Mechanics of Solids, Advanced Course (MSK530)

MSK530 deals with the methods of structural design and strength of materials. The provided theoretical tools of the stress and strain theory, constitutive equations, inelastic response and yield criteria, beam theory, energy methods, Airy’s stress functions, thick wall cylinders, contact stresses, flat plates, and stability provide the necessary knowledge for estimating the performance of materials and structures under mechanical loading. Methods of assessment of the safety margins of complex structures and implementation of the theory on representative cases are described. Design of mechanical and civil engineering structures like wind turbines, buildings, bridges, oil rigs and marine structures, automotive and robotic structures is based on the principles of the mechanics of solids

After completing this course the students will:

• Be able to carry out stress and strain analyses of different load bearing elements and/or constructions
• Understand the response of individual elements to the applied loads in both elastic and plastic regime and after unloading
• Be able to dimension various elements of constructions subjected to a wide range of loading conditions
• Be able to apply the knowledge on calculation of real constructions

The Chapter 1 deals with the fundamental concepts of stress analysis to estimate the strength of materials and structures. The stress vector and tensor in a material element is described and their transformation in a rotated coordinate system is provided. Analytic derivation of the formulae for the principal stresses and the corresponding planes in material elements under triaxial and plane stress conditions is presented and applications in real problems are demonstrated. Differential equations of motion in Cartesian and Polar coordinate system are formulated for analysing the behaviour of solids under various types of stress conditions. This chapter provides the necessary background for understanding the failure criteria of materials and estimate the safety margins of complex structures. The Chapter 2 provides the fundamental concepts of strain analysis to estimate the deformation of materials and structures. The chapter starts with the definition of the normal and shear strains and their correlation with displacements. The strain vector and tensor in a material element are described and their transformation in a rotated coordinate system is provided. The strain compatibility equations are formulated for analysing the behaviour of solids under various types of strain conditions. This chapter aims to provide the necessary background for understanding the terms of stiffness, ductility, strain hardening, strain energy density, etc and to investigate the mechanics of material deformation.  The Chapter 3 describes how the stresses and strains are correlated and how elastic materials respond under multiaxial stress state. The constitutive equations of elasticity for isotropic and anisotropic solids are provided for understanding the effects of the mechanical properties of materials in the mechanical behaviour of structural components. The chapter 3 provides the necessary background for stress and strain analysis of isotropic and composite materials. Chapter 4 deals with the main mechanical properties of the engineering materials. The basic mechanical tests for quality assessment and characterization of materials are described and the effect of the stress, temperature, strain rate, and the surface quality on the material properties is discussed. The chapter focuses on the methods for measuring the elastic modulus, yield stress, ultimate stress, ductility, fracture toughness, and fatigue endurance limit from the tensile test, impact test, and fatigue test. The chapter aims to provide the necessary background for understanding the nature of the mechanical properties and the material selection criteria for structural design. Chapter 5 includes the yield criteria of materials under multiaxial stresses and investigates the capacity of beams to carry bending moments beyond the elastic limit. It provides the theoretical tools for the structural design under multiaxial loading conditions and the estimation of the safety margin of structures against yielding. The methodology for the residual stress distribution on beam cross-sections after unloading from the fully plastic state is described and demonstrated on typical examples. Chapter 6 describes the analysis of beams under bending. Bending induced normal stresses, bending induced shear stresses, and bending induced displacements of beams of compact and thin-wall open cross sections as well as the location of the shear centre are included in the chapter.  Special focus is paid on the analysis of beams on elastic foundation where a new technique based on the Fourier Cosine Integral Transform is presented along with implementation on representative examples. Chapter 7 aims to present the analysis of beams under torsion. The general torsion theory of beams of arbitrary cross-section is presented first, and implementation on beams of elliptical and circular cross section is provided. The method of the Prandtl membrane analogy is presented and torsion of slender and non-slender rectangular, composite, hollow thin-wall single or multicompartment cross-sections is analysed. Analytic formulae for shear stresses and angle of twist versus the torque are derived, and demonstration of the method in characteristic examples is given. Chapter 8 provides the fundamental energy principles for analysing mechanical and structural engineering problems. It starts with the law of conservation of energy (1st law of thermodynamics) and the correlation of the stress and displacement components to the elastic energy density of solids. The principles of the Virtual Work, Minimum Potential Energy and the Castigliano’s theorem are explained in details and formulae for analysing statically determinate and indeterminate structures are derived. Implementation of Energy methods in force and displacement analysis of typical structures is provided. The Energy methods is the necessary background for understanding the fundamentals of numerical methods like Finite and Boundary Elements in Mechanics. Chapter 9 deals with the analysis of flat plates. Cylindrical pure bending, bending in two directions, displacement and stress analysis of rectangular plates under general loading, boundary conditions of plate problems, circular plates under axisymmetric loading, and plates on elastic foundation are included in Chapter 9. A new technique based on Hankel’s integral transform is used to analyse plates on elastic foundation under axisymmetric loads, and implementation of the method on representative examples is provided. Chapter 10 presents a powerful theoretical tool for analytical stress analysis of components and structures. Airy’s Stress Function is based on the equilibrium equations of stresses and the compatibility equations of strains. It is used for obtaining exact solutions in complex engineering problems. With the aid of this method the difference between exact and approximate solutions for bending stresses in beams can be assessed. The investigation of the effect of holes in the structural integrity and the calculation of the hole-induced stress concentration in structural components is provided. Chapter 11 includes the fomulation of the equations for the stress distribution within the wall of thick-wall cylinders under internal and external pressure. With the Airy’s Stress Function stress and displacement analysis is performed and the governing biharmonic equation is derived. The obtained formulae are applied to analyse compount cylinders after shrink-fitting. Implementation of the method on characteristic problems is provided. Chapter 12 deals with local stresses associated with the force transfer from one body to another. Stress and deformation analysis on the contact surface between two bodies and the condition assessment of safe force transfer between solids is provided. The chapter aims to investigate the reasons of material failure in the contact surface and to describe the failure mechanism between rolling bodies in contact. The provided formulae are the necessary tools for designing joints, supports, and foundations. Contact mechanics is the background of the principles of engineering tribology. Chapter 13 is devoted to the understanding of the nature and the consequences of elastic instability of structural members. Classification of elastic bifurcation buckling is provided firstly. With the aid of Euler’s theory, analytical formulation of the governing equations and the boundary conditions for calculating the critical loads causing elastic instability in long, intermediate, and short columns and beams is given. The validity range of Euler’s formula for buckling is discussed and the Johnson’s formula is presented. Formulae for calculating the critical buckling load for beams with geometric imperfection (eccentric loading) and flexural-torsional buckling of beams are derived. Chapter 14 provides an introduction of the Mechanics of composite materials. With the aid of Kirchhoff’s assumption, the Classical Lamination Theory for fiber reinforced laminates is presented analytically. Formulae for stiffness and compliance matrices of multilayered composite materials and the steps for calculating the principal stresses on each layer of a laminate is described. The Tsai –Wu failure criterion is explained, and examples are provided. The Appendix offers a collection of representative Solid Mechanics problems with solutions. The solution of the provided problems require combination of the theory of several chapters of the book and are suitable for the preparation of students for the exam.