考虑剪切变形影响的薄壁钢梁分析方法与应用
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摘要
薄壁钢梁广泛应用于土木建筑、机械、海洋工程和航空等领域中，为保证其安全、适用，建立合理、可靠的薄壁梁力学模型和分析方法十分重要。传统的Euler-Bernoulli梁理论和Timoshenko梁理论以及相应的有限元分析方法得到了广泛应用。Euler-Bernoulli梁理论没有考虑剪切变形的影响，Timoshenko梁理论虽然考虑了剪切变形的影响，但仅考虑横向剪切变形且与Euler-Bernoulli梁理论一样适用于实体梁弯曲性能的分析，不能用于薄壁梁的弯曲与扭转分析。Vlasov发展了开口薄壁梁扭转与弯曲相结合的综合理论，但忽略了剪切变形对构件性能的影响，仅适用于跨高比大于10的薄壁梁。
随着国民经济和城市建设的发展，钢结构在高层、超高层建筑结构以及重型工业厂房等工程中得到了广泛应用。当钢梁的跨高比较小时，剪切变形对其力学性能的影响不能忽略。为了考虑弯曲和扭转所产生的剪切变形对薄壁钢梁力学性能的影响，本文在Vlasov薄壁梁理论的基础上发展了考虑剪切变形影响的一阶薄壁梁理论。由于截面剪心的特性，薄壁钢梁弯曲与扭转是解耦的，可以分别研究。本文分别对薄壁构件的弯曲和扭转进行了分析研究，建立了相应的微分方程。同时在理论分析的基础上，建立了三维梁柱二阶弹塑性单元，并编制了有限元分析程序。具体的工作如下：
首先研究了开口薄壁钢梁的弯曲与约束扭转问题。在假设位移场的基础上，针对开口薄壁钢梁，考虑了薄壁截面中线上剪切变形的影响，给出了弯曲时的应力、变形以及横向剪切系数的表达式，并给出了横向位移与转角的关系。在分析薄壁钢梁约束扭转时，将薄壁截面总转角分为自由翘曲转角和约束剪切转角。在此基础上，推导了约束扭转状态下截面翘曲位移表达式，给出了截面上的翘曲正应力和剪应力的计算方法，并以能量原理为基础得到了扭转剪切系数。扭转剪切系数说明了真实翘曲剪应力在截面上分布的情况。编制了常用薄壁梁截面特性的计算程序。
推导了开口薄壁梁约束扭转的新型微分方程，给出了一阶扭转理论的初参数求解公式以及截面总转角和自由翘曲转角之间的关系，为后面有限单元插值函数的建立奠定了基础。对于开口薄壁截面当构件参数λ（与截面特性和梁长有关）满足特定条件时，St.Venant扭矩可以忽略，此时可以得到开口薄壁截面的约束扭转简化分析方法。开口薄壁截面约束扭转简化分析方法的相关公式和Timoshenko梁理论的公式具有完全相同的表达形式。算例分析表明，本文分析方法可以较好地模拟开口薄壁梁的约束扭转性能，并能分析剪切变形对约束扭转性能的影响。
研究了闭口薄壁截面梁约束扭转问题，提出了闭口薄壁梁的一阶扭转理论。给出了闭口薄壁梁一阶扭转理论中的正应力的表达式。对闭口薄壁截面上的各种扭矩和相应的剪应力进行了探讨。基于能量原理得到了闭口薄壁梁扭转剪切系数的概念与计算方法。推导了闭口截面约束扭转控制微分方程以及相应的初参数求解方法。此微分方程可作为任意截面薄壁梁约束扭转的统一理论。通过算例，分析了由于约束剪切转角产生的剪切变形对闭口薄壁梁扭转性能的影响。
在理论分析的基础上，建立了一种新的开、闭口薄壁梁单元。利用薄壁梁弯曲时横向位移与截面转角的关系以及约束扭转时截面总转角和自由翘曲转角之间的关系建立了薄壁梁单元的位移模式。此位移模式仅需要两个节点，避免了通过在单元内部设置节点建立单元刚度矩阵过程。根据连续介质力学有限变形理论，利用更新的Lagrange方法（UL法）建立了新型梁单元刚度矩阵。新单元能更准确地考虑剪切变形的影响，避免剪切自锁现象的发生，并能包含翘曲变形和弯扭耦合等因素。
为了更准确地考虑P-δ效应的影响，基于UL法提出了考虑P-δ效应的增量方法，此方法弥补了三次插值函数在切线刚度矩阵中产生的误差，同时也避免了利用稳定插值函数推导刚度矩阵需依轴力方向而变化的问题。
假定材料服从Von Mises屈服准则、相关流动法则以及等向强化假设。为了跟踪在截面上和单元长度方向上材料的塑性发展，利用塑性区方法，在薄壁截面曲线坐标s方向划分微段，在单元的长度方向设置Guass积分点，并根据微段中的应力来判断单元塑性发展情况。
最后在新型薄壁梁单元推导的基础上，利用面向对象的C++语言编制了非线性分析程序，并利用数值算例进行了验证。算例分析表明，建立的有限单元模型可以给出更精确的结果，并且避免了剪切自锁现象的发生。
Thin-walled steel beams are widely used in the fields of civil, mechanical, and navalconstructions. In order to guarantee the security and applicability of these steel members, it isvery important to build reasonable and reliable physical model and analysis method. Theclassical Euler-Bernoulli beam theory and Timoshenko beam theory as well as thecorresponding beam element are used extensively in engineering analysis. Euler-Bernoullibeam theory does not consider the effect of shear deformation on the behavior of beams.Timoshenko beam theory includes the effect of shear deformation, but is limited to flexuralanalysis of solid beam as well as Euler-Bernoulli beam theory and cannot be used to flexuraland torsional analysis of thin-walled beam. Vlasov developed an integrated theory which cananalyze the flexural and torsional behavior of thin-walled beam. However, due to theignorance of shear deformation in middle surface, Vlasov theory is only applicable forthin-walled beams with span-height-ratio greater than10.
With the growth of national economy and city construction, steel members are widelyused in heavy structures, such as high-rise, super high-rise and heavy industrial buildings. Inthese buildings, the span-height-ratio of steel beam is small and the effect of sheardeformation becomes obvious. In order to include the effect of shear deformation due toflexure and torsion, in this thesis, a first-order thin-walled beam theory is developed based onVlasov theory. This theory can analyze the flexural and torsional problems of thin-walledbeam. Due to the shear center of thin-walled cross section, the flexural and torsional problems of thin-walled beams can be analyzed respectively. Accordingly, the governing differentialequations are obtained for flexure and torsion of thin-walled beam respectively. Asecond-order elasto-plastic beam element based on the first-order thin-walled beam theory isobtained, and the corresponding code is written. The specific contents are given as follows.
Firstly the first-order thin-walled beam theory is developed for flexural and torsionalproblems of open thin-walled beam. The expressions of flexural stress, deformation andtransverse shear coefficient are given. The relationships between transverse deflection androtation are obtained. When analyzing the restrained torsion of open thin-walled beam, thetotal rotation of section is divided into free warping rotation and restrained shear rotation. Theexpression of warping deformation, the warping normal stress and warping shear stress arederived based on this assumption. And the torsion shear coefficient is then obtained on thebasis of energy principle. Torsion shear coefficient could account for the true shear stressdistribution in the cross section. Program is compiled to calculate the geometric properties ofopen thin-walled section.
The new governing differential equations of the restrained torsion are derived and thecorresponding initial method is given to solve the equations. The relationship between totalrotation and free warping rotation is obtained. A parameter λ, which is associated with thestiffness property of a cross section and the beam length, is introduced to determine thecondition, under which the St. Venant constant is negligible. Consequently a simplifiedmethod of restrained torsion is derived. The simple method has the same style withTimoshenko beam theory. Numerical examples are illustrated to validate the current approachand the results of the current theory are compared with those of some other available methods.The results of comparison show that the current theory predicts the torsion behavior ofthin-walled beam more accurately.
The restrained torsion of closed thin-walled beam is analyzed. The first-order torsiontheory of closed thin-walled beam is proposed. The various stresses in closed cross section are analyzed and divided into categories, and the expressions of these stresses are given. Thetorsion shear coefficient is then obtained on the basis of energy principle. The governingdifferential equations of the restrained torsion of closed thin-walled beam are derived and thecorresponding initial method is given to solve the equation. These equations can be used asunified theory for restrained torsion of arbitrary cross section. The effect of restrained shearrotation on torsional behavior of closed thin-walled beam is investigated through numericalanalysis.
The thin-walled beam element is derived. The displacement mode of thin-walled beamelement is obtained through the relationship between deflection and rotation and that betweentotal rotation and free warping rotation. Only two nodes are needed in this displacement mode,which avoid the deriving process of beam element with internal node. The new beam elementis obtained through the updated Lagrange (UL) method on the basis of finite deformationtheory in continuum mechanics. This new beam element avoids the phenomenon of shearlocking and includes the effects of warping deformation and the coupling of flexure andtorsion.
An incremental method is proposed based on UL method in order to include the P-δeffect more accurately. This method makes up the defect of the cubic interpolation function intangential stiffness matrix and avoids the problem of variation of stiffness matrix derivedfrom stable interpolation function with the direction of axial force.
It is assumed that the Von Mises yield criterion, associated flow rules and isotropichardening model are applied to the material that used in this paper. Plastic-zone method isused and the thin-walled section is divided into a number of finite parts along the s directionand three Guass integral points are placed along the length of thin-walled beam in order totrace the plasticity development in the beam. And the plasticity development can be judged bythe stress condition in finite parts.
Finally, on the basis of derivation of new thin-walled beam element, nonlinear analysis program is compiled by using the object oriented program language C++. And numericalexamples are illustrated to validate the current approach. The results of the current theory arecompared with those of some other available methods. The results of comparison show thatthe current theory provides more accurate results and avoids the phenomenon of shearlocking.
随着国民经济和城市建设的发展，钢结构在高层、超高层建筑结构以及重型工业厂房等工程中得到了广泛应用。当钢梁的跨高比较小时，剪切变形对其力学性能的影响不能忽略。为了考虑弯曲和扭转所产生的剪切变形对薄壁钢梁力学性能的影响，本文在Vlasov薄壁梁理论的基础上发展了考虑剪切变形影响的一阶薄壁梁理论。由于截面剪心的特性，薄壁钢梁弯曲与扭转是解耦的，可以分别研究。本文分别对薄壁构件的弯曲和扭转进行了分析研究，建立了相应的微分方程。同时在理论分析的基础上，建立了三维梁柱二阶弹塑性单元，并编制了有限元分析程序。具体的工作如下：
首先研究了开口薄壁钢梁的弯曲与约束扭转问题。在假设位移场的基础上，针对开口薄壁钢梁，考虑了薄壁截面中线上剪切变形的影响，给出了弯曲时的应力、变形以及横向剪切系数的表达式，并给出了横向位移与转角的关系。在分析薄壁钢梁约束扭转时，将薄壁截面总转角分为自由翘曲转角和约束剪切转角。在此基础上，推导了约束扭转状态下截面翘曲位移表达式，给出了截面上的翘曲正应力和剪应力的计算方法，并以能量原理为基础得到了扭转剪切系数。扭转剪切系数说明了真实翘曲剪应力在截面上分布的情况。编制了常用薄壁梁截面特性的计算程序。
推导了开口薄壁梁约束扭转的新型微分方程，给出了一阶扭转理论的初参数求解公式以及截面总转角和自由翘曲转角之间的关系，为后面有限单元插值函数的建立奠定了基础。对于开口薄壁截面当构件参数λ（与截面特性和梁长有关）满足特定条件时，St.Venant扭矩可以忽略，此时可以得到开口薄壁截面的约束扭转简化分析方法。开口薄壁截面约束扭转简化分析方法的相关公式和Timoshenko梁理论的公式具有完全相同的表达形式。算例分析表明，本文分析方法可以较好地模拟开口薄壁梁的约束扭转性能，并能分析剪切变形对约束扭转性能的影响。
研究了闭口薄壁截面梁约束扭转问题，提出了闭口薄壁梁的一阶扭转理论。给出了闭口薄壁梁一阶扭转理论中的正应力的表达式。对闭口薄壁截面上的各种扭矩和相应的剪应力进行了探讨。基于能量原理得到了闭口薄壁梁扭转剪切系数的概念与计算方法。推导了闭口截面约束扭转控制微分方程以及相应的初参数求解方法。此微分方程可作为任意截面薄壁梁约束扭转的统一理论。通过算例，分析了由于约束剪切转角产生的剪切变形对闭口薄壁梁扭转性能的影响。
在理论分析的基础上，建立了一种新的开、闭口薄壁梁单元。利用薄壁梁弯曲时横向位移与截面转角的关系以及约束扭转时截面总转角和自由翘曲转角之间的关系建立了薄壁梁单元的位移模式。此位移模式仅需要两个节点，避免了通过在单元内部设置节点建立单元刚度矩阵过程。根据连续介质力学有限变形理论，利用更新的Lagrange方法（UL法）建立了新型梁单元刚度矩阵。新单元能更准确地考虑剪切变形的影响，避免剪切自锁现象的发生，并能包含翘曲变形和弯扭耦合等因素。
为了更准确地考虑P-δ效应的影响，基于UL法提出了考虑P-δ效应的增量方法，此方法弥补了三次插值函数在切线刚度矩阵中产生的误差，同时也避免了利用稳定插值函数推导刚度矩阵需依轴力方向而变化的问题。
假定材料服从Von Mises屈服准则、相关流动法则以及等向强化假设。为了跟踪在截面上和单元长度方向上材料的塑性发展，利用塑性区方法，在薄壁截面曲线坐标s方向划分微段，在单元的长度方向设置Guass积分点，并根据微段中的应力来判断单元塑性发展情况。
最后在新型薄壁梁单元推导的基础上，利用面向对象的C++语言编制了非线性分析程序，并利用数值算例进行了验证。算例分析表明，建立的有限单元模型可以给出更精确的结果，并且避免了剪切自锁现象的发生。
Thin-walled steel beams are widely used in the fields of civil, mechanical, and navalconstructions. In order to guarantee the security and applicability of these steel members, it isvery important to build reasonable and reliable physical model and analysis method. Theclassical Euler-Bernoulli beam theory and Timoshenko beam theory as well as thecorresponding beam element are used extensively in engineering analysis. Euler-Bernoullibeam theory does not consider the effect of shear deformation on the behavior of beams.Timoshenko beam theory includes the effect of shear deformation, but is limited to flexuralanalysis of solid beam as well as Euler-Bernoulli beam theory and cannot be used to flexuraland torsional analysis of thin-walled beam. Vlasov developed an integrated theory which cananalyze the flexural and torsional behavior of thin-walled beam. However, due to theignorance of shear deformation in middle surface, Vlasov theory is only applicable forthin-walled beams with span-height-ratio greater than10.
With the growth of national economy and city construction, steel members are widelyused in heavy structures, such as high-rise, super high-rise and heavy industrial buildings. Inthese buildings, the span-height-ratio of steel beam is small and the effect of sheardeformation becomes obvious. In order to include the effect of shear deformation due toflexure and torsion, in this thesis, a first-order thin-walled beam theory is developed based onVlasov theory. This theory can analyze the flexural and torsional problems of thin-walledbeam. Due to the shear center of thin-walled cross section, the flexural and torsional problems of thin-walled beams can be analyzed respectively. Accordingly, the governing differentialequations are obtained for flexure and torsion of thin-walled beam respectively. Asecond-order elasto-plastic beam element based on the first-order thin-walled beam theory isobtained, and the corresponding code is written. The specific contents are given as follows.
Firstly the first-order thin-walled beam theory is developed for flexural and torsionalproblems of open thin-walled beam. The expressions of flexural stress, deformation andtransverse shear coefficient are given. The relationships between transverse deflection androtation are obtained. When analyzing the restrained torsion of open thin-walled beam, thetotal rotation of section is divided into free warping rotation and restrained shear rotation. Theexpression of warping deformation, the warping normal stress and warping shear stress arederived based on this assumption. And the torsion shear coefficient is then obtained on thebasis of energy principle. Torsion shear coefficient could account for the true shear stressdistribution in the cross section. Program is compiled to calculate the geometric properties ofopen thin-walled section.
The new governing differential equations of the restrained torsion are derived and thecorresponding initial method is given to solve the equations. The relationship between totalrotation and free warping rotation is obtained. A parameter λ, which is associated with thestiffness property of a cross section and the beam length, is introduced to determine thecondition, under which the St. Venant constant is negligible. Consequently a simplifiedmethod of restrained torsion is derived. The simple method has the same style withTimoshenko beam theory. Numerical examples are illustrated to validate the current approachand the results of the current theory are compared with those of some other available methods.The results of comparison show that the current theory predicts the torsion behavior ofthin-walled beam more accurately.
The restrained torsion of closed thin-walled beam is analyzed. The first-order torsiontheory of closed thin-walled beam is proposed. The various stresses in closed cross section are analyzed and divided into categories, and the expressions of these stresses are given. Thetorsion shear coefficient is then obtained on the basis of energy principle. The governingdifferential equations of the restrained torsion of closed thin-walled beam are derived and thecorresponding initial method is given to solve the equation. These equations can be used asunified theory for restrained torsion of arbitrary cross section. The effect of restrained shearrotation on torsional behavior of closed thin-walled beam is investigated through numericalanalysis.
The thin-walled beam element is derived. The displacement mode of thin-walled beamelement is obtained through the relationship between deflection and rotation and that betweentotal rotation and free warping rotation. Only two nodes are needed in this displacement mode,which avoid the deriving process of beam element with internal node. The new beam elementis obtained through the updated Lagrange (UL) method on the basis of finite deformationtheory in continuum mechanics. This new beam element avoids the phenomenon of shearlocking and includes the effects of warping deformation and the coupling of flexure andtorsion.
An incremental method is proposed based on UL method in order to include the P-δeffect more accurately. This method makes up the defect of the cubic interpolation function intangential stiffness matrix and avoids the problem of variation of stiffness matrix derivedfrom stable interpolation function with the direction of axial force.
It is assumed that the Von Mises yield criterion, associated flow rules and isotropichardening model are applied to the material that used in this paper. Plastic-zone method isused and the thin-walled section is divided into a number of finite parts along the s directionand three Guass integral points are placed along the length of thin-walled beam in order totrace the plasticity development in the beam. And the plasticity development can be judged bythe stress condition in finite parts.
Finally, on the basis of derivation of new thin-walled beam element, nonlinear analysis program is compiled by using the object oriented program language C++. And numericalexamples are illustrated to validate the current approach. The results of the current theory arecompared with those of some other available methods. The results of comparison show thatthe current theory provides more accurate results and avoids the phenomenon of shearlocking.
引文
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