In this work, the least squares weighted residual method (LSWRM) was used to solve the generalised elastic column buckling problem for the case of pinned ends. Mathematically, the problem solved was a boundary value problem (BVP) represented by a system of three coupled linear ordinary differential equations (ODEs) in terms of three unknown displacement functions and subject to boundary conditions. The least squares residual method used formulated the problem as a variational problem, and reduced it to an algebraic eigenvalue problem which was solved to obtain the characteristic buckling equation. The characteristic stability equation was found to be a cubic polynomial for the general asymmetric sectioned column. The buckling modes were found as coupled flexuraltorsional buckling modes. Two special cases of the problem were studied namely: doubly symmetric and singly symmetric sections. For doubly symmetric sections, the buckling loads and the buckling mode were found to be decoupled and the buckling mode could be flexural or flexural-torsional. For singly symmetric section columns, one of the bucking modes becomes decoupled while the others are coupled. The buckling equation showed the column could fail by either pure flexure or coupled flexural-torsional buckling mode. The results of the present work agree with Timoshenko's results, and other results from the technical literature.

In this study, the Ritz variational method was used to analyze and solve the bending problem of simply supported rectangular Kirchhoff plate subject to transverse hydrostatic load distribution over the entire plate domain. The deflection function was chosen based on double series of infinite terms as coordinate function that satisfy the geometric and force boundary conditions and unknown generalized displacement parameters. Upon substitution into the total potential energy functional for homogeneous, isotropic Kirchhoff plates, and evaluation of the integrals, the total potential energy functional was obtained in terms of the unknown generalized displacement parameters. The principle of minimization of the total potential energy was then applied to determine the unknown displacement parameters. Moment curvature relations were used to find the bending moments. It was found that the deflection functions and the bending moment functions obtained for the plate domain, and the values at the plate center were exactly identical as the solutions obtained by Timoshenko and Woinowsky-Krieger using the Navier series method.

Plates are important structural elements used to model bridge decks, retaining walls, floor slabs, spacecraft panels, aerospace structures, and ship hulls amongst. Plates have been modelled using three dimensional elasticity theory, Reissner's theory, Kirchhoff theory, Shimpi's theory, Von Karman's theory, etc. The resulting plate equations have also been solved using classical and numerical techniques.In this research, the Galerkin-Vlasov variational method was used to present a general formulation of the Kirchhoff plate problem with simply supported edges and under distributed loads. The problem was then solved to obtain the displacements, and the bending moments in a Kirchhoff plate with simply supported edges and under uniform load. Maximum values of the displacement and the bending moments were found to occur at the plate center. The Galerkin Vlasov solutions for a rectangular simply supported Kirchhoff plate carrying uniform load was found to be exactly identical with the Navier double trigonometric series solution.

In this work, the governing differential equations of elastic column buckling represented by a system of three coupled differential equations in the three unknown displacement functions, v(x), w(x) and (x) are solved using the method of Fourier series. The column was pinned at both ends x = 0, x = l. The unknown displacements were assumed to be a Fourier sine series of infinite terms, which was found to satisfy apriori the pinned conditions at the ends and substituted into the governing equations. The governing equations were found to reduce to a system of algebraic eigenvalueeigenvector problem. The buckling equation was found to be a cubic polynomial for the general asymmetric sectioned column. The buckling modes were found as flexural torsional buckling modes. For columns with monosymmetric sections, it was found that the buckling mode could be flexural or flexuraltorsional depending on the root of the cubic polynomial buckling equation which is the smallest. For columns with bisymmetric sections, it was found that the buckling modes are uncoupled and bisymmetric columns could fail by pure flexural buckling about the axes of symmetry or pore torsional buckling. The findings are in excellent agreement with Timoshenko's solutions.

In this study, the single Fourier sine integral transform method was used to solve the elastic buckling problem of Kirchhoff rectangular plates simply supported at two opposite edges x = 0, and x = a and clamped at the other two edges y = 0, and y = b. The problem considered was that for uniaxial uniform compressive load in the x coordinate direction. The single finite Fourier sine integral transformation was applied to the governing fourth order partial differential equation of Kirchhoff plates under uniaxial uniform in-plane compressive loads to convert the problem to a fourth order ordinary differential equation in terms of the finite Fourier sine transform space variables. Solution of the ordinary differential equations yielded the buckling modal shape functions in the Fourier sine transform variables. Enforcement of boundary conditions along the y direction at y = 0, and y = b yielded an algebraic eigenvalue eigenvector problem which was solved to obtain non-trivial solutions. The characteristic buckling equation was obtained by requiring the vanishing of the matrix of coefficients as the transcendental equation involving the buckling load. The buckling load was obtained by solving the transcendental equation for various assumed values of the plate aspect ratios. Critical buckling loads for various values of the plate aspect ratio were found to be identical with classical solutions obtained in the technical literature. The present study thus yielded exact solution for the buckling loads and buckling modes of uniaxially compressed Kirchhoff plates; illustrating the effectiveness of the analytical tool.

Abstract-

It has become very important in the field of concrete technology to develop intelligent models to reduce overdependence on laboratory studies prior to concrete infrastructure designs. In order to achieve this, a database representing the global behavior and performance of concrete mixes is collected and prepared for use. In this research work, an extensive literature search was used to collect 112 concrete mixes corresponding to fly ash and binder ratios (FA/B), coarse aggregate and binder ratios (CAg/B), fine aggregate and binder ratios (FAg/B), 28-day concrete compressive strength (Fc28), and the environmental impact point (P) estimated as a life cycle assessment of greenhouse gas emissions from fly ash- and cement-based concrete. Statistical analysis, linear regression (LNR), and artificial intelligence (AI) studies were conducted on the collected database. The material binder ratios were deployed as input variables to predict Fc28 and P as the response variables. From the collected concrete mix data, it was observed that mixes with a higher cement content produce higher compressive strengths and a higher carbon footprint impact compared to mixes with a lower amount of FA. The results of the LNR and AI modeling showed that LNR performed lower than the AI techniques, with an R2(SSE) of 48.1% (26.5) for Fc and 91.2% (7.9) for P. But ANN, with performance indices of 95.5% (9.4) and 99.1% (2.6) for Fc and P, respectively, outclassed EPR with 90.3% (13.9) and 97.7% (4.2) performance indices for Fc and P, respectively. Taylor’s and variance diagrams were also used to study the behavior of the models for Fc28 and P compared to the measured values. The results show that the ANN and EPR models for Fc28 lie within the RMSE envelop of less than 0.5% and a standard deviation of between 15 MPa and 20 MPa, while the coefficient of determination sector lies between 95% and 99% except for LNR, which lies in the region of less than 80%. In the case of the P models, all the predicted models lie within the RMSE envelop of between 0.5% and 1.0%, a coefficient of determination sector of 95% and above, and a standard deviation between 2.0 and 3.0 points of impact. The variance between measured and modeled values shows that ANN has the best distribution, which agrees with the performance accuracy and fits. Lastly, the ANN learning ability was used to develop a mix design tool used to design sustainable concrete Fc28 based on environmental impact considerations. Doi: 10.28991/CEJ-SP2023-09-03 Full Text: PDF

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