矩形基础地基临塑荷载公式

The formula of subgrade’s approaching plasticity load for rectangular footings

目前土力学教材中给出的临塑荷载公式只适用于条形基础。实际工程中柱下矩形基础比条形基础应用更广泛,对矩形基础地基临塑荷载公式开展研究不仅对土力学学科发展具有理论意义,同时对矩形基础设计具有实践价值。首先通过二重积分得到矩形基底左右两侧区域内向上拉应力在基底边缘处产生的应力;再以条形基底地基中主应力方向建立局部直角坐标系,通过应力叠加得到矩形基底边缘处应力表达式;接着对开平方近似计算式中系数n的不同取值作误差分析;最后将应力表达式代入Coulomb强度准则,利用开平方近似计算式和三角函数级数展开式,通过求导得到矩形基础地基临塑荷载公式。不同类型地基土的算例表明,地基土泊松比越小,矩形基础与条形基础的地基临塑荷载比值越小,利用后者的临塑荷载公式计算前者的临塑荷载不安全。

At present,the formula of subgrade’s approaching plasticity load given in the textbook of soil mechanics is only applicable to strip footings.In practical engineering,rectangular footings under columns are more widely used than strip footings.The research on the formula of subgrade’s approaching plasticity load for rectangular foundation has not only theoretical significance for the development of soil mechanics,but also practical value for the design of rectangular footings.At first,the stress at the edge of the footing caused by the upward tensile stress in the left and right side regions of the rectangular footing is obtained by double integral;Then,the local rectangular coordinate system is established along the principal stress direction in the subgrade under the strip footing,and the stress expression at the edge of the rectangular footing is obtained by stress superposition.Then,an error analysis is carried out on the approximate calculation formula for square root with different values of coefficient n;Finally,the stress expression is substituted into the Coulomb strength criterion,and the formula of subgrade’s approaching plasticity load for rectangular footings is derived with the help of the square root approximation formula and the polar expansion of trigonometric function.The calculation examples of different types of subgrade soil show that the smaller the Poisson’s ratio is,the smaller the ratio of approaching plasticity load of rectangular footing to strip footing is.It is not safe to use the latter’s formula to calculate the former’s approaching plasticity load.