Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.     

饱和多孔弹性Timoshenko悬臂梁的动、静力弯曲

在经典单相Timoshenko梁变形和孔隙流体仅沿饱和多孔弹性梁轴向运动的假定下,基于不可压饱和多孔介质的三维Gurtin型变分原理,首先建立了饱和多孔弹性Timoshenko悬臂梁动力响应的一维数学模型.在若干特殊情形下,该模型可分别退化为饱和多孔弹性梁的Euler-Bernoulli模型、Rayleigh模型和Shear模型等.其次,利用Laplace变换,分析了固定端不可渗透、自由端可渗透的饱和多孔弹性Timoshenko悬臂梁在自由端阶梯载荷作用下的动静力响应,给出了梁自由端处挠度随时间的响应曲线,考察了固相与流相相互作用系数、梁长细比等参数对悬臂梁动静力行为的影响.结果表明:饱和多孔弹性梁的拟静态挠度具有与粘弹性梁挠度类似的蠕变特征.在动力响应中,随着梁长细比的增大,自由端挠度的振动周期和幅值增大,且趋于稳态值的时间增长,而随着两相相互作用系数的增大,梁挠度振动衰减加快,并最终趋于经典单相弹性Timoshenko梁的静态挠度.     

Bending of simply-supported circular timber beam strengthened with fiber reinforced polymer简

The bending behavior of the circular cross-section timber beam strengthened with a fiber reinforced polymer (FRP) sheet is investigated. The tight bonding is on the interface between the surface of a timber beam and the reinforcement layer of the FRP sheet. An analytical expression for the bending moment-curvature relation is presented, and its failure modes are analyzed. The governing equation for nonlinear small deflection of the FRP-strengthened circular timber beam is established, and the corresponding numerical method is given. The bending deformation of the simply-supported circular timber beam strengthened with the carbon fiber reinforced polymer (CFRP) sheet subject to a uniform load is studied numerically. The influence of the angle and thickness of the CFRP layer as well as the timber strength on the bending deflection of the FRP-strengthened circular timber beam is examined. It is revealed that, with the increases of the thickness and angle, the deflection of the CFRP-strengthened circular timber beam is decreased, and its carrying capacity and ductility are increased. However, when the angle of the layer reaches a certain value, the deflection will no longer decrease with the increase of the angle. At the same time, the nonlinear bending moment-curvature relation of the CFRP-strengthened circular timber beam is simplified as an approximate bilinear constitutive model. The approximate deflections of the simply-supported circular timber beam strengthened with the CFRP sheet are obtained. The results are compared with the linearly elastic bending and nonlinear bending models, showing that the mid-span deflections of a CFRP-strengthened circular timber beam with the approximate bilinear constitutive model are greater than those with the nonlinear constitutive model. The results of its stiffness analysis are on the safe side.