沥青路面轴对称动力响应问题中解析解对比

doi:10.13229/j.cnki.jdxbgxb.20240914

摘要/Abstract

摘要：

当前有关路面力学的研究多集中于解析解推导以及影响因素探究上，缺乏不同解析解方法的差异性研究。为此，本文分别以传递矩阵法、刚度矩阵法、波传递法以及反射、透射矩阵法推导出了考虑材料横观各向同性以及层间接触状态的轴对称路面结构动力响应解析解。在此基础上，结合4种路面结构研究了上述4种解析解的计算效率以及适用范围。结果表明：①在矩阵规模上，由小到大依次为反射、透射矩阵法、传递矩阵法、刚度矩阵法以及波传递法；②在求解速度上，传递矩阵法耗时显著少于其余3种解析解方法，且反射、透射矩阵法耗时最长；③在数值稳定性上，传递矩阵法在进行有限厚度的动力响应计算时极易出现数值溢出问题，而其余解析解方法均具有较好的数值稳定性；④在实际应用上，传递矩阵法尤其适用于求解半空间体表面弯沉；刚度矩阵法仅可计算各结构层表面的动力响应；而波传递法和反射、透射矩阵法可求解路面内部任意位置的动力响应。

关键词: 层状体系力学, 传递矩阵, 刚度矩阵, 波传递, 反射、透射矩阵

Abstract:

The current researches on pavement mechanics usually focus on the derivation of analytical solutions and the exploration of the factors affecting them， there are few studies on the difference of analytical solutions. In order to solve this problem， transfer matrix method， stiffness matrix method， wave transfer method and reflection and transmission matrix method were applied in the deduction process of the analytical solutions considering the transverse isotropy of materials and the contact state between layers. On this basis， the calculation efficiency and the range of adaptability of the above four analytical solutions are studied by combining four kinds of pavement structures. The results show that： ①for the matrix scale， reflection and transmission matrix method， transfer matrix method， stiffness matrix method and wave transfer method increase in turn； ② for the solution speed， the transfer matrix method takes much less time than the other three methods， while the reflection and transmission matrix method consumes the most time； ③in the numerical stability， the transfer matrix method is very easy to overflow data when calculating the dynamic response of finite thickness， while the other analytical solution methods have good numerical stability； ④in practical application， the transfer matrix method is particularly suitable for the surface of semi space problems， and the stiffness matrix method can only calculate the dynamic response of the surface of each structural layer， however， the wave transfer method and the reflection and transmission matrix method can solve the dynamic response of any position in the pavement.

Key words: mechanics of the layered system, transfer matrix method, stiffness matrix method, wave transfer method, reflection and transmission matrix method

中图分类号:

U416.2

尚桦雨,范海山,张军辉. 沥青路面轴对称动力响应问题中解析解对比[J]. 吉林大学学报(工学版), 2026, 56(3): 758-771.

Hua-yu SHANG,Hai-shan FAN,Jun-hui ZHANG. Comparative of analytical solutions for axial symmetric dynamic response of asphalt pavements[J]. Journal of Jilin University(Engineering and Technology Edition), 2026, 56(3): 758-771.

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结构层	E_v/MPa	n	μ_v（=μ_h）	ρ/（kg?m^-3）	h/m	α_x
路面	7 000	0.3	0.25	2 400	0.18	0.8
基层	10 000	0.5	0.30	2 300	0.35	0.0
底基层	400	1.0	0.35	2 200	0.20	0.0
土基	100	1.0	0.35	1 600	3.00	-

计算方法	路面结构	传递/反射透射矩阵/矩阵元素计算	构建矩阵方程并求解	数值逆变换	总耗时
传递矩阵法	A	76.3	54.4	28.9	160.9
	B	计算失败，计算结果为NAN
	C	80.2	54.7	28.9	191.1
	D	计算失败，计算结果为NAN
刚度矩阵法	A	423.4	121.3	30.2	590.7
	B	398.2	116.8	27.9	557.3
	C	554.4	141.4	30.1	741.5
	D	531.5	138.6	27.8	712.8
波传递法	A	123.2	396.6	31.0	643.1
	B	113.5	401.8	28.2	629.1
	C	158.4	496.4	29.7	821.7
	D	147.7	499.7	28.0	803.9
反射、透射矩阵法	A	76.8	555.1	30.5	677.8
	B	70.4	559.2	29.1	674.3
	C	101.3	745.4	31.6	894.1
	D	95.5	756.1	29.1	895.8

计算方法	计算速度	数值稳定性	求解位置	适应范围
传递矩阵法	快	较稳定	路面表面	半空间
刚度矩阵法	一般	稳定	各结构层表面	半空间有限厚度
波传递法	较慢	稳定	任意位置	半空间有限厚度
反射、透射矩阵法	较慢	稳定	任意位置	半空间有限厚度