ANALYTICAL SOLUTION OF INTERNAL FORCE OF PARABOLIC ARCH WITH ELASTIC SUPPORTS CONSIDERING ELASTIC COMPRESSION
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摘要: 为解决非理想边界约束的拱结构内力理论计算问题，将非理想边界约束简化为弹性支承，基于弹性中心法对其力法方程进行简化，考虑弹性压缩的影响，采用精确曲线积分，推导了竖向移动荷载作用下计算弹性支承抛物线拱刚臂长度、常变位、载变位和内力的解析解，研究了弯压刚度比、矢跨比和水平弹性约束对支承处水平推力的影响规律，以及水平弹性约束对拱轴内力分布的影响。研究表明：该文提出的解析解物理概念清晰、正确可靠，可以显式地明确呈现弹性支承相关参数对内力计算的影响过程；不考虑拱肋弹性压缩影响导致的水平推力计算误差随弯压刚度的增大而增大，拱趾支承为刚性约束时误差最大，可以达到27.8%；水平弹性支承对拱轴内力分布和水平推力具有显著的调控作用；水平推力影响系数随矢跨比的增大呈非线性增大，当弯压刚度比为1.93、水平弹性支承柔度系数为0.02时，常见矢跨比对应的水平推力影响系数在0.15左右。Abstract: In order to solve the theoretical calculation problem of the internal force of arch structures with non-ideal boundary constraints, this paper simplifies the non-ideal boundary constraints as elastic support, and simplifies the force method equation based on the elastic center method. The elastic compression is considered and the precise curve integral method is adopted. The analytical solutions are derived for rigid arm length, constant displacement, load displacement and internal force of parabolic arch with elastic support and under vertical moving load. The influences of flexural compression stiffness ratio, rise span ratio and horizontal elastic restraint on horizontal thrust at the support are studied. The influence of horizontal elastic restraint on internal force distribution of arch axis is also studied. The results show that the analytical expression proposed in this paper has a clear physical concept, is correct and reliable, and can explicitly show the influence process of elastic support parameters on internal force calculation. The calculation error of horizontal thrust increases with the increase of bending and flexural compression stiffness, ignoring the influence of elastic compression of arch rib. The calculation error of horizontal thrust is the largest when the arch toe support is a rigid constraint, which can reach 27.8%. Horizontal elastic support can significantly change the distribution of horizontal thrust and arch axis internal force at the support. The influence coefficient of horizontal thrust increases nonlinearly with the increase of rise span ratio. The influence coefficient of horizontal thrust corresponding to the common rise span ratio is about 0.15 when the flexural stiffness ratio is 1.93 and the flexibility coefficient of horizontal elastic support is 0.02.
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图 3 坐标转换后的弹性中心法基本体系
Figure 3. Basic system of elastic center method after coordinate transformation
图 4 不同荷载作用位置的内力分布图(f/l=1/4.8)
Figure 4. Distribution of internal forces at different loading positions (f/l=1/4.8)
图 5 不同参数影响下水平推力影响系数ψ的变化曲线
Figure 5. Variation curve of influence coefficient of horizontal thrust ψ under different parameters
图 6 不同柔度系数水平弹性支承抛物线拱内力分布图
Figure 6. Distribution of internal forces of parabolic arches with different flexibility coefficients of horizontal elastic supports
图 7 内力特征值随水平弹性支承柔度变化曲线
Figure 7. Variation curve of internal force characteristic value with flexibility coefficient of horizontal elastic support
表 1 赘余力和单位外荷载作用下基本结构的内力
Table 1. Superfluous force and internal forces under the effect of redundant forces or unit external load
内力 赘 余力 移动单位荷载 ${x_1}$ $ {x_2} $ ${x_3}$ $\xi < {\xi _{\text{P}}}$ $\xi \geqslant {\xi _{\text{P}}}$ 弯矩 ${\overline M_1}$ 1 ${\overline M_2}$ $f{\xi ^2} - {y_{\text{S}}}$ ${\overline M_3}$ $x$ ${M_{\text{P}}}$ 0 ${M_{\text{P}}}$ $ \mp P{l_1}\left( {\xi - {\xi _{\text{P}}}} \right)$ 轴力 $ {\overline N_1} $ 0 $ {\overline N_2} $ $\cos \varphi $ $ {\overline N_3} $ $ - \sin \varphi $ $ {N_{\text{P}}} $ 0 $ {N_{\text{P}}} $ $ \pm P\sin \varphi $ 剪力 ${\overline Q_1}$ 0 ${\overline Q_2}$ $\sin \varphi $ ${\overline Q_3}$ $\cos \varphi $ ${Q_{\text{P}}}$ 0 ${Q_{\text{P}}}$ $ \mp P\cos \varphi $ 注：正负半轴上拱轴切线与水平线的夹角$\varphi $异号；当单移动荷载位于右边拱时，表中移动单位荷载MP、NP、QP表达式的正负号分别取−、+、−；当单移动荷载位于左边拱时，表中移动单位荷载MP、NP、QP表达式的正负号分别取+、−、+”。 
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表 2 刚臂长度、常变位及载变位显式解
Table 2. Explicit solution of rigid arm length, constant displacement and load displacement
参数分类 参数 参数表达式 刚臂长度 ${y_{\text{S}}}$ $\dfrac{{{C_{{\text{S1}}}}\sqrt {1 + {a^2}}+ 2{\zeta _1}}}{{{C_{{\text{S2}}}}\sqrt {1 + {a^2}} + 2{\zeta _1}}}f$ 常变位 ${\delta _{11}}$ $\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot ( { {C_1}\sqrt {1 + {a^2} } + 2{\zeta _1} } )$ ${\delta _{22}}$ $\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot \left[ { {C_2}\sqrt {1 + {a^2} } + 2{ {\left( {f - { {y_{\text{S} } } } } \right)}^2}\zeta + 2l_1^2{\zeta _2} } \right]$ ${\delta _{33}}$ $\dfrac{ { {l_1} } }{ {a{{EI} } } } \cdot ( { {C_3}\sqrt {1 + {a^2} } + 2{\zeta _1}l_1^2 + 2{\zeta _2}l_1^2} )$ 载变位 ${\varDelta _{ {\text{1P} } } }$ $- \dfrac{ {Pl_1^2} }{ {a{{EI} } } } \cdot [ { {C_{ {\text{1P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{1P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {1 - {\xi _{\text{P} } } } \right){\zeta _1} } ]$ ${\varDelta _{ {\text{2P} } } }$ $- \dfrac{ {P{l_1} } }{ {a{{EI} } } } \cdot [ { {C_{ {\text{2P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{2P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {f - {y_{\rm{S} } } } \right)\left( {1 - {\xi _{\text{P} } } } \right){l_1}{\zeta _2} }]$ ${\varDelta _{ {\text{3P} } } }$ $- \dfrac{ {P{l_1} } }{ {a{{EI} } } } \cdot [ { {C_{ {\text{3P} } } }\sqrt {1 + {a^2} } + {D_{ {\text{3P} } } }\sqrt {1 + { {\left( {a{\xi _{\text{P} } } } \right)}^2} } + \left( {1 - {\xi _{\text{P} } } } \right){\zeta _1}l_1^2 + {\zeta _3}l_1^2} ]$ 注：当移动荷载作用在左半拱时即x轴负半轴时，将载变位中的ξP替换为−ξP，且Δ3P取正号。
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表 3 典型截面数值对比表
Table 3. Comparison of results of typical sections
算例 内力 截面位置 荷载作用位置ξP=0.0 荷载作用位置ξP=0.5 本文解 有限元解 相对误差/(%) 本文解 有限元解 相对误差/(%) 算例1(f/l=1/4) 弯矩M/(N·m) ξ=−0.5 −10.858 −10.803 0.503 −11.932 −11.918 0.116 ξ=0.0 28.029 28.202 0.614 −7.714 −7.635 1.023 ξ=0.5 −10.858 −10.803 0.503 34.946 34.981 0.102 轴力N/N ξ=−0.5 1.045 1.042 0.282 0.551 0.550 0.237 ξ=0.0 0.918 0.915 0.358 0.535 0.533 0.280 ξ=0.5 1.045 1.042 0.282 0.405/0.853 0.404/0.851 0.342/0.159 剪力Q/N ξ=−0.5 0.037 0.038 3.832 −0.093 −0.093 0.773 ξ=0.0 0.500/−0.500 0.500/−0.500 0.000/0.000 0.163 0.163 0.047 ξ=0.5 −0.037 −0.038 3.832 0.385/−0.509 0.384/−0.510 0.152/0.112 算例2(f/l=1/5) 弯矩M/(N·m) ξ=−0.5 −10.982 −10.906 0.697 −11.609 −11.586 0.198 ξ=0.0 27.730 27.986 0.920 −7.558 −7.437 1.612 ξ=0.5 −10.982 −10.906 0.697 34.745 34.794 0.141 轴力N/N ξ=−0.5 1.253 1.251 0.463 0.678 0.675 0.401 ξ=0.0 1.153 1.147 0.544 0.665 0.662 0.445 ξ=0.5 1.257 1.251 0.463 0.558/0.929 0.555/0.926 0.500/0.298 剪力Q/N ξ=−0.5 0.036 0.038 6.214 −0.097 −0.096 1.206 ξ=0.0 0.500/−0.500 0.500/−0.500 0.000/0.000 0.161 0.161 0.056 ξ=0.5 −0.036 −0.038 6.214 0.397/−0.532 0.396/−0.533 0.253/0.187 算例3(f/l=1/6) 弯矩M/(N·m) ξ=−0.5 −11.046 −10.944 0.922 −11.398 −11.363 0.307 ξ=0.0 27.567 27.92 1.278 −7.456 −7.283 2.347 ξ=0.5 −11.046 −10.944 0.922 34.604 34.668 0.185 轴力N/N ξ=−0.5 1.475 1.465 0.682 0.805 0.800 0.605 ξ=0.0 1.388 1.378 0.764 0.795 0.790 0.650 ξ=0.5 1.475 1.465 0.682 0.704/1.020 0.699/1.015 0.702/0.483 剪力Q/N ξ=−0.5 0.035 0.039 8.329 −0.100 −0.098 1.735 ξ=0.0 0.500/−0.500 0.500/−0.500 0.000/0.000 0.160 0.160 0.063 ξ=0.5 −0.035 −0.039 8.329 0.403/−0.545 0.402/−0.547 0.379/0.279 注：“/”左右两侧的数值分别表示左截面和右截面的数据。
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