Numerical and energy methods are used to dynamically analyze beams and complex structures. Hamilton’s principle gives exact results but cannot be easily applied in frames and complex structures. Lagrange’s equations can easily be applied in complex structures by lumping the continuous masses at selected nodes. However, this would alter the mass distribution of the system, thus introducing errors in the results of the dynamic analysis. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. This was achieved by simulating a beam with uniformly distributed mass with the force equilibrium equations. The lumped mass structures were simulated with the equations of motion. The continuous systems were analyzed using the Hamilton’s principle and the vector of nodal forces {P} causing vibration obtained. The nodal forces and displacements were then substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. When the stiffness distribution of the system was modified by means of these stiffness modification factors, it was possible to predict accurately the natural fundamental frequencies of the lumped mass encastre beam irrespective of the position or number of lumped masses.

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Coarse aggregates gradings have varying effects on the properties of concrete. Coarse aggregates of sizes 10mm, 20mm and 25mm were used for this study. Sieve analysis was conducted on these aggregates. Their coefficients of curvature (Cc) ranged between 1.01 and 1.19, while the coefficients of uniformity (Cu) ranged between 1.24 and 1.73, showing that they are all uniformly graded. 10mm and 25mm coarse aggregates were mixed in various proportions to produce five different concrete mixes (concrete C1, C2, C3, C4 and C5). Concrete C6 was produced with 100% 20mm and was used as a check. Slump test was performed for each mixture, C1 gave the highest slump height of 88mm. Compressive strengths of the cast concrete cubes were determined after 7- and 28-days curing periods. C4 gave maximum compressive strength of 26.33N/mm2 at 28 days. Increase in the percentages of 25mm aggregates resulted in higher compressive strength and decreased workability.

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This paper evaluates nonlinear free vibrations of Levy plates using Weak-Form variational principle in algebraic polynomial displacement functions. The energy functional of the plate problem was formulated using Weak-Form variational technique on the integral function of the Von Karman thin plate differential equations. The displacement functions were developed based on static deflection configurations of orthogonal beam network. The process of repeated direct integration on compatibility equation was used to determine the algebraic expressions for stress function. The amplitude of deflection which directly influences the geometric nonlinearity of the plate was determined using integration process on energy functional based on static equilibrium equations. The modal combination method was used to develop the stiffness and mass matrices respectively from the expressions of energy functional based on dynamic equilibrium equations. The numerical values of amplitudes of deflection at various aspect ratios were computed. Also, the first four nonlinear natural frequencies at various aspect ratios were numerically computed. The validation of the present study's results using the results from previous work found in literature shows satisfactory convergence, with an absolute mean error of 0.186 %. Conclusively, the application of Weak-Form variational principle in polynomial displacement functions provides satisfactory approximation to nonlinear dynamic analysis of Levy plates.