混凝土等效断裂韧度的计算模式与钢筋混凝土梁裂缝的诊断方法
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摘要
混凝土断裂引起的结构裂缝会破坏结构的整体性,形成结构内部的力学断面,导致结构应力状态的恶化,由此引起的桥梁坍塌、建筑毁坏等重大工程事故不断发生。梁是结构中最基本的构件,钢筋混凝土梁中普遍存在的裂缝,严重威胁钢筋混凝土结构的安全性。因此,深入研究混凝土的断裂机理以及钢筋混凝土梁的裂缝诊断,对保证大型土木工程结构的安全性,具有重要的理论意义和工程价值。
本文全面系统地开展混凝土等效断裂韧度的计算模式和钢筋混凝土梁的裂缝诊断方法的研究,主要研究工作与创新成果包括以下几个方面:
(1)基于混凝土断裂力学与复合材料力学,建立了求解混凝土等效断裂韧度的解析计算模式。运用修正的剪滞理论,建立了带裂缝混凝土梁的分层剪滞模型;采用虚拟裂缝模型,修正了求解混凝土断裂韧度的应力边界条件;结合剪滞模型的应力、位移边界条件,得到了分层解析模型各子层的位移和应力的分布函数;运用能量法则,建立了求解混凝土等效断裂韧度的解析表达式。运用所建立的解析模型对8根带裂缝试件的试验数据进行数值仿真分析,得到了混凝土等效断裂韧度的解析解。计算结果表明,4根体积系列试件的混凝土等效断裂韧度的均方差和变异系数分别小于0.0398和0.0384,4根高度系列试件的混凝土等效断裂韧度的均方差和变异系数分别小于0.0394和0.0363,与相关试验对应的数值解相比,解析解的均方差和变异系数更小,且具有更好的鲁棒性。因此,所建立的解析计算模式为求解混凝土等效断裂韧度提供了一种可靠的实用解析方法。
(2)考虑混凝土主裂缝亚临界扩展长度及虚拟裂缝区粘聚力,利用J积分与应力强度因子的解析关系,建立了基于小波基无单元方法求解混凝土等效断裂韧度的数值计算模式。运用三阶B样条函数作为小波基函数,取代了传统无单元方法中的多项式基,建立了小波基无单元方法;结合J积分原理,推导了混凝土等效断裂韧度的数值计算模式。研究了不同基底小波分辨率对混凝土等效断裂韧度计算值的影响。运用所建立的数值计算模式对8根带裂缝试件的试验数据进行数值仿真分析,得到了混凝土等效断裂韧度的数值解。结果表明,在相同基底小波分辨率下,4根体积系列试件计算值的均方差和变异系数分别小于0.0244和0.0231,4根高度系列试件计算值的均方差和变异系数分别小于0.0384和0.0362,计算结果的一致性好;与解析解相比,8个试件的计算值的最大相对误差均在5%左右,计算结果的精度高;且不同基底小波分辨率对计算结果影响不大。因此,所建立的数值计算方法是求解混凝土等效断裂韧度的一种可靠的高精度数值方法。
(3)基于断裂力学理论,考虑钢筋的影响因素,运用应力强度因子与应变能的解析关系,得到了钢筋混凝土梁单元由于直裂缝或斜裂缝所产生的附加应变能的解析表达式,结合虚功原理,推导出了含直裂缝或斜裂缝的钢筋混凝土梁的单元刚度矩阵。制作了含不同直或斜裂缝损伤工况的模型梁,通过模态试验验证了所建立的含裂缝梁单元刚度矩阵的适用性。试验结果显示,裂缝使得梁的频率降低,且裂缝对第二阶频率的影响大于第一阶频率,这与既有研究结论相吻合,说明了试验的有效性。运用所建立的含裂缝梁的单元刚度矩阵对试验梁进行了模态分析,计算结果显示,频率计算值与试验值的最大相对误差仅为2.76%,计算振型与对应实测振型的MAC值均大于0.9,显示了二者较好的相关性,充分说明所建立的含裂缝钢筋混凝土梁的单元刚度矩阵,可以用于结构有限元计算,并且具有良好的计算精度。
(4)运用线弹性断裂理论,结合时域内的位移响应,建立了钢筋混凝土梁直裂缝的时域两阶段诊断方法。首先,基于时域内的位移响应,结合正则化技术和假设检验,建立了用平均曲率变化比表示的损伤指标,用于诊断直裂缝的位置;其次,结合线弹性断裂力学中应变能释放率、应力强度因子与裂缝深度的解析关系式,建立了直裂缝深度的时域模型,用于识别裂缝的深度。运用所建立的带直裂缝梁的单元刚度矩阵对裂缝梁进行了仿真分析,进而验证了所建立方法的有效性,结果表明,该方法能够在较高的置信区间内准确识别钢筋混凝土梁中多处裂缝的位置,当正则化参量的阈值为1.0时,成功诊断了80%的裂缝位置;通过结合损伤指标图,裂缝位置诊断成功率达到了100%;运用所建立的尺寸诊断模型,也可以在一定精度内诊断出裂缝的深度。通过对含测量噪声的损伤指标与深度模型的诊断结果的分析表明,该方法在5%噪声的影响下仍具有良好的鲁棒性。因此,所建立的时域两阶段诊断方法是诊断钢筋混凝土梁直裂缝位置和深度的一种高效方法。
(5)建立了基于遗传神经网络的钢筋混凝土梁斜裂缝的频域诊断方法。运用遗传算法,优化了BP神经网络的拓扑结构、权值和阈值;用归一化的损伤前后的频率变化比、同阶频率平方变化比和振型分量作为损伤指标,对所构造的遗传神经网络进行训练,建立了基于遗传神经网络的钢筋混凝土梁斜裂缝的频域诊断方法,用以识别斜裂缝的位置、深度和角度。运用所建立的带斜裂缝梁的单元刚度矩阵进行了模态分析,验证了所建立的频域诊断方法的有效性进行,结果表明,该方法可以成功诊断出斜裂缝的位置、深度和角度,且诊断精度高、鲁棒性强;在5%测量噪声影响下,对斜裂缝位置诊断的正确率为100%,对斜裂缝深度的诊断误差在25%以内,对斜裂缝角度的诊断误差在12%以内。因此,所建立的基于遗传神经网络的频域诊断方法是诊断钢筋混凝土梁中斜裂缝的一种有效方法。
The integrity of a structure may be destructed by cracks induced by fracture of concrete, some mechanical fracture surfaces may be formed within the structure, and the stress condition of the structure may be deteriorated. This may induce serious engineering accidents as collapse of bridges and destroy of buildings constantly. Beam is the most basic member in structures. The cracks existed within reinforced concrete (RC) beam generally may threaten the safety of RC structures. Therefore, systematical studies on fracture mechanism of concrete and crack detection of RC beam have important theoretical significance and engineering value for guaranteeing the safety of large civil structures.
The researches on calculation model of equivalent fracture toughness of concrete and detection method of cracks of RC beam are performed systematically in this dissertation. The main research work and innovative achievements are summarized as follows:
(1) Based on fracture mechanics of concrete and composite material mechanics, the analytical model is built to solve the equivalent fracture toughness of concrete. The modified shear lag model is used to build the layered shear lag model of the concrete specimen with a crack. The fictitious crack model is used to modify the boundary condition of stress of the analytical model. Combined with other boundary condition of stress and displacement, displacement and stress functions of layers of the analytical model are obtained. Using energy method, the analytical expression of the equivalent fracture toughness of concrete is obtained. Analytical solutions of the equivalent fracture toughness of concrete are obtained using the analytical calculation model with the experimental data of 8 specimens. The calculated results show that mean square errors and variation coefficients of the calculated values with different volumes are lower than 0.0398 and 0.0384 respectively, and those with different heights are lower than 0.0394 and 0.0363 respectively. Being lower mean square errors and variation coefficients, the analytical solutions are more robust than the numerical solutions based on the coherent experiment. Therefore, the analytical model provides a reliable practical analytical method to solve the equivalent fracture toughness of concrete.
(2) Considering the length of the subcritical growth of the main crack and the cohesive strength of the fictitious crack of concrete, and using the analytical relationship between the J integral and the stress intensity factor, a numerical algorithm based on the element free Galerkin method with wavelet basis to solve the equivalent fracture toughness of concrete is built. The third order B spline function is taken as the wavelet function to substitute the traditional polynomial basis in the element free Galerkin method, which is used to build the element free Galerkin method with wavelet basis. The numerical model to solve the equivalent fracture toughness of concrete is deduced using J integral theory. The influence of different resolution of wavelet basis on the calculated values is investigated. The calculated results for 8 specimens with crack show that in the case of identical resolution of wavelet basis, the mean square derivations and the coefficients of variation of the calculated values of the 4 volume series specimens are smaller than 0.0244 and 0.0231 respectively, and the those of the 4 height series specimens are smaller than 0.0384 and 0.0362, so to have good consistency. By comparing with the analytical solutions, the maximum relative error of the calculated values of 8 specimens is about 5%, so the precision of the calculated results is high. And the influence of different resolution of wavelet basis on the calculated results is little. Therefore, the element free Galerkin method with wavelet basis is a reliable numerical algorithm with high precision to solve the equivalent fracture toughness of concrete.
(3) Based on fracture mechanics, the analytical relationship of the stress intensity factor and strain energy is used to obtain the analytical expression of additional strain energy induced by straight or diagonal crack in element of RC beam. Considering the influence of steel bar, the virtual work principle is used to deduce the stiffness matrix of element with a straight or diagonal crack of RC beam. Some test beams with different straight or diagonal cracks damage cases are fabricated, whose results of model test verify the applicability of the stiffness matrix of the element with a straight or diagonal crack. The tested results show that frequencies of test beams are reduced and the influence of the second order frequency is bigger than the influence of the first frequency for cracks, which is consistent with conclusions given by others. This proofs the model test is valid. The modal analysis for test beams is given based on the deduced elemental stiffness matrix with a crack. The calculated results show that the maximum value of relative error between the computational frequencies and experiment frequencies is only 2.76% and the values of MAC of correlative model shapes between experimental and calculation are bigger than 0.9. This verifies good relativity between the tested and calculated modal shapes. Therefore, the deduced elemental stiffness matrix of RC beam with a crack can be used in the calculation of infinite element and the calculation precision is higher.
(4) A two-stage detection method in time domain for straight cracks of RC beam is established using the linear elastic fracture theory and displacement responses of RC beam in time domain. Firstly, normalized method and hypothesis testing are used to build a damage index with relation to the mean curvature change ratio to locate cracks based on displacement responses in time domain. Secondly, the analytical relationships of the strain energy release rate, stress intensity factor and depth of crack are used to obtain a crack depth model in time domain to detect the depth of crack. The stiffness matrix of cracked element is used to simulate the cracked RC beams to verify the feasibility of the algorithm of straight crack detection. The calculated results show that the algorithm could locate cracks precisely with high confidence interval. 80% of locations of cracks are detected successfully when the threshold of the normalized parameter is put to 1.0. Combining the diagram of damage index, all of locations of cracks can be detected precisely. Depths of cracks can be detected in a certain accurate degree. Meanwhile the contaminated displacement responses are used to analyze the robustness of the damage index and the crack depth model, which demonstrates that the method is robust against even if 5% noise level. Therefore, the two-stage detection method in time domain is a method with high efficiency to detect the depths and locations of straight cracks in RC beam.
(5) A genetic neural network-based detection method in frequency domain for diagonal cracks within RC beams is built. The topological structure, weight and threshold of the BP neural network are optimized using the genetic algorithm. Normalized ratios of variations of the frequencies, normalized ratios of variations of the square of same order frequencies and components of mode of the cracked RC beam are used as the damage indices to train the genetic neural network. The method based on the genetic neural network is built to detect the location, depth and angle of the diagonal crack. The stiffness matrix of cracked element is used to obtain frequencies and model shapes of the cracked RC beams to verify the feasibility of the method of diagonal crack detection. The numerical simulation shows that the method detects the location, depth and angle of the diagonal crack successfully with a high detection precision and robust. In the 5% noise level, the successful rate of location detection is 100%, and the detection error of crack depth is controlled in 25%, and the detection error of crack angle is controlled in 12%. Therefore, the genetic neural network-based detection method in frequency domain is an effective method to detect the diagonal cracks within RC beams.
本文全面系统地开展混凝土等效断裂韧度的计算模式和钢筋混凝土梁的裂缝诊断方法的研究,主要研究工作与创新成果包括以下几个方面:
(1)基于混凝土断裂力学与复合材料力学,建立了求解混凝土等效断裂韧度的解析计算模式。运用修正的剪滞理论,建立了带裂缝混凝土梁的分层剪滞模型;采用虚拟裂缝模型,修正了求解混凝土断裂韧度的应力边界条件;结合剪滞模型的应力、位移边界条件,得到了分层解析模型各子层的位移和应力的分布函数;运用能量法则,建立了求解混凝土等效断裂韧度的解析表达式。运用所建立的解析模型对8根带裂缝试件的试验数据进行数值仿真分析,得到了混凝土等效断裂韧度的解析解。计算结果表明,4根体积系列试件的混凝土等效断裂韧度的均方差和变异系数分别小于0.0398和0.0384,4根高度系列试件的混凝土等效断裂韧度的均方差和变异系数分别小于0.0394和0.0363,与相关试验对应的数值解相比,解析解的均方差和变异系数更小,且具有更好的鲁棒性。因此,所建立的解析计算模式为求解混凝土等效断裂韧度提供了一种可靠的实用解析方法。
(2)考虑混凝土主裂缝亚临界扩展长度及虚拟裂缝区粘聚力,利用J积分与应力强度因子的解析关系,建立了基于小波基无单元方法求解混凝土等效断裂韧度的数值计算模式。运用三阶B样条函数作为小波基函数,取代了传统无单元方法中的多项式基,建立了小波基无单元方法;结合J积分原理,推导了混凝土等效断裂韧度的数值计算模式。研究了不同基底小波分辨率对混凝土等效断裂韧度计算值的影响。运用所建立的数值计算模式对8根带裂缝试件的试验数据进行数值仿真分析,得到了混凝土等效断裂韧度的数值解。结果表明,在相同基底小波分辨率下,4根体积系列试件计算值的均方差和变异系数分别小于0.0244和0.0231,4根高度系列试件计算值的均方差和变异系数分别小于0.0384和0.0362,计算结果的一致性好;与解析解相比,8个试件的计算值的最大相对误差均在5%左右,计算结果的精度高;且不同基底小波分辨率对计算结果影响不大。因此,所建立的数值计算方法是求解混凝土等效断裂韧度的一种可靠的高精度数值方法。
(3)基于断裂力学理论,考虑钢筋的影响因素,运用应力强度因子与应变能的解析关系,得到了钢筋混凝土梁单元由于直裂缝或斜裂缝所产生的附加应变能的解析表达式,结合虚功原理,推导出了含直裂缝或斜裂缝的钢筋混凝土梁的单元刚度矩阵。制作了含不同直或斜裂缝损伤工况的模型梁,通过模态试验验证了所建立的含裂缝梁单元刚度矩阵的适用性。试验结果显示,裂缝使得梁的频率降低,且裂缝对第二阶频率的影响大于第一阶频率,这与既有研究结论相吻合,说明了试验的有效性。运用所建立的含裂缝梁的单元刚度矩阵对试验梁进行了模态分析,计算结果显示,频率计算值与试验值的最大相对误差仅为2.76%,计算振型与对应实测振型的MAC值均大于0.9,显示了二者较好的相关性,充分说明所建立的含裂缝钢筋混凝土梁的单元刚度矩阵,可以用于结构有限元计算,并且具有良好的计算精度。
(4)运用线弹性断裂理论,结合时域内的位移响应,建立了钢筋混凝土梁直裂缝的时域两阶段诊断方法。首先,基于时域内的位移响应,结合正则化技术和假设检验,建立了用平均曲率变化比表示的损伤指标,用于诊断直裂缝的位置;其次,结合线弹性断裂力学中应变能释放率、应力强度因子与裂缝深度的解析关系式,建立了直裂缝深度的时域模型,用于识别裂缝的深度。运用所建立的带直裂缝梁的单元刚度矩阵对裂缝梁进行了仿真分析,进而验证了所建立方法的有效性,结果表明,该方法能够在较高的置信区间内准确识别钢筋混凝土梁中多处裂缝的位置,当正则化参量的阈值为1.0时,成功诊断了80%的裂缝位置;通过结合损伤指标图,裂缝位置诊断成功率达到了100%;运用所建立的尺寸诊断模型,也可以在一定精度内诊断出裂缝的深度。通过对含测量噪声的损伤指标与深度模型的诊断结果的分析表明,该方法在5%噪声的影响下仍具有良好的鲁棒性。因此,所建立的时域两阶段诊断方法是诊断钢筋混凝土梁直裂缝位置和深度的一种高效方法。
(5)建立了基于遗传神经网络的钢筋混凝土梁斜裂缝的频域诊断方法。运用遗传算法,优化了BP神经网络的拓扑结构、权值和阈值;用归一化的损伤前后的频率变化比、同阶频率平方变化比和振型分量作为损伤指标,对所构造的遗传神经网络进行训练,建立了基于遗传神经网络的钢筋混凝土梁斜裂缝的频域诊断方法,用以识别斜裂缝的位置、深度和角度。运用所建立的带斜裂缝梁的单元刚度矩阵进行了模态分析,验证了所建立的频域诊断方法的有效性进行,结果表明,该方法可以成功诊断出斜裂缝的位置、深度和角度,且诊断精度高、鲁棒性强;在5%测量噪声影响下,对斜裂缝位置诊断的正确率为100%,对斜裂缝深度的诊断误差在25%以内,对斜裂缝角度的诊断误差在12%以内。因此,所建立的基于遗传神经网络的频域诊断方法是诊断钢筋混凝土梁中斜裂缝的一种有效方法。
The integrity of a structure may be destructed by cracks induced by fracture of concrete, some mechanical fracture surfaces may be formed within the structure, and the stress condition of the structure may be deteriorated. This may induce serious engineering accidents as collapse of bridges and destroy of buildings constantly. Beam is the most basic member in structures. The cracks existed within reinforced concrete (RC) beam generally may threaten the safety of RC structures. Therefore, systematical studies on fracture mechanism of concrete and crack detection of RC beam have important theoretical significance and engineering value for guaranteeing the safety of large civil structures.
The researches on calculation model of equivalent fracture toughness of concrete and detection method of cracks of RC beam are performed systematically in this dissertation. The main research work and innovative achievements are summarized as follows:
(1) Based on fracture mechanics of concrete and composite material mechanics, the analytical model is built to solve the equivalent fracture toughness of concrete. The modified shear lag model is used to build the layered shear lag model of the concrete specimen with a crack. The fictitious crack model is used to modify the boundary condition of stress of the analytical model. Combined with other boundary condition of stress and displacement, displacement and stress functions of layers of the analytical model are obtained. Using energy method, the analytical expression of the equivalent fracture toughness of concrete is obtained. Analytical solutions of the equivalent fracture toughness of concrete are obtained using the analytical calculation model with the experimental data of 8 specimens. The calculated results show that mean square errors and variation coefficients of the calculated values with different volumes are lower than 0.0398 and 0.0384 respectively, and those with different heights are lower than 0.0394 and 0.0363 respectively. Being lower mean square errors and variation coefficients, the analytical solutions are more robust than the numerical solutions based on the coherent experiment. Therefore, the analytical model provides a reliable practical analytical method to solve the equivalent fracture toughness of concrete.
(2) Considering the length of the subcritical growth of the main crack and the cohesive strength of the fictitious crack of concrete, and using the analytical relationship between the J integral and the stress intensity factor, a numerical algorithm based on the element free Galerkin method with wavelet basis to solve the equivalent fracture toughness of concrete is built. The third order B spline function is taken as the wavelet function to substitute the traditional polynomial basis in the element free Galerkin method, which is used to build the element free Galerkin method with wavelet basis. The numerical model to solve the equivalent fracture toughness of concrete is deduced using J integral theory. The influence of different resolution of wavelet basis on the calculated values is investigated. The calculated results for 8 specimens with crack show that in the case of identical resolution of wavelet basis, the mean square derivations and the coefficients of variation of the calculated values of the 4 volume series specimens are smaller than 0.0244 and 0.0231 respectively, and the those of the 4 height series specimens are smaller than 0.0384 and 0.0362, so to have good consistency. By comparing with the analytical solutions, the maximum relative error of the calculated values of 8 specimens is about 5%, so the precision of the calculated results is high. And the influence of different resolution of wavelet basis on the calculated results is little. Therefore, the element free Galerkin method with wavelet basis is a reliable numerical algorithm with high precision to solve the equivalent fracture toughness of concrete.
(3) Based on fracture mechanics, the analytical relationship of the stress intensity factor and strain energy is used to obtain the analytical expression of additional strain energy induced by straight or diagonal crack in element of RC beam. Considering the influence of steel bar, the virtual work principle is used to deduce the stiffness matrix of element with a straight or diagonal crack of RC beam. Some test beams with different straight or diagonal cracks damage cases are fabricated, whose results of model test verify the applicability of the stiffness matrix of the element with a straight or diagonal crack. The tested results show that frequencies of test beams are reduced and the influence of the second order frequency is bigger than the influence of the first frequency for cracks, which is consistent with conclusions given by others. This proofs the model test is valid. The modal analysis for test beams is given based on the deduced elemental stiffness matrix with a crack. The calculated results show that the maximum value of relative error between the computational frequencies and experiment frequencies is only 2.76% and the values of MAC of correlative model shapes between experimental and calculation are bigger than 0.9. This verifies good relativity between the tested and calculated modal shapes. Therefore, the deduced elemental stiffness matrix of RC beam with a crack can be used in the calculation of infinite element and the calculation precision is higher.
(4) A two-stage detection method in time domain for straight cracks of RC beam is established using the linear elastic fracture theory and displacement responses of RC beam in time domain. Firstly, normalized method and hypothesis testing are used to build a damage index with relation to the mean curvature change ratio to locate cracks based on displacement responses in time domain. Secondly, the analytical relationships of the strain energy release rate, stress intensity factor and depth of crack are used to obtain a crack depth model in time domain to detect the depth of crack. The stiffness matrix of cracked element is used to simulate the cracked RC beams to verify the feasibility of the algorithm of straight crack detection. The calculated results show that the algorithm could locate cracks precisely with high confidence interval. 80% of locations of cracks are detected successfully when the threshold of the normalized parameter is put to 1.0. Combining the diagram of damage index, all of locations of cracks can be detected precisely. Depths of cracks can be detected in a certain accurate degree. Meanwhile the contaminated displacement responses are used to analyze the robustness of the damage index and the crack depth model, which demonstrates that the method is robust against even if 5% noise level. Therefore, the two-stage detection method in time domain is a method with high efficiency to detect the depths and locations of straight cracks in RC beam.
(5) A genetic neural network-based detection method in frequency domain for diagonal cracks within RC beams is built. The topological structure, weight and threshold of the BP neural network are optimized using the genetic algorithm. Normalized ratios of variations of the frequencies, normalized ratios of variations of the square of same order frequencies and components of mode of the cracked RC beam are used as the damage indices to train the genetic neural network. The method based on the genetic neural network is built to detect the location, depth and angle of the diagonal crack. The stiffness matrix of cracked element is used to obtain frequencies and model shapes of the cracked RC beams to verify the feasibility of the method of diagonal crack detection. The numerical simulation shows that the method detects the location, depth and angle of the diagonal crack successfully with a high detection precision and robust. In the 5% noise level, the successful rate of location detection is 100%, and the detection error of crack depth is controlled in 25%, and the detection error of crack angle is controlled in 12%. Therefore, the genetic neural network-based detection method in frequency domain is an effective method to detect the diagonal cracks within RC beams.
引文
[1] Kaplan M F. Crack propagation and the fracture of concrete. Journal of American Concrete Institute, 1961, 58(5): 591~610.
[2] Hillerborg A, Modeer M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 1976, 6(6): 773~782.
[3] Bazant Z P, Oh B H. Crack band theory for fracture of Concrete. Material and Structures, 1983, 16(93): 155~177.
[4] Jenq Y S, Shah S P. Two parameter fracture model for concrete. Journal of Engineering Mechanics, 1985, 111(10): 1227~1241.
[5] Jenq Y S, Shah S P. Determination of fracture parameters of plain concrete using three~point bend tests. Materials and Structures, 1990, 23(138): 457~460.
[6] Bazant Z P, Kazemi M T. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. International Journal of Fracture, 1990, 44: 111~131.
[7] 徐世烺,赵国藩. 混凝土断裂力学研究. 大连:大连理工大学出版社,1991.
[8] Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials, 1994, 18(2): 89~101.
[9] Xu Shilang, Reinhardt H W. Determination of double~K criterion for crack propagation in quasi~brittle fracture. International Journal of Fracture, 1999, 98(2): 111~193.
[10] Xu Shilang, Reinhardt H W. A simplified method for determining double-K fracture parameters for three-point bending tests. International Journal of Fracture, 2000, 104(2):181~209.
[11] 吴智敏, 王金来, 徐世烺, 刘毅. 基于虚拟裂缝模型的混凝土等效断裂韧度. 工程力学, 2000, 17(1): 99~104.
[12] Carpinteri A, Chiaia B, Cornetti P. A scale-invariant Cohesive crack model for quasi~brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207~217.
[13] 徐世烺, 吴智敏, 丁生根. 砼双 K 断裂参数的实用解析方法. 工程力学, 2003, 20(3): 54~61.
[14] Kuang K S, Akmaludd C. Crack detection and vertical deflection monitoring in concrete beams using plastic optical fiber sensors. Measurement Science and Technology, 2003, 14(2): 205~216.
[15] Park S, Ahmad S. Multiple crack detection of concrete structures using impedance-based structural health monitoring techniques. Experimental Mechanics, 2006, 46(5): 609-618
[16] Chen G D, Sun S S. Crack detection of a full-scale reinforced concrete girder with a distributed cable sensor. Smart Materials and Structures, 2005, 14 (3):88-97.
[17] 李忠献, 刘永光. 时域内梁式结构的两阶段裂缝诊断方法. 天津大学学报 2007, 35(4).
[18] 李忠献, 刘永光. 基于遗传神经网络与模态应变能的斜裂缝两阶段诊断方法. 工程力学, 已录用待发表.
[19] Oyadiji S O, Zhong S C. Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform. Mechanical Systems and Signal Processing, 2007, 21( 4) :1853-1884.
[20] Lee Y S, Chung M J. A study on crack detection using eigenfrequency test data. Computers and Structures, 2000, 77(3): 327-342.
[21] 莱昂哈特. 钢筋混凝土结构裂缝与变形的验算. 北京:水利电力出版社, 1983.
[22] Xu Shilang, Reinhardt H W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture. International Journal of Fracture,1998,92(3): 71~99.
[23] Xu Shilang, Reinhardt H W. Determination of double-K criterion for crack propagation in quasi~brittle fracture. Part I: Experimental investigation of crack propagation. International Journal of Fracture, 1999, 98(2): 1l1~149.
[24] Xu Shilang, Reinhardt H W. Determination of double-K criterion for crack propagation in quasi-brittle fracture. PartⅡ: An analytical evaluating and a practical measuring method for three~point bending notched beams. International Journal of Fracture, 1999, 98(2): 151~177.
[25] Reihardt H W, Xu Shilang. Crack extension resistance based on the cohesive force in concrete. Engineering Fracture Mechanics, 1999, 64(5): 563~587.
[26] Swartz S E. Validity of compliance calibration to cracked concrete beams in bending. Experimental Mechanics, 1984, 24(2): 129~134.
[27] Karihaloo B L, Nallathambi P. An improved effective crack model for the determination of fracture toughness K of concrete. Cement and Concrete Research, 1989, 19(4): 603~610.
[28] Karihaloo B L, Nallathambi P. Effective crack model for the determination of fracture toughness of concrete. Engineering Fracture Mechanics, 1990, 35(4/5): 637~645.
[29] 金南国,金贤玉, 沈毅. 早龄期混凝土断裂性能和微观结构的试验研究. 浙江大学学报:工学版,2005,39(9):1374~1377.
[30] Tang Xuesong, Zhang Jianren, Li Chuanxi. Damage analysis and numerical simulation for failure process of a reinforced concrete arch structure. Computers and Structures, 2005, 83(31): 2609~2631.
[31] Navalurkar R K, Cheng T H. True fracture energy of concrete. ACI Materials Journal, 1999, 96(2): 213~225.
[32] Beaumont P W, Aleszka J C. Cracking and toughening of concrete and polymer concrete dispersed with short steel wires. Journal of Materials Science, 1978, 13(8): 1749~1760.
[33] Hanson J H, Ingraffea A R. Using numerical simulations to compare the fracture toughness values for concrete from the size-effect, two-parameter and fictitious crack models. Engineering Fracture Mechanics, 2003, 70(7): 1015~1027
[34] Reis J M, De O R, Ferreira, A J. A NDT assessment of fracture mechanics properties of fiber reinforced polymer concrete. Polymer Testing, 2003, 22(4): 395~401.
[35] Lambert D E, Ross C A. Strain rate effects on dynamic fracture and strength. International Journal of Impact Engineering, 2000, 24(10): 985~989.
[36] Bazant Z P. Statistical prediction of fracture parameters of concrete and implications for choice of testing standard. Cement and Concrete Research, 2002, 32(4): 529~556.
[37] Tang Tianxi, Shah S P. Fracture mechanics and size effect of concrete in tension. Journal of Structural Engineering, 1992, 118(11): 3169~3185.
[38] Yang S C, Tang Tianxi. A systematic method to determine optimal specimen geometry for measuring concrete fracture properties by the peak-load method. Cement, Concrete and Aggregates, 1998, 20(2): 278~287.
[39] Vipulanandan C, Dharmarajan N. Critical crack tip opening displacement for polymer composites. Engineering Fracture Mechanics, 1989, 33(3): 409~419.
[40] 李忠献, 刘永光. 基于虚拟裂缝模型求解混凝土等效断裂韧度的实用解析方法. 工程力学,2006, 23(11): 91~98.
[41] Wu Zhimin,Yang Shutong, Hu Xiaozhi. An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams. Engineering Fracture Mechanics, 2006, 73(15): 2166~2191.
[42] Abdalla H M, Karihaloo B L. A method for constructing the bilinear tension softening diagram of concrete corresponding to its true fracture energy. Magazine of Concrete Research, 2004, 56(10): 597~604.
[43] Larsson R. Generalized fictitious crack model based on plastic localization and discontinuous approximation. International Journal for Numerical Methods in Engineering, 1995, 38(18): 3167~3188.
[44] Wu Z J, Bailey C G. Fracture resistance of a cracked concrete beam post-strengthened with FRP sheets. International Journal of Fracture, 2005, 135(1~4):35~49.
[45] Lin Gao, Qi Congshan, Zhou Hongtao. Analytical approach of the fictitious crack model for concrete. Engineering Fracture Mechanics, 1994, 47(2): 269~280.
[46] Reinhardt H W. Crack softening zone in plain concrete under static loading. Cement and Concrete Research, 1985, 15(1): 42~52.
[47] Rabczuk T, Akkermann J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5~6): 1327~1354.
[48] 周瑞忠, 姚志雄, 石成恩. 活性粉末混凝土为基底材料的断裂和疲劳试验研究. 水力发电学报, 2005, 24(6): 40~44.
[49] 王承强, 郑长良. 混凝土断裂力学虚拟裂缝模型的半解析有限元法. 应用数学和力学, 2004, 25(11): 1157~1162.
[50] 王承强, 郑长良. 裂纹扩展过程中线性内聚力模型计算的半解析有限元法.计算力学学报, 2006, 23(2): 146~151.
[51] 刘海成, 吴智敏, 宋玉普. 碾压混凝土拱坝诱导缝损伤开裂准则研究. 水力发电学报, 2004, 23(5): 22~27.
[52] 赵艳华, 徐世烺, 吴智敏. 混凝土结构裂缝扩展的双 G 准则. 土木工程学报, 2004, 37(10):13~18,51,91.
[53] Fantilli A P, Vallini P. Strains in steel bars of reinforced concrete elements subjected to repeated loads. Journal of Strain Analysis for Engineering Design, 2004, 39(5): 447~457.
[54] Olesen J F. Fictitious crack propagation in fiber-reinforced concrete beams. Journal of Engineering Mechanics, 2001, 127(3): 272~280.
[55] Gajer G, Dux P F. Simplified nonorthogonal crack model for concrete. Journal of Structural Engineering, 1991, 117(1): 149~164. 90
[56] 周元德, 张楚汉, 金峰. 混凝土断裂的三维旋转裂缝模型研究. 工程力学, 2004, 21(5): 1~4.
[57] Bazant Z P, Caner F C. Microplane model M5 with kinematic and static constraints for concrete fracture and an elasticity. I: Theory. Journal of Engineering Mechanics, 2005, 131(1): 31~40.
[58] 曲蒙, 金南国, 金贤玉. 早龄期混凝土三点弯曲试验梁断裂的数值分析. 浙江大学学报:工学版, 2006, 40(7): 1224~1229.
[59] Berton S, Bolander J E. Crack band model of fracture in irregular lattices. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7172~7181.
[60] 周伟, 常晓林, 刘杏红. 基于温度应力仿真分析的碾压混凝土重力坝诱导缝开裂研究. 岩石力学与工程学报, 2006, 25(1): 122~127.
[61] Cervenka J, Bazant, Z P, Wierer M. Equivalent localization element for crack band approach to mesh-sensitivity in microplane model. International Journal for Numerical Methods in Engineering, 2005, 62(5): 700~726.
[62] Bae Y H, Leet J H, Yoon Y S. Prediction of shear strength in high~strength concrete beams considering size effect. Magazine of Concrete Research, 2006, 58(4):193~200.
[63] Carpinteri A, Chiaia B, Cornetti, P. A scale-invariant cohesive crack model for quasi-brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207~217.
[64] 张子明, 郭兴文. 水化热引起的大体积混凝土墙应力与开裂分析. 河海大学学报:自然科学版, 2002, 30(5): 12~16.
[65] Ozbolt J, Reinhardt H W. Numerical study of mixed~mode fracture in concrete. International Journal of Fracture, 2002, 118(2):145~161.
[66] Gajer G. Crack band based model for FEM analysis of concrete structures. Journal of Structural Engineering, 1990, 116(6): 1696~1714.
[67] Khalfallah S. Cracking analysis of reinforced concrete tensioned members. Structural Concrete, 2006, 7(3): 111~116.
[68] 徐世烺,吴智敏,丁生根. 砼双 K 断裂参数的实用解析方法. 工程力学,2003, 20(3): 54~61.
[69] 吴智敏, 徐世烺、王金来. 基于虚拟裂缝模型的砼双 K 断裂参数,水利学报,1999(7): 12~16
[70] 吴智敏, 徐世烺, 卢喜经. 试件初始缝高比对砼双 K 断裂参数的影响. 水利学报, 2000, 4: 35~39.
[71] 吴智敏, 徐世烺, 王金来. 三点弯曲梁研究混凝土双 K 断裂参数及其尺寸效应. 水力发电学报, 2000, 4: 16~23.
[72] 吴智敏,徐世烺,卢喜经. 砼非标准三点弯曲梁试件的双 K 断裂参数. 中国工程科学,2001, 4: 76~81.
[73] 吴智敏,徐世烺,刘毅. 试件初始缝长对砼双 K 断裂参数的影响. 水利学报,2000, 4: 57~62.
[74] 吴智敏,徐世烺,高洪波. 楔入劈拉法研究混凝土双 K 断裂参数及其尺寸效应. 岩石力学与工程学报,2001,20(增 2): 1499-1504.
[75] 吴智敏,徐世烺,刘红艳. 骨料最大粒径对砼双 K 断裂参数的影响. 大连理工大学学报,2000,40(3): 358~361.
[76] 赵志方、徐世烺. 混凝土软化本构曲线形状对双 K 断裂参数的影响. 土木工程学报, 2001, 34(5): 29~34.
[77] 吴智敏,徐世烺,刘佳毅. 光弹贴片法研究裂缝扩展和双 K 断裂参数的尺寸效应. 水利学报, 2001, 4, 34~39.
[78] 赵志方,徐世烺,周厚贵. 混凝土软化本构关系对双 K 断裂参数的影响. 工程力学, 2002 , 19(4):149~154.
[79] 刘佳毅. 混凝土双 K 断裂参数及尺寸效应:[硕士学位论文]. 大连: 大连理工大学,2000.
[80] Xu Shilang, Reinhardt H W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture. International Journal of Fracture, 1998, 92(1): 71~99.
[81] Reinhardt H W, Xu Shilang. Crack extension resistance based on the cohesive force in concrete. Engineering Fracture Mechanics, 1999,64(5): 563~587.
[82] 李忠献, 刘永光. I-II 复合型裂缝断裂角的多尺度建模与分析. 计算力学学报,已录用待发表.
[83] 曾庆敦, 刘永光, 温靖赟. 含裂缝钢纤维混凝土破坏分析的一种新方法. 工程力学, 2005, 22(2): 120~123.
[84] 江见鲸, 陆新征, 叶列平. 混凝土结构有限元分析. 北京:清华大学出版社, 2005.
[85] 宋玉普. 多种混凝土破坏准则和本构关系. 北京:中国水利水电出版社,2002.
[86] 谢和平. 岩石混凝土损伤力学. 北京:中国矿业大学出版社,1998.
[87] 陆新征, 江见鲸. 利用无网格方法分析钢筋混凝土梁开裂问题. 工程力学, 2004, 21(2): 24~28.
[88] 陆新征, 江见鲸. 利用无网格方法分析斜拉破坏钢筋混凝土梁. 计算力学学报, 2004, 21(6): 701~705.
[89] 唐春安, 朱万成. 混凝土损伤与断裂-数值试验. 北京:科学出版社,2003.
[90] Chondros T G, Dimarogonas A D. Identification of cracks in welded joints of complex structures. Journal of Sound and Vibration, 1980, 69(4): 531~538.
[91] Narkis Y. Identification of crack location in vibrating simple supported beams. Journal of Sound and Vibration, 1994, 172: 549~558.
[92] Masoud A, Jarrad M A. Effect of crack depth on natural frequency of prestressed fixed–fixed beam. Journal of Sound and Vibration, 1998,214(2): 201~212.
[93] Bamnios Y, Douka E, Trochidis A. Crack identification in beam structures using mechanical impedance. Journal of Sound and Vibration, 2002, 256 (2) : 287~297.
[94] Zhao J, Dewolf J T. Sensitivity study for vibrational parameters used in damage detection. Journal of Structure Engineering, 1999,125(4): 410~416.
[95] Shifrin E I, Ruotolo R. Natural frequencies of a beam with an arbitrary number of cracks. Journal of Sound and Vibration, 1999, 222 (3): 409~423.
[96] Ostachowicz W, Krawczak M. On modeling of structural stiffness loss due to damage. Key Engineering Materials, 2001, 204: 185~200.
[97] Adams, R D, Cawley P, Pye C J. A Vibration Technique for Nondestructively Assessing the Integrity of Structures, Journal of Mechanical Engineering. Science, 1978, 20: 93~100.
[98] Ruotolo R, Surace C. Damage assessment of multiple cracked beams: numerical results and experimental validation. Journal of Sound and Vibration, 1997, 206 (4): 567–588.
[99] Nandwana B P, Maiti S K. Modeling of vibration of beam in presence of inclined edge or internal crack for its possible detection based on frequency measurements. Engineering Fracture Mechanics, 1997, 58(3): 193~205.
[100] Patil D P. Modeling to facilitate detection of single and multiple cracks in beams of various geometries based on transverse vibration frequencies. Engineering Fracture Mechanics, 1999, 60(2): 231~240.
[101] Gounaris G D, Dimarogonas A D. A finite element of a cracked prismatic beam in Structural analysis. Computers and Structures, 1988, 28(3): 309~313.
[102] Dimarogonas A D, Paipetis S A. Analytical Methods in Rotor Dynamics. Computers and Structures, 1996, 35(3): 217~226.
[103] Papadopoulos C A, Dimarogonas A D. Coupled longitudinal and bending vibration of rotating shaft with an open crack. Journal of Sound and Vibration, 1987,117 (1) : 81~93.
[104] Sukumar N, Moes N B, Belytschko M T. Extended finite element method for three-dimensional crack modeling. International Journal for Numerical Methods in Engineering, 2000, 48 (11): 1549–1570.
[105] Cuitino L A, Saaverda P N. Crack detection and vibration behaviour of cracked beams. Computers and Structures, 2001, 79 (12): 1451–1459.
[106] Saavedra P N. A frequency divider with crack of variable division ratio. Engineering Fracture Mechanics, 2001, 62(3): 13~27.
[107] Krawczuk M. A rectangular plate finite element with an open crack. Computers and Structures , 1993, 46(3), 487~493.
[108] Krawczuk M, Ostachowicz W. A finite plate element for dynamic analysis of a crack plate. Computer Methods in Applied Mechanics and Engineering, 1994, 115(1): 67~78.
[109] Dharmarajua N, Tiwaria R, Talukdarb S. Development of a novel hybrid reduction scheme for identification of an open crack model in a beam. Mechanical Systems and Signal Processing,2005, 19 (3): 633~657.
[110] Wu D, Law S S. Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection. International Journal of Solids and Structures, 2004, 41 (16): 4321~4336.
[111] Kim J T, Stubbs N. Crack detection in beam-type structures using frequency data. Journal of Sound and Vibration, 2003, 259(1): 145~160.
[112] Gudmunson P. Eigenfrequency changes of structures due to cracks, notches or other geometrical changes. Journal of Mechanics and Physics of Solids, 1982, 30(5): 339~353.
[113] Matveev V V, Bovsunovsky A P. Vibration-based diagnostics of fatigue damage of beam-like structures. Journal of Sound Vibration, 2002, 249(1): 23–40.
[114] Chondros T G, Dimarogonas A D, Yao J. A continuous cracked beam vibration theory. Journal of Sound Vibration, 1998, 215(1): 17~34.
[115] Rizos P F, Aspragathos N, Dimarogona s A D. Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 1990, 138(3), 381~388.
[116] Hassiotis S. Identification of damage using natural frequencies and Markov parameters. Computers and Structures, 2000, 74(3): 65~73.
[117] 王术新, 姜哲. 裂缝悬臂梁的振动特性及其裂缝参数识别. 振动与冲击, 2003, 22(3): 83~85.
[118] Ertugrul C, Sadettin O, Murat L. An analysis of cracked beam structure using impact echo method. NDT&E International, 2005, 38 (5): 368~373.
[119] Kim M B. Damage identification of beam structure based on vibration responses. Journal of Vibration and Shock, 2003, 25(1): 86~103.
[120] Kim T, Stubbs N. Improved damage identification method based on modal information, Journal of Sound and Vibration, 2002, 252 (2): 223~238.
[121] Kim T J, Ryu Y R, Cho H M. Damage identification in beam-type structures: frequency-based method vs mode-shape based method. Engineering Structures, 2003, 25 (1): 57~67.
[122] Patil D P, Maiti S K. Experimental verification of a method of detection of multiple cracks in beams based on frequency measurements. Journal of Sound and Vibration, 2005, 281(2): 439~451.
[123] Cornwell P, Doebling S W, Farrar C R. Application of the strain energy damage detection method to plate-like structures. Journal of Sound and Vibration, 1999, 224 (2): 567~588.
[124] Hadjileontiadis L J, Douka E, Trochidisc A. Fractal dimension analysis for crack identification in beam structures. Mechanical Systems and Signal Processing, 2005,19 (3): 659~674.
[125] 任宜春, 易伟建. 基于小波分析的梁裂缝识别研究. 计算力学学报, 2005, 22(4): 399~404.
[126] Quek S, Wang Q. Sensitivity analysis of crack detection in beams by the wavelet technique. Journal of Mechanical Sciences, 2001, 43(12): 2899~2910.
[127] Wang Q, Deng X. Damage detection with spatial wavelets. International Journal of Solids and Structures, 1999, 36(23): 3443~3468.
[128] Hong J C, Kim Y Y, Lee H C. Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam. International Journal of Solids and Structures, 2002, 39(7): 1803~1816.
[129] Quek S T, Wang Q. Practical issues in the detection of damage in beams using wavelets. Smart Materials and Structures, 2001, 10 (5) : 1009~1017.
[130] Tian Jiayong, Li Zheng, Su Xianyue. Crack detection in beams by wavelet analysis of transient flexural waves. Journal of Sound and Vibration, 2003, 261(4): 715~727.
[131] Quek S, Wang Q, Zhang L. Sensitivity analysis of crack detection in beams by wavelet techniques. International Journal of Mechanical Sciences, 2001, 43 (12): 2899~2910.
[132] Liew K.M, Wang Q. Application of wavelet theory for crack identification in structures. Journal of Engineering Mechanics, 1998, 124(7): 52~57.
[133] Lam H F, Leea Y Y, Suna H Y. Application of the spatial wavelet transform and Bayesian approach to the crack detection of a partially obstructed beam. Thin-Walled Structures, 2005, 43(1): 1~21.
[134] Sung D U, Kim C G, Hong C S. Monitoring of impact damages in composite laminates using wavelet transform. Composites Part B: engineering, 2002, 33 (1): 35~43.
[135] Li Bing, Chen Xuefeng, He Zhengjia. Detection of crack location and size in structures using wavelet finite element methods. Journal of Sound and Vibration, 2005, 285(4): 767~782.
[136] Carpinteri A, Cornetti P, Barpi F. Cohesive crack model description of ductile to brittle size-scale transition:dimensional analysis vs. renormalization group theory. Engineering Fracture Mechanics, 2003, 70(14): 1809-1839.
[137] 王铎. 断裂力学. 哈尔滨:哈尔滨工业大学出版社,1989.
[138] 曾庆敦. 复合材料的细观破坏机制与强度. 北京:科学出版社,2002.
[139] 尹双增. 断裂、损伤理论及应用. 北京:清华大学出版社,1992.
[140] Nayroles B, Touzot G, Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Computer Mechanics, 1992, 10(3): 307~318.
[141] Belytschko T, Lu Y Y, Gu L. Element free Galerkin methods. International Journal of Numerical Methods in Engineering. 1994, 37(5): 229~256.
[142] Lu Y Y, Beltschko T. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and engineering, 1994, 113(3): 397~414.
[143] Krongauz Y, Belytschko T. EFG approximation with discontinuous derivatives. International Journal for Numerical methods in Engineering. 1998, 41(7): 1215~1233.
[144] Lu Y Y, Belytschko T, Tabbara M. Element-free Galerkin methods for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and engineering, 1995, 126(7) :131~153.
[145] Belytschko T, Tabbara M. Dynamic fracture using element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1996, 39(5): 923~938.
[146] Belytschko T, Gu L, Lu Y Y. Fracture and crack growth by element free Galerkin methods. Modelling and Simulation in Materials Science and Engineering, 1994, 2(3): 519~534.
[147] Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkinmethods. Engineering Fracture Mechanics, 1995, 51(2): 295~315.
[148] Fleming M, Belytschko T. Enriched Element-Free Galerkin Methods for crack tip fields. International Journal for Numerical Methods in Engineering, 1997, 40(8): 1483~1504.
[149] Krysl P, Belytschko T. Analysis of thin shells by element-free Galerkin method. International Journal of Solids and Structures, 1995, 33(20): 3057~3080.
[150] Hegen D. Element-free Galerkin methods in combination with finite element approaches. Computer Methods in Applied Mechanics and Engineering, 1996, 135(1): 143~166.
[151] 冯象初,甘小冰,宋国乡. 数值泛函与小波理论. 西安:西安电子科技大学出版社,2003.
[152] Han J G, Ren W X, Sun Z S. Wavelet packet based damage identification of beam structures. International Journal of Solids and Structures, 2005, 42(26): 6610-6627.
[153] Ovanesova A V, Suarez L E. Applications of wavelet transforms to damage detection in frame structures. Engineering Structures, 2004, 26(1): 39~49.
[154] Yam L H,Yan Y J,Jiang J S. Vibration-based damage detection for composite structures using wavelet transform and neural network identification. Composite Structures, 2003, 60(4): 403~412.
[155] Newland D E. Wavelet analysis of vibration, Part 2: Wavelet maps. Journal of Vibration and Acoustics, 1994, 116(4): 417~425.
[156] Han Jiangang, Ren Weixin, Huang Yihe. A spline wavelet finite-element method in structural mechanics. International Journal for Numerical Methods in Engineering, 2006, 66(1): 166~190.
[157] Rathsfeld A. Wavelet algorithm for the boundary element solution of a geodetic boundary value problem. Computer Methods in Applied Mechanics and Engineering, 1998, 157(3):267~287.
[158] 张雄,刘岩. 无网格法. 北京:清华大学出版社,2004.
[159] Zhang Hong, Zhang Xuanbing, Ge Xiurun. Application of wavelet theory in research on weight function of meshless method. Applied Mathematics and Mechanics (English Edition) , 2005, 26(5): 662~666.
[160] 吴琛,周瑞忠. 小波基无单元法及其与有限元法的比较. 工程力学, 2006, 23(4): 28~32.
[161] 周瑞忠,周小平,缪圆冰. 小波基无单元法及其工程应用. 工程力学, 2003, 20(6): 70~74.
[162] 缪圆冰. 小波无单元方法及其应用:[硕士学位论文]. 福州:福州大学, 2001.
[163] 周先雁,刘希,沈蒲生. 用含裂纹的梁单元识别混凝土框架结构损伤. 振动工程学报,1999,12(1),116～119.
[164] 钱管良,顾松年. 在线振动监测与故障诊断的一种新途径. 固体力学学报, 1990,11(3):217～228.
[165] 周先雁,沈蒲生,刘希. 用断裂动力学对混凝土平面杆系进行破损评估. 湖南大学学报, 1998, 25(1): 66~70.
[166] Nahvi H, Jabbari M. Crack detection in beams using experimental modal data and finite element model. International Journal of Mechanical Sciences, 2005, 47(10): 1477~1497.
[167] 铁摩辛柯. 材料力学. 北京:科学出版社, 1978.
[168] 中国航空研究院. 应力强度因子手册. 北京:科学出版社, 1981.
[169] 伍必庆, 张青喜. 道路建筑材料. 北京: 清华大学出版社, 2006.
[170] Allemang R J. The modal assurance criterion - twenty years of use and abuse. Journal of Sound and Vibration, 2003, 37(8): 14~23.
[171] 任伟新,林友勤,彭雪林. 大跨度斜拉桥环境振动试验与分析. 实验力学, 2006, 21(4): 418～426.
[172] Choi S, Stubbs N. Damage identification in structures using the time-domain response. Journal of sound and vibration, 2004, 275(3): 577~590.
[173] 苏金明, 阮沈勇, 王永利. MATLAB 工程数学. 北京: 电子工业出版社, 2005.
[174] Leonard J, Kramer M A. Improvement of the back propagation algorithm for training neural networks. Computers and Chemical Engineering, 1990, 14(3): 337~341.
[175] 姜绍飞. 基于神经网络的结构优化与损伤检测. 北京: 科学出版社, 2002.
[176] Masri S F, Nakamura M, Chassiakos A G. Neural network approach to detection of changes in structural parmaters. Journal of Engineering Mechanics, 1996, 122(4): 350~360.
[177] Waszczyszyn Z, Ziemianski L. Neural networks in mechanics of structures and materials-new results and prospects of applications. Computers and Structures, 2001, 79(22): 2261~2276.
[178] Kao C Y, Hung S L. Detection of structural damage via free vibration responses generated by approximating artificial neural networks. Computers and Structures, 2003, 81(28): 2631-2644.
[179] Yam H L, Yan Y J, Jiang J S. Vibration-based damage detection for composite structures using wavelet transform and neural network identification. Composite Structures, 2003, 60(4): 403~412.
[180] 朱宏平, 千力. 利用振动模态测量值和神经网络方法的结构损伤识别研究. 计算力学学报, 2005, 22(2): 193~196.
[181] Fang X, Luo H, Tang J. Structural damage detection using neural network with learning rate improvement. Computers and Structures, 2005, 83(25): 2150~2161.
[182] 韩大建, 杨炳尧, 颜全胜. 用人工神经网络方法评估桥梁缺损状况. 华南理工大学学报, 2004, 32(9): 72~76.
[183] Sahin M, Shenoi R A. Quantification and localization of damage in beam-like structures by using artificial neural networks with experimental validation. Engineering Structures, 2003, 25(14): 1785~1802.
[184] 罗跃纲, 张松鹤, 闻邦椿. 基于频率下降率的结构损伤自适应神经网络识别. 中国安全科学学报, 2005, 15(5): 13~16.
[185] Lee J J, Lee J W, Yi J H, et al. Neural networks-based damage detection for bridges considering errors in baseline finite element models [J]. Journal of Sound and Vibration, 2005, 280(3-5): 555~578.
[186] 李忠献, 杨晓明, 丁 阳. 应用人工神经网络技术的大型斜拉桥子结构损伤识别研究. 地震工程与工程振动, 2003, 23(3): 92~99.
[187] Zweiri Y H, Seneviratne L D, Althoefer K. Stability analysis of a three-term back propagation algorithm. Neural Networks, 2005, 18(10): 1341~1347.
[188] 孙宗光, 高赞明, 倪一清. 基于神经网络的损伤构件及损伤程度识别. 工程力学, 2006, 23(2): 18～22.
[189] 何 翔, 李守巨, 刘迎曦. 基于遗传神经网络的坝基岩体渗透系数识别. 岩石力学与工程学报, 2004, 23(5): 751～757.
[190] 马祥森, 史治宇. 基于 GA-BP 神经网络的结构损伤位置识别. 振动工程学报, 2004, 17(4): 453～456.
[191] 雷英杰, 张善文, 李续武等. MATLAB 遗传算法工具箱及应用. 西安: 西安电子科技大学出版社, 2005.
[2] Hillerborg A, Modeer M, Petersson P E. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research, 1976, 6(6): 773~782.
[3] Bazant Z P, Oh B H. Crack band theory for fracture of Concrete. Material and Structures, 1983, 16(93): 155~177.
[4] Jenq Y S, Shah S P. Two parameter fracture model for concrete. Journal of Engineering Mechanics, 1985, 111(10): 1227~1241.
[5] Jenq Y S, Shah S P. Determination of fracture parameters of plain concrete using three~point bend tests. Materials and Structures, 1990, 23(138): 457~460.
[6] Bazant Z P, Kazemi M T. Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. International Journal of Fracture, 1990, 44: 111~131.
[7] 徐世烺,赵国藩. 混凝土断裂力学研究. 大连:大连理工大学出版社,1991.
[8] Carpinteri A. Fractal nature of material microstructure and size effects on apparent mechanical properties. Mechanics of Materials, 1994, 18(2): 89~101.
[9] Xu Shilang, Reinhardt H W. Determination of double~K criterion for crack propagation in quasi~brittle fracture. International Journal of Fracture, 1999, 98(2): 111~193.
[10] Xu Shilang, Reinhardt H W. A simplified method for determining double-K fracture parameters for three-point bending tests. International Journal of Fracture, 2000, 104(2):181~209.
[11] 吴智敏, 王金来, 徐世烺, 刘毅. 基于虚拟裂缝模型的混凝土等效断裂韧度. 工程力学, 2000, 17(1): 99~104.
[12] Carpinteri A, Chiaia B, Cornetti P. A scale-invariant Cohesive crack model for quasi~brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207~217.
[13] 徐世烺, 吴智敏, 丁生根. 砼双 K 断裂参数的实用解析方法. 工程力学, 2003, 20(3): 54~61.
[14] Kuang K S, Akmaludd C. Crack detection and vertical deflection monitoring in concrete beams using plastic optical fiber sensors. Measurement Science and Technology, 2003, 14(2): 205~216.
[15] Park S, Ahmad S. Multiple crack detection of concrete structures using impedance-based structural health monitoring techniques. Experimental Mechanics, 2006, 46(5): 609-618
[16] Chen G D, Sun S S. Crack detection of a full-scale reinforced concrete girder with a distributed cable sensor. Smart Materials and Structures, 2005, 14 (3):88-97.
[17] 李忠献, 刘永光. 时域内梁式结构的两阶段裂缝诊断方法. 天津大学学报 2007, 35(4).
[18] 李忠献, 刘永光. 基于遗传神经网络与模态应变能的斜裂缝两阶段诊断方法. 工程力学, 已录用待发表.
[19] Oyadiji S O, Zhong S C. Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform. Mechanical Systems and Signal Processing, 2007, 21( 4) :1853-1884.
[20] Lee Y S, Chung M J. A study on crack detection using eigenfrequency test data. Computers and Structures, 2000, 77(3): 327-342.
[21] 莱昂哈特. 钢筋混凝土结构裂缝与变形的验算. 北京:水利电力出版社, 1983.
[22] Xu Shilang, Reinhardt H W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture. International Journal of Fracture,1998,92(3): 71~99.
[23] Xu Shilang, Reinhardt H W. Determination of double-K criterion for crack propagation in quasi~brittle fracture. Part I: Experimental investigation of crack propagation. International Journal of Fracture, 1999, 98(2): 1l1~149.
[24] Xu Shilang, Reinhardt H W. Determination of double-K criterion for crack propagation in quasi-brittle fracture. PartⅡ: An analytical evaluating and a practical measuring method for three~point bending notched beams. International Journal of Fracture, 1999, 98(2): 151~177.
[25] Reihardt H W, Xu Shilang. Crack extension resistance based on the cohesive force in concrete. Engineering Fracture Mechanics, 1999, 64(5): 563~587.
[26] Swartz S E. Validity of compliance calibration to cracked concrete beams in bending. Experimental Mechanics, 1984, 24(2): 129~134.
[27] Karihaloo B L, Nallathambi P. An improved effective crack model for the determination of fracture toughness K of concrete. Cement and Concrete Research, 1989, 19(4): 603~610.
[28] Karihaloo B L, Nallathambi P. Effective crack model for the determination of fracture toughness of concrete. Engineering Fracture Mechanics, 1990, 35(4/5): 637~645.
[29] 金南国,金贤玉, 沈毅. 早龄期混凝土断裂性能和微观结构的试验研究. 浙江大学学报:工学版,2005,39(9):1374~1377.
[30] Tang Xuesong, Zhang Jianren, Li Chuanxi. Damage analysis and numerical simulation for failure process of a reinforced concrete arch structure. Computers and Structures, 2005, 83(31): 2609~2631.
[31] Navalurkar R K, Cheng T H. True fracture energy of concrete. ACI Materials Journal, 1999, 96(2): 213~225.
[32] Beaumont P W, Aleszka J C. Cracking and toughening of concrete and polymer concrete dispersed with short steel wires. Journal of Materials Science, 1978, 13(8): 1749~1760.
[33] Hanson J H, Ingraffea A R. Using numerical simulations to compare the fracture toughness values for concrete from the size-effect, two-parameter and fictitious crack models. Engineering Fracture Mechanics, 2003, 70(7): 1015~1027
[34] Reis J M, De O R, Ferreira, A J. A NDT assessment of fracture mechanics properties of fiber reinforced polymer concrete. Polymer Testing, 2003, 22(4): 395~401.
[35] Lambert D E, Ross C A. Strain rate effects on dynamic fracture and strength. International Journal of Impact Engineering, 2000, 24(10): 985~989.
[36] Bazant Z P. Statistical prediction of fracture parameters of concrete and implications for choice of testing standard. Cement and Concrete Research, 2002, 32(4): 529~556.
[37] Tang Tianxi, Shah S P. Fracture mechanics and size effect of concrete in tension. Journal of Structural Engineering, 1992, 118(11): 3169~3185.
[38] Yang S C, Tang Tianxi. A systematic method to determine optimal specimen geometry for measuring concrete fracture properties by the peak-load method. Cement, Concrete and Aggregates, 1998, 20(2): 278~287.
[39] Vipulanandan C, Dharmarajan N. Critical crack tip opening displacement for polymer composites. Engineering Fracture Mechanics, 1989, 33(3): 409~419.
[40] 李忠献, 刘永光. 基于虚拟裂缝模型求解混凝土等效断裂韧度的实用解析方法. 工程力学,2006, 23(11): 91~98.
[41] Wu Zhimin,Yang Shutong, Hu Xiaozhi. An analytical model to predict the effective fracture toughness of concrete for three-point bending notched beams. Engineering Fracture Mechanics, 2006, 73(15): 2166~2191.
[42] Abdalla H M, Karihaloo B L. A method for constructing the bilinear tension softening diagram of concrete corresponding to its true fracture energy. Magazine of Concrete Research, 2004, 56(10): 597~604.
[43] Larsson R. Generalized fictitious crack model based on plastic localization and discontinuous approximation. International Journal for Numerical Methods in Engineering, 1995, 38(18): 3167~3188.
[44] Wu Z J, Bailey C G. Fracture resistance of a cracked concrete beam post-strengthened with FRP sheets. International Journal of Fracture, 2005, 135(1~4):35~49.
[45] Lin Gao, Qi Congshan, Zhou Hongtao. Analytical approach of the fictitious crack model for concrete. Engineering Fracture Mechanics, 1994, 47(2): 269~280.
[46] Reinhardt H W. Crack softening zone in plain concrete under static loading. Cement and Concrete Research, 1985, 15(1): 42~52.
[47] Rabczuk T, Akkermann J. A numerical model for reinforced concrete structures. International Journal of Solids and Structures, 2005, 42(5~6): 1327~1354.
[48] 周瑞忠, 姚志雄, 石成恩. 活性粉末混凝土为基底材料的断裂和疲劳试验研究. 水力发电学报, 2005, 24(6): 40~44.
[49] 王承强, 郑长良. 混凝土断裂力学虚拟裂缝模型的半解析有限元法. 应用数学和力学, 2004, 25(11): 1157~1162.
[50] 王承强, 郑长良. 裂纹扩展过程中线性内聚力模型计算的半解析有限元法.计算力学学报, 2006, 23(2): 146~151.
[51] 刘海成, 吴智敏, 宋玉普. 碾压混凝土拱坝诱导缝损伤开裂准则研究. 水力发电学报, 2004, 23(5): 22~27.
[52] 赵艳华, 徐世烺, 吴智敏. 混凝土结构裂缝扩展的双 G 准则. 土木工程学报, 2004, 37(10):13~18,51,91.
[53] Fantilli A P, Vallini P. Strains in steel bars of reinforced concrete elements subjected to repeated loads. Journal of Strain Analysis for Engineering Design, 2004, 39(5): 447~457.
[54] Olesen J F. Fictitious crack propagation in fiber-reinforced concrete beams. Journal of Engineering Mechanics, 2001, 127(3): 272~280.
[55] Gajer G, Dux P F. Simplified nonorthogonal crack model for concrete. Journal of Structural Engineering, 1991, 117(1): 149~164. 90
[56] 周元德, 张楚汉, 金峰. 混凝土断裂的三维旋转裂缝模型研究. 工程力学, 2004, 21(5): 1~4.
[57] Bazant Z P, Caner F C. Microplane model M5 with kinematic and static constraints for concrete fracture and an elasticity. I: Theory. Journal of Engineering Mechanics, 2005, 131(1): 31~40.
[58] 曲蒙, 金南国, 金贤玉. 早龄期混凝土三点弯曲试验梁断裂的数值分析. 浙江大学学报:工学版, 2006, 40(7): 1224~1229.
[59] Berton S, Bolander J E. Crack band model of fracture in irregular lattices. Computer Methods in Applied Mechanics and Engineering, 2006, 195(52): 7172~7181.
[60] 周伟, 常晓林, 刘杏红. 基于温度应力仿真分析的碾压混凝土重力坝诱导缝开裂研究. 岩石力学与工程学报, 2006, 25(1): 122~127.
[61] Cervenka J, Bazant, Z P, Wierer M. Equivalent localization element for crack band approach to mesh-sensitivity in microplane model. International Journal for Numerical Methods in Engineering, 2005, 62(5): 700~726.
[62] Bae Y H, Leet J H, Yoon Y S. Prediction of shear strength in high~strength concrete beams considering size effect. Magazine of Concrete Research, 2006, 58(4):193~200.
[63] Carpinteri A, Chiaia B, Cornetti, P. A scale-invariant cohesive crack model for quasi-brittle materials. Engineering Fracture Mechanics, 2002, 69(2): 207~217.
[64] 张子明, 郭兴文. 水化热引起的大体积混凝土墙应力与开裂分析. 河海大学学报:自然科学版, 2002, 30(5): 12~16.
[65] Ozbolt J, Reinhardt H W. Numerical study of mixed~mode fracture in concrete. International Journal of Fracture, 2002, 118(2):145~161.
[66] Gajer G. Crack band based model for FEM analysis of concrete structures. Journal of Structural Engineering, 1990, 116(6): 1696~1714.
[67] Khalfallah S. Cracking analysis of reinforced concrete tensioned members. Structural Concrete, 2006, 7(3): 111~116.
[68] 徐世烺,吴智敏,丁生根. 砼双 K 断裂参数的实用解析方法. 工程力学,2003, 20(3): 54~61.
[69] 吴智敏, 徐世烺、王金来. 基于虚拟裂缝模型的砼双 K 断裂参数,水利学报,1999(7): 12~16
[70] 吴智敏, 徐世烺, 卢喜经. 试件初始缝高比对砼双 K 断裂参数的影响. 水利学报, 2000, 4: 35~39.
[71] 吴智敏, 徐世烺, 王金来. 三点弯曲梁研究混凝土双 K 断裂参数及其尺寸效应. 水力发电学报, 2000, 4: 16~23.
[72] 吴智敏,徐世烺,卢喜经. 砼非标准三点弯曲梁试件的双 K 断裂参数. 中国工程科学,2001, 4: 76~81.
[73] 吴智敏,徐世烺,刘毅. 试件初始缝长对砼双 K 断裂参数的影响. 水利学报,2000, 4: 57~62.
[74] 吴智敏,徐世烺,高洪波. 楔入劈拉法研究混凝土双 K 断裂参数及其尺寸效应. 岩石力学与工程学报,2001,20(增 2): 1499-1504.
[75] 吴智敏,徐世烺,刘红艳. 骨料最大粒径对砼双 K 断裂参数的影响. 大连理工大学学报,2000,40(3): 358~361.
[76] 赵志方、徐世烺. 混凝土软化本构曲线形状对双 K 断裂参数的影响. 土木工程学报, 2001, 34(5): 29~34.
[77] 吴智敏,徐世烺,刘佳毅. 光弹贴片法研究裂缝扩展和双 K 断裂参数的尺寸效应. 水利学报, 2001, 4, 34~39.
[78] 赵志方,徐世烺,周厚贵. 混凝土软化本构关系对双 K 断裂参数的影响. 工程力学, 2002 , 19(4):149~154.
[79] 刘佳毅. 混凝土双 K 断裂参数及尺寸效应:[硕士学位论文]. 大连: 大连理工大学,2000.
[80] Xu Shilang, Reinhardt H W. Crack extension resistance and fracture properties of quasi-brittle softening materials like concrete based on the complete process of fracture. International Journal of Fracture, 1998, 92(1): 71~99.
[81] Reinhardt H W, Xu Shilang. Crack extension resistance based on the cohesive force in concrete. Engineering Fracture Mechanics, 1999,64(5): 563~587.
[82] 李忠献, 刘永光. I-II 复合型裂缝断裂角的多尺度建模与分析. 计算力学学报,已录用待发表.
[83] 曾庆敦, 刘永光, 温靖赟. 含裂缝钢纤维混凝土破坏分析的一种新方法. 工程力学, 2005, 22(2): 120~123.
[84] 江见鲸, 陆新征, 叶列平. 混凝土结构有限元分析. 北京:清华大学出版社, 2005.
[85] 宋玉普. 多种混凝土破坏准则和本构关系. 北京:中国水利水电出版社,2002.
[86] 谢和平. 岩石混凝土损伤力学. 北京:中国矿业大学出版社,1998.
[87] 陆新征, 江见鲸. 利用无网格方法分析钢筋混凝土梁开裂问题. 工程力学, 2004, 21(2): 24~28.
[88] 陆新征, 江见鲸. 利用无网格方法分析斜拉破坏钢筋混凝土梁. 计算力学学报, 2004, 21(6): 701~705.
[89] 唐春安, 朱万成. 混凝土损伤与断裂-数值试验. 北京:科学出版社,2003.
[90] Chondros T G, Dimarogonas A D. Identification of cracks in welded joints of complex structures. Journal of Sound and Vibration, 1980, 69(4): 531~538.
[91] Narkis Y. Identification of crack location in vibrating simple supported beams. Journal of Sound and Vibration, 1994, 172: 549~558.
[92] Masoud A, Jarrad M A. Effect of crack depth on natural frequency of prestressed fixed–fixed beam. Journal of Sound and Vibration, 1998,214(2): 201~212.
[93] Bamnios Y, Douka E, Trochidis A. Crack identification in beam structures using mechanical impedance. Journal of Sound and Vibration, 2002, 256 (2) : 287~297.
[94] Zhao J, Dewolf J T. Sensitivity study for vibrational parameters used in damage detection. Journal of Structure Engineering, 1999,125(4): 410~416.
[95] Shifrin E I, Ruotolo R. Natural frequencies of a beam with an arbitrary number of cracks. Journal of Sound and Vibration, 1999, 222 (3): 409~423.
[96] Ostachowicz W, Krawczak M. On modeling of structural stiffness loss due to damage. Key Engineering Materials, 2001, 204: 185~200.
[97] Adams, R D, Cawley P, Pye C J. A Vibration Technique for Nondestructively Assessing the Integrity of Structures, Journal of Mechanical Engineering. Science, 1978, 20: 93~100.
[98] Ruotolo R, Surace C. Damage assessment of multiple cracked beams: numerical results and experimental validation. Journal of Sound and Vibration, 1997, 206 (4): 567–588.
[99] Nandwana B P, Maiti S K. Modeling of vibration of beam in presence of inclined edge or internal crack for its possible detection based on frequency measurements. Engineering Fracture Mechanics, 1997, 58(3): 193~205.
[100] Patil D P. Modeling to facilitate detection of single and multiple cracks in beams of various geometries based on transverse vibration frequencies. Engineering Fracture Mechanics, 1999, 60(2): 231~240.
[101] Gounaris G D, Dimarogonas A D. A finite element of a cracked prismatic beam in Structural analysis. Computers and Structures, 1988, 28(3): 309~313.
[102] Dimarogonas A D, Paipetis S A. Analytical Methods in Rotor Dynamics. Computers and Structures, 1996, 35(3): 217~226.
[103] Papadopoulos C A, Dimarogonas A D. Coupled longitudinal and bending vibration of rotating shaft with an open crack. Journal of Sound and Vibration, 1987,117 (1) : 81~93.
[104] Sukumar N, Moes N B, Belytschko M T. Extended finite element method for three-dimensional crack modeling. International Journal for Numerical Methods in Engineering, 2000, 48 (11): 1549–1570.
[105] Cuitino L A, Saaverda P N. Crack detection and vibration behaviour of cracked beams. Computers and Structures, 2001, 79 (12): 1451–1459.
[106] Saavedra P N. A frequency divider with crack of variable division ratio. Engineering Fracture Mechanics, 2001, 62(3): 13~27.
[107] Krawczuk M. A rectangular plate finite element with an open crack. Computers and Structures , 1993, 46(3), 487~493.
[108] Krawczuk M, Ostachowicz W. A finite plate element for dynamic analysis of a crack plate. Computer Methods in Applied Mechanics and Engineering, 1994, 115(1): 67~78.
[109] Dharmarajua N, Tiwaria R, Talukdarb S. Development of a novel hybrid reduction scheme for identification of an open crack model in a beam. Mechanical Systems and Signal Processing,2005, 19 (3): 633~657.
[110] Wu D, Law S S. Anisotropic damage model for an inclined crack in thick plate and sensitivity study for its detection. International Journal of Solids and Structures, 2004, 41 (16): 4321~4336.
[111] Kim J T, Stubbs N. Crack detection in beam-type structures using frequency data. Journal of Sound and Vibration, 2003, 259(1): 145~160.
[112] Gudmunson P. Eigenfrequency changes of structures due to cracks, notches or other geometrical changes. Journal of Mechanics and Physics of Solids, 1982, 30(5): 339~353.
[113] Matveev V V, Bovsunovsky A P. Vibration-based diagnostics of fatigue damage of beam-like structures. Journal of Sound Vibration, 2002, 249(1): 23–40.
[114] Chondros T G, Dimarogonas A D, Yao J. A continuous cracked beam vibration theory. Journal of Sound Vibration, 1998, 215(1): 17~34.
[115] Rizos P F, Aspragathos N, Dimarogona s A D. Identification of crack location and magnitude in a cantilever beam from the vibration modes. Journal of Sound and Vibration, 1990, 138(3), 381~388.
[116] Hassiotis S. Identification of damage using natural frequencies and Markov parameters. Computers and Structures, 2000, 74(3): 65~73.
[117] 王术新, 姜哲. 裂缝悬臂梁的振动特性及其裂缝参数识别. 振动与冲击, 2003, 22(3): 83~85.
[118] Ertugrul C, Sadettin O, Murat L. An analysis of cracked beam structure using impact echo method. NDT&E International, 2005, 38 (5): 368~373.
[119] Kim M B. Damage identification of beam structure based on vibration responses. Journal of Vibration and Shock, 2003, 25(1): 86~103.
[120] Kim T, Stubbs N. Improved damage identification method based on modal information, Journal of Sound and Vibration, 2002, 252 (2): 223~238.
[121] Kim T J, Ryu Y R, Cho H M. Damage identification in beam-type structures: frequency-based method vs mode-shape based method. Engineering Structures, 2003, 25 (1): 57~67.
[122] Patil D P, Maiti S K. Experimental verification of a method of detection of multiple cracks in beams based on frequency measurements. Journal of Sound and Vibration, 2005, 281(2): 439~451.
[123] Cornwell P, Doebling S W, Farrar C R. Application of the strain energy damage detection method to plate-like structures. Journal of Sound and Vibration, 1999, 224 (2): 567~588.
[124] Hadjileontiadis L J, Douka E, Trochidisc A. Fractal dimension analysis for crack identification in beam structures. Mechanical Systems and Signal Processing, 2005,19 (3): 659~674.
[125] 任宜春, 易伟建. 基于小波分析的梁裂缝识别研究. 计算力学学报, 2005, 22(4): 399~404.
[126] Quek S, Wang Q. Sensitivity analysis of crack detection in beams by the wavelet technique. Journal of Mechanical Sciences, 2001, 43(12): 2899~2910.
[127] Wang Q, Deng X. Damage detection with spatial wavelets. International Journal of Solids and Structures, 1999, 36(23): 3443~3468.
[128] Hong J C, Kim Y Y, Lee H C. Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam. International Journal of Solids and Structures, 2002, 39(7): 1803~1816.
[129] Quek S T, Wang Q. Practical issues in the detection of damage in beams using wavelets. Smart Materials and Structures, 2001, 10 (5) : 1009~1017.
[130] Tian Jiayong, Li Zheng, Su Xianyue. Crack detection in beams by wavelet analysis of transient flexural waves. Journal of Sound and Vibration, 2003, 261(4): 715~727.
[131] Quek S, Wang Q, Zhang L. Sensitivity analysis of crack detection in beams by wavelet techniques. International Journal of Mechanical Sciences, 2001, 43 (12): 2899~2910.
[132] Liew K.M, Wang Q. Application of wavelet theory for crack identification in structures. Journal of Engineering Mechanics, 1998, 124(7): 52~57.
[133] Lam H F, Leea Y Y, Suna H Y. Application of the spatial wavelet transform and Bayesian approach to the crack detection of a partially obstructed beam. Thin-Walled Structures, 2005, 43(1): 1~21.
[134] Sung D U, Kim C G, Hong C S. Monitoring of impact damages in composite laminates using wavelet transform. Composites Part B: engineering, 2002, 33 (1): 35~43.
[135] Li Bing, Chen Xuefeng, He Zhengjia. Detection of crack location and size in structures using wavelet finite element methods. Journal of Sound and Vibration, 2005, 285(4): 767~782.
[136] Carpinteri A, Cornetti P, Barpi F. Cohesive crack model description of ductile to brittle size-scale transition:dimensional analysis vs. renormalization group theory. Engineering Fracture Mechanics, 2003, 70(14): 1809-1839.
[137] 王铎. 断裂力学. 哈尔滨:哈尔滨工业大学出版社,1989.
[138] 曾庆敦. 复合材料的细观破坏机制与强度. 北京:科学出版社,2002.
[139] 尹双增. 断裂、损伤理论及应用. 北京:清华大学出版社,1992.
[140] Nayroles B, Touzot G, Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Computer Mechanics, 1992, 10(3): 307~318.
[141] Belytschko T, Lu Y Y, Gu L. Element free Galerkin methods. International Journal of Numerical Methods in Engineering. 1994, 37(5): 229~256.
[142] Lu Y Y, Beltschko T. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and engineering, 1994, 113(3): 397~414.
[143] Krongauz Y, Belytschko T. EFG approximation with discontinuous derivatives. International Journal for Numerical methods in Engineering. 1998, 41(7): 1215~1233.
[144] Lu Y Y, Belytschko T, Tabbara M. Element-free Galerkin methods for wave propagation and dynamic fracture. Computer Methods in Applied Mechanics and engineering, 1995, 126(7) :131~153.
[145] Belytschko T, Tabbara M. Dynamic fracture using element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1996, 39(5): 923~938.
[146] Belytschko T, Gu L, Lu Y Y. Fracture and crack growth by element free Galerkin methods. Modelling and Simulation in Materials Science and Engineering, 1994, 2(3): 519~534.
[147] Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkinmethods. Engineering Fracture Mechanics, 1995, 51(2): 295~315.
[148] Fleming M, Belytschko T. Enriched Element-Free Galerkin Methods for crack tip fields. International Journal for Numerical Methods in Engineering, 1997, 40(8): 1483~1504.
[149] Krysl P, Belytschko T. Analysis of thin shells by element-free Galerkin method. International Journal of Solids and Structures, 1995, 33(20): 3057~3080.
[150] Hegen D. Element-free Galerkin methods in combination with finite element approaches. Computer Methods in Applied Mechanics and Engineering, 1996, 135(1): 143~166.
[151] 冯象初,甘小冰,宋国乡. 数值泛函与小波理论. 西安:西安电子科技大学出版社,2003.
[152] Han J G, Ren W X, Sun Z S. Wavelet packet based damage identification of beam structures. International Journal of Solids and Structures, 2005, 42(26): 6610-6627.
[153] Ovanesova A V, Suarez L E. Applications of wavelet transforms to damage detection in frame structures. Engineering Structures, 2004, 26(1): 39~49.
[154] Yam L H,Yan Y J,Jiang J S. Vibration-based damage detection for composite structures using wavelet transform and neural network identification. Composite Structures, 2003, 60(4): 403~412.
[155] Newland D E. Wavelet analysis of vibration, Part 2: Wavelet maps. Journal of Vibration and Acoustics, 1994, 116(4): 417~425.
[156] Han Jiangang, Ren Weixin, Huang Yihe. A spline wavelet finite-element method in structural mechanics. International Journal for Numerical Methods in Engineering, 2006, 66(1): 166~190.
[157] Rathsfeld A. Wavelet algorithm for the boundary element solution of a geodetic boundary value problem. Computer Methods in Applied Mechanics and Engineering, 1998, 157(3):267~287.
[158] 张雄,刘岩. 无网格法. 北京:清华大学出版社,2004.
[159] Zhang Hong, Zhang Xuanbing, Ge Xiurun. Application of wavelet theory in research on weight function of meshless method. Applied Mathematics and Mechanics (English Edition) , 2005, 26(5): 662~666.
[160] 吴琛,周瑞忠. 小波基无单元法及其与有限元法的比较. 工程力学, 2006, 23(4): 28~32.
[161] 周瑞忠,周小平,缪圆冰. 小波基无单元法及其工程应用. 工程力学, 2003, 20(6): 70~74.
[162] 缪圆冰. 小波无单元方法及其应用:[硕士学位论文]. 福州:福州大学, 2001.
[163] 周先雁,刘希,沈蒲生. 用含裂纹的梁单元识别混凝土框架结构损伤. 振动工程学报,1999,12(1),116～119.
[164] 钱管良,顾松年. 在线振动监测与故障诊断的一种新途径. 固体力学学报, 1990,11(3):217～228.
[165] 周先雁,沈蒲生,刘希. 用断裂动力学对混凝土平面杆系进行破损评估. 湖南大学学报, 1998, 25(1): 66~70.
[166] Nahvi H, Jabbari M. Crack detection in beams using experimental modal data and finite element model. International Journal of Mechanical Sciences, 2005, 47(10): 1477~1497.
[167] 铁摩辛柯. 材料力学. 北京:科学出版社, 1978.
[168] 中国航空研究院. 应力强度因子手册. 北京:科学出版社, 1981.
[169] 伍必庆, 张青喜. 道路建筑材料. 北京: 清华大学出版社, 2006.
[170] Allemang R J. The modal assurance criterion - twenty years of use and abuse. Journal of Sound and Vibration, 2003, 37(8): 14~23.
[171] 任伟新,林友勤,彭雪林. 大跨度斜拉桥环境振动试验与分析. 实验力学, 2006, 21(4): 418～426.
[172] Choi S, Stubbs N. Damage identification in structures using the time-domain response. Journal of sound and vibration, 2004, 275(3): 577~590.
[173] 苏金明, 阮沈勇, 王永利. MATLAB 工程数学. 北京: 电子工业出版社, 2005.
[174] Leonard J, Kramer M A. Improvement of the back propagation algorithm for training neural networks. Computers and Chemical Engineering, 1990, 14(3): 337~341.
[175] 姜绍飞. 基于神经网络的结构优化与损伤检测. 北京: 科学出版社, 2002.
[176] Masri S F, Nakamura M, Chassiakos A G. Neural network approach to detection of changes in structural parmaters. Journal of Engineering Mechanics, 1996, 122(4): 350~360.
[177] Waszczyszyn Z, Ziemianski L. Neural networks in mechanics of structures and materials-new results and prospects of applications. Computers and Structures, 2001, 79(22): 2261~2276.
[178] Kao C Y, Hung S L. Detection of structural damage via free vibration responses generated by approximating artificial neural networks. Computers and Structures, 2003, 81(28): 2631-2644.
[179] Yam H L, Yan Y J, Jiang J S. Vibration-based damage detection for composite structures using wavelet transform and neural network identification. Composite Structures, 2003, 60(4): 403~412.
[180] 朱宏平, 千力. 利用振动模态测量值和神经网络方法的结构损伤识别研究. 计算力学学报, 2005, 22(2): 193~196.
[181] Fang X, Luo H, Tang J. Structural damage detection using neural network with learning rate improvement. Computers and Structures, 2005, 83(25): 2150~2161.
[182] 韩大建, 杨炳尧, 颜全胜. 用人工神经网络方法评估桥梁缺损状况. 华南理工大学学报, 2004, 32(9): 72~76.
[183] Sahin M, Shenoi R A. Quantification and localization of damage in beam-like structures by using artificial neural networks with experimental validation. Engineering Structures, 2003, 25(14): 1785~1802.
[184] 罗跃纲, 张松鹤, 闻邦椿. 基于频率下降率的结构损伤自适应神经网络识别. 中国安全科学学报, 2005, 15(5): 13~16.
[185] Lee J J, Lee J W, Yi J H, et al. Neural networks-based damage detection for bridges considering errors in baseline finite element models [J]. Journal of Sound and Vibration, 2005, 280(3-5): 555~578.
[186] 李忠献, 杨晓明, 丁 阳. 应用人工神经网络技术的大型斜拉桥子结构损伤识别研究. 地震工程与工程振动, 2003, 23(3): 92~99.
[187] Zweiri Y H, Seneviratne L D, Althoefer K. Stability analysis of a three-term back propagation algorithm. Neural Networks, 2005, 18(10): 1341~1347.
[188] 孙宗光, 高赞明, 倪一清. 基于神经网络的损伤构件及损伤程度识别. 工程力学, 2006, 23(2): 18～22.
[189] 何 翔, 李守巨, 刘迎曦. 基于遗传神经网络的坝基岩体渗透系数识别. 岩石力学与工程学报, 2004, 23(5): 751～757.
[190] 马祥森, 史治宇. 基于 GA-BP 神经网络的结构损伤位置识别. 振动工程学报, 2004, 17(4): 453～456.
[191] 雷英杰, 张善文, 李续武等. MATLAB 遗传算法工具箱及应用. 西安: 西安电子科技大学出版社, 2005.