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This book covers the essential topics for a second-level course in strength of materials or mechanics of materials, with an emphasis on techniques that are useful for mechanical design. Design typically involves an initial conceptual stage during which many options are considered. At this stage, quick approximate analytical methods are crucial in determining which of the initial proposals are feasible. The ideal would be to get within 30% with a few lines of calculation. The designer also needs to develop experience as to the kinds of features in the geometry or the loading that are most likely to lead to critical conditions.

With this in mind, the author tries wherever possible to give a physical and even an intuitive interpretation to the problems under investigation. For example, students are encouraged to estimate the location of weak and strong bending axes and the resulting neutral axis of bending before performing calculations, and the author discusses ways of getting good accuracy with a simple one degree of freedom Rayleigh-Ritz approximation. Students are also encouraged to develop a feeling for structural deformation by performing simple experiments in their outside environment, such as estimating the radius to which an initially straight bar can be bent without producing permanent deformation, or convincing themselves of the dramatic difference between torsional and bending stiffness for a thin-walled open beam section by trying to bend and then twist a structural steel beam by hand-applied loads at one end.

In choosing dimensions for mechanical components, designers will expect to be guided by criteria of minimum weight, which with elementary calculations, generally leads to a thin-walled structure as an optimal solution. This consideration motivates the emphasis on thin-walled structures, but also demands that students be introduced to the limits imposed by structural instability. Emphasis is also placed on the effect of manufacturing errors on such highly-designed structures - for example, the effect of load misalignment on a beam with a large ratio between principal stiffness and the large magnification of initial alignment or loading errors in a strut below, but not too far below the buckling load.

Additional material can be found on http://extras.springer.com/ .

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The following ISBNs are associated with this title:

ISBN - 10:9400702949

ISBN - 13:9789400702943

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Table of Contents

**Introduction.- **1.1 The Engineering design process;1.2 Design optimization;1.2.1 Predicting the behaviour of the component.- 1.2.2 Approximate solutions.- 1.3 Relative magnitude of different effects.- 1.4 Formulating and solving problems.- 1.4.1 Use of procedures.- 1.4.2 Inverse problems.- 1.4.3 Physical uniqueness and existence arguments.- 1.5 Review of elementary mechanics of materials.- 1.5.1 Definition of stress components.- 1.5.2 Transformation of stress components.- 1.5.3 Displacement and strain.- 1.5.4 Hooke's law.- 1.5.5 Bending of beams.- 1.5.6 Torsion of circular bars.- 1.6 Summary.- Problems.- **2 Material Behaviour and Failure.- **2.1 Transformation of stresses.- 2.1.1 Review of two-dimensional results.- 2.1.2 Principal stresses in three dimensions.- 2.2 Failure theories for isotropic materials.- 2.2.1 The failure surface.- 2.2.2 The shape of the failure envelope.- 2.2.3 Ductile failure (yielding).- 2.2.4 Brittle failure.- 2.3 Cyclic loading and fatigue.- 2.3.1 Experimental data.- 2.3.2 Statistics and the size effect.- 2.3.3 Factors influencing the design stress.- 2.3.4 Effect of combined stresses.- 2.3.5 Effect of a superposed mean stress.- 2.3.6 Summary of the design process.- 2.4 Summary.- Problems.- **3 Energy Methods.- **3.1 Work done on loading and unloading.- 3.2 Strain energy.- 3.3 Load-displacement relations.- 3.3.1 Beams with continuously varying bending moments.- 3.3.2 Axial loading and torsion.- 3.3.3 Combined loading.- 3.3.4 More general expressions for strain energy.- 3.3.5 Strain energy associated with shear forces in beams.- 3.4 Potential energy.- 3.5 The principle of stationary potential energy.- 3.5.1 Potential energy due to an external force.- 3.5.2 Problems with several degrees of freedom.- 3.5.3 Non-linear problems.- 3.6 The Rayleigh-Ritz method.- 3.6.1 Improving the accuracy.- 3.6.2 Improving the back of the envelope approximation.- 3.7 Castigliano's first theorem.- 3.8 Linear elastic systems.- 3.8.1 Strain energy.- 3.8.2 Bounds on the coefficients.- 3.8.3 Use of the reciprocal theorem.- 3.9 The stiffness matrix.- 3.9.1 Structures consisting of beams.- 3.9.2 Assembly of the stiffness matrix.- 3.10 Castigliano's second theorem.- 3.10.1 Use of the theorem.- 3.10.2 Dummy loads.- 3.10.3 Unit load method.- 3.10.4 Formal procedure for using Castigliano's second theorem.- 3.10.5 Statically indeterminate problems.- 3.10.6 Three-dimensional problems.- 3.11 Summary.- Problems.- **4 Unsymmetrical Bending.- **4.1 Stress distribution in bending.- 4.1.1 Bending about the *x*-axis only.- 4.1.2 Bending about the *y*-axis only.- 4.1.3 Generalized bending.- 4.1.4 Force resultants.- 4.1.5 Uncoupled problems.- 4.1.6 Coupled problems.- 4.2 Displacements of the beam.- 4.3 Second moments of area.- 4.3.1 Finding the centroid.- 4.3.2 The parallel axis theorem.- 4.3.3 Thin-walled sections.- 4.4 Further properties of second moments.- 4.4.1 Coordinate transformation.- 4.4.2 Mohr's circle of second moments.- 4.4.3 Solution of unsymmetrical bending problems in principal coordinates.- 4.4.4 Design estimates for the behaviour of unsymmetrical sections.- 4.4.5 Errors due to misalignment.- 4.5 Summary.- Problems.- **5 Non-linear and Elastic-Plastic Bending.- **5.1 Kinematics of bending.- 5.2 Elastic-plastic constitutive behaviour.- 5.2.1 Unloading and reloading.- 5.2.2 Yield during reversed loading.- 5.2.3 Elastic-perfectly plastic material.- 5.3 Stress fields in non-linear and inelastic bending.- 5.3.1 Force and moment resultants.- 5.4 Pure bending about an axis of symmetry.- 5.4.1 Symmetric problems for elastic-perfectly plastic materials.- 5.4.2 Fully plastic moment and shape factor.- 5.5 Bending of a symmetric section about an orthogonal axis.- 5.5.1 The fully plastic case.- 5.5.2 Non-zero axial force.- 5.5.3 The partially plastic solution.- 5.6 Unsymmetrical plastic bending.- 5.7 Unloading, springback and residual stress.- 5.7.1 Springback and residual curvature.- 5.7.2 Reloading and shakedown.- 5.8 Limit analysis in the design of beams.- 5.8.1 Plastic hinges.- 5.8.2 Indeterminate problems.- 5.9 Summary.- Problems.- **6 Shear and Torsion of Thin-walled Beams.- **6.1 Derivation of the shear stress formula.- 6.1.1 Choice of cut and direction of the shear stress.- 6.1.2 Location and magnitude of the maximum shear stress.- 6.1.3 Welds, rivets and bolts.- 6.1.4 Curved sections.- 6.2 Shear centre.- 6.2.1 Finding the shear centre.- 6.3 Unsymmetrical sections.- 6.3.1 Shear stress for an unsymmetrical section.- 6.3.2 Determining the shear centre.- 6.4 Closed sections.- 6.4.1 Determination of the shear stress distribution.- 6.5 Pure torsion of closed thin-walled sections.- 6.5.1 Torsional stiffness.- 6.5.2 Design considerations in torsion.- 6.6 Finding the shear centre for a closed section.- 6.6.1 Twist due to a shear force.- 6.6.2 Multicell sections.- 6.7 Torsion of thin-walled open sections.- 6.7.1 Loading of an open section away from its shear centre.- 6.8 Summary.- Problems.- **7 Beams on Elastic Foundations;**7.1 The governing equation.- 7.1.1 Solution of the governing equation.- 7.2 The homogeneous solution.- 7.2.1 The semi-infinite beam.- 7.3 Localized nature of the solution.- 7.4 Concentrated force on an infinite beam.- 7.4.1 More general loading of the infinite beam.- 7.5 The particular solution.- 7.5.1 Uniform loading.- 7.5.2 Discontinuous loads.- 7.6 Finite beams.- 7.7 Short beams.- 7.8 Summary.- Problems.- **8 Membrane Stresses in Axisymmetric Shells.- **8.1 The meridional stress.- 8.1.1 Choice of cut.- 8.2 The circumferential stress.- 8.2.1 The radii of curvature.- 8.2.2 Sign conventions.- 8.3 Self-weight.- 8.4 Relative magnitudes of different loads.- 8.5 Strains and Displacements.- 8.5.1 Discontinuities.- 8.6 Summary.- Problems.- **9 Axisymmetric Bending of Cylindrical Shells.- **9.1 Bending stresses and moments.- 9.2 Deformation of the shell.- 9.3 Equilibrium of the shell element.- 9.4 The governing equation.- 9.4.1 Solution strategy.- 9.5 Localized loading of the shell.- 9.6 Shell transition regions.- 9.6.1 The cylinder/cone transition.- 9.6.2 Reinforcing rings.- 9.7 Thermal stresses.- 9.8 The ASME pressure vessel code.- 9.9 Summary.- Problems.- **10 Thick-walled Cylinders and Disks.- **10.1 Solution method.- 10.1.1 Stress components and the equilibrium condition.- 10.1.2 Strain, displacement and compatibility.- 10.1.3 The elastic constitutive law.- 10.2 The thin circular disk.- 10.3 Cylindrical pressure vessels.- 10.4 Composite cylinders, limits and fits.- 10.4.1 Solution procedure.- 10.4.2 Limits and fits.- 10.5 Plastic deformation of disks and cylinders.- 10.5.1 First yield.- 10.5.2 The fully-plastic solution.- 10.5.3 Elastic-plastic problems.- 10.5.4 Other failure modes.- 10.5.5 Unloading and residual stresses.- 10.6 Summary.- Problems.- **11 Curved Beams.- **11.1 The governing equation.- 11.1.1 Rectangular and circular cross sections.- 11.1.2 The bending moment.- 11.1.3 Composite cross sections.- 11.1.4 Axial loading.- 11.2 Radial stresses.- 11.3 Distortion of the cross section.- 11.4 Range of application of the theory.- 11.5 Summary.- Problems.- **12 Elastic Stability.- **12.1 Uniform beam in compression.- 12.2 Effect of initial perturbations.- 12.2.1 Eigenfunction expansions.- 12.3 Effect of lateral load (beam-columns).- 12.4 Indeterminate problems.- 12.5 Suppressing low-order modes ..- 12.6 Beams on elastic foundations.- 12.6.1 Axisymmetric buckling of cylindrical shells.- 12.6.2 Whirling of shafts.- 12.7 Energy methods.- 12.7.1 Energy methods in beam problems.- 12.7.2 The uniform beam in compression.- 12.7.3 Inhomogeneous problems.- 12.8 Quick estimates for the buckling force.- 12.9 Summary.- Problems.- **A The Finite Element Method.- **A.1 Approximation.- A.1.1 The 'best' approximation.- A.1.2 Choice of weight functions.- A.1.3 Piecewise approximations.- A.2 Axial loading ..- A.2.1 The structural mechanics approach.- A.2.2 Assembly of the global stiffness matrix.- A.2.3 The nodal forces.- A.2.4 The Rayleigh-Ritz approach.- A.2.5 Direct evaluation of the matrix equation.- A.3 Solution of differential equations.- A.4 Finite element solutions for the bending of beams.- A.4.1 Nodal forces and moments.- A.5 Two and three-dimensional problems.- A.6 Computational considerations.- A.6.1 Data storage considerations.- A.7 Use of the finite element method in design.- A.8 Summary.- Problems.- **B Properties of Areas.- C Stress Concentration Factors.- D Answers to Even Numbered Problems.- Index.**

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