; 8.5.3 Comparison of open and closed sections; 8.5.4 Torsion of combined open and closed sections; 8.5.5 Torsion of multi-cellular sections; 8.5.6 Problems; 8.6 Coupled bending-torsion problems; 8.6.1 Problems; 8.7 Warping of thin-walled beams under torsion;8.7.1 Kinematic description; 8.7.2 Stress-strain relations; 8.7.3 Warping of open sections; 8.7.4 Problems; 8.7.5 Warping of closed sections; 8.7.6 Warping of multi-cellular sections; 8.8 Equivalence of the shear and twist centers; 8.9 Non-uniform torsion; 8.9.1 Non-uniform torsion: a classical approach; 8.9.2 Problems; 8.10 Structural idealization; 8.10.1 Lumping the thin-walled section into sheet and stringer components 8.10.2 Axial stress in the stringers; 8.10.3 Shear flow in the sheet components; 8.10.4 Torsion of sheet-stringer sections; 8.10.5 Problems; Part III Energy and variational methods9 Virtual Work Principles; 9.1 Introduction; 9.2 Equilibrium and work fundamentals; 9.2.1 Static equilibrium conditions; 9.2.2 Concept of mechanical work; 9.3 Principle of virtual work; 9.3.1 Principle of virtual work for a single particle; 9.3.2 Kinematically admissible virtual displacements; 9.3.3 Use of infinitesimals as virtual displacements; 9.3.4 Principle of virtual work for a system of particles; 9.3.5 Application of the principle of virtual work to mechanical systems; 9.3.6 Application of the principle of virtual work to trusses; 9.3.7 Generalized coordinates and forces; 9.3.8 Problems; 9.4 Principle of complementary virtual work; 9.4.1 Compatibility equations for a planar truss; 9.4.2 Principle of complementary virtual work for trusses; 9.4.3 Complementary virtual work; 9.4.4 Problems; 9.5 Unit load method for trusses; 9.5.1 Statement of the unit load method for trusses; 9.5.2 Application to trusses; 9.5.3 Problems .; 9.6 Unit load method for beams; 9.6.1 Beam deflection due to bending; 9.6.2 Beam deflection due to torsion; 9.6.3 Application to beam problems; 9.6.4 Deflections of beams with unsymmetric cross sections; 9.6.5 Problems; 9.7 Application of the unit load method to hyperstatic problems; 9.7.1 Force method for trusses; 9.7.2 Force method for beams; 9.7.3 Combined truss and beam problems; 9.7.4 Multiple redundancies; 9.7.5 Problems; 10 Energy Methods; 10.1 Conservative forces;10.1.1 Potential for internal and external forces; 10.1.2 Calculation of the potential functions; 10.2 Principle of minimum total potential energy;10.2.1 Nonconservative external forces; 10.3 Strain energy in springs; 10.3.1 Rectilinear springs; 10.3.2 Torsional springs; 10.4 Problems; 10.5 Strain energy in beams; 10.5.1 Beam under axial loads; 10.5.2 Beam under transverse loads;10.5.3 Beam under torsional loads; 10.6 Strain energy in solids; 10.6.1 General three-dimensional stress state; 10.6.2 Beams under multi-axis bending and axial load; 10.7 Applications to trusses; 10.7.1 Problems;10.7.2 Development of a finite element formulation for trusses;10.7.3 Problems;10.7.4 Applications to beams;10.8 Principle of minimum complementary energy;10.8.1 The potential of the prescribed displacements;10.8.2 Constitutive laws for elastic materials; 10.8.3 The principle of minimum complementary energy; 10.8.4 The principle of least work; 10.8.5 Examples using the PMCP/LWP; 10.8.6 Problems; 10.9 Energy theorems;10.9.1 Clapeyron's theorem; 10.9.2 Castigliano's first theorem; 10.9.3 Crotti-Engesser theorem; 10.9.4 Castigliano's second theorem; 10.9.5 Applications of energy theorems; 10.9.6 The dummy load method; 10.9.7 Unit load method revisited; 10.9.8 Conclusions; 10.9.9 Problems; 10.10Reciprocity theorems; 10.10.1Betti's theorem; 10.10.2Maxwell's theorem; 10.10.3Problems; 11 Variational and Approximate Solutions; 11.1 Approach; 11.2 Approximations based on the principle of minimum total potential energy; 11.2.1 Application to a bending of a beam; 11.2.2 Examples; 11.2.3 Problems; 11.3 The strong and weak statements of equilibrium; 11.3.1 The weak form for beams under axial loads; 11.3.2 Approximate solutions for beams under axial loads; 11.3.3 Problems; 11.3.4 The weak form for beams under transverse loads; 11.3.5 Approximate solutions for beams under transverse loads; 11.3.6 Problems 11.3.7 Equivalence with energy principles;11.3.8 The principle of minimum total potential energy; 11.3.9 Treatment of the boundary conditions; 11.4 Formal procedures for the derivation of approximate solutions; 11.4.1 Basic approximations; 11.4.2 Principle of virtual work; 11.4.3 The principle of minimum total potential energy; 11.4.4 Examples; 11.4.5 Problems; 11.5 A Finite Element formulation for beams; 11.5.1 Formulation of an Euler-Bernoulli beam element; 11.5.2 Examples; 11.5.3 Summary; 11.5.4 Problems; 12 Variational and Energy Principles; 12.1 Variational formulation of beam problems; 12.2 Mathematical Preliminaries; 12.2.1 Stationary point of a function; 12.2.2 Lagrange multiplier method; 12.2.3 Stationary point of a definite integral; 12.3 Variational and Energy Principles; 12.3.1 Review of the equations of linear elasticity; 12.3.2 The principle of virtual work;12.3.3 The principle of complementary virtual work; 12.3.4 The strain energy density function; 12.3.5 The stress energy density function; 12.3.6 The principle of minimum total potential energy; 12.3.7 The principle of minimum complementary energy; 12.3.8 Examples; 12.3.9 The principle of least work;12.3.10Hu-Washizu's principle; 12.3.11Hellinger-Reissner's principle; 12.3.12Problems; 12.4 Applications of Variational and Energy Principles; 12.4.1 The shear lag problem; 12.4.2 The non-uniform torsion problem; 12.4.3 The non-uniform torsion problem (closed sections); 12.4.4 The non-uniform torsion problem (open sections); 12.4.5 Problems; Part IV Advanced topics; 13 Introduction to Plasticity and Thermal Stresses; 13.1 Yielding under combined loading; 13.1.1 Introduction to yield criteria; 13.1.2 Tresca's criterion; 13.1.3 Von Mises' criterion; 13.1.4 Problems; 13.2 Applications of yield criteria to structural problems; 13.2.1 Problems; 13.2.2 Plastic bending; 13.2.3 Problems; 13.2.4 Plastic torsion; 13.2.5 Examples; 13.3 Thermal stresses in structures; 13.3.1 The direct method; 13.3.2 Examples; 13.3.3 Problems; 13.3.4 The constraint method; 13.4 Application to bars and trusses; 13.4.1 Examples; 13.4.2 Problems; 13.4.3 Application to beams; 13.4.4 Examples; 13.4.5 Problems; 14 Buckling of Beams; 14.1 Rigid bar with root torsional spring; 14.1.1 Analysis of a perfect system; 14.1.2 Analysis of an imperfect system; 14.2 Buckling of beams; 14.2.1 Equilibrium equations; 14.2.2 Buckling of a pinned-pinned beam (Equilibrium approach); 14.2.3 Buckling of a pinned-pinned beam (Imperfection approach); 14.2.4 Work done by the axial force; 14.2.5 Buckling of a pinned-pinned beam (Energy approach); 14.2.6 Examples; 14.2.7 Buckling of beams with various end conditions; 14.2.8 Problems; 14.3 Buckling of sandwich beams; 15 Shearing Deformations in Beams; 15.1 Introduction; 15.1.1 A simplified approach; 15.1.2 An equilibrium approach; 15.1.3 Examples; 15.1.4 Problems; 15.2 Shear deformable beams: an energy approach; 15.2.1 Shearing effects on static deflections; 15.2.2 Examples; 15.2.3 Shearing effects on buckling; 15.2.4 Shearing and rotary inertia effects on vibration; 15.2.5 Problems; 16 Kirchhoff Plate Theory; 16.1 Governing equations of Kirchhoff plate theory; 16.1.1 Kirchhoff assumptions; 16.1.2 Stress resultant; 16.1.3 Equilibrium equations; 16.1.4 Constitutive laws; 16.1.5 Summary of Kirchhoff plate theory; 16.2 The bending problem; 16.2.1 Typical boundary conditions; 16.2.2 Simple plate bending solutions; 16.2.3 Problems; 16.3 Anisotropic plates; 16.3.1 Laminated composite plates; 16.3.2 Constitutive laws for laminated composite plates; 16.3.3 The in-plane stiffness matrix; 16.3.4 Problems; 16.3.5 The bending stiffness matrix; 16.3.6 The coupling stiffness matrix; 16.3.7 Problems; 16.3.8 Directionally stiffened plates; 16.3.9 Governing equations for anisotropic plates; 16.4 Solution techniques for rectangular plates; 16.4.1 Navier's solution for simply supported plates; 16.4.2 Examples; 16.4.3 Problems; 16.4.4 Levy's solution; 16.4.5 Problems; 16.5 Circular plates; 16.5.1 Governing equations for the bending of circular plates; 16.5.2 Circular plates subjected to loading presenting circular symmetry; 16.5.3 Examples; 16.5.4 Problems; 16.5.5 Circular plates subjected to arbitrary loading; 16.5.6 Examples;;16.5.7 Problems 16.6 Energy formulation of Kirchhoff plate theory; 16.6.1 The virtual work done by internal forces and moments; 16.6.2 The virtual work done by the applied loads; 16.6.3 The principle of virtual work for Kirchhoff plates; 16.6.4 The principle of minimum total potential energy for Kirchhoff plates; 16.6.5 Energy based approximate solutions for Kirchhoff plates; 16.6.6 Examples;16.6.7 Problems; 16.7 Shear deformable plates; 16.7.1 Problems;16.8 Buckling of plates; 16.8.1 Problems; 17 Appendix: Mathematical Tools;17.1 Coordinate systems and transformations; 17.1.1 The rotation matrix; 17.1.2 Rotation of vector components; 17.1.3 The rotation matrix in two dimensions; 17.1.4 Rotation of vector components in two dimensions; 17.2 Least-square solution of linear systems with redundant equations; 17.3 Arrays and matrices; 17.3.1 Partial derivatives of a linear form; 17.3.2 Partial derivatives of a quadratic form; 17.4 Orthogonality properties of trigonometric functions; References; Index

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