Nonexistence of Solution for Horizontally Rigid Half-Space

The Westergaard expressions for stresses and displacements in a half-space of Westergaard material are shown to be invalid because the vertical shear stress does not vanish at the plane boundary of the half-space. The same inadequacy is discovered in the solution for a horizontally rigid cross-anisotropic half-space deduced from Michell’s expressions. A surprising conclusion is drawn, namely, that there is currently no exact elastic solution for the stresses and displacements in a horizontally rigid cross-anisotropic half-space. The Westergaard theory should cease to be regarded as an exact elastic solution for a problem of the theory of elasticity.     

Displacements and Stresses Due to a Uniform Vertical Circular Load in an Inhomogeneous Cross-Anisotropic Half-Space

In this work, we present the solutions for displacements and stresses along the centerline of a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space with its Young’s and shear moduli varying exponentially with depth. The planes of cross anisotropy are assumed to be parallel to the horizontal surface. The presented solutions can be directly integrated from the point load solution in a cylindrical coordinate system, which were derived by the writers. However, the resulting integrals of the circular solution for displacements and stresses cannot be given in closed form; hence, numerical integrations are required. For a homogeneous cross-anisotropic half-space, the numerical results agree very well with the exact solutions of Hanson and Puja, published in 1996. Two examples are given to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement and vertical normal stress in the inhomogeneous isotropic/cross-anisotropic soils subjected to a uniform vertical circular load acting on the surface. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice.     

Planar Bending of Sandwich Beams with Transverse Loads off the Centroidal Axis

Conventional analysis methods for beams do not distinguish between transverse loads that are applied at the beam centroidal axis and those acting either above or below the centroidal axis. In contrast, this paper formulates a sandwich beam finite element solution which models the effect of load height relative to the centroidal axis. Towards this goal, the governing equilibrium equations and associated boundary conditions are derived based on a Timoshenko beam formulation for the core material. Special shape functions satisfying the homogeneous form of the equilibrium equations are derived and subsequently used to formulate exact stiffness matrices. By omitting the stiffness terms related to the faces, the formulation for a homogeneous Timoshenko beam can be recovered. Also, the Euler–Bernouilli counterpart of the formulation is recovered as a limiting case of the current Timoshenko beam formulation. Effects of load height relative to the centroid are observed to have similarities with those induced by axial forces in beam-columns. For a simply supported beam, downward acting loads located below the centroidal axis are found to induce a stiffening effect while those acting above the centroidal axis are found to induce a softening effect, resulting in higher transverse displacements.     

Large Deflection Stability of Slender Beam-Columns with Semirigid Connections: Elastica Approach

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.     

Lagrangian Approach to Structural Collapse Simulation

Computer analysis of structures has traditionally been carried out using the displacement method combined with an incremental iterative scheme for nonlinear problems. In this paper, a Lagrangian approach is developed, which is a mixed method, where besides displacements, the stress resultants and other variables of state are primary unknowns. The method can potentially be used for the analysis of collapse of structures subjected to severe vibrations resulting from shocks or dynamic loads. The evolution of the structural state in time is provided a weak formulation using Hamilton’s principle. It is shown that a certain class of structures, known as reciprocal structures, has a mixed Lagrangian formulation in terms of displacements and internal forces. The form of the Lagrangian is invariant under finite displacements and can be used in geometric nonlinear analysis. For numerical solution, a discrete variational integrator is derived starting from the weak formulation. This integrator inherits the energy and momentum conservation characteristics for conservative systems and the contractivity of dissipative systems. The integration of each step is a constrained minimization problem and it is solved using an augmented Lagrangian algorithm. In contrast to the traditional displacement-based method, the Lagrangian method provides a generalized formulation which clearly separates the modeling of components from the numerical solution. Phenomenological models of components, essential to simulate collapse, can be incorporated without having to implement model-specific incremental state determination algorithms. The state variables are determined at the global level by the optimization method.     

Green's Function of Thermoelastic Mixed Boundary Value Problem for an Elliptic Hole

The static thermoelastic mixed boundary value problem is studied, the one modeling a heat source and a heat sink acting in an infinite plane that contains a partially debonded rigid elliptic inclusion. By employing the conformal mapping technique and the traction-free Green's function for the exterior of the ellipse, the Green's function is obtained for the mixed boundary value problem with a finite number of debondings occurring on the interface between the rigid inclusion and the elastic matrix. Adiabatic and isothermal boundary conditions are considered. The stress distribution is computed on the interface and on the x-axis. The stress-intensity factors are obtained for the case when the ellipse reduces to a crack.     

Free-Edge and Ply Cracking Effect in Angle-Ply Laminated Composites Subjected to In-Plane Loads

This paper presents a semianalytical method for the prediction of interlaminar stresses and displacements near the free edges and ply cracks in general angle-ply laminates subjected to biaxial extensions and/or in plane shear deformation. The method is based on a state space representation of the three-dimensional equations of elasticity. Numerical solutions are obtained by using layer refinement in the through thickness direction and Fourier series expansion in the other directions. By this approach, an angle-ply laminate may be composed of an arbitrary number of monoclinic layers and each layer may have different material property and thickness. This method guarantees continuous fields of all interlaminar stresses across interfaces between material layers. Numerical results are compared with those obtained from other methods. It is found that the theory provides a satisfactory approximation to the stress singularities near the free edges and ply cracks. Numerical solutions for antisymmetric laminates under extension and general laminates under shearing are new in the literature and can be used as benchmarks for validating new models.     

Time-Dependent Response of an Axially Loaded Elastic Bar in a Multilayered Poroelastic Medium

This paper considers load transfer from an axially loaded long elastic bar into a multilayered poroelastic half-space. The problem is analyzed by decomposing the bar-half-space system into an extended half-space governed by Biot’s theory of poroelasticity and a one-dimensional fictitious bar. The interaction problem is formulated in the Laplace transform domain. Vertical displacement of the bar is approximated by an exponential series with a set of arbitrary functions. The arbitrary functions are determined by using a variational method. The vertical displacement influence function of a multilayered half-space subjected to a buried uniform vertical patch load is required in the variational formulation. The required influence function is obtained by employing a previously developed exact stiffness matrix method. Time domain solutions are computed by using a numerical Laplace inversion scheme. Selected numerical results are presented to portray the influence of the bar length–radius ratio, layer configuration, poroelastic material parameters, and loading time history on the time dependent response of a bar.     

Asymmetric Dynamic Green’s Functions in a Two-Layered Transversely Isotropic Half-Space

By virtue of a complete representation using two displacement potentials, an analytical derivation of the elastodynamic Green’s functions for a transversely isotropic layer underlain by a transversely isotropic half-space is presented. Three-dimensional point-load and patch-load Green’s functions for stresses and displacements are given in the complex-plane line-integral representations. The formulation includes a complete set of transformed stress-potential and displacement-potential relations in the framework of Fourier expansions and Hankel integral transforms, that is useful in a variety of elastodynamic as well as elastostatic problems. For the numerical computation of the integrals, a robust and effective methodology is laid out. Comparisons with the existing numerical solutions for a two-layered transversely isotropic half-space under static surface load, and a homogeneous transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacement and stress Green’s functions are presented to portray the dependence of the response of the two-layered half-space on the frequency of excitation and the role of the upper layer.     

Comparison between Coupled and Uncoupled Consolidation Analysis of a Rigid Sphere in a Porous Elastic Infinite Space

Linear consolidation analyses are usually treated either by means of Terzaghi-Rendulic uncoupled theory or Biot’s consolidation theory. In this note, the problem of consolidation displacements around an axially loaded sphere was considered. It is demonstrated that both the uncoupled analysis and the coupled analysis give the same governing equation for pore fluid pressure dissipation with time. A simplified procedure for deriving transient strain components is illustrated. A general solution for time-dependent displacements is obtained using uncoupled consolidation analysis. Close agreement is evident between the new approximate uncoupled analysis solution and the existing coupled analysis solution with a maximum error of less than 0.5%.     

Fracture normal to a bimaterial interface: Effects of plasticity on crack-tip shielding and amplification

The problem of a crack approaching a bimaterial interface is considered in this paper. Attention is focused on an interface between two elastoplastic solids whose elastic properties are identical and whose plastic properties are different. For the case of a crack approaching a bimaterial interface perpendicularly, it is shown by recourse to detailed finite element analyses that the near-tip “driving force” for fracture is strongly influenced by whether the crack approaches the interfaces from the lower strength or the higher strength materials. Specifically, it is demonstrated that the crack-tip is “shielded” from the remote loads when it approaches the interface from the weaker material, and that the effective J-integral at the crack tip is greater than the remote J when it approaches the interface from the stronger material. This plasticity effect determines whether a crack approaching the bimaterial interface continues to advance through the interface or is arrested before penetrating the interface. These theoretical findings are substantiated using controlled experiments of fatigue crack growth perpendicular to a ferrite—austenite bimaterial interface. The effect of the non-singular T-stress, acting parallel to the crack plane, on shielding and amplification of the stress fields is also discussed.     

Heat Source in Infinite Plane with Elliptic Rigid Inclusion and Hole

The plane thermoelastic problems of a stationary heat source in an infinite plane with an elliptic rigid inclusion and an elliptic hole are analyzed under thermally adiabatic and isothermal boundary conditions. The problems of an elliptic rigid inclusion are derived for the following cases: (1) the case that there are rigid-body displacement and rotation; and (2) the case that there is no rigid-body displacement or rotation. To analyze these problems, the following three fundamental solutions are derived: Problem A, in which a point heat source exists within an infinite domain; Problem B, in which the inclusion has a small amount of rotation; and Problem C, in which the inclusion is subjected to concentrated loads. Two cases can be obtained by superimposing these fundamental solutions. For the hole problem, the fundamental solution (Green's function) is also derived. In analysis, the complex stress functions, the mapping function, and the thermal dislocation method are used. The complex stress functions are obtained as a closed form. For analytic examples, the stress distributions are shown under thermally adiabatic and isothermal boundary conditions. For the crack problem, the stress intensity factors are shown for the location of the heat source.     

Analytical Solution for Plane Trusses with Equidistant Supports

The type of plane truss considered in this paper is the continuous Warren truss resting on equidistant roller supports, subjected to transverse nodal loads. In order to uncouple the governing equation, we need to replace the original structure, which does not possess cyclic periodicity, by an equivalent system that is cyclic biperiodic. By applying the U-transformation twice, the governing equation for the equivalent system can be uncoupled and become a set of single degree of freedom equations, which leads to the explicit form solution for the supporting reaction and nodal displacement. The expressions of the solution include two numbers of substructures and supports. As an example, a Warren truss with six substructures and four supports subjected to a concentrated load acting at its center node in the symmetric line of the original structure is worked out by means of the formulas obtained in the present study. It is shown that the result is exact.     

Some considerations on fatigue crack closure at near-threshold stress intensities due to fracture surface morphology

Summary It is noted that at near-threshold levels, in addition to the role of plasticity-and oxide-induced crackclosure, fracture surface roughness or morphology may promote significant closure effects in plane strain, as similarly noted by Minakawa and McEvily.This is considered to result from the fact that, where maximum plastic zones sizes are small compared to the grain size, fatigue crack growth proceeds by a single shear decohesion mechanism (Stage I) with associated Mode II+I displacements. The resulting serrated or faceted fracture surfaces (“microstructurally-sensitive growth”) coupled with Mode II crack tip displacements thus induce high closure loads (i.e., K _cl/K _max ~0.5) by wedging the crack open at discrete contact points. At higher growth rates where the plastic zone encompasses many grains, striation growthvia alternating or simultaneous shear mechanisms (Stage II) produces a more planar fracture surface, with pure Mode I displacements, and a corresponding reduction in closure loads. Such concepts of roughness-induced closure are shown to be consistent with observations of the role of coarse grain sizes in reducing near-threshold crack growth rates at low load ratios and of the absence of this effect at high load ratios. R. O. RITCHIE and S. SURESH, both formerly with Massachusetts Institute of Technology     

Simplified Approach for the Seismic Response of a Pile Foundation

Pseudostatic approaches for the seismic analysis of pile foundations are attractive for practicing engineers because they are simple when compared to difficult and more complex dynamic analyses. To evaluate the internal response of piles subjected to earthquake loading, a simplified approach based on the “p-y” subgrade reaction method has been developed. The method involves two main steps: first, a site response analysis is carried out to obtain the free-field ground displacements along the pile. Next, a static load analysis is carried out for the pile, subjected to the computed free-field ground displacements and the static loading at the pile head. A pseudostatic push over analysis is adopted to simulate the behavior of piles subjected to both lateral soil movements and static loadings at the pile head. The single pile or the pile group interact with the surrounding soil by means of hyperbolic p-y curves. The solution derived first for the single pile, was extended to the case of a pile group by empirical multipliers, which account for reduced resistance and stiffness due to pile-soil-pile interaction. Numerical results obtained by the proposed simplified approach were compared with experimental and numerical results reported in literature. It has been shown that this procedure can be used successfully for determining the response of a pile foundation to “inertial” loading caused by the lateral forces imposed on the superstructure and “kinematic” loading caused by the ground movements developed during an earthquake.     

Nonlinear Dynamics of Structures Via Residual Flexibility of Components

An approach for the analysis of systems comprising multiple components subjected to dynamic loading is presented. It allows for an efficient treatment, stepwise in time, of linear and nonlinear connections between components. The constraint forces at the junctions of the components are computed directly without the synthesis of component modes of the determination of system modes. This is accomplished by expressing the displacements at the junction coordinates of the components in terms of the retained unconstrained normal modes and the residual flexibility of the unretained modes, in conjunction with a Newmark algorithm representation of nodal kinematics within a time step. This leads to a set of junction-sized equations, similar in form to that of the flexibility formulation in statics, in terms of the unknown junction forces. For the linear problem, the connection forces are solved for directly. For the nonlinear problem, the connection forces are determined in an iterative manner. The approach is applied to a problem involving the dynamic response of a Mini-Pressurized Logistic Module (MPLM) rack in a Space Shuttle liftoff event. The results of the proposed approach are compared with pertinent results derived by relying on component-mode synthesis.     

Green's Functions for Two-and-a-Half-Dimensional Elastodynamic Problems

This paper presents a fully analytical solution, together with explicit expressions, for the steady-state response of a homogeneous 3D space subjected to a spatially sinusoidal, harmonic line load. In the literature, this problem is often referred to as the two-and-a-half-dimensional fundamental solution or two-and-a-half-dimensional Green's functions. These equations are of great usefulness in the formulation of 3D elastodynamic problems by means of integral transforms methods and∕or boundary elements. The final expressions are validated here by applying the equations to the problem of a 3D point load, for which the solution is known in analytical form.     

Displacement-Based and Two-Field Mixed Variational Formulations for Composite Beams with Shear Lag

Displacement-based and two-field mixed beam elements are proposed for the linear analysis of steel–concrete composite beams with shear lag and deformable shear connection. The kinematics of the shear lag relies on a parabolic shear warping function of uniform shape along the slab. These assumptions are verified by comparing a closed-form solution of the composite beam problem with the results provided by the ABAQUS code. Moreover, three displacement-based finite elements and two mixed elements where both variables, forces, and displacements are approximated within the elements are developed especially for very coarse discretizations. All models neglect uplift and consider shear connectors using distributed interface elements. Locking problems that arise in the 10 degrees-of-freedom (DOF) displacement-based element which ensures the lowest regularity required by the problem are detected. Then, a locking-free element which relies on a reduced integration and a scaling factor method is proposed and analyzed for fine mesh discretizations. Energy errors and convergence rates of the proposed elements are illustrated while numerical examples dealing with a fixed-end steel–concrete composite beam and a simply supported concrete Tee beam are considered to confirm the validity of the closed-form solution and illustrate the performance of the proposed elements, especially of the ones with 10 and 13 DOF.     

Experimental Methods for Estimating In Situ Tensile Force in Tie-Rods

Several methods have been proposed for estimating the actual tensile forces in metallic tie-rods inserted into the masonry arches and vaults of historic buildings. Static and dynamic methods based on the experimental measurement of the vertical displacements caused by concentrated loads, accelerations, and the fundamental frequencies and periods of free vibrations have been applied to arrive at a more precise evaluation of the axial tensile forces acting on the tie-rod under examination. Each of these methods, with its corresponding experimental procedure, presents different advantages and disadvantages with regard to operating procedures, equipment requirements, and accuracy range in the determination of tie-rod stress. In this paper the writers propose a simple method and experimental procedure based on a single static test. The reference structural system consists of a bending moment-resistant tie beam with unknown restraint conditions at its ends. Therefore, the proposed method does not require any assumptions when modeling the rod's extremities, nor does it discount the increase in pull induced by the transverse load as irrelevant. The relevant experimental data are represented by three vertical displacements under a concentrated load and the strain variations measured in three sections of the rod. The reliability of the method has been verified through laboratory tests using tie-rods set within a sufficiently stiff metal framework in which the tension is known through continuous monitoring by means of a load cell. The correlation of the measurements with the actual tensile strength was checked; laboratory tests show good agreement between the analytical estimates and experimental measurements.     

Transverse cracking in fiber-reinforced brittle matrix,cross-ply laminates

The topic addressed in this paper is transverse cracking in the matrix of the 90° layers of a cross-ply laminate loaded in tension. Several aspects of the problem are considered, including conditions for the onset of matrix cracking, the evolution of crack spacing, the compliance of the cracked laminate, and the overall strain contributed by residual stress when matrix cracking occurs. The heart of the analysis is the plane strain problem for a doubly periodic array of cracks in the 90° layers. A fairly complete solution to this problem is presented based on finite element calculations. In addition, a useful, accurate closed form representation is also included. This solution permits the estimation of compliance change and strain due to release of residual stress. It can also be used to predict the energy release rate of cracks tunneling through the matrix. In turn, this energy release rate can be used to predict both the onset of matrix cracking and the evolution of crack spacing in the 90° layers as a function of applied stress. All these results are used to construct overall stress-strain behavior of a laminate undergoing matrix cracking in the presence of initial residual stress.