Phillip L. Gould

Thin Plates and Shells

In the preceding chapter, several cases of thin elastic plates loaded in the plane of the plate were analyzed using plane stress or plane strain equations. The resulting deformations are confined to the plane of the plate in accordance with the plane strain assumption. If such a plate is loaded transverse to the plane, the plate bends and the deflection of the surface in the normal direction is predominant, similar to a gridwork of beams. Correspondingly a distributed transverse loading can be efficiently resisted by an initially curved thin surface structure such as a cylindrical or spherical tank. The initial curvature mobilizes in-plane or membrane resistance that can be far more efficient than the bending resistance provided by flat plates. Several classical problems for the bending of rectangular and circular plates and for the membrane behavior of thin shells of revolution are presented as an introduction to this extensive topic.

Stress analysis of aortic valve leaflets with smoothed geometrical data

Abstract A method of smoothing the geometrical data obtained from photogrammetric processing of silicone rubber molds of human aortic heart valves under pressure is proposed. An estimation of the magnitude and orientation of the maximum stresses which would be encountered in an aortic prosthesis is an important component of the design criteria, especially when the prosthesis is to be a leaflet type, central flow device physically resembling the actual valve. The stress patterns are important for both strength and fatigue design requirements. To this end, it is felt that the best description of the stress distribution can be garnered by studying the actual valve in situ . This is accomplished by performing stress analyses on molds made from freshly excised leaflets. The cited references show increasing refinement and reflect the progressive improvement in the analysis methods of structural mechanics. In this paper, further improvements in the analysis procedure which enhance the reliability and repeatability of the results are described. Based on the smoothed data which is felt to be consistent with thin shell theory, a finite element analysis is conducted. Stress patterns consistent with anatomical observations are obtained and stress levels in excess of those previously reported are found. Also, the stress distribution in individual leaflets of the same valves are investigated and the non-coronary leaflet is found to carry the highest maximum stress.

Stress analysis of the human aortic valve

Abstract The role of the stress analysis of a human aortic valve is examined in relation to the design of a prosthetic valve. Using experimentally determined information to approximate the in situ loading conditions, geometry and material properties, the analysis is conducted using a thin shell finite element model. Results for in-plane stresses based on preliminary geometrical information indicate that the stress analysis and the corresponding strength provision will be important in the subsequent design and manufacture of the prosthetic valve.

In vivo stresses in the human left ventricular wall: Analysis accounting for the irregular 3-dimensional geometry and comparison with idealised geometry analyses

Abstract The in vivo stresses in the left ventricular wall are of interest to a cardiologist in evaluating the state of response or adjustment of the left ventricle to a heart disease. To determine the instantaneous in vivo stresses, we employ data consisting cineangiocardiographic determination (in single plane antero-posterior projection) of the instantaneous dimensions of the left ventricle and the left ventricular and chamber pressure. At each frame, an irregular planor outline of the LV is obtained from the X-ray film; each half of the frame is revolved into an irregular shell chamber of revolution, which is analysed for the wall stresses (due to the chamber pressure loading) by finite element procedure. The characteristic features of the finite element procedure are: (i) the left ventricular shell is divided into shell ring elements; the meridian of each element is represented by a 4th order polynomial passed through closely spaced points on the cineangiographically obtained single plane contour of the left ventricle, (ii) the geometrical properties of the shell are found from the derivatives of the equation of the meridional curve, (iii) the governing equilibrium equations of a shell ring element include effects of bending and transverse shear deformation, (iv) Displacement method of solution is employed; this is essentially an application of Ritz method utilizing the principle of minimum total potential energy, (v) the high order displacement polynomials that are employed in the analysis, provide good approximations for the displacements as well as for the stresses. The wall stress results obtained from finite element analysis are compared with the ventricular wall stresses obtained by Mirsky and Ghista (thick shell and exact elasticity, respectively) idealised geometry (ellipsoidal shaped) models of the left ventricle. The peak stress states obtained from the two models are nearly the same, with the idealised geometry models underestimating the peak circumferential stress at the apex. So the finite element analysis of the left ventricle is effectively able to incorporate the effect of variations in the curvature of the left ventricular wall boundary on the stress distribution across the wall thickness and on, the level of stress concentrations.

Nonlinear analysis of 3D seismic interaction of soil–pile–structure systems and application

Abstract To investigate more precisely the seismic response of interactive soil–pile–structure systems, a three-dimensional finite element subsystem methodology with an advanced plasticity-based constitutive model for soils has been developed. The structure subsystem is represented by space frame elements while the pile–soil subsystem is idealized as an assemblage of solid elements. By means of the δ *-version of the hierarchical single surface (HiSS) modelling approach for cyclic behavior of soft clays, tangent matrices of the soil properties are formulated with distinct constitutive laws for individual stress–strain regimes such as virgin loading, unloading, and reloading. A successive-coupling, incremental solution scheme in the time domain is created to take account of both inertial and kinematic soil–pile–structure interactions simultaneously. The seismic inputs can be any combination of three-dimensional motions. The proposed methodology may be used to analyse the seismic responses of structures for different support excitations such as rigid ground motion, interactive pile-foundation motion, and, for long span structures, non-uniform free-field motion.

Time-Dependent Effects

The strains in an elastic body may be computed from a specified displacement field using the equations of compatibility, regardless of whether the displacements arise from static or dynamic excitation. The corresponding stresses and, indeed, the displacements themselves may be dependent on the rate characteristics of the loading function. Therefore, the time derivatives of the displacements, i.e., velocities and accelerations, enter into these equations.

Validation of a simplified mathematical model for the stress analysis of human aortic heart valves

Abstract Validation studies to justify comparatively simple modeling of human aortic heart valves for the purposes of stress analysis are described. The effects of including the adjacent portion of the aorta along with the leaflet in the mathematical model and of incorporating geometric nonlinearities into the formulation are examined and it is concluded that a linear analysis of the leaflet alone is conservative from the standpoint of the determination of the magnitude of the stresses within the leaflet.

Effects of the January 2010 Haitian Earthquake on Selected Electrical Equipment

The focus of this survey was to collect data on the performance of mechanical and electrical systems at selected critical facilities in Haiti. First-hand observations confirmed that nonstructural elements that are well anchored and/or laterally restrained will perform well during a moderate seismic event. However, the investigation also revealed that many critical institutions in Haiti did not utilize state-of-the-art engineering design or construction practices when installing nonstructural equipment that turned out to be crucial to their post-earthquake operations. The survey team believes that absent or poorly implemented seismic anchorage of nonstructural elements hampered the ability to restore essential systems to operation after the event.

A local–global FE model for nonlinear analysis of column-supported shells of revolution

A local-global finite element model suitable for the analysis of column-supported shells is presented. The model combines, in a single analysis, axisymmetric shell elements, general shell elements, and column elements. Axisymmetric or rotational shell elements are employed in the axisymmetric portion of the tower in which the nonaxisymmetric behavior in loading and deformation is accounted for by including appropriate Fourier harmonics. The column-supported area is modeled using individual column elements. The local zone is identified as the region consisting of the column elements and the general shell elements inserted between the column-support area and the axisymmetric shell portion. In addition, any deviation from the axisymmetric geometry of the shell, such as imperfections or cut-outs, can be easily included in the local zone. An example which uses the proposed procedure as well as a commercial program is presented.

On the resonant component of the response of single degree-of-freedom systems under random loading

Abstract A simple expression for the resonant component of the response variance of a single degree-of-freedom system under wind pressure loading is derived. For slowly varying excitation spectral densities, low structural damping and relatively wide excitation frequency bands, the expression gives a good estimate of the response variance. The derivation substantiates a frequently used approximation in the analysis of single-degree-of-freedom system under wind loading.