Equilibrium Equations Of Beams Research Articles

Analytical solution for composite layered beam subjected to uniformly distributed load

ABSTRACTThe article presents an analytical theory for multilayered composite beams subjected to transverse uniformly distributed loads. The formulation is based on a layerwise model characterized by third-order approximation of the axial displacements and fourth-order approximation of the transverse displacements. The layerwise kinematical model is rewritten in terms of generalized variables. The beam equilibrium equations, expressed in terms of stress resultant, allow writing the boundary value governing problem. The layerwise fields are obtained by postprocessing steps. The main advantage is to ensure the accuracy level associated to the layerwise formulations preserving the computational efficiency of the equivalent-single-layer theories.

Reliability Analysis for Flexural Capacity of Recycled Aggregate Concrete Beams

Concrete made of recycled coarse aggregate (RCA) derived from multiple sources can have relative variation in mechanical properties compared with natural aggregate concrete (NAC), resulting in a lower reliability of a recycled aggregate concrete (RAC) beam than that of NAC beam. In this paper, a method is proposed to improve the reliability of beams made with RAC by increasing the reinforcement ratio. According to the equilibrium equations of beams subjected to bending moment, the increase in reinforcement ratio can be achieved by decreasing the RAC design strength. Same mean compressive strengths are assumed for RAC and NAC and four factors are considered including concrete grade, reinforcement grade, reinforcement ratio and the ratio of external loads in reliability analysis. Using NAC beams as a reference and keeping the target reliability index the same as that of RAC beams, the analysis results show that RAC design strength decreases with increase in coefficient of variation (CoV) of RAC compressive strength. When the CoV of RAC compressive strength is 20%, the RAC design strengths are 0.93 and 0.90 times of the NAC design strength with a 95% confidence level for grade C30 and C40 concrete, respectively.

Refined zigzag theory for homogeneous, laminated composite, and sandwich beams derived from Reissner’s mixed variational principle

A highly accurate and computationally attractive shear-deformation theory for homogeneous, laminated composite, and sandwich laminates is developed for the linearly elastic analysis of planar beams. The theory is derived using the kinematic assumptions of Refined Zigzag Theory (RZT) and a two-step procedure that implements Reissner’s Mixed Variational Theorem (RMVT). The basic expression for the transverse-shear stress that satisfies a priori the equlibrium conditions along the layer interfaces is obtained from Cauchy’s equilibrium equations. The resulting transverse-shear stress consists of second-order derivatives of the two rotation variables of the theory, which subsequently are restated as the unknown stress functions. As the first step in fulfilling RMVT, the Lagrange-multiplier functional is minimized with respect to the unknown stress functions, resulting in the stress functions consisting of first-order derivatives of the kinematic variables. Subsequently, the second term of RMVT is minimized, producing four beam equilibrium equations and consistent boundary conditions. For any number of material layers the new theory maintains only four kinematic variables. The theory is labeled RZT(m), where the superscript (m) stands for mixed formulation. The RZT(m) can accurately model the axial stretch, bending, and transverse-shear deformations, without shear-correction factors. Analytic solutions are derived for simply supported beams subjected to transverse-normal and transverse-shear tractions on the top and bottom surfaces. It is demonstrated that RZT(m) has a wide range of applicability which includes sandwich construction and the laminates with embedded thin compliant layers that can potentially model progression of delaminations. The main advantage of RZT(m) over RZT is in the superior predictions of transverse-shear stresses that are obtained directly from the low-order transverse-shear strain measures of the theory without resorting to a post-processing integration procedure. Importantly, the methodology can be readily extended to plate theory, and it can be applied effectively for developing simple and efficient C0-continuous finite elements.

Refined Beam Element for Second Order Analysis of Latticed Shells

Based on equilibrium equation of beam, the displacement interpolating functions with shear effect of spatial beam elements which are used to simulate the structure members of latticed shells are deduced. The different displacement interpolating functions in compression and tension spatial beam-column elements are unified by the method of Maclaurin series expansion, and the unified expressions which are used to simulate structure members are equivalent to those expressed by stability functions. Numerical analyses results indicate that the second-order elastic analysis method for beam structures proposed in this paper, which can perfectly incarnate the second-order effects and the geometrical nonlinearity of the single-layer cylindrical reticulated shell, is of better accurateness and higher effectiveness.