楊成永 馬文輝 韓薛果 程霖
摘? ?要:以矩形板的Navier解為基礎(chǔ),采用帶補(bǔ)充項(xiàng)的傅里葉級(jí)數(shù)作為撓度函數(shù),研究了局部均布荷載作用下四邊支承矩形薄板的彎曲問題. 推導(dǎo)了確定待定系數(shù)的線性代數(shù)方程組,給出了簡(jiǎn)支邊和固支邊不同組合條件下的統(tǒng)一計(jì)算公式. 討論了帶補(bǔ)充項(xiàng)法級(jí)數(shù)解的收斂速度,并與疊加法級(jí)數(shù)解及有限元數(shù)值解分別進(jìn)行了精度和計(jì)算量的對(duì)比. 結(jié)果表明,帶補(bǔ)充項(xiàng)法的級(jí)數(shù)解達(dá)到收斂的級(jí)數(shù)項(xiàng)數(shù)約為40項(xiàng). 帶補(bǔ)充項(xiàng)法的級(jí)數(shù)解與疊加法級(jí)數(shù)解具有同樣的求解精度. 有限元解隨網(wǎng)格的細(xì)分,計(jì)算結(jié)果逐漸接近級(jí)數(shù)法解. 級(jí)數(shù)解法的計(jì)算量與有限元解法相比是微不足道的. 研究成果適于進(jìn)行構(gòu)筑物頂板受局部均布荷載作用的結(jié)構(gòu)計(jì)算.
關(guān)鍵詞:矩形板;四邊支承;局部均布荷載;級(jí)數(shù)解;求解精度
中圖分類號(hào):U411? ? ? ? ? ? ? ? ? ? ? ? ? ?文獻(xiàn)標(biāo)志碼:A
Internal Force Calculation of Four Edges Supported Rectangular
Plates under Local Uniformly Distributed Load
YANG Chengyong,MA Wenhui?,HAN Xueguo,CHENG Lin
(School of Civil Engineering,Beijing Jiaotong University,Beijing 100044,China)
Abstract:On the basis of Naviers solution to rectangular plates, the bending problem was studied for the four edges supported thin plates under local uniformly distributed load, where the double Fourier series with additional terms was adopted as the deflection function of the plates. Linear algebraic equations for solving the undetermined coefficients were derived. A unified solution was obtained to the rectangular plates with clamped and simply supported edges. The rate of convergence was discussed on the solution of the series method with additional terms. The proposed method was compared both with superposition series method on accuracy, and with finite element numerical method on computational cost. The results show that 40 terms should be employed for a convergence of the series. The method with additional terms shows the same accuracy of solution as superposition series method does. The solution by finite element method gradually approaches that by the series method as the mesh gets finer and finer. In comparison with finite element method, the computational time by the series method is negligible. This work is applicable for structural analysis of the top plates of underground buildings under truck wheel pressure.
Key words:rectangular plate;four supported edges;locally uniformly distributed load;series solution;solution accuracy
地鐵、熱力和燃?xì)獾鹊叵鹿こ讨?,地下?gòu)筑物的頂板多為四邊支承的薄板,板上常承受局部均布荷載如汽車輪壓作用. 為了確定像汽車輪壓這類荷載在板內(nèi)產(chǎn)生的最大撓度和內(nèi)力,需要進(jìn)行任意位置局部均布荷載作用下?lián)隙群蛢?nèi)力的計(jì)算.
對(duì)四邊支承的矩形薄板問題,可以從四邊簡(jiǎn)支板的Navier解出發(fā),采用疊加方法[1-2]或加補(bǔ)充項(xiàng)的方法[3-4]解決. 如:蔡長安等[5-6]以帶附加補(bǔ)充項(xiàng)的Fourier級(jí)數(shù)作為撓度函數(shù),求解了Winkler地基及Pasternak地基上自由邊矩形板的彎曲問題. 許琪樓等[7-8]采用一種能滿足自由角點(diǎn)條件的撓度表達(dá)式,解決了二鄰邊支承二鄰邊自由矩形板和二鄰邊及對(duì)角點(diǎn)支承矩形板的彎曲問題. 他們還采用疊加方法[9-10],提出了四邊支承矩形板及一邊固定一角點(diǎn)或二角點(diǎn)支承的矩形板的統(tǒng)一求解方法. 岳建勇等[11-12]采用一種雙三角級(jí)數(shù)形式的撓度函數(shù),得到了三邊固定一邊自由及兩對(duì)邊固定兩對(duì)邊自由矩形板的精確解. 鐘陽等[13]在辛幾何空間中利用分離變量法推導(dǎo)出了四邊固支彈性矩形薄板的精確解析表達(dá)式. 于天崇等[14]假定矩形板的抗彎剛度沿板的寬度方向按照一般冪函數(shù)形式變化,研究了四邊簡(jiǎn)支一對(duì)邊受彎作用下面內(nèi)變剛度矩形板的彎曲問題. 肖閃閃等[15]采用載荷疊加法研究了集中載荷下四邊固支正交各向異性矩形板的線性彎曲,并討論了經(jīng)典Kirchhoff薄板假設(shè)對(duì)于集中載荷的適用性.
4.4.2? ?與文獻(xiàn)[16]對(duì)比
文獻(xiàn)[16]列出了四邊簡(jiǎn)支板中央受局部均布荷載作用時(shí)彎矩的計(jì)算系數(shù). 為與其對(duì)比并避免查表計(jì)算中的插值,取泊松比μ = 0,輪壓x方向分布長度c = 2 m,y方向分布長度d = 1 m,其余參數(shù)采用4.2節(jié)的數(shù)據(jù),板四邊均簡(jiǎn)支.
根據(jù) =? = 1.4, =? = 0.4, =? = 0.2,按文獻(xiàn)[16] 中表4-29查得計(jì)算彎矩Mx的系數(shù)為0.148 0,My的系數(shù)為0.130 8. 然后有
計(jì)算結(jié)果列于表3.
由表3看出,兩種方法的結(jié)果,前3位有效數(shù)字相同. 由于文獻(xiàn)[16]表格的有效數(shù)字是4位,可以認(rèn)為表3中兩種結(jié)果是一致的.
4.5? ?計(jì)算結(jié)果與有限元對(duì)比
采用4.2節(jié)的計(jì)算參數(shù),板左邊(x = 0邊)及前邊(y = 0邊)固支,其余兩邊簡(jiǎn)支. 按相同的參數(shù)和邊界條件采用ANSYS軟件SHELL63號(hào)單元,劃分3種粗細(xì)不同網(wǎng)格進(jìn)行計(jì)算. 計(jì)算結(jié)果列于表4.
由表4可看出:
1)隨有限元網(wǎng)格的加密,計(jì)算結(jié)果逐漸趨于本文的級(jí)數(shù)解. 由此可說明,本文級(jí)數(shù)解是四邊支承板變形問題的理論解或精確解. 當(dāng)有限元網(wǎng)格細(xì)到5 mm × 5 mm時(shí),撓度及彎矩有5位有效數(shù)字與級(jí)數(shù)解相同. 可以認(rèn)為這時(shí)數(shù)值解與級(jí)數(shù)解基本一致.
2)表4中的軟件運(yùn)行所用時(shí)間是從數(shù)據(jù)輸入到輸出全部的計(jì)算機(jī)運(yùn)行時(shí)間. 要達(dá)到較高的精度,有限元需要花費(fèi)的計(jì)算機(jī)時(shí)間大大高于級(jí)數(shù)解. 就本算例來說,相差達(dá)10萬倍以上. 需要注意的是,本算例中,有限元在5 mm × 5 mm網(wǎng)格時(shí),需要求解的方程組的階數(shù),不少于(x方向節(jié)點(diǎn)數(shù)5/0.005) × (y方向節(jié)點(diǎn)數(shù)7/0.005) × 6個(gè)自由度 = 840萬;而級(jí)數(shù)解取40項(xiàng)時(shí)需要求解的方程組的階數(shù)僅為40 × 2個(gè)固支邊 = 80.
5? ?結(jié)束語
采用帶補(bǔ)充項(xiàng)的撓度函數(shù),研究了四邊支承矩形薄板的彎曲問題. 給出了局部均布荷載作用下簡(jiǎn)支邊和固支邊不同組合條件下的統(tǒng)一計(jì)算公式.
對(duì)比計(jì)算表明,以Navier解為基礎(chǔ)帶補(bǔ)充項(xiàng)的傅里葉級(jí)數(shù)解,達(dá)到收斂的級(jí)數(shù)項(xiàng)數(shù)約為40項(xiàng). 該級(jí)數(shù)解與其他采用疊加法得到的傅里葉級(jí)數(shù)解,具有同樣的求解精度. 與有限元數(shù)值法相比,級(jí)數(shù)解的計(jì)算量十分微小.
值得一提的是,式(3)若換成滿布荷載、線荷載和集中力相應(yīng)的傅里葉系數(shù),本文方法也適用.
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