中文摘要：

针对预应力混凝土结构中空间曲线形状预应力筋的预应力损失计算,运用摩擦力原理和空间微分几何关系,推导出了在张拉阶段预应力筋空间曲线段应力损失的计算公式;结合预应力工程中常用的2类空间曲线,即圆柱面上的圆弧曲线和投影面上为双抛物线的形式,详细地给出了空间弯曲角φ和曲线长s的实用计算公式;对这2类曲线预应力筋两端张拉且布筋形状非对称时的情况,给出了确定理论不动点（应力平衡点）的计算公式。研究结果表明：在给出的工程算例中,从计算开始点到理论不动点处的弧长,公式法和有限元法的计算结果分别为23.554、23.520 m,相差0.15%;理论不动点处有效预应力值的计算结果分别是868.48、912.28 MPa,相差4.8%。其产生误差的原因包括在给出公式计算中各分段计算四舍五入等误差的积累,MIDAS有限元软件在空间预应力筋形状输入时存在的微小误差,以及建模划分梁单元时的疏密程度造成的误差等。给出的计算方法可解决这2类非对称布置空间曲线筋预应力损失的计算问题,能较为便利地计算出这2类空间多曲线布筋形式的弧长和有效预应力,其计算结果可靠,表明了公式的通用性和实用性。

英文摘要：

Aimed at the calculation of prestress loss of prestressing tendons with space curve layout in prestressed concrete structure, based on the friction principle and space differential geometry relationship, formulas were derived to be used to calculate the stress loss in tensioning stage. For the two commonly used space curves of circular arc on a cylindrical surface and double parabolas in the projection planes in prestressing engineering, the practical formulas were given to calculate the space bending angle φ and space curve length s. In view of the situation of asymmetrical arrangement of tendons when tensioning was imposed on both ends, the calculating expression was also obtained to determine the fixed point （the stress balance point） in theory in prestressing tendons. The results show that in the given engineering example, the value of curve length from the beginning point of calculation to the fixed point in theory is 23. 554 m by formulasmethod, and 23. 520 m by finite element method. The difference between the two results is 0.15%. The value of effective prestress at the fixed point in theory is 868.48 MPa by formulas method, and 912.28 MPa by finite element method, The difference between the two results is 4.8%. The reasons for the error include the rounding error accumulation in data processing in each section by the formula method, the slight difference between the real space curve and that into the MIDAS Civil software and the density difference of beam element division in modeling. The loss of prestress, the values of efficient prestress and curve length can be conveniently obtained by the proposed formulas, and the calculation values are reliable, which means that the formula method is universal and practical.