(4.2.1-1)

 式中：

                         M —— 质量矩阵；

                          C —— 阻尼矩阵；

                    Q(ut) —— 弹塑性恢复力；

 —— 在时间 时的各节点的加速度、速度、位移；

                          Rt —— 节点上的动力荷载。

                                                                                                                       (4.2.1-2)

                                                                                      (4.2.1-3)

可重新整理上面公式如下：

                                                                                    (4.2.1-4)

                                                                                                      (4.2.1-5)

位移、速度、加速度增量为：

                                                                                                                                                                        (4.2.1-6)

                                                                                                              (4.2.1-7)

                                                                                                                         (4.2.1-8)

使用完全牛顿-拉普森计算的迭代增量为：

                                                                                                                                    (4.2.1-9)

                                                                                                                                (4.2.1-10)

因此在时间t+Δt时的第(i)次迭代计算时的位移、速度、加速度为：

                                                                                                                                                         (4.2.1-11)

                                                                                                 (4.2.1-12)

                                                                                         (4.2.1-13)

在时间t+Δt时的第(i)次迭代计算时的非线性运动方程如下：

                                                                                                                         (4.2.1-14)

将(4.2.1-12)和(4.2.1-13)代入(4.2.1-14)可得对于位移增量δu(i)的平衡方程：

                                                                                                                                                                      (4.2.1-15)

式中：

                         KEff —— 有效刚度矩阵；

                       ΔPEff —— 各迭代计算中的有效荷载向量；

                   K(i)t+Δt —— 包含非线性单元的切线刚度矩阵；

                          δu(i)—— 各迭代计算中的位移增量向量；

                               β —— Newmark-β法时间积分参数。

                                                                                                                         (4.2.1-16)

                                                                                       (4.2.1-17)