Som Soni - Academia.edu (original) (raw)

Papers by Som Soni

Research paper thumbnail of Minimum weight design of pressure vessel with constraints on stiffness and strength

A new design method of composite pressure vessel with constraints on stiffness and strength is pr... more A new design method of composite pressure vessel with constraints on stiffness and strength is proposed in this paper. A netting analysis approach is used to develop an optimization procedure. Filament wound pressure vessels are assumed to have adjacent ({+-}{phi}) angle lay ups. It is proved that laminate of two layer orientations has minimum weight. The additional constraint on strength of the first layer forming vessel`s dome is used. Minimum lamination weight is determined from the condition of active execution of two constraints. Two examples are given to obtain optimum layer orientations, thicknesses and materials. Pressure vessel without change in cylindrical diameter or length can be made. For comparison purpose, calculations of stresses are done in orthotropic material using classical lamination theory. Matrix degrades at 30 to 50% of ultimate load without fiber failure. It is allowable because elastomeric liners are used to prevent leakage due to matrix cracking.

Research paper thumbnail of Effect of transverse shear and rotatory inertia on the forced axisymmetric response of linearly tapered circular plates

International Journal of Mechanical Sciences, 1995

Research paper thumbnail of Nonmetallic Materials Division Nonmetallic Materials Division FOR THE COMMANDER

0 % S/ &.LECTE 0i

Research paper thumbnail of AFRL-RX-WP-TP-2012-0367 Analysis and Modeling of Small Crack Detectionin Pressurized Fuselages for Structural Health Monitoring Applications ( Preprint )

This effort explores the evolution and characterization of structural cracks in aircraft fuselage... more This effort explores the evolution and characterization of structural cracks in aircraft fuselage structures in which the loads are varying. During flight, an aircraft fuselage skin and structure are subjected to varied cyclic loads, which can cause embedded cracks and other damage features to change their characteristics due to loading effects. The current research uses finite element modeling and experimental techniques to characterize the behavior of cracks under different static loads, with the goal of understanding the interaction of ultrasonic energy with opening-closing crack features. Specimen testing under tensile loads were considered, where crack detection and crack characterization were studied for bonded piezoelectric sensing and guided ultrasonic waves useful in structural health monitoring applications. The results suggest that crack detection and crack sizing accuracy can be impacted by load-induced, crack openingclosure effects, where linear elastic loading of the s...

Research paper thumbnail of Prediction of the response of metal matrix composite laminates under multiaxial loading

In this paper, a simple micromechanics model is proposed to predict the response of metal matrix ... more In this paper, a simple micromechanics model is proposed to predict the response of metal matrix composites under multiaxial loading. The model includes the effects of residual thermal stresses, interphasial yielding and matrix plasticity. In this work, the concentric cylinders model (CCM) developed by Pagano and Tandon has been modified to include effects that are commonly observed in metal matrix composites (MMC). The matrix region is divided into five layers, and the stresses are determined in each of these layers and the fiber and interphase regions using the CCM. Interfacial debonding is modeled using a cylindrical interphase region and evaluating the yielding behavior of this region under thermo-mechanical loading. The nonlinear response of the MMC is predicted by considering progressive yielding of the various matrix layers. An iterative scheme is used to predict the onset and progression of plasticity in each matrix region. At any applied external load (strain), the volume a...

Research paper thumbnail of Crack Detection in High Strain Aerospace Applications1

Detecting through-thickness fatigue cracks in a geometrically constrained structure is a challeng... more Detecting through-thickness fatigue cracks in a geometrically constrained structure is a challenging structural health monitoring (SHM) problem due to potentially degraded sensor performance. Fatigue cracks are typically found in aircraft structures during visual inspections and non-destructive testing (NDT); however, there exists a real need to detect damage between NDT intervals. Over the last decade, a significant amount of research effort has been focused on developing “hot spot” approaches to monitor areas of structures known to have damage using Lamb waves generated from surface-mounted lead zirconate titanate (PZT) transducers. This research is focused on evaluating an SHM approach for detecting fatigue cracks in a “hot spot” that takes into account tight geometric constraints and changes in sensor performance due to high strain levels and high-cycle fatigue, so as to maintain a high probability of damage detection. In order to account for changes in PZT performance due to hi...

Research paper thumbnail of Models for Studying Free-edge Effects

Composite Materials Series, 1989

Abstract A historical/technical review is given, depicting the development of analytical models o... more Abstract A historical/technical review is given, depicting the development of analytical models of the free-edge delamination phenomenon. Emphasis is placed on the role and importance of the free-edge problem in laminate elasticity in fostering an understanding of interlaminar stresses and their influence on composite response. From the early modeling work of Hayashi in 1967, and the definitive experiments of Foye and Baker in 1970, we trace analytical developments over the past two decades. The concept of ply elasticity or effective-modulus representation is discussed, as well as its consequences in laminate modeling. The first elasticity solution of the free-edge problem by Pipes and Pagano using finite differences is then described. This solution has been very instrumental in defining the general character of the interlaminar stress field in the neighborhood of a free edge. We then provide results for the elementary modeling of the effect of stacking sequence on laminate response and derivation of simplified equations to either optimize or minimize this effect in test specimens. Thence, a description of a model based upon the concept of a plate on a smooth foundation is given to capture the essence of the distribution of interlaminar normal stress, its boundary layer zone, and its implication on initiation of delamination. This model leads naturally to the formulation of a variational theorem for laminates that provides an accurate way to compute composite stresses. The chapter culminates with a derivation of the global-local model, which provides a practical way to describe the stress field in a multi-layered composite laminate, and a review of some recent modeling activities that have their basis in the concepts given earlier here. Numerous results are shown for stress fields within laminates which have both practical and theoretical importance.

Research paper thumbnail of Analysis and Modeling of Small Crack Detection in Pressurized Fuselages for Structural Health Monitoring Applications (Preprint)

Abstract : This effort explores the evolution and characterization of structural cracks in aircra... more Abstract : This effort explores the evolution and characterization of structural cracks in aircraft fuselage structures in which the loads are varying. During flight, an aircraft fuselage skin and structure are subjected to varied cyclic loads, which can cause embedded cracks and other damage features to change their characteristics due to loading effects. The current research uses finite element modeling and experimental techniques to characterize the behavior of cracks under different static loads, with the goal of understanding the interaction of ultrasonic energy with opening-closing crack features. Specimen testing under tensile loads were considered, where crack detection and crack characterization were studied for bonded piezoelectric sensing and guided ultrasonic waves useful in structural health monitoring applications. The results suggest that crack detection and crack sizing accuracy can be impacted by load-induced, crack opening-closure effects, where linear elastic loading of the structure resulted in linear changes in the ultrasonic signal response.

Research paper thumbnail of Failure Mechanisms of Composite Structures

... S. R. Soni, SC Tan, G. P. Tandon, T. Hsiao, L. Roundy, ' ... Barbara Woolsey, and Lisa Wi... more ... S. R. Soni, SC Tan, G. P. Tandon, T. Hsiao, L. Roundy, ' ... Barbara Woolsey, and Lisa Wilson [., T"IC (AdTech Systems Research Inc.) ~JAN 0 5 1990 JAN5 99IIFinal ... Laboratory MIPR No.: Barbara Woolsey, and Lisa Wilson FY1457-87-N-5057 (AdTech Systems Research Inc.) ...

Research paper thumbnail of Thermo-Stress Analysis of a Plated-Through-Hole in a Composite Printed Wiring Board

Journal of Thermoplastic Composite Materials, 1993

ABSTRACT

Research paper thumbnail of Vibrations of non-uniform rectangular plates: A spline technique method of solution

Journal of Sound and Vibration, 1974

The fourth order differential equation governing the transverse motion of an elastic rectangular ... more The fourth order differential equation governing the transverse motion of an elastic rectangular plate of variable thickness has been solved, by using the quintic spline interpolation technique. An algorithm for computing the solution of this differential equation is presented, for the case of equal intervals. Frequencies, mode shapes and moments for doubly symmetric, antisymmetric-symmetric and second symmetric-symmetric modes of vibration are presented for various cases of boundary conditions.

Research paper thumbnail of Resonant response of orthotropic cylindrical shells

Journal of Sound and Vibration, 1977

The resonant response of simply supported, thin and thick, orthotropic cylindrical shells is dete... more The resonant response of simply supported, thin and thick, orthotropic cylindrical shells is determined by using modal analysis, for three levels of hysteretic damping. This response is significantly changed if the larger of the elastic constants associated with normal stresses refers to the meridional, instead of the circumferential, direction. These changes may be related to changes in the relevant natural frequency. The causes of the latter are studied by considering the modal amplitudes and the contributions to the strain energy function associated with the various stress resultants. On leave from the Indian Space Research Organization, Trivandrum, India.

Research paper thumbnail of Experimental and Analytical Studies On the Onset of Delamination in Laminated Composites

Journal of Composite Materials, 1984

A large number of laminates prone to delamination under uniaxial loading have been tested. The on... more A large number of laminates prone to delamination under uniaxial loading have been tested. The onset of delamination is determined on the basis of acoustic emission data. A two parameter failure criterion, i.e. average interlaminar normal stress failure criterion on σ=, has been used in conjunction with the stress analysis done by applying the global local model [5]. A comparison between the experimental and the predicted edge delamination threshold strengths has been made. For the class of laminates studied, the onset of edge delamination load can be accurately predicted.

Research paper thumbnail of Vibrations of orthotropic rectangular plates with edges possessing different rotational flexibility and subjected to in-plane forces

Computers & Structures, 1978

Research paper thumbnail of Errors in response calculations for non-classically damped structures

Earthquake Engineering & Structural Dynamics, 1977

The classical normal mode method of determining response is extremely useful for practical calcul... more The classical normal mode method of determining response is extremely useful for practical calculations, but depends upon the damping matrix being orthogonal with respect to the modal vectors. Approximations that allow the method to be used when this condition is not satisfied have been suggested; the simplest approach is to neglect off-diagonal terms in the triple matrix product formed from the damping and modal matrices. In this paper the errors in response caused by this approximation are determined for several simple structures for a wide range of damping parameters and different types of excitation. Based on these results a criterion, relating modal damping and natural frequencies, is formulated; if this is satisfied, the errors in response caused by this diagonalization procedure are within acceptable limits.

Research paper thumbnail of Global-Local Laminate Variational Model

Solid Mechanics and Its Applications, 1994

Research paper thumbnail of Elastic Response of Composite Laminates

Mechanics of Composite Materials, 1983

Research paper thumbnail of 3-Dimensional Response of Composites

Ia. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS Unclassified Za. SECURITY CLASSIFICAT... more Ia. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS Unclassified Za. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABILITY OF REPORT 2b. DECLASSIFICATION/DOWNGRADING SCHEDULE Approved for public release; distribution is unlimited. 4. PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S) N/A AFWAL-TR-88-4242 6a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION * St(If appkable) Air Force Wright Aeronautical Laboratories A SMaterials Laboratory (AFWAL/MLBM) 6c. ADDRESS (City, State, and ZIP Code) 7b. ADDRESS (city, State, and ZIP Code) 1342 N. Fairfield Road Dayton OH 45432 Wright-Patterson AFB OH 45433-6533 go. NAME OF FUNDING /SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (If applicable) I F33615-85-C-5034 ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. NO. NO. ACCESSION NO. 62102F 2419 01 93 11. TITLE (Include Security Classification)

Research paper thumbnail of Torsional vibrations of shells of revolution of variable thickness

The Journal of the Acoustical Society of America, 1973

Free torsional vibrations of conical and cylindrical shells of thickness varying as a power of di... more Free torsional vibrations of conical and cylindrical shells of thickness varying as a power of distance have been discussed. The solutions depend upon a linear differential equation of second order. The governing equation is solved for shells with both edges fixed. The numerical values of the frequency parameter for the first three modes of vibration are computed for shells of linearly and parabolically varying thickness for several different ratios of terminal radii.

Research paper thumbnail of Global-local laminate variational model

International Journal of Solids and Structures

Research paper thumbnail of Minimum weight design of pressure vessel with constraints on stiffness and strength

A new design method of composite pressure vessel with constraints on stiffness and strength is pr... more A new design method of composite pressure vessel with constraints on stiffness and strength is proposed in this paper. A netting analysis approach is used to develop an optimization procedure. Filament wound pressure vessels are assumed to have adjacent ({+-}{phi}) angle lay ups. It is proved that laminate of two layer orientations has minimum weight. The additional constraint on strength of the first layer forming vessel`s dome is used. Minimum lamination weight is determined from the condition of active execution of two constraints. Two examples are given to obtain optimum layer orientations, thicknesses and materials. Pressure vessel without change in cylindrical diameter or length can be made. For comparison purpose, calculations of stresses are done in orthotropic material using classical lamination theory. Matrix degrades at 30 to 50% of ultimate load without fiber failure. It is allowable because elastomeric liners are used to prevent leakage due to matrix cracking.

Research paper thumbnail of Effect of transverse shear and rotatory inertia on the forced axisymmetric response of linearly tapered circular plates

International Journal of Mechanical Sciences, 1995

Research paper thumbnail of Nonmetallic Materials Division Nonmetallic Materials Division FOR THE COMMANDER

0 % S/ &.LECTE 0i

Research paper thumbnail of AFRL-RX-WP-TP-2012-0367 Analysis and Modeling of Small Crack Detectionin Pressurized Fuselages for Structural Health Monitoring Applications ( Preprint )

This effort explores the evolution and characterization of structural cracks in aircraft fuselage... more This effort explores the evolution and characterization of structural cracks in aircraft fuselage structures in which the loads are varying. During flight, an aircraft fuselage skin and structure are subjected to varied cyclic loads, which can cause embedded cracks and other damage features to change their characteristics due to loading effects. The current research uses finite element modeling and experimental techniques to characterize the behavior of cracks under different static loads, with the goal of understanding the interaction of ultrasonic energy with opening-closing crack features. Specimen testing under tensile loads were considered, where crack detection and crack characterization were studied for bonded piezoelectric sensing and guided ultrasonic waves useful in structural health monitoring applications. The results suggest that crack detection and crack sizing accuracy can be impacted by load-induced, crack openingclosure effects, where linear elastic loading of the s...

Research paper thumbnail of Prediction of the response of metal matrix composite laminates under multiaxial loading

In this paper, a simple micromechanics model is proposed to predict the response of metal matrix ... more In this paper, a simple micromechanics model is proposed to predict the response of metal matrix composites under multiaxial loading. The model includes the effects of residual thermal stresses, interphasial yielding and matrix plasticity. In this work, the concentric cylinders model (CCM) developed by Pagano and Tandon has been modified to include effects that are commonly observed in metal matrix composites (MMC). The matrix region is divided into five layers, and the stresses are determined in each of these layers and the fiber and interphase regions using the CCM. Interfacial debonding is modeled using a cylindrical interphase region and evaluating the yielding behavior of this region under thermo-mechanical loading. The nonlinear response of the MMC is predicted by considering progressive yielding of the various matrix layers. An iterative scheme is used to predict the onset and progression of plasticity in each matrix region. At any applied external load (strain), the volume a...

Research paper thumbnail of Crack Detection in High Strain Aerospace Applications1

Detecting through-thickness fatigue cracks in a geometrically constrained structure is a challeng... more Detecting through-thickness fatigue cracks in a geometrically constrained structure is a challenging structural health monitoring (SHM) problem due to potentially degraded sensor performance. Fatigue cracks are typically found in aircraft structures during visual inspections and non-destructive testing (NDT); however, there exists a real need to detect damage between NDT intervals. Over the last decade, a significant amount of research effort has been focused on developing “hot spot” approaches to monitor areas of structures known to have damage using Lamb waves generated from surface-mounted lead zirconate titanate (PZT) transducers. This research is focused on evaluating an SHM approach for detecting fatigue cracks in a “hot spot” that takes into account tight geometric constraints and changes in sensor performance due to high strain levels and high-cycle fatigue, so as to maintain a high probability of damage detection. In order to account for changes in PZT performance due to hi...

Research paper thumbnail of Models for Studying Free-edge Effects

Composite Materials Series, 1989

Abstract A historical/technical review is given, depicting the development of analytical models o... more Abstract A historical/technical review is given, depicting the development of analytical models of the free-edge delamination phenomenon. Emphasis is placed on the role and importance of the free-edge problem in laminate elasticity in fostering an understanding of interlaminar stresses and their influence on composite response. From the early modeling work of Hayashi in 1967, and the definitive experiments of Foye and Baker in 1970, we trace analytical developments over the past two decades. The concept of ply elasticity or effective-modulus representation is discussed, as well as its consequences in laminate modeling. The first elasticity solution of the free-edge problem by Pipes and Pagano using finite differences is then described. This solution has been very instrumental in defining the general character of the interlaminar stress field in the neighborhood of a free edge. We then provide results for the elementary modeling of the effect of stacking sequence on laminate response and derivation of simplified equations to either optimize or minimize this effect in test specimens. Thence, a description of a model based upon the concept of a plate on a smooth foundation is given to capture the essence of the distribution of interlaminar normal stress, its boundary layer zone, and its implication on initiation of delamination. This model leads naturally to the formulation of a variational theorem for laminates that provides an accurate way to compute composite stresses. The chapter culminates with a derivation of the global-local model, which provides a practical way to describe the stress field in a multi-layered composite laminate, and a review of some recent modeling activities that have their basis in the concepts given earlier here. Numerous results are shown for stress fields within laminates which have both practical and theoretical importance.

Research paper thumbnail of Analysis and Modeling of Small Crack Detection in Pressurized Fuselages for Structural Health Monitoring Applications (Preprint)

Abstract : This effort explores the evolution and characterization of structural cracks in aircra... more Abstract : This effort explores the evolution and characterization of structural cracks in aircraft fuselage structures in which the loads are varying. During flight, an aircraft fuselage skin and structure are subjected to varied cyclic loads, which can cause embedded cracks and other damage features to change their characteristics due to loading effects. The current research uses finite element modeling and experimental techniques to characterize the behavior of cracks under different static loads, with the goal of understanding the interaction of ultrasonic energy with opening-closing crack features. Specimen testing under tensile loads were considered, where crack detection and crack characterization were studied for bonded piezoelectric sensing and guided ultrasonic waves useful in structural health monitoring applications. The results suggest that crack detection and crack sizing accuracy can be impacted by load-induced, crack opening-closure effects, where linear elastic loading of the structure resulted in linear changes in the ultrasonic signal response.

Research paper thumbnail of Failure Mechanisms of Composite Structures

... S. R. Soni, SC Tan, G. P. Tandon, T. Hsiao, L. Roundy, ' ... Barbara Woolsey, and Lisa Wi... more ... S. R. Soni, SC Tan, G. P. Tandon, T. Hsiao, L. Roundy, ' ... Barbara Woolsey, and Lisa Wilson [., T"IC (AdTech Systems Research Inc.) ~JAN 0 5 1990 JAN5 99IIFinal ... Laboratory MIPR No.: Barbara Woolsey, and Lisa Wilson FY1457-87-N-5057 (AdTech Systems Research Inc.) ...

Research paper thumbnail of Thermo-Stress Analysis of a Plated-Through-Hole in a Composite Printed Wiring Board

Journal of Thermoplastic Composite Materials, 1993

ABSTRACT

Research paper thumbnail of Vibrations of non-uniform rectangular plates: A spline technique method of solution

Journal of Sound and Vibration, 1974

The fourth order differential equation governing the transverse motion of an elastic rectangular ... more The fourth order differential equation governing the transverse motion of an elastic rectangular plate of variable thickness has been solved, by using the quintic spline interpolation technique. An algorithm for computing the solution of this differential equation is presented, for the case of equal intervals. Frequencies, mode shapes and moments for doubly symmetric, antisymmetric-symmetric and second symmetric-symmetric modes of vibration are presented for various cases of boundary conditions.

Research paper thumbnail of Resonant response of orthotropic cylindrical shells

Journal of Sound and Vibration, 1977

The resonant response of simply supported, thin and thick, orthotropic cylindrical shells is dete... more The resonant response of simply supported, thin and thick, orthotropic cylindrical shells is determined by using modal analysis, for three levels of hysteretic damping. This response is significantly changed if the larger of the elastic constants associated with normal stresses refers to the meridional, instead of the circumferential, direction. These changes may be related to changes in the relevant natural frequency. The causes of the latter are studied by considering the modal amplitudes and the contributions to the strain energy function associated with the various stress resultants. On leave from the Indian Space Research Organization, Trivandrum, India.

Research paper thumbnail of Experimental and Analytical Studies On the Onset of Delamination in Laminated Composites

Journal of Composite Materials, 1984

A large number of laminates prone to delamination under uniaxial loading have been tested. The on... more A large number of laminates prone to delamination under uniaxial loading have been tested. The onset of delamination is determined on the basis of acoustic emission data. A two parameter failure criterion, i.e. average interlaminar normal stress failure criterion on σ=, has been used in conjunction with the stress analysis done by applying the global local model [5]. A comparison between the experimental and the predicted edge delamination threshold strengths has been made. For the class of laminates studied, the onset of edge delamination load can be accurately predicted.

Research paper thumbnail of Vibrations of orthotropic rectangular plates with edges possessing different rotational flexibility and subjected to in-plane forces

Computers & Structures, 1978

Research paper thumbnail of Errors in response calculations for non-classically damped structures

Earthquake Engineering & Structural Dynamics, 1977

The classical normal mode method of determining response is extremely useful for practical calcul... more The classical normal mode method of determining response is extremely useful for practical calculations, but depends upon the damping matrix being orthogonal with respect to the modal vectors. Approximations that allow the method to be used when this condition is not satisfied have been suggested; the simplest approach is to neglect off-diagonal terms in the triple matrix product formed from the damping and modal matrices. In this paper the errors in response caused by this approximation are determined for several simple structures for a wide range of damping parameters and different types of excitation. Based on these results a criterion, relating modal damping and natural frequencies, is formulated; if this is satisfied, the errors in response caused by this diagonalization procedure are within acceptable limits.

Research paper thumbnail of Global-Local Laminate Variational Model

Solid Mechanics and Its Applications, 1994

Research paper thumbnail of Elastic Response of Composite Laminates

Mechanics of Composite Materials, 1983

Research paper thumbnail of 3-Dimensional Response of Composites

Ia. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS Unclassified Za. SECURITY CLASSIFICAT... more Ia. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS Unclassified Za. SECURITY CLASSIFICATION AUTHORITY 3. DISTRIBUTION/AVAILABILITY OF REPORT 2b. DECLASSIFICATION/DOWNGRADING SCHEDULE Approved for public release; distribution is unlimited. 4. PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBER(S) N/A AFWAL-TR-88-4242 6a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION * St(If appkable) Air Force Wright Aeronautical Laboratories A SMaterials Laboratory (AFWAL/MLBM) 6c. ADDRESS (City, State, and ZIP Code) 7b. ADDRESS (city, State, and ZIP Code) 1342 N. Fairfield Road Dayton OH 45432 Wright-Patterson AFB OH 45433-6533 go. NAME OF FUNDING /SPONSORING 8b. OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (If applicable) I F33615-85-C-5034 ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. NO. NO. ACCESSION NO. 62102F 2419 01 93 11. TITLE (Include Security Classification)

Research paper thumbnail of Torsional vibrations of shells of revolution of variable thickness

The Journal of the Acoustical Society of America, 1973

Free torsional vibrations of conical and cylindrical shells of thickness varying as a power of di... more Free torsional vibrations of conical and cylindrical shells of thickness varying as a power of distance have been discussed. The solutions depend upon a linear differential equation of second order. The governing equation is solved for shells with both edges fixed. The numerical values of the frequency parameter for the first three modes of vibration are computed for shells of linearly and parabolically varying thickness for several different ratios of terminal radii.

Research paper thumbnail of Global-local laminate variational model

International Journal of Solids and Structures