Exact, Hierarchical Solutions for Localized Loadings in Isotropic, Laminated, and Sandwich Shells

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
E. Carrera ◽  
G. Giunta
Keyword(s):  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Application Area ◽  
Spherical Shells ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Transverse Pressure ◽  
Shell Models ◽  
Stress And Deformation

This paper presents closed form solutions for simply supported cylindrical and spherical shells subjected to uniform localized distributions of transverse pressure and bending moment. These distributions have been expanded in terms of Fourier’s series for which Navier type “exact” solutions have been found for the governing differential equations of the employed shell theories. Shells made of isotropic materials, composites laminates, and sandwich have been analyzed. Carrera’s unified formulation has been adopted in order to implement a large variety of two-dimensional theories. Classical, refined, zigzag, layerwise, and mixed theories are compared in order to evaluate the stress and deformation variables. Conclusions are drawn with respect to the accuracy of the various theories for the considered loadings and layouts. The importance of the refined shell models in order to describe accurately the three-dimensional stress state in the neighborhood of the localized loading application area is outlined.

Bending of an Infinite Beam on an Elastic Foundation

1937 ◽  
Vol 4 (1) ◽  
pp. A1-A7 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Elastic Foundation ◽  
Poisson Ratio ◽  
Elementary Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Infinite Beam ◽  
A Value

Abstract The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

Soil Parameters for Design with the 3D PLAXIS Hardening Soil Model

2019 ◽  
Vol 2673 (10) ◽  
pp. 708-713
Author(s):  
Jeremy T. Bowers ◽  
Mark C. Webb ◽  
Jesse L. Beaver
Keyword(s):  
Interaction Analysis ◽  
Three Dimensional ◽  
Constitutive Models ◽  
Soil Parameters ◽  
Buried Structures ◽  
Closed Form Solutions ◽  
Soil Model ◽  
Wide Range ◽  
Stress And Deformation ◽  
Hardening Soil Model

The design and analysis of buried structures presents difficulties that cannot often be solved by closed-form solutions. Finite element methods (FEM) have increasingly become the tool of choice for advanced soil-structure interaction analysis, with three-dimensional FEM being required for irregular non-plane-strain cases. To accurately capture the stress and deformation of soils, complex material constitutive models are required. Several input parameters to these models must be determined from expensive soil testing, which is impractical for most applications. For two-dimensional FEM, good approximations of these parameters for a wide range of placed backfill soils have been developed and used in practice for many years in the computer program CANDE. It is the purpose of this paper to take these parameters, developed by Selig for use in CANDE, and convert them to equivalent parameters for the three-dimensional PLAXIS computer program’s Hardening Soil model.

On the Displacement of a Two-Dimensional Eshelby Inclusion of Elliptic Cylindrical Shape

2017 ◽  
Vol 84 (7) ◽  
Author(s):  
Xiaoqing Jin ◽  
Xiangning Zhang ◽  
Pu Li ◽  
Zheng Xu ◽  
Yumei Hu ◽  
...  
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Companion Paper ◽  
Stress And Strain ◽  
Cylindrical Shape ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Strain Fields ◽  
Closed Form Solutions ◽  
Eshelby Inclusion

In a companion paper, we have obtained the closed-form solutions to the stress and strain fields of a two-dimensional Eshelby inclusion. The current work is concerned with the complementary formulation of the displacement. All the formulae are derived in explicit closed-form, based on the degenerate case of a three-dimensional (3D) ellipsoidal inclusion. A benchmark example is provided to validate the present analytical solutions. In conjunction with our previous study, a complete elasticity solution to the classical elliptic cylindrical inclusion is hence documented in Cartesian coordinates for the convenience of engineering applications.

An Interferometer Based Experimental Technique to Evaluate Large Strains and Springback on Sheet Metal

2013 ◽  
Author(s):  
Luis Rafael Sanchez ◽  
Shannon Peterson ◽  
Carl G. Simonsen ◽  
Abrar Satar
Keyword(s):  
Sheet Metal ◽  
Three Dimensional ◽  
Bending Moment ◽  
Large Strains ◽  
Two Dimensional ◽  
Strain Gages ◽  
Additional Information ◽  
Software Algorithms ◽  
Dimensional Measurements ◽  
Vertical Scanning Interferometry

A technique was successfully developed to measure large tensile, compressive strains, springback and strain reversal effects on sheet metal bent to small radii. Vertical Scanning Interferometry (VSI) was used to measure three dimensional data from surfaces with sides varying from 160 nm to 2 mm. Software algorithms were utilized to determine surface topography maps from three-dimensional curved locations and to represent them in a two dimensional plane. Fine reference marks were engraved on both sides of sample. The sample was bent /unbent to small radii under a pure bending moment. Outer strains were calculated from VSI two-dimensional measurements of the original and final lengths between the reference marks. Strain gages, applied at locations close to the reference marks, gave additional information at the elasto-plastic range. Experimental data collected included bending moment as a function of strain, 3-D curvature profiles, springback and reverse bending effects. The technique was proved useful for the experimental evaluation and theoretical validation of bending and springback properties of sheet metal. Experimental results for aluminum and steel alloys are presented.

Exact Elasticity Solution for Laminated Anisotropic Cylindrical Shells

1993 ◽  
Vol 60 (1) ◽  
pp. 41-47 ◽  
Author(s):  
K. Bhaskar ◽  
T. K. Varadan
Keyword(s):  
Shell Theory ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Elasticity Solution ◽  
Transverse Loading ◽  
Cylindrical Bending ◽  
Simply Supported ◽  
Classical Shell Theory ◽  
Three Dimensional Elasticity ◽  
Anisotropic Cylindrical Shell

An exact three-dimensional elasticity solution is obtained for cylindrical bending of simply-supported laminated anisotropic cylindrical shell strips subjected to transverse loading. Displacements and stresses are presented for different angle-ply layups and radius-to-thickness ratios, so as to serve as useful benchmark results for the assessment of various two-dimensional shell theories. Finally, in the light of these results, the accuracy of the Love-type classical shell theory is examined.

Service Factor Assessment of a Great Lakes Bulk Carrier Incorporating the Effects of Hydroelasticity

2009 ◽  
Vol 46 (02) ◽  
pp. 116-121
Author(s):  
Spyridon E. Hirdaris ◽  
Norbert Bakkers ◽  
Nigel White ◽  
Pandeli Temarel
Keyword(s):  
Rigid Body ◽  
Great Lakes ◽  
Body Wave ◽  
Three Dimensional ◽  
Bending Moment ◽  
Suez Canal ◽  
Two Dimensional ◽  
Bulk Carrier ◽  
Service Factor

This paper presents a summary of an investigation into the effects of hull flexibility when deriving an equivalent service factor for a single passage of a Great Lakes Bulk Carrier from the Canadian Great Lakes to China. induced bending moment predicted using traditional three-dimensional rigid body hydrodynamic methods is augmented due to the effects of springing and whipping by including allowances based on two-dimensional hydroelasticity predictions across a range of headings and sea states. The analysis results are correlated with full scale measurements that are available for this ship. By combining the long term "rigid body" wave-bending moment with the effects of hydroelasticity, a suitable service factor is derived for a Great Lakes Bulk Carrier traveling from the Canadian Great Lakes to China via the Suez Canal.

Plastic Limit Loads for Pipe-in-Pipes With Circumferential Through-Wall Cracks Based on Finite Element Analyses

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Se-Chang Kim ◽  
Jae-Boong Choi ◽  
Hyun-Su Kim ◽  
Nam-Su Huh ◽  
Kyunghoon Kim
Keyword(s):  
Finite Element ◽  
Extreme Environments ◽  
Three Dimensional ◽  
Bending Moment ◽  
Piping System ◽  
Plastic Limit ◽  
Limit Loads ◽  
Closed Form Solutions ◽  
Integrity Assessment ◽  
Three Dimensional Finite Element

Pipe-in-pipes (PIPs) are generally applied to the extreme environments such as deep-sea and next-generation reactors due to their functionality and robustness. Thus, it is important to estimate the fracture behaviors of PIPs for integrity assessment of this unique piping system. In this work, the plastic collapse behaviors of PIPs with circumferential through-wall cracks (TWCs) are investigated based on three-dimensional finite element (FE) limit analysis, where the crack is assumed to be located at the inner pipe of PIPs. As for loading conditions, internal pressure, axial tension, and global bending moment are considered. In particular, the bending restraint effect induced by interconnection between the inner and outer pipes of PIPs is quantified through the FE analyses considering a practical range of geometries of PIPs. Based on the FE analysis results, the tabular and closed-form solutions of the plastic limit loads of the circumferential through-wall cracked PIPs are proposed, and then, validated against numerical simulations.

A Comparison of Two-Dimensional and Three-Dimensional Hydroelasticity Theories Including the Effect of Slamming

1991 ◽  
Vol 205 (1) ◽  
pp. 3-15 ◽  
Author(s):  
S Aksu ◽  
W G Price ◽  
P Temarel
Keyword(s):  
Steady State ◽  
Shear Force ◽  
Statistical Data ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Flexible Bodies ◽  
Strip Theory ◽  
Head Waves

The behaviour of slender and non-slender flexible bodies travelling in irregular seaways is examined. This is achieved by using a two-dimensional (2D) and a three-dimensional (3D) theory. These theories are based on different assumptions and mathematical models, though both are capable of assessing the influence of transient loadings caused by slamming. The two-dimensional theory is restricted to steady state and transient vertical responses (motion, distortion, bending moment, shear force) in irregular head waves, whereas the three-dimensional theory allows calculations of both vertical responses and transverse responses (motion, distortion, bending moment, shear force, twist) in head and oblique waves. Time-domain simulations of the responses (steady state and transient) are generated from which statistical data are determined. For a slender uniform barge structure travelling in head seas, the response simulations and statistical data evaluated by the two theories show favourable agreement. However, for a non-slender uniform barge differences between predictions arise with the two-dimensional strip theory eventually failing, while the three-dimensional approach remains effective and its versatility is further demonstrated by predicting the slamming behaviour of a flexible barge structure travelling at arbitrary heading in an irregular seaway.

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