非理想邊界拱的面內(nèi)失穩(wěn)模式與屈曲荷載

康厚軍+易壯鵬+曾有藝

摘要：將拱結(jié)構(gòu)中既非固結(jié)也非鉸支的非理想邊界考慮為沿不同方向的具有一定剛度的彈性約束，利用變形幾何關(guān)系和能量變分原理推導(dǎo)了拱的非線性平衡方程.以圓弧拱為例建立了徑向均布荷載下外荷載與結(jié)構(gòu)內(nèi)力、徑向位移之間的關(guān)系，通過定義拱的深淺參數(shù)和臨界約束剛度進(jìn)行分析并得到了跳躍屈曲、分岔屈曲等的發(fā)生條件及存在區(qū)間.本文方法所得屈曲路徑和屈曲荷載與有限元法所得結(jié)論吻合良好，且用數(shù)值法分析了不同約束剛度時(shí)的屈曲路徑和臨界荷載.結(jié)果表明，臨界深淺參數(shù)和臨界約束剛度對(duì)圓弧拱的屈曲模式及屈曲臨界荷載影響顯著.

關(guān)鍵詞：屈曲;圓弧拱;非理想邊界;分岔屈曲;變分原理

中圖分類號(hào)：O343.9 文獻(xiàn)標(biāo)識(shí)碼：A

Planar Buckling Mode and Critical Load

for Arch Structure with Non-ideal Boundary

KANG Hou-jun1， YI Zhuang-peng2， ZENG You-yi2

（1. College of Civil Engineering， Hunan Univ，Changsha，Hunan410082， China;

2.School of Civil Engineering and Architecture， Changsha Univ of Science and Technology，Changsha，Hunan410114， China）

Abstract： The non-ideal boundary conditions （neither fixed nor hinged） of arch structure were considered as an elastic constraint with certain stiffness in different directions， and the nonlinear equilibrium equation was determined by using deformation geometric relation and energy variation principle. A circular arch under radial uniform load was taken as an example to establish the relationships between the external load and the internal force， and the radial displacement. By defining the shallowness and critical constraint stiffness， the snap-through buckling and bifurcation buckling were studied and the occurrence condition and distribution range were investigated. The buckling path and critical buckling load in the proposed method were in good agreement with the results from the finite element method. And the numerical method was used to study the buckling path and critical buckling load for different stiffness of elastic constraint. The results show that the critical shallowness and the critical constraint stiffness play a fundamental role in the buckling mode and critical buckling load for circular arch.

Key words： buckling; circular arch; non-ideal boundary;bifurcation buckling; variation principle

拱結(jié)構(gòu)[1]在土木、機(jī)械和航天航空等領(lǐng)域應(yīng)用廣泛.拱作為一種基本結(jié)構(gòu)構(gòu)件具有優(yōu)良的受力特性，其力學(xué)特性受到國(guó)內(nèi)外學(xué)者[2-3]廣泛關(guān)注.如周期激勵(lì)下內(nèi)共振[4-6]時(shí)的分岔和混沌特性，沖擊荷載作用下彈性淺拱的跳躍屈曲[7]等.靜力方面，近年來，Pi等[8-10]采用解析法與有限元法對(duì)各種荷載與邊界條件下拱結(jié)構(gòu)的非線性屈曲特性進(jìn)行了深入系統(tǒng)的研究.衛(wèi)星等[11]探討了多種參數(shù)對(duì)拱結(jié)構(gòu)考慮2階效應(yīng)時(shí)彈性屈曲特性的影響.程鵬和童根樹[12]綜述了徑向均布荷載下圓弧拱的面內(nèi)屈曲特性.郭彥林等[13]提出了壓彎圓弧拱平面內(nèi)穩(wěn)定承載力的設(shè)計(jì)建議公式.

這些文獻(xiàn)側(cè)重于研究邊界為理想固結(jié)或鉸支時(shí)拱的力學(xué)性能.結(jié)構(gòu)的復(fù)雜分析在很多情況下需考慮復(fù)雜邊界，如：大跨系桿拱橋中系桿將兩端連起來，系桿與豎向彈性支座、地基的作用可抽象為軸向、徑向彈性約束;機(jī)械拱臂或曲臂與相鄰結(jié)構(gòu)彈性連接，共同受力，可將其考慮為彈性約束;彈性地基上的拱型結(jié)構(gòu)在外荷載作用下邊界考慮為彈性更加合理.本文以圓弧拱為例，將非理想邊界考慮為徑向、軸向彈性約束，通過能量變分原理[14]建立非線性平衡方程，得到外荷載與結(jié)構(gòu)內(nèi)力、位移的關(guān)系，并分析屈曲模式與結(jié)構(gòu)重要參數(shù)的關(guān)系.

1基本方程與穩(wěn)定分析

1.1變形幾何關(guān)系

圖1（a）所示圓弧坐標(biāo)下的圓弧拱，徑向均布荷載q增至一定程度時(shí)會(huì)發(fā)生分岔屈曲（圖1（b））或跳躍屈曲（圖1（c）），2Θ為展開角，R為半徑，θ為角坐標(biāo)，kvi， kwi（i=±Θ）分別為兩端徑向、軸向彈性約束剛度.屈曲前變形呈現(xiàn)非線性，求解屈曲荷載及變形時(shí)需考慮屈曲前非線性的影響.圖1（a）中拱上任意一點(diǎn)P總的軸向應(yīng)變?chǔ)臥=εm+εb，其中薄膜應(yīng)變?chǔ)舖和彎曲應(yīng)變?chǔ)舃分別為：