有限条法分析双参数弹性地基上板的静动响应
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摘要
弹性地基上板是土木工程中较常见的一种结构形式,应用非常广泛,如建筑物的基础,机场跑道,公路路面,试验台等,其分析方法得到广大理论工作者和工程人员的重视。随着现代工程建设的发展,对基础板的分析提出了更高的要求。
本文采用有限条法和弹簧体系分析了弹性地基上板的静动响应,并将结果与有限元线法、有限条法、级数精确解等进行比较,证明本文方法具有足够的精确性与实际的可行性。有限条法是一种半解析单元法,是解析方法与数值方法相结合的一种方法。它兼有数值方法和解析法的优点,因而在许多工程分析中得到应用。计算中本文应用Matlab数学软件编制出求解该问题的有限条程序,程序适用性较强,可以求解双参数弹性地基上任意边界条件组合下任意竖向荷载(包括板面上任意位置的均布荷载、集中荷载以及两者的组合)作用下的板内任意点的位移场、内力场以及弹性地基板的自由振动频率。本文利用该程序计算大量的算例,对所得结果进行了分析与研究,并总结了各种边界条件下板与地基相互作用的规律,即探讨了地基参数k、t以及荷载q等不同因素对于双参数地基上的板挠度、内力和自由振动频率的影响。给出了双参数地基上板在荷载作用下的内力分布图和挠度变化的表面三维图,从而可以从总体上直观的观察到板的内力分布和挠度变化的规律。本文最后计算了弹性地基上不同宽厚比板的中点挠度值,并与中厚板理论的加权残值解进行比较,根据其相对的误差绝对值,对弹性地基上薄板和中厚板尝试进行了大致的界定。
ThePlateonelasticfoundationisacommonmodelforseveraltypesofcivilengineeringstructures, which has received considerable attention from researchers and engineers. Thebuildingfoundation,runways ofairports, roadways andtest-bedareusuallymodeledas platesonelasticfoundation.Withtherapiddevelopments inthefieldofconstruction,theanalysis oftheplatesonelasticfoundationisofvitalimportance.
Aprocedure incorporating the finite strip method and a spring system has been used totreat the static and dynamic response of plates resting on elastic foundation. The finite stripmethod results were compared with other methods such as finite element method of lines,finite element analysis and closed solutions, showing the finite strip method is correct andfeasible. Finite strip method is a semi-analytical numerical method which has advantages ofthe analytical methods and numerical methods. Finite strip method is a procedureincorporating analytical method and numerical method which was used widely in theconstructional analysis. In this article, a program about the finite strip method is compiled byusing Matlab mathematics software to treat the static and dynamic response of the plates ontwo-parameter elasticfoundation.Theprogram has broad applicability,whichcan analyzethedisplacement fields and the internal force fields of plates on two-parameter elastic foundationwith arbitrary boundary and arbitrary loading conditions(contains uniform line load,concentrated load or combined loads on arbitrary position of plates). This program is alsoapplied on the dynamical analysis of plates on two-parameter elastic foundation, the naturefrequencyof plates with arbitraryboundarywas researched. Analyzingand researchingin theresultsoftheplate’sdeflection、bendingmomentandthenaturefrequencyfromcalculatingamass of examples by the program. The laws of interactions of plates and foundation witharbitrary boundary are summarized. Namely, discuss the influence of the foundation’sparametersk,t,theloadqandotherfactorsonthedeflection、bendingmomentandthenaturefrequency of plates on two-parameter elastic foundation. The distributed chart of bendingmoment and the surface 3-D chart of deflection are given, from which we can visuallyobserve the distributing of bending moment and the variational directions of deflection as awhole. The article gives the results of rectangular plates under various width -thickness ratiosand the results are compared with the Reissner’s plate’s solution which was given by aweighted residual method. At last the article gives a rough limitation between thin and thickplatesaccordingtotheabsolutevaluesoftheirrelativeerrors.
本文采用有限条法和弹簧体系分析了弹性地基上板的静动响应,并将结果与有限元线法、有限条法、级数精确解等进行比较,证明本文方法具有足够的精确性与实际的可行性。有限条法是一种半解析单元法,是解析方法与数值方法相结合的一种方法。它兼有数值方法和解析法的优点,因而在许多工程分析中得到应用。计算中本文应用Matlab数学软件编制出求解该问题的有限条程序,程序适用性较强,可以求解双参数弹性地基上任意边界条件组合下任意竖向荷载(包括板面上任意位置的均布荷载、集中荷载以及两者的组合)作用下的板内任意点的位移场、内力场以及弹性地基板的自由振动频率。本文利用该程序计算大量的算例,对所得结果进行了分析与研究,并总结了各种边界条件下板与地基相互作用的规律,即探讨了地基参数k、t以及荷载q等不同因素对于双参数地基上的板挠度、内力和自由振动频率的影响。给出了双参数地基上板在荷载作用下的内力分布图和挠度变化的表面三维图,从而可以从总体上直观的观察到板的内力分布和挠度变化的规律。本文最后计算了弹性地基上不同宽厚比板的中点挠度值,并与中厚板理论的加权残值解进行比较,根据其相对的误差绝对值,对弹性地基上薄板和中厚板尝试进行了大致的界定。
ThePlateonelasticfoundationisacommonmodelforseveraltypesofcivilengineeringstructures, which has received considerable attention from researchers and engineers. Thebuildingfoundation,runways ofairports, roadways andtest-bedareusuallymodeledas platesonelasticfoundation.Withtherapiddevelopments inthefieldofconstruction,theanalysis oftheplatesonelasticfoundationisofvitalimportance.
Aprocedure incorporating the finite strip method and a spring system has been used totreat the static and dynamic response of plates resting on elastic foundation. The finite stripmethod results were compared with other methods such as finite element method of lines,finite element analysis and closed solutions, showing the finite strip method is correct andfeasible. Finite strip method is a semi-analytical numerical method which has advantages ofthe analytical methods and numerical methods. Finite strip method is a procedureincorporating analytical method and numerical method which was used widely in theconstructional analysis. In this article, a program about the finite strip method is compiled byusing Matlab mathematics software to treat the static and dynamic response of the plates ontwo-parameter elasticfoundation.Theprogram has broad applicability,whichcan analyzethedisplacement fields and the internal force fields of plates on two-parameter elastic foundationwith arbitrary boundary and arbitrary loading conditions(contains uniform line load,concentrated load or combined loads on arbitrary position of plates). This program is alsoapplied on the dynamical analysis of plates on two-parameter elastic foundation, the naturefrequencyof plates with arbitraryboundarywas researched. Analyzingand researchingin theresultsoftheplate’sdeflection、bendingmomentandthenaturefrequencyfromcalculatingamass of examples by the program. The laws of interactions of plates and foundation witharbitrary boundary are summarized. Namely, discuss the influence of the foundation’sparametersk,t,theloadqandotherfactorsonthedeflection、bendingmomentandthenaturefrequency of plates on two-parameter elastic foundation. The distributed chart of bendingmoment and the surface 3-D chart of deflection are given, from which we can visuallyobserve the distributing of bending moment and the variational directions of deflection as awhole. The article gives the results of rectangular plates under various width -thickness ratiosand the results are compared with the Reissner’s plate’s solution which was given by aweighted residual method. At last the article gives a rough limitation between thin and thickplatesaccordingtotheabsolutevaluesoftheirrelativeerrors.
引文
[1] Cheung Y K,Zienkie wicz O C. Plates and tanks on elastic foundations –Anapplication on finite element method.International Journal of Solids Structure,1965,1(4): 451-461
[2] Balas J,Sladek V,Sladek J.The boundary integral equation method for platesresting on a two-parameter foundation. ZAMM,1984, 64(4):137-146
[3] Katsikadelis J T,Kallivokas L F.Clamped plates on Pasternak-type elasticfoundation by the boundary element method.Journal of Applied Mechanics,1986,53(4):909-917
[4] Wang J G, Huang M K.Fundamental solutions and boundary integral equationsfor Reissner's plates on two parameter foundations.International Journal ofSolids Structure,1992,29(8):1233-1239
[5] Liew K M,Han J-B, Xiao Z M.Differential quadrature method for Mindlinplates of Winkler foundations.International Journal of Mechanical Sciences,1996, 38(4):405-421
[6] Han J B,Liew K M.Numerical differential quadrature method for mindlin plateson two-parameter foundations.International Journal of Mechanics Science,1997,39(9): 977-989
[7] Huang M H.Analysis of plate resting on elastic supports and elastic foundationby finite strip method.Computers and Structures, 2001,79(29-30):2547-2557
[8] 陈树坚,彭育池.多种地基假定下弹性基础板的样条有线元计算.中山大学学报,1980,8(2):15-23
[9] 张建辉.双参数地基的无单元法.岩土工程学报,2005,27(1):777-779
[10] 崔敏文.文克勒地基上板的有限条法.建筑结构学报,1992,13(2):49-55
[11] 秦荣,何昌如.弹性地基薄板的静力和动力分析.土木工程学报,1988,21(3):71-80
[12] 李刚,罗世平,尚守平.Vlazov地基上的Reissner板相互作用的加权残值分析法.湖南大学学报:自然科学版,2006,33(1):20-24
[13] Liu Qiong,Wang YuanHang.A triangular finite element in the study of finiteplate on elastic foundation. Proceeding of the Second International Conferenceon Soft Solid Engineer. Nanjing: Hehai University Press,1996,351-356
[14] 吴 波 . 弹 性 地 基 上 不 规 则 厚 板 的 荷 载 应 力 分 析 . 重 庆 交 通 学 院 学报,1992,4(1):1-10
[15] 张绍川,吴波.Winkler 弹性地基上异型中厚板的应力分析.武汉城市建设院学报,1995,12(2):63-69
[16] 王元汉,邱先敏,张佑启.弹性地基板的等参有线元法计算.岩土工程学报,1998,20(4): 7-11
[17] 曹广德,李同春.半无限弹性地基上任意形状弹性基础板的有限元分析.东北水利水电,2002,20(1):3-5
[18] 凌道盛,陈云敏,丁皓江.任意基础板的有限元分析.岩土工程学报,2000,22(4):450-455
[19] 王 祥 来 ,李 同 春 .异 形 弹 性 基 础 板 的 有 限 元 分 析 .水 利 水 运 工 程 学报,2002,1(1):49-51
[20] 何芳社,黄义,郭雅云.双参数弹性地基板的动力问题.西安建筑科技大学学报,2006,38(1):130-133
[21] 杨瑞生,黄炎,潘军.双参数弹性地基上板的自由振动.动力学与控制学报,2004,2(1):92-95
[22] 黄炎,廖瑛,谢燕.双参数弹性地基上受压的正交异形板的自由振动.工程力学,2006,23(6): 46-49
[23] Kanaka K,Raju,Hinton E.Natural frequencies and modes of rhombic Mindlinplates. Earthquake Engineering and Structural Dynamics,1980,8(1):55-62
[24] Ganesan N,Nagaraja Rao.Vibration analysis of moderately thick skew plates bya variational approach.Journal of Sound and Vibration,1985,101(7):117-119
[25] McGee O G,Leissa A W.Three-dimensional free vibrations of thick skewedcantilevered plates.Journal of Sound and Vibration,1991,144(1):305-322
[26] Wang G,Hsu C T.Static and dynamic analysis of arbitrary quadrilateral flexuralplates by B-3 spline functions.International Journal of Solids and Structures,1994, 31(5): 657-667
[27] Karunasena W,Liew K M,Al-Bermani F G A.Natural frequencies of thickarbitrary quadrilateral plates using the pb–2 Ritz method.Journal of Sound andVibration,1996,196(4):371-385
[28] Karunasena W,Kitipornchai S,Al-Bermani F G A.Free vibration ofcantilevered arbitrary triangular Mindlin plates.International Journal ofMechanical Sciences,1996,38(4):431-442
[29] Barik M,Mukhopadhyay M.Finite element free flexural vibration analysis ofarbitrary plates.Finite Elements in Analysis and Design,1998,29(2):137-151
[30] 武兰河.任意四边形厚板的自由振动.力学季刊,2001,22(2): 258-263
[31] 武兰河,赵永茂,李延强.求解任意形状厚板自由振动的微分容积法.应用力学学报,2003,20(1):149-153
[32] 贺国京,陈大鹏.动态有限元法及其在薄板弯曲振动中的应用.长沙铁道学院报,1999,17(1):1-6
[33] 熊渊博,龙述尧.用局部Petrov-Galerkin法分析薄板自由振动.力学季刊,2004,25(4):577-582
[34] 钟阳,周福霖,张永山.弹性地基上四边自由矩形薄板振动分析的Kantorovich法.振动与冲击,2007,26(3):33-36
[35] 王博,寇磊,徐建国.大型渡槽有限条法动力建模研究.世界地震工程,2007,23(1): 73-79
[36] 向俊,赫丹,曾庆元.横向有限条与无砟轨道板段单元的车轨系统竖向振动分析法.铁道学报,2007,29(4):64-69
[37] 张多新,王卫平,施兴会.加劲柱壳结构外压屈曲问题的柱壳有限条元法.水力发电,2006,32(10):47-49
[38] 杨开云,张多新,白新理.柱壳有限条元法在预应力U 型薄壳渡槽稳定性分析中的应用.水利学报,2006,37(5):598-602
[39] 方自虎.三维退化壳条分析钢筋混凝土板壳.工程力学,2006,23(7):165-169
[40] 慕光波,王连广.钢骨高强混凝土受弯构件非线性分析.东北大学学报,2006,27(5):571-574
[41] 陈 培 奇 ,李 忠 献 .箱 梁 横 向 内 力 的 有 限 条 法 .天 津 城 市 建 设 学 院 学报,2003,9(3):168-172
[42] 谢 友 军 , 孙 冬 梅 . 高 层 剪 力 墙 分 析 的 超 级 有 限 条 法 . 燕 山 大 学 学报,2002,26(2):171-173
[43] 程远胜,张佑启,区达光.有限条法分析板在移动车载作用下的动响应.应用数学与力学,2002,23(5):453-458
[44] 张佑启.结构分析有限条法.北京:人民交通出版社,1980,68-197
[45] 严人觉,王贻荪,韩清宇.动力基础半空间理论概论.北京:中国建筑工业出版社,1981,205-220
[46] 李永彪,张德澄.双参数弹性地基上中厚板弯曲问题的有限元线法分析.土木工程学报,2002,35(5):87-92
[47] 陈军,陈北雁.弹性地基上的任意四边形板的计算.见:第一届全国解析与数值结合法会议论文集.长沙:湖南大学出版社,1990,460-467
[48] 铁摩辛柯,沃诺斯基.板壳理论.北京:科学出版社,1977,206-217
[49] 生 跃 , 黄 义 . 双 参 数 弹 性 地 基 上 自 由 边 矩 形 . 应 用 数 学 与 力学,1987,8(4):317-329
[2] Balas J,Sladek V,Sladek J.The boundary integral equation method for platesresting on a two-parameter foundation. ZAMM,1984, 64(4):137-146
[3] Katsikadelis J T,Kallivokas L F.Clamped plates on Pasternak-type elasticfoundation by the boundary element method.Journal of Applied Mechanics,1986,53(4):909-917
[4] Wang J G, Huang M K.Fundamental solutions and boundary integral equationsfor Reissner's plates on two parameter foundations.International Journal ofSolids Structure,1992,29(8):1233-1239
[5] Liew K M,Han J-B, Xiao Z M.Differential quadrature method for Mindlinplates of Winkler foundations.International Journal of Mechanical Sciences,1996, 38(4):405-421
[6] Han J B,Liew K M.Numerical differential quadrature method for mindlin plateson two-parameter foundations.International Journal of Mechanics Science,1997,39(9): 977-989
[7] Huang M H.Analysis of plate resting on elastic supports and elastic foundationby finite strip method.Computers and Structures, 2001,79(29-30):2547-2557
[8] 陈树坚,彭育池.多种地基假定下弹性基础板的样条有线元计算.中山大学学报,1980,8(2):15-23
[9] 张建辉.双参数地基的无单元法.岩土工程学报,2005,27(1):777-779
[10] 崔敏文.文克勒地基上板的有限条法.建筑结构学报,1992,13(2):49-55
[11] 秦荣,何昌如.弹性地基薄板的静力和动力分析.土木工程学报,1988,21(3):71-80
[12] 李刚,罗世平,尚守平.Vlazov地基上的Reissner板相互作用的加权残值分析法.湖南大学学报:自然科学版,2006,33(1):20-24
[13] Liu Qiong,Wang YuanHang.A triangular finite element in the study of finiteplate on elastic foundation. Proceeding of the Second International Conferenceon Soft Solid Engineer. Nanjing: Hehai University Press,1996,351-356
[14] 吴 波 . 弹 性 地 基 上 不 规 则 厚 板 的 荷 载 应 力 分 析 . 重 庆 交 通 学 院 学报,1992,4(1):1-10
[15] 张绍川,吴波.Winkler 弹性地基上异型中厚板的应力分析.武汉城市建设院学报,1995,12(2):63-69
[16] 王元汉,邱先敏,张佑启.弹性地基板的等参有线元法计算.岩土工程学报,1998,20(4): 7-11
[17] 曹广德,李同春.半无限弹性地基上任意形状弹性基础板的有限元分析.东北水利水电,2002,20(1):3-5
[18] 凌道盛,陈云敏,丁皓江.任意基础板的有限元分析.岩土工程学报,2000,22(4):450-455
[19] 王 祥 来 ,李 同 春 .异 形 弹 性 基 础 板 的 有 限 元 分 析 .水 利 水 运 工 程 学报,2002,1(1):49-51
[20] 何芳社,黄义,郭雅云.双参数弹性地基板的动力问题.西安建筑科技大学学报,2006,38(1):130-133
[21] 杨瑞生,黄炎,潘军.双参数弹性地基上板的自由振动.动力学与控制学报,2004,2(1):92-95
[22] 黄炎,廖瑛,谢燕.双参数弹性地基上受压的正交异形板的自由振动.工程力学,2006,23(6): 46-49
[23] Kanaka K,Raju,Hinton E.Natural frequencies and modes of rhombic Mindlinplates. Earthquake Engineering and Structural Dynamics,1980,8(1):55-62
[24] Ganesan N,Nagaraja Rao.Vibration analysis of moderately thick skew plates bya variational approach.Journal of Sound and Vibration,1985,101(7):117-119
[25] McGee O G,Leissa A W.Three-dimensional free vibrations of thick skewedcantilevered plates.Journal of Sound and Vibration,1991,144(1):305-322
[26] Wang G,Hsu C T.Static and dynamic analysis of arbitrary quadrilateral flexuralplates by B-3 spline functions.International Journal of Solids and Structures,1994, 31(5): 657-667
[27] Karunasena W,Liew K M,Al-Bermani F G A.Natural frequencies of thickarbitrary quadrilateral plates using the pb–2 Ritz method.Journal of Sound andVibration,1996,196(4):371-385
[28] Karunasena W,Kitipornchai S,Al-Bermani F G A.Free vibration ofcantilevered arbitrary triangular Mindlin plates.International Journal ofMechanical Sciences,1996,38(4):431-442
[29] Barik M,Mukhopadhyay M.Finite element free flexural vibration analysis ofarbitrary plates.Finite Elements in Analysis and Design,1998,29(2):137-151
[30] 武兰河.任意四边形厚板的自由振动.力学季刊,2001,22(2): 258-263
[31] 武兰河,赵永茂,李延强.求解任意形状厚板自由振动的微分容积法.应用力学学报,2003,20(1):149-153
[32] 贺国京,陈大鹏.动态有限元法及其在薄板弯曲振动中的应用.长沙铁道学院报,1999,17(1):1-6
[33] 熊渊博,龙述尧.用局部Petrov-Galerkin法分析薄板自由振动.力学季刊,2004,25(4):577-582
[34] 钟阳,周福霖,张永山.弹性地基上四边自由矩形薄板振动分析的Kantorovich法.振动与冲击,2007,26(3):33-36
[35] 王博,寇磊,徐建国.大型渡槽有限条法动力建模研究.世界地震工程,2007,23(1): 73-79
[36] 向俊,赫丹,曾庆元.横向有限条与无砟轨道板段单元的车轨系统竖向振动分析法.铁道学报,2007,29(4):64-69
[37] 张多新,王卫平,施兴会.加劲柱壳结构外压屈曲问题的柱壳有限条元法.水力发电,2006,32(10):47-49
[38] 杨开云,张多新,白新理.柱壳有限条元法在预应力U 型薄壳渡槽稳定性分析中的应用.水利学报,2006,37(5):598-602
[39] 方自虎.三维退化壳条分析钢筋混凝土板壳.工程力学,2006,23(7):165-169
[40] 慕光波,王连广.钢骨高强混凝土受弯构件非线性分析.东北大学学报,2006,27(5):571-574
[41] 陈 培 奇 ,李 忠 献 .箱 梁 横 向 内 力 的 有 限 条 法 .天 津 城 市 建 设 学 院 学报,2003,9(3):168-172
[42] 谢 友 军 , 孙 冬 梅 . 高 层 剪 力 墙 分 析 的 超 级 有 限 条 法 . 燕 山 大 学 学报,2002,26(2):171-173
[43] 程远胜,张佑启,区达光.有限条法分析板在移动车载作用下的动响应.应用数学与力学,2002,23(5):453-458
[44] 张佑启.结构分析有限条法.北京:人民交通出版社,1980,68-197
[45] 严人觉,王贻荪,韩清宇.动力基础半空间理论概论.北京:中国建筑工业出版社,1981,205-220
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