Analytical Solutions for Thick, Doubly Curved, Laminated Shells

Jun, Fan; Zhang, Jie

doi:10.1061/(asce)0733-9399(1992)118:7(1338)

Cited by 83 publications

(78 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Analytical solutions for three-dimensional deformation, stress and free vibration of thick, doubly curved, laminated shells were obtained by Fan and Zhang [30]. Beakou and Touratier [31] developed a rectangular "nite element for analyzing composite multilayered shallow shells in statics, vibration and buckling with interface continuity and lateral surface conditions of transverse 1010 shear stresses represented by cosine functions without regard to the interfacial transverse normal stress.…”

Section: Introductionmentioning

confidence: 99%

Comparison of Natural Frequencies of Laminates by 3-D Theory, Part Ii: Curved Panels

Chern

Chao

2000

Journal of Sound and Vibration

View full text Add to dashboard Cite

Section: Introductionmentioning

confidence: 99%

Comparison of Natural Frequencies of Laminates by 3-D Theory, Part Ii: Curved Panels

Chern

Chao

2000

Journal of Sound and Vibration

View full text Add to dashboard Cite

“…In Table 3, we have compared nondimensional deflection,w (0.5a, 0.5b, 0.5h), at the centroid of the top surface of the shell for a = b, a/h = 100 and 10, and different values of R/a found using the TSNDT with that given by the analytical 3D LET solution of Fan and Zhang [45]. There is at most 2.74% difference between the two sets of results for R/a = 1 and a/h = 10, and this difference decreases with an increase in R/a.…”

Section: Spherical Shellmentioning

confidence: 99%

“…They found that natural frequencies increase with an increase in the R 2 /R 1 ratio for a given R 2 /l 2 ratio and decrease with an increase in the R 2 /l 2 ratio for a given R 2 /R 1 ratio, where l 1 and l 2 are the two planform lengths. Fan and Zhang [45], Wu et al [46], and Huang et al [47] computed analytical solutions for static deformations of laminated doubly curved shells and reported the centroidal displacement of shells for different curvatures. They found that the displacement decreases with an increase in the curvature for a fixed aspect ratio.…”

Section: Introductionmentioning

confidence: 99%

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

Shah

Batra

2016

Mechanics of Advanced Materials and Structures

View full text Add to dashboard Cite

We analyze static infinitesimal deformations of doubly curved shells using a third-order shear and normal deformable theory (TSNDT) and delineate effects of the curvilinear length/thickness ratio, R/a, a/h, radius of curvature/curvilinear length, and the ratio of the two principal radii on through-the-thickness stresses, strain energies of the in-plane and the transverse shear and normal deformations, and strain energies of stretching and bending deformations for loads that include uniform normal tractions on a major surface and equal and opposite tangential tractions on the two major surfaces. In the TSNDT the three displacement components at a point are represented as complete polynomials of degree three in the thickness coordinate. Advantages of the TSNDT include not needing a shear correction factor, allowing stresses for monolithic shells to be computed from the constitutive relation and the shell theory displacements, and considering general tractions on bounding surfaces. For laminated shells we use an equivalent single layer TSNDT and find the in-plane stresses from the constitutive relations and the transverse stresses with a onestep stress recovery scheme. The in-house developed finite element software is first verified by comparing displacements and stresses in the shell computed from it with those from either analytical or numerical solutions of the corresponding 3D problems. The strain energy of a spherical shell is found to approach that of a plate when R/a exceeds 10. For a thick clamped shell of aspect ratio 5 subjected to uniform normal traction on the outer surface, the in-plane and the transverse deformations contribute equally to the total strain energy for R/a greater than 5. However, for a cantilever shell of aspect ratio 5 subjected to equal and opposite uniform tangential tractions on the two major surfaces, the strain energy of in-plane deformations equals 95-98% of the total strain energy. Numerical results presented herein for several problems provide insights into different deformation modes, help designers decide when to consider effects of transverse deformations, and use the TSNDT for optimizing doubly curved shells.

show abstract

“…In another paper, Qatu and Leissa (1993) investigated the vibrations of shallow shells with two adjacent edges clamped and the others free by using the Ritz method. Fan and Zhang (1992) …”

Section: Doubly Curved Shellsmentioning

confidence: 99%

Nonlinear vibrations of doubly-curved cross-ply shallow shells

Alhazza

Nayfeh

2001

19th AIAA Applied Aerodynamics Conference

View full text Add to dashboard Cite

The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von Kármán-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented.We used two approaches to study the responses of shells to a primary resonance: a direct approach and a discretization approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency-and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared.For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating 2n + (2n) 2 nonlinear first-order ordinary-differential equations simultaneously, where n is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results.The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown.Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo...

show abstract

Analytical Solutions for Thick, Doubly Curved, Laminated Shells

Cited by 83 publications

References 8 publications

Comparison of Natural Frequencies of Laminates by 3-D Theory, Part Ii: Curved Panels

Comparison of Natural Frequencies of Laminates by 3-D Theory, Part Ii: Curved Panels

Stretching and bending deformations due to normal and shear tractions of doubly curved shells using third-order shear and normal deformable theory

Nonlinear vibrations of doubly-curved cross-ply shallow shells

Contact Info

Product

Resources

About