考虑剪力滞效应的薄壁结构特性分析
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摘要
薄壁结构在弯曲荷载作用下存在剪力滞效应。国内外对剪力滞效应已有很多研究,也取得了一些成果,解决了部分实际工程问题,但仍然有许多问题还没有完全得到解决。本文针对∏形截面主梁斜拉桥、普通混凝土箱梁、波形钢腹板组合箱梁等结构,研究了剪力滞效应对薄壁结构静动力特性的影响,主要进行了以下几方面的研究。
(1)采用有限单元法建立模型,分析了一斜拉桥∏形截面主梁的剪力滞效应的分布规律,并将有限元结果同实桥的实测值进行了对比,验证了有限元模型的正确性。详细分析了斜拉桥兀形主梁的预应力、横坡、小纵梁等参数对剪力滞效应的影响,并分析了剪力滞效应对∏形主梁斜拉桥动力特性的影响。从分析结果可知∏形截面斜拉桥主梁根部截面的剪力滞效应比跨中截面更为显著;其主梁横截面轴向应力横向分布曲线存在多个拐点,剪力滞效应很难采用单个参数来描述;预应力对∏形截面斜拉桥主梁轴向应力的横向分布规律有明显影响,计算时不应忽略预应力束的影响;桥面横坡也是影响∏形截面主梁斜拉桥剪力滞效应的一个重要因素,尤其是对顶板中央处的纵向正应力影响较大;在桥面板沿纵向增设小纵梁是减小剪力滞效应的一种有效方法,高度较低的纵梁即可显著改善应力分布:首次分析了剪力滞效应对∏形截面主梁斜拉桥动力特性的影响,结果表明:考虑剪力滞效应后主梁的振动频率有所降低,且随着频率阶次的升高,降低越明显。
(2)采用能量变分法推导了考虑剪力滞效应的简支箱梁自振基频解析解,算例表明该方法比一般梁理论计算精度有显著提高,且计算简单方便。在此基础上,又推导了简支箱梁同时考虑剪力滞效应以及剪切变形的各阶频率显式计算公式,进一步提高了计算精度;对影响箱梁自振频率的参数进行了详尽的分析,结果表明:考虑剪力滞效应以及考虑剪切变形均使简支箱梁的振动基频降低,且对于箱梁的高阶振动频率降低程度尤为明显,影响频率降低程度的参数主要有:箱梁的顶底板抗弯惯性矩占整个截面的抗弯惯性矩的比重(I_s/I)、箱梁的跨宽比(l/26)、箱形截面的剪切系数k以及跨高比(l/r)等参数,而其中箱梁的宽跨比是影响其降低程度的最重要因素。
(3)推导了简支波形钢腹板组合箱梁的振动频率公式,得到了考虑剪切变形及剪力滞效应的波形钢腹板组合箱梁各阶自振频率的解析解;对一试验波形钢腹板组合箱梁进行了动力测试,得到了该试验波形钢腹板组合箱梁的实测自振频率,并与简单梁理论、本文理论公式以及三维有限元模型的计算频率进行对比。对比结果表明剪力滞效应及剪切变形对波形钢腹板组合箱梁的振动频率影响较大,考虑剪力滞及剪切变形影响后的波形钢腹板组合箱梁的振动频率有所降低,相对于普通混凝土箱梁降低程度更为明显,且降低程度随着计算频率的阶次的增加而迅速增加。而本文提出的理论公式可考虑剪力滞及剪切变形效应影响,且计算简单,精度较高。
(4)为简支箱梁在简谐荷载作用下的强迫振动提出了一种实用计算方法,该方法可以考虑剪力滞效应的影响。在现有初等梁受迫振动理论的基础上,推导了考虑剪力滞效应的简支箱梁简谐振动荷载作用下跨中截面的动应力放大系数,并将其与实体有限元结果进行比较,验证了其计算精度;针对地震荷载作用下的箱梁动力响应,根据反应谱法以及时程分析法,首次提出了等效弹性模量法以考虑其影响,文中算例验证了等效弹性模量法的可行性。
(5)采用能量变分法,推导了考虑剪力滞效应的悬臂箱梁稳定临界力的显式解,并通过算例将其与初等梁理论以及实体有限元算得的临界力进行比较,结果表明该公式精度较高,且公式形式简单,计算方便。考虑剪力滞效应后,悬臂矩形梁的稳定临界力有所降低,即初等梁理论计算的结果是偏于不安全,而本文的计算公式与初等梁理论一样简单,但计算的精度相对于初等梁理论有所提高。
There will be shear lag effect existing in the thin-walled structure when it is suffered from bending load. Many eyes are focused on shear lag effect and some achievements have been obtained. But there are still some engineering problems not solved wholly. Based onΠ-shaped cross section cable-stayed bridge, concrete box beam and composite box girder with corrugated steel webs, the effect of shear lag on the static and dynamic character of thin-walled structure were studied. The main achievements are as followings.
(1) With the help of finite element method, the shear lag effect of aΠ-shaped cross section cable-stayed bridge was discussed. The credibility of the finite-element model was validated by comparison of its results with the test results of the bridge. A parametric study using finite element analysis was carried out, in which the parameters such as prestress, transverse slope of bridge deck and little longitudinal beams were considered. Also, the effect of shear lag on the dynamic character ofΠ-shaped cross section cable-stayed bridge was discussed. The results show that shear lag effect is more serious at the root of theΠ-shaped cross section girder than that at its mid-span. The axial stress distribution curve of all sections has several knee points and so it is difficult to describe the shear lag with one parameter. Presstress and transverse slope of the girder have much effect on the shear lag, and it should not be neglected during calculation. And transverse slope is another important factor that influencing the shear lag, especially for the longitudinal normal stress near the middle area of the deck. It also shows that longitudinal girders can improve the stress distribution obviously. The analysis of dynamic character indicates that the vibration frequencies will decrease due to shear lag effect. The decrease will be more obviously with the increasing of frequency order.
(2) Based on energy variational principle, a new method to analyze the vibration of simply supported box girder was proposed, in which the effect of shear lag was considered, and the closed-form solutions of the fundamental frequency were derived. The example shows that the accuracy and effectiveness of this new method is much improved compared with the conventional beam theory. Furthermore, the frequency formula was extended to calculate all orders of frequencies of box girder, in which both the effect of shear lag and shear deformation were taken into consideration, and the accuracy was improved. The parameters affecting the vibration frequencies were discussed in detail. The parametric study shows that the natural frequencies of simply supported box girder will decrease when the effects of shear lag or shear deformation is taken into account, especially for the high order frequencies' calculation. The main parameters that affect the frequencies decrease include inertia moment ratio(I_s/I), the width span ratio(l/2b), shear factor(k) and span high ratio(l/r), in which the width span ratio is the most important parameter.
(3) An analytic formula of the natural frequencies of simply supported box girder with corrugated steel webs was deduced, in which both the effects of shear lag and shear deformation were taken into consideration. The natural frequencies of a composite box girder model with corrugated steel webs were tested and compared with the results of the elementary beam theory, formula of this dissertation and the 3D finite element model. The results show that the effects of shear lag and shear deformation on vibration are so pronounced that it should not be neglected in the frequencies calculation. The frequencies of composite box girder with corrugated steel webs will decrease when the effect of shear lag and shear deformation are taken into consideration. Compared with the concrete box girder, the effect on composite box girder is more pronounced. And the effects increase sharply with the increase of the frequency order. The formula proposed in this dissertation can take the effects of shear lag and shear deformation into consideration; furthermore, it is simple and accurate.
(4) A practical method was proposed to calculate the forced vibration of simply supported box girder bridge, and the effect of shear lag is included in this method. Based on the forced vibration theory of elementary beam, the dynamic stress amplification factor of a simply supported box girder under harmonic load was deduced, in which the effect of shear lag was considered. The comparison of it with the results of solid finite element model shows the method of this dissertation is feasible. As for the earthquake calculation, equivalent elastic modulus method was proposed to analyze the effect of shear lag according to response spectrum method and time history method. Example shows that the equivalent elastic modulus method is feasible.
(5) Based on energy variational principle, the closed-form formula of stability critical force of cantilever box girder was deduced, in which the effect of shear lag was included. A cantilever box section pier was taken as example, and the comparison between elementary beam theory and solid finite element model shows that the formula is more simple and accurate. The critical force will decrease when shear lag is taken into consideration, so elementary beam theory would lead to non-conservative result. The formula introduced in this dissertation can improve the accuracy while the formula is as simple as elementary beam theory's.
(1)采用有限单元法建立模型,分析了一斜拉桥∏形截面主梁的剪力滞效应的分布规律,并将有限元结果同实桥的实测值进行了对比,验证了有限元模型的正确性。详细分析了斜拉桥兀形主梁的预应力、横坡、小纵梁等参数对剪力滞效应的影响,并分析了剪力滞效应对∏形主梁斜拉桥动力特性的影响。从分析结果可知∏形截面斜拉桥主梁根部截面的剪力滞效应比跨中截面更为显著;其主梁横截面轴向应力横向分布曲线存在多个拐点,剪力滞效应很难采用单个参数来描述;预应力对∏形截面斜拉桥主梁轴向应力的横向分布规律有明显影响,计算时不应忽略预应力束的影响;桥面横坡也是影响∏形截面主梁斜拉桥剪力滞效应的一个重要因素,尤其是对顶板中央处的纵向正应力影响较大;在桥面板沿纵向增设小纵梁是减小剪力滞效应的一种有效方法,高度较低的纵梁即可显著改善应力分布:首次分析了剪力滞效应对∏形截面主梁斜拉桥动力特性的影响,结果表明:考虑剪力滞效应后主梁的振动频率有所降低,且随着频率阶次的升高,降低越明显。
(2)采用能量变分法推导了考虑剪力滞效应的简支箱梁自振基频解析解,算例表明该方法比一般梁理论计算精度有显著提高,且计算简单方便。在此基础上,又推导了简支箱梁同时考虑剪力滞效应以及剪切变形的各阶频率显式计算公式,进一步提高了计算精度;对影响箱梁自振频率的参数进行了详尽的分析,结果表明:考虑剪力滞效应以及考虑剪切变形均使简支箱梁的振动基频降低,且对于箱梁的高阶振动频率降低程度尤为明显,影响频率降低程度的参数主要有:箱梁的顶底板抗弯惯性矩占整个截面的抗弯惯性矩的比重(I_s/I)、箱梁的跨宽比(l/26)、箱形截面的剪切系数k以及跨高比(l/r)等参数,而其中箱梁的宽跨比是影响其降低程度的最重要因素。
(3)推导了简支波形钢腹板组合箱梁的振动频率公式,得到了考虑剪切变形及剪力滞效应的波形钢腹板组合箱梁各阶自振频率的解析解;对一试验波形钢腹板组合箱梁进行了动力测试,得到了该试验波形钢腹板组合箱梁的实测自振频率,并与简单梁理论、本文理论公式以及三维有限元模型的计算频率进行对比。对比结果表明剪力滞效应及剪切变形对波形钢腹板组合箱梁的振动频率影响较大,考虑剪力滞及剪切变形影响后的波形钢腹板组合箱梁的振动频率有所降低,相对于普通混凝土箱梁降低程度更为明显,且降低程度随着计算频率的阶次的增加而迅速增加。而本文提出的理论公式可考虑剪力滞及剪切变形效应影响,且计算简单,精度较高。
(4)为简支箱梁在简谐荷载作用下的强迫振动提出了一种实用计算方法,该方法可以考虑剪力滞效应的影响。在现有初等梁受迫振动理论的基础上,推导了考虑剪力滞效应的简支箱梁简谐振动荷载作用下跨中截面的动应力放大系数,并将其与实体有限元结果进行比较,验证了其计算精度;针对地震荷载作用下的箱梁动力响应,根据反应谱法以及时程分析法,首次提出了等效弹性模量法以考虑其影响,文中算例验证了等效弹性模量法的可行性。
(5)采用能量变分法,推导了考虑剪力滞效应的悬臂箱梁稳定临界力的显式解,并通过算例将其与初等梁理论以及实体有限元算得的临界力进行比较,结果表明该公式精度较高,且公式形式简单,计算方便。考虑剪力滞效应后,悬臂矩形梁的稳定临界力有所降低,即初等梁理论计算的结果是偏于不安全,而本文的计算公式与初等梁理论一样简单,但计算的精度相对于初等梁理论有所提高。
There will be shear lag effect existing in the thin-walled structure when it is suffered from bending load. Many eyes are focused on shear lag effect and some achievements have been obtained. But there are still some engineering problems not solved wholly. Based onΠ-shaped cross section cable-stayed bridge, concrete box beam and composite box girder with corrugated steel webs, the effect of shear lag on the static and dynamic character of thin-walled structure were studied. The main achievements are as followings.
(1) With the help of finite element method, the shear lag effect of aΠ-shaped cross section cable-stayed bridge was discussed. The credibility of the finite-element model was validated by comparison of its results with the test results of the bridge. A parametric study using finite element analysis was carried out, in which the parameters such as prestress, transverse slope of bridge deck and little longitudinal beams were considered. Also, the effect of shear lag on the dynamic character ofΠ-shaped cross section cable-stayed bridge was discussed. The results show that shear lag effect is more serious at the root of theΠ-shaped cross section girder than that at its mid-span. The axial stress distribution curve of all sections has several knee points and so it is difficult to describe the shear lag with one parameter. Presstress and transverse slope of the girder have much effect on the shear lag, and it should not be neglected during calculation. And transverse slope is another important factor that influencing the shear lag, especially for the longitudinal normal stress near the middle area of the deck. It also shows that longitudinal girders can improve the stress distribution obviously. The analysis of dynamic character indicates that the vibration frequencies will decrease due to shear lag effect. The decrease will be more obviously with the increasing of frequency order.
(2) Based on energy variational principle, a new method to analyze the vibration of simply supported box girder was proposed, in which the effect of shear lag was considered, and the closed-form solutions of the fundamental frequency were derived. The example shows that the accuracy and effectiveness of this new method is much improved compared with the conventional beam theory. Furthermore, the frequency formula was extended to calculate all orders of frequencies of box girder, in which both the effect of shear lag and shear deformation were taken into consideration, and the accuracy was improved. The parameters affecting the vibration frequencies were discussed in detail. The parametric study shows that the natural frequencies of simply supported box girder will decrease when the effects of shear lag or shear deformation is taken into account, especially for the high order frequencies' calculation. The main parameters that affect the frequencies decrease include inertia moment ratio(I_s/I), the width span ratio(l/2b), shear factor(k) and span high ratio(l/r), in which the width span ratio is the most important parameter.
(3) An analytic formula of the natural frequencies of simply supported box girder with corrugated steel webs was deduced, in which both the effects of shear lag and shear deformation were taken into consideration. The natural frequencies of a composite box girder model with corrugated steel webs were tested and compared with the results of the elementary beam theory, formula of this dissertation and the 3D finite element model. The results show that the effects of shear lag and shear deformation on vibration are so pronounced that it should not be neglected in the frequencies calculation. The frequencies of composite box girder with corrugated steel webs will decrease when the effect of shear lag and shear deformation are taken into consideration. Compared with the concrete box girder, the effect on composite box girder is more pronounced. And the effects increase sharply with the increase of the frequency order. The formula proposed in this dissertation can take the effects of shear lag and shear deformation into consideration; furthermore, it is simple and accurate.
(4) A practical method was proposed to calculate the forced vibration of simply supported box girder bridge, and the effect of shear lag is included in this method. Based on the forced vibration theory of elementary beam, the dynamic stress amplification factor of a simply supported box girder under harmonic load was deduced, in which the effect of shear lag was considered. The comparison of it with the results of solid finite element model shows the method of this dissertation is feasible. As for the earthquake calculation, equivalent elastic modulus method was proposed to analyze the effect of shear lag according to response spectrum method and time history method. Example shows that the equivalent elastic modulus method is feasible.
(5) Based on energy variational principle, the closed-form formula of stability critical force of cantilever box girder was deduced, in which the effect of shear lag was included. A cantilever box section pier was taken as example, and the comparison between elementary beam theory and solid finite element model shows that the formula is more simple and accurate. The critical force will decrease when shear lag is taken into consideration, so elementary beam theory would lead to non-conservative result. The formula introduced in this dissertation can improve the accuracy while the formula is as simple as elementary beam theory's.
引文
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