土木工程英語翻譯.doc

翻譯6 Lateral buckling of beams6.1 IntroductionIn the discussion given in Chapter 5 of the in-plane behaviour of beams, it was assumed that when a beam is loaded in its stiffer principal plane, it deflects only in that plane. If the beam does not have sufficient lateral stiffness or lateral support to ensure that this is so, then it may buckle out of the plane of loading, as shown in Fig. 6.1. The load at which this buckling occurs may be substantially less than the beams in-plane load carrying capacity, as indicated in Fig. 6.2.6.梁的側(cè)面翹曲6.1 說明在第五章關(guān)于梁的平面內(nèi)性能的討論中，假定梁按剛性主平面放置時，梁僅在該平面內(nèi)傾斜。如果梁沒有足夠的側(cè)向剛度或側(cè)面支撐，梁會發(fā)生平面外屈曲，如圖6.1所示。如圖6.2所示，當(dāng)發(fā)生平面外屈曲時梁的承載能力會大大減小。For an idealized perfectly straight elastic beam, there are no out-of-plane deformations until the applied moment M reaches the elastic buckling moment M0b, when the beam buckles by deflecting laterally and twisting, as shown in Fig. 6.1. These two deformations are interdependent: when the beam deflects laterally, the applied moment has a component which exerts a torque about the deflected longitudinal axis which causes the beam to twist. This behaviour, which is important for long unrestrained I-beams whose resistances to lateral bending and torsion are low, is called elastic flexural-torsional buckling.作為一個理想的彈性直梁，當(dāng)施加彎矩達到彈性屈曲彎矩時，梁才會發(fā)生側(cè)向彎曲和扭轉(zhuǎn)變形，發(fā)生平面外屈曲，如圖6.1所示。這兩種變形是相互聯(lián)系的：當(dāng)梁側(cè)向傾斜時，所承受的彎矩會對側(cè)向梁軸產(chǎn)生扭矩并引起梁扭轉(zhuǎn)。這種特性，對于抵抗側(cè)向彎曲和扭轉(zhuǎn)能力差的無限制I形梁來說很重要，被成為彎扭屈曲。The failure of a perfectly straight slender beam is initiated when the addi-tional stresses induced by elastic buckling cause first yield. However, a per-fectly straight beam of intermediate slenderness may yield before the elastic buckling moment is reached, because of the combined effects of the in-plane bending stresses and any residual stresses, and may subsequently buckle in- elastically, as indicated in Fig. 6.2. For very stocky beams, the inelastic buckling moment may be higher than the in-plane plastic collapse moment in which case the moment capacity of the beam is not affected by lateral buckling.理想彈性直梁的屈服始于因為因為彈性屈曲產(chǎn)生的附加應(yīng)力導(dǎo)致的屈服，然而，受平面內(nèi)彎曲應(yīng)力和殘余應(yīng)力的影響，理想彈性直梁的中間部位可能在到達屈服彎矩前先行屈服，并發(fā)生塑形彎曲，如圖6.2所示。對于短梁，其非彈性屈曲彎矩會大于平面內(nèi)塑形破壞彎矩，受彎承載力不由側(cè)向屈曲控制。In this chapter, the behaviour and design of beams which fail by lateral buckling and yielding are discussed. It is assumed that local buckling of the compression flange or of the web (which is dealt with in Chapter 4) does not occur. The behaviour and design of beams bent about both principal axes, and of beams with axial loads, are discussed in Chapter 7.在本章，將講述由側(cè)向屈曲和屈服引起破壞的梁的性能和設(shè)計方法。假設(shè)第四章中討論的局部屈曲不會發(fā)生。第七章將討論軸壓及壓彎構(gòu)件的性能和設(shè)計方法。6.2 Elastic beams6.2.1 BUCKLING OF STRAIGHT BEAMS 6.2.1.1 Simply supported beams with equal end momentsA perfectly straight elastic beam which is loaded by equal and opposite end moments is shown in Fig. 6.3. The beam is simply supported at its ends so that lateral deflection and twist rotation are prevented, while the flange ends are free to rotate in horizontal planes so that the beam ends are free to warp (see section 10.8.3). The beam will buckle at a moment A/0b when a deflected and twisted equilibrium position, such as that shown in Fig. 6.3, is possible. It is shown in section 6.10.1.1 that this position is given by where is the undetermined magnitude of the central deflection, and that the elastic buckling moment is given by （6.2）Wherewhere EIy is the minor axis flexural rigidity, GJ the torsional rigidity, and E/w the warping rigidity of the beam. Equation 6.3 shows that the resistance to buckling depends on the geometric mean of the flexural resistance and the torsional resistance.6.2.彈性梁6.2.1 直梁的屈曲6.2.1.1 端彎矩相等的簡支梁如圖6.3所示，一個承受相等梁端彎矩的理想彈性直梁。梁端簡支側(cè)向彎曲和扭轉(zhuǎn)不會發(fā)生，因為端部可以在平面內(nèi)自由轉(zhuǎn)動從而不限制梁端轉(zhuǎn)角。當(dāng)側(cè)移和扭轉(zhuǎn)達到平衡時，在作用下梁會彎曲，如圖6.3所示。這種情況在6.10.1.1中給出公式：為梁跨中撓度，大小未知，彈性屈曲彎矩計算公式為： （6.2）為側(cè)向彎曲剛度，為扭轉(zhuǎn)剛度，為翹曲剛度。公式6.3表示梁的抗屈曲能力取決于臨界彎矩以及臨界扭矩。Equation 6.3 ignores the effects of the major axis curvature and produces conservative estimates of the elastic buckling moment equal to times the true value. This correction factor, which is just less than unity for most beam sections but may be significantly less than unity for column sections, is usually neglected in design. Nevertheless, its value approaches zero as Iy approaches Ix so that the true elastic buckling moment approaches infinity. Thus an I-beam in uniform bending about its weak axis does not buckle, which is intuitively obvious. Research 1 has indicated that in some other cases the correction .factor may be close to unity, and that it is prudent to ignore the effect of major axis curvature.公式6.3忽略了強軸曲率，并且保守估計梁的屈曲彎矩等于真實彎矩乘以。這種修正，在大多數(shù)梁截面設(shè)計中是可以忽略的，柱的設(shè)計中則不然。它把梁的和當(dāng)作零處理，使得梁的彈性屈曲彎矩接近無窮大。很明顯，I型鋼梁不會繞弱軸屈曲。研究【1】表明在其他情形下修正值接近真實值，忽略強軸曲率是可以的。6.2.1.2 Beams with unequal end momentsA simply supported beam with unequal major axis end moments M and as shown in Fig. 6.4a. It is shown in section 6.10.1.2 that the value of the end Jimoment Mab at elastic flexural-torsional buckling can be expressed in the .-form of （6.4）in which the moment modification factor which accounts for the effect of the non-uniform distribution of the major axis bending moment can be closely approximated byor by6.2.1.2 端彎矩不等的簡支梁如圖6.4.a所示，一簡支梁承受端彎矩M和，6.10.1.2所示，彎扭屈曲梁端彎矩公式為： （6.4）表明強軸彎矩不均勻分配作用影響的修正系數(shù)近似表示為：或者These approximations form the basis of a very simple method of predicting the buckling of the segments of a beam which is loaded only by concentrated . loads applied through transverse members preventing local lateral deflection and twist rotation. In this case, each segment between load points may be treated as a beam with unequal end moments, and its elastic buckling moment may be estimated by using equation 6.4 and either equation 6.5 or 6.6 and by taking L as the segment length. Each buckling moment so calculated corre-sponds to a particular buckling load parameter for the complete load set, and the lowest of these parameters gives a conservative approximation of the ac-tual buckling load parameter. This simple method ignores any buckling in-teractions between the segments. A more accurate method which accounts for these interactions is discussed in section 6.6.2.2.這種近似是一種預(yù)測集中荷載作用下梁段的屈曲情況簡單方法，是為了防止截面發(fā)生側(cè)向位移或扭轉(zhuǎn)。在這種情況下，集中荷載作用點之間的梁可以被當(dāng)作承受不相等梁端彎矩的梁，其屈曲彎矩可以用公式6.4及6.5或6.6來表示，L為集中荷載作用點之間的梁段長度。每一個估計的屈曲彎矩對應(yīng)所有荷載中的一個具體屈曲荷載，其中的最小值給出了實際屈曲荷載系數(shù)的保守估計。這種簡化忽略了梁端間屈曲的相互影響。在6.6.2.2中將討論不忽略相互影響的更精確的方法。6.2.1.3 Beams with central concentrated loadsA simply supported beam with a central concentrated load Q acting at a distance above the centroidal axis of the beam is shown in Fig. 6.5a. When the beam buckles by deflecting laterally and twisting, the line of action of the load moves with the central cross-section, but remains vertical, as shown in Fig. 6.5c. The case when the load acts above the centroid is more dangerous than that of centroidal loading because of the additional torque which increases the twisting of the beam and decreases its resistance to buckling.如圖6.5.a所示，一個承受跨中集中荷載Q的簡支梁，Q作用位置距離梁中軸線為。當(dāng)梁發(fā)生側(cè)向偏移和扭轉(zhuǎn)時，荷載作用線隨著跨中截面的的旋轉(zhuǎn)而移動，但仍保持垂直，如圖6.5.C所示。這種荷載不過形心的情況比荷載過形心更加危險，因為附加扭矩增加了粱所受的扭矩，降低了它的抵抗屈曲能力。 It is shown in section 6.10.1.3 that the dimensionless buckling load varies as shown in Fig. 6.6 with the beam parameter and the dimensionless height of the point of application of the load given by (6.7)6.10.1.3中所示梁的屈曲荷載的無量綱系數(shù)同圖6.6中梁的系數(shù)，施加荷載的無量綱系數(shù)計算公式為： (6.7)For centroidal loading (), the elastic buckling moment in-creases with the beam parameter K in much the same way as does the buckling moment of beams with equal and opposite end moments (see equation 6.3). Thus the elastic buckling moment can be written in the formin which the moment modification factor which accounts for the effect of the non-uniform distribution of major axis bending moment is approximately equal to 1.35.當(dāng)承受過重心的荷載（即）,彈性屈曲彎矩同承受梁端等大異號彎矩梁的屈曲彎矩一樣要乘以系數(shù)K(見6.3節(jié))，其彈性屈曲彎矩公式為：這表明強軸不均勻分配彎矩作用影響的彎矩修正系數(shù)近似等于1.35。The elastic buckling load also varies with the load height parameter s, and although the resistance to buckling is high when the load acts below the centroidal axis, it decreases significantly as the point of application rises, as shown in Fig. 6.6. For equal flanged I-beams, the parameter can be trans-formed into梁的屈曲彎矩隨著梁的荷載高度系數(shù)的變化而變化，當(dāng)荷載作用在梁的軸線以下時，梁的抗屈曲能力越高，當(dāng)荷載作用點升高時，抗屈曲能力明顯降低，如圖6.6所示。當(dāng)梁的上下翼緣相等時，適用于公式：where is the distance between flange centroids. The variation of the buckling load with lytjds is shown by the solid lines in Fig. 6,6, and it can be seen that the differences between top and bottom flange loading increase with the beam parameter K. This effect is therefore more important for deep beam-type sections of short span than for shallow column- type sections of long span. Approximate expressions for the variations of the moment modification factor am with the beam parameter K which account for the dimensionless load heightfor equal flanged I-beams are given in 2. Alternatively, the maximum moment at elastic buckling may be approximated by usingand , in which 表示翼緣形心間的距離，如圖6.6中所示，屈曲彎矩隨著的變化而變化。介于最高點和最低點之間的翼緣荷載隨著梁系數(shù)k的增加而增加。比起長跨短柱型截面來，這種影響在短跨深梁型截面中更加重要。彎矩修正系數(shù)隨著可以代表梁的無量綱系數(shù)作用的梁的系數(shù)K的變化而變化，在上下翼緣相等的I型梁中，它的近似表述在【2】中給出?；蛘撸旱淖畲笄鷱澗亟七m用于公式：當(dāng)時，6.2.1.4 Other loading conditionsThe effect of the distribution of the applied load along the length of a simply supported beam on its elastic buckling strength has been investigated numeri-cally by many methods, including those discussed in 3-5. A particularly powerful computer method is the finite element method 6-10, while the finite integral method 11, 12, which allows accurate numerical solutions of the coupled minor axis bending and torsion equations to be obtained, has been used extensively. Many particular cases have been studied 13 -16, and tabulations of elastic buckling loads are available 2, 3, 5, 13, 15, 17, as is a user-friendly computer program 18 for analysing elastic flexural-torsional buckling.6.2.1.4 其他荷載情況承受沿直線分布荷載的簡支梁的荷載分配情況對彈性屈曲長度的影響有很多種方法考慮，包括【3-5】中討論的。一種比較有效的方法是有限元法【6-10】，還有有限積分法【11，12】，其中有限積分法能夠得到弱軸屈曲和扭轉(zhuǎn)的精確解，因而被廣泛應(yīng)用。許多特殊情況適用于【13-16】，【2，3，5，13，15，17】適用于彈性屈曲荷載，【18】是一個可以分析彎扭屈曲的比較好的電腦程序。Some approximate solutions for the maximum moments elastic buckling of simply supported beams which are loaded along their centroidal axes are given in Fig. 6.7 by the moment modification factors in the equation （6.12）求承受沿軸線分布荷載的簡支梁的最大彈性屈曲彎矩的很多解決辦法在圖6.7中給出，彎矩修正系數(shù)公式為： （6.12）It can be seen that the more dangerous loadings are those which produce more nearly constant distributions of major axis bending moment, and that the worst case is that of equal and opposite end moments for which .可見危險荷載布置是那些產(chǎn)生恒定的強軸彎曲彎矩的情況，承受等大異號彎矩，的情況為最不利情況。For other beam loadings than those shown in Fig. 6.7, the moment modi-fication factor may be approximated by usingin which Mm is the maximum moment,，the moments at the quarter points, and the moment at the mid-point of the beam.對于圖6.7之外的情況，彎矩修正系數(shù)適用于公式:為最大彎矩，是1/4位置的彎矩，為梁跨中彎矩。6.2.1.5 CantileversThe support conditions of cantilevers differ from those of simply supported beams in that a cantilever is usually completely fixed at one end and com-pletely free at the other. The elastic buckling solution for a cantilever in uniform bending caused by an end moment M which rotates with the end of the cantilever 16 can be obtained from the solution given by equations 6.2 and 6.3 for simply supported beams by replacing the beam length L by twice the cantilever length 2L, whenceThis procedure is similar to the effective length method used to obtain the buckling load of a cantilever column (see Fig. 3.14).6.2.1.5懸臂梁懸臂梁的支撐情況與簡支梁不同，一端自由，一端固結(jié)。關(guān)于存在梁端彎矩引起的均勻彎曲的懸臂梁的彈性屈曲問題的解決，梁端轉(zhuǎn)角可以用針對簡支梁的公式6.2和6.3來求解，只要把長度L換為懸臂梁的長度的兩倍，此處：這種處理方法跟應(yīng)用于獲得懸臂柱屈曲荷載的有效長度法原理相同。（見圖3.14）Cantilevers with other loading conditions are not so easily analysed, but numerical solutions are available 16, 19-21. The particular case of a canti-lever with an end concentrated load Q is discussed in section 6.10.1.4, and plots of the dimensionless elastic buckling moments for bottom flange, centroidal, and top flange loading are given in Fig. 6.8, together with plots of the dimensionless elastic buckling moments of can-tilevers with uniformly distributed loads q.受其它形式荷載的懸臂梁就不這么容易求解了?！?6，19-21】是幾種使用的數(shù)學(xué)方法。6.10.1.4講述了受端部集中荷載的的懸臂梁的求解公式。如圖6.8所示上翼緣受中心荷載，下翼緣要乘以無量綱屈曲彎矩系數(shù)，受均布荷載q的懸臂梁無量綱屈曲彎矩系數(shù)為。6.2.2 BENDING AND TWISTING OF CROOKED BEAMSReal beams are not perfectly straight, but have small initial curvatures and twists which cause them to bend and twist at the beginning of loading. If a simply supported beam with equal and opposite end moments M has an initial curvature and twist which are given byin which the central initial lack of straightness and twist rotation are related bythen the deformations of the beam are given byin whichas shown in section 6.10.2. The variations of the dimensionless central de-flection and twist are shown in Fig. 6.9, and it can be seen that deformation begins at the commencement of loading, and increases rapidly as the elastic buckling moment M is approached.6.2.2 曲梁的彎曲與扭轉(zhuǎn)曲梁并不是理想直線，而是存在初始彎曲和初始扭曲，導(dǎo)致荷載剛作用便會發(fā)生彎曲和扭轉(zhuǎn)。如6.10.2節(jié)所示，如果作用有等大異號端彎矩M的簡支梁存在初彎曲和初扭轉(zhuǎn)，便有公式：初始彎曲缺陷和扭轉(zhuǎn)角之間滿足關(guān)系式：梁的變形符合公式：其中：圖6.9顯示了梁的中心撓度系數(shù)以及扭轉(zhuǎn)系數(shù)，可以看出變形始于荷載剛剛作用，隨著彎矩接近屈曲彎矩變形迅速增加。The simple load-deformation relationships of equations 6.17 and 6.18 are of the same forms as those of equations 3.6 and 3.7 for compression members with sinusoidal initial curvature. It follows that the Southwell plot technique for extrapolating the elastic buckling loads of compression members from experimental measurements (see section 3.2.2) may also be used for beams.關(guān)于公式6.17和6.18之間負(fù)荷變形的簡單關(guān)系同公式3.6和3.7之間的關(guān)系一樣，曲率為正弦曲線形式。通過針對受壓構(gòu)件的實驗得出結(jié)論它服從索斯維爾繪圖法，這同樣適用與梁。As the deformations increase with the applied moments M, so do the stresses. It is shown in section 6.10.2 that the limiting moment at which a beam without residual stresses first yields is given byin which is the nominal first yield moment, when the central lack of straightness is given by變形隨著施加彎矩的增加而增加，壓力也是。如6.10.2所示，一個沒有初次屈服產(chǎn)生殘余應(yīng)力的梁的限制彎矩服從公式：其中是名義初始屈曲彎矩，中心曲率給出公式：in which Noy is given by equation 6.11. Equation 6.19 is similar to equations 3.9 and 3.11 for the limiting axial force in an elastic compression member. The variation of the dimensionless limiting moment is shown in Fig. 6.10, in which the ratio plotted along the horizontal axis is equivalent to the modified slenderness ratio used in Fig. 3.4 for an elastic compression member. Figure 6.10 shows that the limiting moments of short beams ap-proach the yield moment My, while for long beams the limiting moments approach the elastic buckling moment .其中在公式6.11中給出解答，公式6.19同公式3.9和3.11一樣，是為了限制彈性壓縮構(gòu)件的軸力。無量綱限制彎矩的變化如圖6.10所示。沿著水平軸比率等同于圖3.4中彈性壓縮構(gòu)件的修正長細(xì)比。圖6.10顯示，短梁的限制彎矩接近屈服彎矩My，而長梁的限制彎矩接近彈性屈曲彎矩。6.3 Inelastic beamsThe solution for the buckling moment of a perfectly straight simply supported I-beam with equal end moments given by equations 6.2 and 6.3 is only valid while the beam remains elastic. In a short span beam, yielding occurs before the ultimate moment is reached, and significant portions of the beams are inelastic when buckling commences. The effective rigidities of these inelastic portions are reduced by yielding, and consequently, the buckling moment is also reduced.6.3.非彈性梁公式6.2和6.3中給出的承受等大異號彎矩的理想直簡支梁的屈曲彎矩的求解方法僅適用于彈性梁。對于短跨梁，屈服在達到極限彎矩前達到，當(dāng)梁開始屈曲時，梁的截面一部分處于塑性狀態(tài)。隨著屈服的進行，塑性部分的有效剛度逐漸減少，相應(yīng)的，屈曲彎矩也逐漸減少。For beams with equal and opposite end moments（）, the distribution of yield across the section does not vary along the beam, and when there are no residual stresses, the inelastic buckling moment can be calculated from a modified form of equation 6.3 as對于承受等大異號彎矩的梁來說（），截面的屈服部分并不沿梁而變化，當(dāng)沒有參與應(yīng)力的時候塑性屈曲彎矩可以由公式6.3的改進形式公式6.21來計算：in which the subscripted quantities ( )e are the reduced inelastic rigidities which are effective at buckling. Estimates of these rigidities can be obtained by using the tangent moduli of elasticity (see section 3.3.1) which are appropriate to the varying stress levels throughout the section. Thus the values of E and G are used in the elastic areas, while the strain-hardening moduli and are used in the yielded and strain-hardened areas (see section 3.3.4). When the effective rigidities calculated in this way are used in equation 6.21, a lower bound estimate of the buckling moment is determined (section 3.3,3). The variation of the dimensionless buckling moment with the ratio of a typical stress-relieved rolled steel section is shown in Fig. 6.2. In the inelastic range, the buckling moment increases almost linearly with decreasing slen-derness from the first yield moment to the full plastic moment，which is reached after the flanges are fully yielded, and buckling is controlled by the strain-hardening moduli，.其中下標(biāo)表示非彈性剛度的減少，是屈曲時的有效值。減少幅度的估算值可以由近似反應(yīng)截面應(yīng)力水平變化的切線彈性模量來得到（見3.3.1節(jié)）。E和G在彈性階段中應(yīng)用，而應(yīng)變硬化模量和在屈服階段和應(yīng)變硬化階段應(yīng)用（見3.3.4節(jié)）。按這種方法計算的有效剛度適用于公式6.21，一個小范圍的屈曲彎矩是由此決定的。如圖6.2，屈曲系數(shù)隨著應(yīng)力消除后的鋼截面比率的變化而變化。在塑性階段，屈曲彎矩隨著長細(xì)比的減小迅速從初始屈服彎矩直線上升為全塑性彎矩，這在翼緣全部屈服之后達到，屈曲由應(yīng)變硬化模量和控制。The inelastic buckling moment of a beam with residual stresses can be obtained in a similar manner, except that the pattern of yielding is not sym-metrical about the section major axis, so that a modified form of equation 6.69 for a monosymmetric I-beam must be used instead of equation 6.21. The jnelastic buckling moment varies markedly with both the magnitude and the ; distribution of the residual stresses. The moment at which inelastic buckling initiates depends mainly on the magnitude of the residual compressive stresses at the compression flange tips, where yielding causes significant reductions in the effective rigi