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.

Rice Husk Ash (RHA) has been found as a potential partial replacement for cement in concrete. This study attempts to make an evidence based evaluation of the sustainability and benefits of RHA utilisation as partial replacement of cement in road concrete works within Anambra State of Nigeria. The ashes of the rice husks collected from different locations were characterised. Direct interviews were conducted among the rice mill personnel and experts in the construction companies. The values of SiO2 + Fe2O3 + Al2O3 for the four analysed RHA samples ranged from 78.9% to 80.3% as revealed by the X-ray fluorescence analysis. This confirms that they are pozzolanic. The highest observed 28th day compressive strength of concrete was 41.8 N/mm2 for the concrete containing 10% RHA. Beyond the 10% replacement level, the compressive strength dropped below the control values. The result of the Analytical Hierarchy Process (AHP) analysis displayed the highest option preference of 40.3% for utilising RHA in road construction. These show that utilising RHA for road concrete works would be a sustainable option. 10% replacement of cement with RHA was recommended for optimum performance based on the compressive strengths of the tested RHA based concretes.

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.

Continuous systems are sometimes analysed as lumped masses connected by massless elements. This reduces the structure’s degree of freedom and therefore simplifies the analysis. However this over simplification introduces an error in the analysis and the results are therefore approximate. In this work sections of the vibrating beam were isolated and the equations of the forces causing vibration obtained using the Hamilton’s principle. These forces were applied to the nodes of an equivalent lumped mass beam and the stiffness modification needed for it to behave as a continuous beam obtained. The beam’s stiffness was modified using a set of stiffness modification factors to . It was observed that by applying these factors in the dynamic analysis of the beam using the Lagrange’s equation, we obtain the exact values of the fundamental frequency irrespective of the way the mass of the beam was lumped. From this work we observed that in order to obtain an accurate dynamic response from a lumped mass beam there is need to modify the stiffness composition of the system and no linear modification of the stiffness distribution of lumped mass beams can cause them to be dynamically equivalent to the continuous beams. This is so because the values of the modification factors obtained for each beam segment were not equal. The stiffness modification factors were obtained for elements at different sections of the beam

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.