S. Reaz Ahmed

Numerical solution of both ends fixed deep beams

This paper presents the exact solution of a two-dimensional elastic problem with mixed boundary conditions. It gives the complete solution in terms of displacement and stress components of both ends fixed deep beams, subjected to uniformly distributed compressive loading on the top edge. A new mathematical model has been used to formulate the problem. In this analysis both the governing equations and boundary conditions are expressed in terms of a new potential function Ψ which is defined in terms of displacement components. Finite-difference technique is used for the discretization of the bi-harmonic equation and the associated boundary conditions. All the parameters of interest in the solution are presented in the form of graphs. Finally, a comparison is made to ascertain how the elementary solutions differ with those of the present numerical solutions. The computed solutions are found to be highly accurate and rational, and thus establish the reliability and practicability of this numerical approach.

A finite-difference scheme for mixed boundary value problems of arbitrary-shaped elastic bodies

Abstract This paper presents a program based on a finite-difference technique, which solves plane stress and plane strain problems of arbitrary shaped elastic bodies with mixed boundary conditions. A new formulation of governing equations in terms of the displacement potential function ψ, as introduced by Uddin (Finite difference solution of two-dimensional elastic problems with mixed boundary conditions, MSc Thesis, Carleton University, Canada, 1966), has been used. This formulation has the capability to handle problems of mixed boundary condition, which is beyond the ability of the conventional formulations in terms of Airys stress function φ. Results found with this program for classical problems are in very good agreement with known solutions. This program can handle practical boundary conditions very efficiently.

Displacement potential solution of a deep stiffened cantilever beam of orthotropic composite material

An analytical solution of the elastic field of a deep stiffened cantilever beam of orthotropic composite material is presented in the paper. The cantilever beam is subjected to a parabolic shear loading at its free lateral end and the two opposing longitudinal edges are stiffened. Unidirectional fibre-reinforced composite is considered for the present analysis where the fibres are assumed to be directed along the beam length. Following a new development, the present mixed-boundary-value elastic problem is formulated in terms of a single potential function defined in terms of the associated displacement components. This formulation reduces the problem to the solution of a single fourth-order partial differential equation of equilibrium and is capable of dealing with mixed modes of boundary conditions appropriately. The solution is obtained in the form of an infinite series. Results of different stress and displacement components at different sections of the composite beam are presented numerically in the form of graphs. Finally, in an attempt to check the reliability as well as the accuracy of the present solution, the problem is solved by using two standard numerical methods of solution. A comparison of the results shows that the analytical and numerical solutions of the present problem are in good agreement and thus establishes the soundness as well as the reliability of the present displacement potential approach to solution of the elastic field of orthotropic composite structures.

Displacement Potential Based Finite Difference Solution to Elastic Field in a Cantilever Beam of Orthotropic Composite.

This paper concerns the finite difference solution to the elastic field in a cantilever beam of unidirectional orthotropic composite material. The mixed boundary value plane elastic problem is formulated in terms of a single displacement potential function. The numerical scheme developed is demonstrated for a particular case of a boron/epoxy cantilever beam for two different loading conditions. Numerical results of different stress and displacement components at different sections of the beam are presented in the form of graphs. With a view to verifying the finite difference results, solutions are also obtained by finite element method. The comparison of the results ensures that the present displacement potential based finite difference scheme is relatively simple, reliable, and accurate enough to analyze the elastic field in orthotropic composite structures under mixed boundary conditions.

Influence of Wall Thickness on the Ultrasonic Evaluation of Small Closed Surface Cracks and Quantitative NDE

A new ultrasonic evaluation method has been developed for the quantitative characterization of smaller tight cracks under no load condition. In this approach, the ultrasonic testing is considered as an inverse problem to determine simultaneously the crack size and the extent of crack closure. The present paper is on the analysis of the influence of wall thickness on the evaluation of closed surface cracks from the inaccessible side of the wall. Based on experimental observation, the evaluation algorithm has been extended for the characterization of cracks with different wall thickness from the measured angle beam response. Evaluated results of both open and tightly closed small cracks with different wall thicknesses verify the method as a powerful tool for detecting and characterizing small surface cracks quantitatively.

Investigation of elastic field of a short orthotropic composite column by using finite-difference technique

Abstract The elastic field of a short column of orthotropic composite material under a linearly varying distributed load is analysed using displacement potential formulation. The finite-difference technique is used to obtain the numerical solutions of different parameters of the elastic field. The differential equations associated with the boundary conditions and the governing equation are approximated by their corresponding finite-difference forms, which are then applied, respectively, to the boundaries and the interior nodal points of the discretized column. To demonstrate the elastic behaviour of the fibre reinforced composite column, the results of displacements and stresses at different sections are presented in a graphical form.

Analytical solution of a stiffened orthotropic plate using alternative displacement potential approach

Abstract Analytical solution of elastic fields of a stiffened composite plate subjected to axial tension and pure bending is obtained using alternative displacement potential approach. In this approach, the problem is formulated in terms of a single potential function, defined in terms of the displacement components, which satisfies a single differential equation of equilibrium. All the parameters of the elastic field are expressed into algebraic equations using Fourier series. Assuming a trigonometric function as the displacement potential solution to the present problem, the mixed-boundary conditions at the stiffened edges are automatically satisfied. The resulting states of stress and displacement in the composite plate due to axial tension and pure bending are discussed in details. The effect of fibre orientation on the elastic field is also investigated.

Analytical solution of short guided orthotropic composite columns under eccentric loading using displacement potential formulation

Abstract In this paper, short guided columns of orthotropic composite materials are considered in order to analyse their elastic field due to an eccentric compressive loading. The supporting edge of the column is rigidly fixed and fibres are directed along the axis of the column. By a new approach of displacement potential formulation, the present mixed boundary value elastic problem is expressed in terms of a single potential function. Displacement and stress components are derived into infinite series using Fourier integral with the coincided boundary conditions along with the physical boundary conditions. Some numerical results of different stress and displacement components at different sections of the column are presented in the form of graphs. The effect of column aspect ratio and material orthotropy on the solution of stresses and displacements at a particular section is also analysed. Finally, analytical results obtained by the present formulation are compared to those of finite-element and finite-difference methods.

Selecting Suitable Probes Distances for Sizing Deep Surface Cracks Using the DCPD Technique

In this study, the way to enhance the sensitivity of evaluating deep surface cracks by DCPD technique using four probes is considered. The potential drops across two-dimensional cracks having different depths are analyzed by the three-dimensional finite-element method. The effect of the distance between current input and output probes and the distance between measuring probes on the change in potential drops are analyzed for a wide range of crack depths. By extending the distance between current input and output probes, the change in potential drop with the change in the depth of deeper crack becomes large. But the voltage of potential drop becomes small to measure. Finally, the way to select the appropriate distances between the probes for the measuring sensor is shown from the viewpoints of sensitivity and the required current.

Generalized mathematical model for the solution of mixed-boundary-value elastic problems

This paper describes a new mathematical formulation, specifically suitable for finite-difference analysis of stresses and displacements of three-dimensional mixed-boundary-value elastic problems. Earlier, mathematical models of elasticity were very deficient in handling three-dimensional practical stress problems. In the present model, a new scheme of reduction of unknowns is used to formulate the three-dimensional problem in terms of a single potential function, defined in terms of the three displacement components. Compared to the conventional models, the present model provides numerical solution of higher accuracy in a shorter period of computational time. The application of the potential function formulation is demonstrated here through a number of classical problems of solid mechanics, and the results are compared with the available solutions in the literature. The comparison of the results establishes the rationality of the present approach.