In the classical finite-element (FE) formulation for beams, plates, and shells infinitesimal rotations are used as nodal coordinates. As a result, beams, plates, and shells are not considered as isoparametric elements. Rigid body motion of these nonisoparametric elements does not result in zero strains and exact modeling of the rigid body inertia using these elements cannot be obtained. In this chapter, a formulation for the large reference displacement and small deformation analysis of deformable bodies using nonisoparametric finite elements is presented. This formulation, in which infinitesimal rotations are used as nodal coordinates, leads to exact modeling of the rigid body dynamics and results in zero strains under an arbitrary rigid body motion. It is crucial in this formulation that the assumed displacement field of the element can describe an arbitrary rigid body translation. Using this property and an intermediate element coordinate system, a concept similar to the parallel axis theorem used in rigid body dynamics can be applied to obtain an exact modeling of the rigid body inertia for deformable bodies that have complex geometrical shapes. More discussion on the use of the parallel axis theorem in modeling the inertia of rigid bodies with complex geometry is presented in Chapter 8 of this book. It is recommended that the reader reviews the basic materials presented in Chapter 8 in order to recognize that the coordinate systems used to develop the large displacement FE/FFR formulation presented in this chapter are the same as the coordinate systems used to model the complex geometry in the case of rigid body dynamics.

There are two main concerns regarding the use of the classical finite-element (FE) formulations in the large deformation and rotation analysis of flexible multibody systems. First, in the classical FE literature on beams, plates, and shells, infinitesimal rotations are used as nodal coordinates. Such a use of coordinates does not lead to the exact modeling of a simple rigid body motion. Second, lumped mass techniques are used in many FE formulations and computer programs to describe the inertia of the deformable bodies. As will be demonstrated in this chapter, such a lumped mass representation of the inertia also does not lead to exact modeling of the equations of motion of the rigid bodies.

While a body-fixed coordinate system is commonly employed as a reference for rigid components, a floating coordinate system is suggested for deformable bodies that undergo large rotations. When dealing with rigid body systems, the kinematics of the body is completely described by the kinematics of its coordinate system because the particles of a rigid body do not move with respect to a body-fixed coordinate system. The local position of a particle on the body can then be described in terms of fixed components along the axes of this moving coordinate system. In deformable bodies, on the other hand, particles move with respect to the selected body coordinate system, and therefore, a distinction is made between the kinematics of the coordinate system and the body kinematics.

In the virtual prototyping, durability analysis, and design processes, accurate computer modeling of a large number of physics and engineering systems is necessary. For such systems that consist of interconnected bodies, developing credible computer models requires the use of accurate geometry description as well as the analysis techniques described in this book. Nonetheless, virtual prototyping, durability analysis, and product design are currently performed in many industry sectors using three different incompatible systems: computer-aided design (CAD) system for creating the geometry, finite element (FE) software for developing the analysis mesh, and multibody system (MBS) software for constructing and numerically solving the differential/algebraic equations (DAEs) of constrained systems. The use of the three-software technology has resulted in unreliable stress and durability results, significant waste of engineering time and efforts, misrepresentation of significant model details, and significant economic loss.

The expressions for the elastic strain energy and its volumetric and deviatoric parts are derived for three-dimensional states of stress and strain. Betti's reciprocal theorem of linear elasticity is formulated, which yields the Maxwell coefficients, frequently used in structural mechanics. Castigliano's theorem is formulated and applied to axially loaded rods and trusses, twisted bars, and bent beams and frames. The principle of virtual work and the variational principle of linear elasticity are introduced. The differential equation of the deformed shape of the bent beam is derived from the consideration of the principle of virtual work. The approximate Rayleigh–Ritz method is introduced and applied to selected problems of structural mechanics. An introduction to the finite element method in the analysis of beam bending, torsion, and axial loading is then presented. The corresponding stiffness matrices and load vectors are derived for each element and are assembled into the global stiffness matrix and load vector of the entire structure.

Two-dimensional problems of plane stress and plane strain in polar coordinates, both axisymmetric and non-axisymmetric, are considered. Among axisymmetric problems, the bending of a curved beam by two end couples and the problem of a pressurized hollow disk or cylinder are analyzed. Among non-axisymmetric problems, solutions are derived for problems of bending of a curved cantilever beam by a vertical force, loading of a circular hole in an infinite medium,concentrated vertical and tangential forces at the boundary of a half-plane, and a semi-elliptical pressure distribution over the boundary of a half-space. The problems of diametral compression of a circular disk (Michell problem), stretching of a large plate weakened by a small circular hole (Kirsch problem), stretching of a large plate strengthened by a small circular inhomogeneity, and spinning of a circular disk are also analyzed and discussed. The chapter ends with an analysis of the stress field near a crack tip under symmetric and antisymmetric remote loadings, the stress and displacement fields around an edge dislocation in an infinite medium, and around a concentrated force in an infinite plate.

A brief coverage of the mechanics of contact problems is presented. The governing equations for three-dimensional axisymmetric elasticity problems in cylindrical coordinates are first formulated, which is followed by the solutions to classical problems of a concentrated force within an infinite medium (Kelvin problem), and a concentrated force at the boundary of a half-space (Boussinesq problem). The stress fields in a half-space loaded by an elliptical and a uniform pressure distribution over a circular portion of its boundary are presented. Indentation by a spherical ball and by a cylindrical circular indenter are analyzed. The second part of the chapter is devoted to Hertzian contact problems. The nonlinear force–displacement relation is derived for elastic contact of two spherical bodies pressed against each other by two opposite forces. The elastic contact of two circular cylinders is also considered. The contact pressure and the maximum shear stress are determined. The approach of the centers of the cylinders requires the consideration of the local contact stresses, as well as the stresses within the bulk of each cylinder.

The generalized Hooke's law is introduced, which represents six linear relations between the stress and strain components in the case of small elastic deformations. For isotropic materials, only two independent elastic constants appear in these stress–strain relations. Each longitudinal strain component depends linearly on the three orthogonal components of the normal stress; the relationship involves two constants: Young's modulus of elasticity and Poisson's coefficient of lateral contraction. Each shear strain component is proportional to the corresponding shear stress component; the shear modulus relates the two. The volumetric strain is proportional to the mean normal stress, with the elastic bulk modulus relating the two. The inverted form of the generalized Hooke's law is derived, which expresses the stress components as a linear combination of strain components. Lamé elastic constants appear in these relations. The Duhamel–Neumann law of linear thermoelasticity is formulated, which incorporates the effects of temperature on stresses and strains. The Beltrami–Michell compatibility equations with and without temperature effects are derived.

The analysis of normal and shear stresses in a cantilever beam bent by a transverse force is presented. The stress function is introduced and the governing Poisson-type partial differential equation and the accompanying boundary conditions are derived for simply and multiply connected cross sections of a prismatic beam. The exact solution to the boundary value problem is presented for circular, semi-circular, hollow-circular, elliptical, and rectangular cross sections. Approximate, but sufficiently accurate, formulas for shear stresses in thin-walled open and thin-walled closed cross sections, including multicell cross sections, are derived and applied to different profiles of interest in structural engineering. The determination of the shear center of thin-walled profiles, which is the point through which the transverse load must pass in order to have bending without torsion, is discussed in detail. The sectorial coordinate is introduced and conveniently used in this analysis. The formulas are derived with respect to the principal and non-principal centroidal axes of the cross section.

A survey of failure criteria for brittle and ductile materials is presented. The maximum principal stress and the maximum principal strain criterion for brittle materials are introduced. The Tresca maximum shear stress and the von Mises energy criterion for ductile materials are formulated and applied to study the onset of plastic yield in thin-walled tubes and other structural members. The Mohr failure criterion is based on the consideration of Mohr's circles. The Coulomb–Mohr criterion for geomaterials incorporates the normal and shear stress, and the coefficient of internal friction. According to Drucker–Prager’s criterion, plastic yield occurs when the shear stress on octahedral planes overcomes the cohesive and frictional resistance to sliding. The fracture mechanics based failure criterion takes into account the presence of cracks. Failure occurs if the release of potential energy accompanying the crack growth is sufficient to supply the increase of the surface energy of expanded crack faces. The fracture criterion is also formulated in terms of the stress intensity factor K, whose critical value is the fracture toughness of the material.

The components of the infinitesimal strain tensor are defined, which represent measures of the relative length changes (longitudinal strains or dilatations) and the angle changes (shear strains) at a considered material point with respect to the chosen coordinate axes. The principal strains (maximum and minimum dilatations) and the maximum shear strains are determined, as well as the areal and volumetric strains. The expressions for the strain components are derived in terms of the spatial gradients of the displacement components. The Saint-Venant compatibility equations are introduced which assure the existence of single-valued displacements associated with a given strain field. The matrix of local material rotations, which accompany the strain components in producing the displacement gradient matrix, is defined. The determination of the displacement components by integration of the strain components is discussed.

The representation of the stress and strain tensors and the formulation of the boundary-value problem of linear elasticity in cylindrical coordinates is considered. The Cauchy equations of equilibrium, expressed in terms of stresses, the strain–displacement relations, the compatibility equations, the generalized Hooke's law, and the Navier equations of equilibrium, expressed in terms of displacements, are all cast in cylindrical coordinates. The axisymmetric boundary-value problem of a pressurized hollow cylinder with either open or closed ends is formulated and solved. The results are used to obtain the elastic fields for a pressurized circular hole in an infinite medium, and to solve a cylindrical shrink-fit problem. A pressurized hollow sphere and a spherical shrink-fit problem are also considered to illustrate the solution procedure in the case of problems with spherical symmetry.

Antiplane shear is a type of deformation in which the only nonvanishing displacement component is the out-of-plane displacement, orthogonal to the (x,y) plane. The corresponding nonvanishing shear stresses are within that plane. For this type of deformation, the displacement is a harmonic function of (x,y), satisfying the Laplace's equation. We solve and discuss the problems of antiplane shear of a circular annulus, a concentrated line force along the surface of a half-space, antiplane shear of a medium weakened by a circular or an elliptical hole, and the problem of a medium strengthened by a circular inhomogeneity. The stress field near a crack tip under remote antiplane shear loading is derived, as well as the stress field around a screw dislocation in infinite and semi-infinite media. The stresses produced by a screw dislocation near a circular hole or a circular inhomogeneity in an infinite homogeneous medium, and the stresses produced by a screw dislocation in the vicinity of a bimaterial interface are examined.