土木工程英语翻译

6 Lateral buckling of beams

6.1 Introduction

In 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 beam's in-plane load carrying capacity, as indicated in Fig. 6.2.

6.梁的侧面翘曲

6.1 说明

在第五章关于梁的平面内机能的评论辩论中，假定梁按刚性主平面放置时，梁仅在该平面内倾斜。假如梁没有足够的侧向刚度或侧面支撑，梁会产生平面外愚蠢，如图6.1所示。如图6.2所示，当产生平面外愚蠢时梁的承载才能会大年夜大年夜减小。

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.

作为一个幻想的弹性直梁，当施加弯矩达到弹性愚蠢弯矩时，梁才会产生侧向曲折和扭改变形，产生平面外愚蠢，如图6.1所示。这两种变形是互相接洽的：当梁侧向倾斜时，所遭受的弯矩会对侧向梁轴产生扭矩并引起梁扭转。这种特点，关于抗击侧向曲折和扭转才能差的无穷制I形梁来说专门重要，被成为弯扭愚蠢。

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 Mp in which case the moment capacity of the beam is not affected by lateral buckling.

幻想弹性直梁的屈从始于因为因为弹性愚蠢产生的附加应力导致的屈从，然而，受平面内曲折应力和残存应力的阻碍，幻想弹性直梁的中心部位可能在达到屈从弯矩前先行屈从，并产生塑形曲折，如图6.2所示。关于短梁，其非弹性愚蠢弯矩会大年夜于平面内塑形破坏弯矩，受弯承载力不由侧向愚蠢操纵。

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，将讲述由侧向愚蠢和屈从引起破坏的梁的机能和设计方法。假设第四章中评论辩论的局部愚蠢可不能产生。第七章将评论辩论轴压及压弯构件的机能和设计方法。

6.2 Elastic beams

6.2.1 BUCKLING OF STRAIGHT BEAMS

6.2.1.1 Simply supported beams with equal end moments

A 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

MobMyz, （6.2） Where

where 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(2EIy/L2) and the torsional resistance(GJ2EIW/L2).

6.2.弹性梁

6.2.1 直梁的愚蠢

6.2.1.1 端弯矩相等的简支梁

如图6.3所示，一个遭受相等梁端弯矩的幻想弹性直梁。梁端简支侧向曲折和扭转可不能产生，因为端部能够在平面内自由迁移转变从而不限制梁端转角。当侧移和扭传达到均衡时，在Mob感化下梁会曲折，如图6.3所示。这种情形在6.10.1.1中给出公式：

为梁跨中挠度，大年夜小未知，弹性愚蠢弯矩运算公式为：MobMyz, （6.2）

EIy为侧向曲折刚度，GJ为扭转刚度，EIW为翘曲刚度。公式6.3表示梁的抗愚蠢才能取决于临界弯矩(2EIy/L2)以及临界扭矩(GJ2EIW/L2)。

Equation 6.3 ignores the effects of the major axis curvature d2vdz2MobEIX and produces conservative estimates of the elastic buckling moment equal to

1EIy/EIX1GJ2EIW/L22EIX 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忽视了强轴曲率d2vdz2MobEIX，同时保守估量梁的愚蠢弯矩等于真实弯矩乘以

1EIy/EIX1GJ2EIW/L22EIX。这种修改，在大年夜多半梁

截面设计中是能够忽视的，柱的设计中则不然。它把梁的Iy和IX算作零处理，使得梁的弹性愚蠢弯矩接近无穷大年夜。专门明显，I型钢梁可不能绕弱轴愚蠢。研究【1】注解在其他情形下修改值接近真实值，忽视强轴曲率是能够的。

6.2.1.2 Beams with unequal end moments

A 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

MobmMyz （6.4） in which the moment modification factor mwhich accounts for the effect of the non-uniform distribution of the major axis bending moment can be closely approximated by

or by

6.2.1.2 端弯矩不等的简支梁

如图6.4.a所示，一简支梁遭受端弯矩M和M，6.10.1.2所示，弯扭愚蠢梁端弯矩公式为： MobmMyz （6.4）

注解强轴弯矩不平均分派感化阻碍的修改系数m近似表示为：

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.

这种近似是一种推测集中荷载感化下梁段的愚蠢情形简单方法，是为了防止截面产生侧向位移或扭转。在这种情形下，集中荷载感化点之间的梁能够被算作遭受不相等梁端弯矩的梁，其愚蠢弯矩能够用公式6.4及6.5或6.6来表示，L为集中荷载感化点之间的梁段长度。每一个估量的愚蠢弯矩对应所有荷载中的一个具体愚蠢荷载，个中的最小值给出了实际愚蠢荷载系数的保守估量。这种简化忽视了梁端间愚蠢的互相阻碍。在6.6.2.2中将评论辩论不忽视互相阻碍的更精确的方法。 6.2.1.3 Beams with central concentrated loads

A simply supported beam with a central concentrated load Q acting at a distance

yQabove 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 QyQL/2 which increases the twisting of the beam and decreases its resistance to buckling.

如图6.5.a所示，一个遭受跨中集中荷载Q的简支梁，Q感化地位距离梁中轴线为yQ。当梁产生侧向偏移和扭转时，荷载感化线跟着跨中截面的的扭转而移动，但仍保持垂直，如图6.5.C所示。这种荷载只是形心的情形比荷载过形心加倍危险，因为附加扭矩QyQL/2增长了粱所受的扭矩，降低了它的抗击愚蠢才能。

It is shown in section 6.10.1.3 that the dimensionless buckling load

QL2/(EIyGJ)varies as shown in Fig. 6.6 with the beam parameter

K(2EIW/GJL2)and the dimensionless height  of the point of application of the load

given by

yQL(EIyGJ) (6.7)

6.10.1.3中所示梁的愚蠢荷载的无量纲系数QL2/(EIyGJ)同图6.6中梁的系数

K(2EIW/GJL2)，施加荷载的无量纲系数运算公式为：

yQL(EIyGJ) (6.7)

For centroidal loading (0), the elastic buckling moment MobQL/4 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 form

in

which the moment modification factor mwhich accounts for the effect of the non-uniform

distribution of major axis bending moment is approximately equal to 1.35.

当遭受过重心的荷载（即0）,弹性愚蠢弯矩MobQL/4同遭受梁端等大年夜异号弯矩梁的愚蠢弯矩一样要乘以系数K(见6.3节)，其弹性愚蠢弯矩公式为：

这注解强轴不平均分派弯矩感化阻碍的弯矩修改系数m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

梁的愚蠢弯矩跟着梁的荷载高度系数的变更而变更，当荷载感化在梁的轴线以下时，梁的抗愚蠢才能越高，当荷载感化点升高时，抗愚蠢才能明显降低，如图6.6所示。当梁的高低翼缘相等时，有用于公式：

where df is the distance between flange centroids. The variation of the buckling load

with lyt^jds is shown by the solid lines in Fig. 6,6, and it can be seen that the differences between top2yQ/df and bottom (2yQ/df1) 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 height2yQ/dffor equal flanged I-beams are given in [2]. Alternatively, the maximum moment at elastic buckling M0bQL/4 may be approximated by using

and

m1.35, in which

df表示翼缘形心间的距离，如图6.6中所示，愚蠢弯矩跟着2yQ/df的变更而变更。介于最高点(2yQ/df1)和最低点(2yQ/df1)之间的翼缘荷载跟着梁系数k的增长而增长。比起长跨短柱型截面来，这种阻碍在短跨深梁型截面中加倍重要。弯矩修改系数m跟着能够代表梁的无量纲系数2yQ/df感化的梁的系数K的变更而变更，在高低翼缘相等的I型梁中，它的近似表述在【2】中给出。或者，梁的最大年夜愚蠢弯矩

M0bQL/4近似有用于公式：

当时m1.35，

6.2.1.

4 Other loading conditions

The 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 其他荷载情形

遭受沿直线分布荷载的简支梁的荷载分派情形对弹性愚蠢长度的阻碍有专门多种方法推敲，包含【3-5】中评论辩论的。一种比较有效的方法是有限元法【6-10】，还有有限积分法【11，12】，个中有限积分法能够或许获得弱轴愚蠢和扭转的精确解，因而被广泛应用。专门多专门情形有用于【13-16】，【2，3，5，13，15，17】有用于弹性愚蠢荷载，【18】是一个能够分析弯扭愚蠢的比较好的电脑法度榜样。

Some approximate solutions for the maximum moments Mob elastic buckling of simply supported beams which are loaded along their centroidal axes are given in Fig. 6.7 by the moment modification factors min the equation

MobmMyz （6.12） 求遭受沿轴线分布荷载的简支梁的最大年夜弹性愚蠢弯矩Mob的专门多解决方法在图6.7中给出，弯矩修改系数m公式为：

MobmMyz （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 m1.

可见危险荷载安排是那些产生恒定的强轴曲折弯矩的情形，遭受等大年夜异号弯矩，m1的情形为最晦气情形。

For other beam loadings than those shown in Fig. 6.7, the moment modi-fication factor

mmay be approximated by using

in

which Mm is the maximum moment,M2,M4，the moments at the quarter points, and M3the moment at the mid-point of the beam.

关于图6.7之外的情形，弯矩修改系数m有用于公式:

Mm为最大年夜弯矩，M2,M4是1/4地位的弯矩，M3为梁跨中弯矩。 6.2.1.5 Cantilevers

The 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 L 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, whence

This 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悬臂梁

悬臂梁的支撑情形与简支梁不合，一端自由，一端凝集。关于存在梁端弯矩引起的平均曲折的悬臂梁的弹性愚蠢问题的解决，梁端转角L能够用针对简支梁的公式6.2和6.3来求解，只要把长度L换为悬臂梁的长度的两倍，此处：

这种处理方法跟应用于获得悬臂柱愚蠢荷载的有效长度法道理雷同。（见图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 QL2/(EIyGJ) for bottom flange, centroidal, and top flange loading are given in Fig. 6.8, together with plots of the dimensionless elastic buckling moments

qL3/2(EIyGJ)of can-tilevers with uniformly distributed loads q.

受其它情势荷载的悬臂梁就不这么轻易求解了。【16，19-21】是几种应用的数学方法。6.10.1.4讲述了受端部集中荷载的的悬臂梁的求解公式。如图6.8所示上翼缘受中间荷载，下翼缘要乘以无量纲愚蠢弯矩系数QL2/(EIyGJ)，受均布荷载q的悬臂梁无量纲愚蠢弯矩系数为qL3/2(EIyGJ)。

6.2.2 BENDING AND TWISTING OF CROOKED BEAMS

Real 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 by

in

which the central initial lack of straightness 0and twist rotation 0are related by

then the deformations of the beam are given by

in which

as shown in section 6.10.2. The variations of the dimensionless central de-flection /0

and twist /0are 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 曲梁的曲折与扭转

曲梁并不是幻想直线，而是存在初始曲折和初始扭曲，导致荷载刚感化便会产生曲折和扭转。如6.10.2节所示，假如感化有等大年夜异号端弯矩M的简支梁存在初曲折和初扭转，便有公式：

初始曲折缺点0和扭转角0之间知足关系式：

梁的变形相符公式：

个中：

图6.9显示了梁的中间挠度系数/0以及扭转系数/0，能够看出变形始于荷载方才感化，跟着弯矩接近愚蠢弯矩变形灵敏增长。

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.

关于公式6.17和6.18之间负荷变形的简单关系同公式3.6和3.7之间的关系一样，曲率为正弦曲线情势。经由过程针对受压构件的实验得出结论它屈从索斯维尔画图法，这同样有用与梁。

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 ML at which a beam without residual stresses first yields is given by

in

which MYfyZXis the nominal first yield moment, when the central lack of straightness0 is given by

变形跟着施加弯矩的增长而增长，压力也是。如6.10.2所示，一个没有初次屈从产生残存应力的梁的限制弯矩ML屈从公式：

个中MYfyZX是名义初始愚蠢弯矩，中襟曲率0给出公式：

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 momentML/MY is shown in Fig. 6.10, in which the ratio

(MY/Myz) 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 MYZ.

个中N0y在公式6.11中给出解答，公式6.19同公式3.9和3.11一样，是为了限制弹性紧缩构件的轴力。无量纲限制弯矩ML/MY的变更如图6.10所示。沿着程度轴比率(MY/Myz)等同于图3.4中弹性紧缩构件的修改长细比。图6.10显示，短梁的限制弯矩接近屈从弯矩My，而长梁的限制弯矩接近弹性愚蠢弯矩MYZ。

6.3 Inelastic beams

The solution for the buckling moment Myzof 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.非弹性梁

公式6.2和6.3中给出的遭受等大年夜异号弯矩的幻想直简支梁的愚蠢弯矩Myz的求解方法仅有用于弹性梁。关于短跨梁，屈从在达到极限弯矩前达到，当梁开端愚蠢时，梁的截面一部分处于塑性状况。跟着屈从的进行，塑性部分的有效刚度逐步削减，响应的，愚蠢弯矩也逐步削减。

For beams with equal and opposite end moments（m1）, 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

关于遭受等大年夜异号弯矩的梁来说（m1），截面的屈从部分并不沿梁而变更，当没有介入应力的时刻塑性愚蠢弯矩能够由公式6.3的改进情势公式6.21来运算：

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 ESt and GStare 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 M/MY with the ratio L/ryof 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

MYfyZxto the full plastic momentMpfySX，which is reached after the flanges are fully yielded, and buckling is controlled by the strain-hardening moduliESt，GSt.

个中下标()e表示非弹性刚度的削减，是愚蠢时的有效值。削减幅度的估算值能够由近似反响截面应力程度变更的切线弹性模量来获得（见3.3.1节）。E和G在弹性时期中应用，而应变硬化模量ESt和GSt在屈从时期和应变硬化时期应用（见3.3.4节）。按这种方法运算的有效刚度有用于公式6.21，一个小范畴的愚蠢弯矩是由此决定的。如图6.2，愚蠢系数M/MY跟着应力清除后的钢截面比率L/ry的变更而变更。在塑性

时期，愚蠢弯矩跟着长细比的减小灵敏从初始屈从弯矩MYfyZx直线上升为全塑性弯矩MpfySX，这在翼缘全部屈从之后达到，愚蠢由应变硬化模量ESt和GSt操纵。

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 rigidities (EIy)e and (EIW)e， The flange tip residual stresses are comparatively high in hot-rolled beams, especially those with high ratios of fiange to web area, and so inelastic buckling is initiated comparatively early in ; these beams, as shown in Fig. 6.2. The residual stresses in hot-rolled beams decrease away from the flange tips (see Fig. 3.9 for example), and so the extent of yielding increases and the effective rigidities steadily decrease as the applied moment increases. Because of this, the inelastic buckling moment decreases in an approximately linear fashion as the slenderness increases, as shown in ' Fig. 6.2.

有残存应力的梁的塑性愚蠢弯矩能够有一种类似的方法获得，然则屈从模式关于主轴不平均，必须用修改公式6.69来代替6.21来解决高低翼缘等宽的I型梁。塑性愚蠢弯矩专门明显受残存应力的大年夜小和分布的阻碍。塑性愚蠢弯矩刚开端重要受压翼缘残存压应变大年夜小的阻碍，因为屈从会引起有效刚度(EIy)e和(EIW)e的减小。在热轧型钢梁中压翼缘边沿残存应力比较高，专门是压翼缘较宽的梁，如图6.2所示在这些梁中非弹性愚蠢开端的比较早。热轧型钢梁的残存应力离翼缘越远越小，因此跟着所受弯矩的增长，屈从范畴增长，有效刚度逐步减小。如图6.2所示，跟着长细比减小，非弹性愚蠢弯矩近似直线降低。

In beams fabricated by welding flange plates to web plates, the compressive residual stresses at the flange tips, which increase with the welding heat input, are usually somewhat smaller than those in hot-rolled beams, and so the initiation of inelastic buckling is delayed, as shown in Fig. 6.2. However, the variations of the residual stresses across the flanges are

nearly uniform in welded beams, and so, once flange yielding is initiated, it spreads quickly through the flange with little increase in moment. This causes large reductions in the inelastic buckling moments of stocky beams, as indicated in Fig. 6.2.

关于把翼缘焊接在腹板上的组合梁，跟着焊接传入热量而增长的翼缘板的残存压应变平日小于热轧型钢梁，因此如图6.2所示，塑性愚蠢开端的比较晚。然而，在焊接梁中翼缘残存应力的变更是平均的，因此一旦翼缘屈从开端，那么弯矩仅增长少许屈从便会贯穿全截面。如图6.2所示，这就引起了短梁非弹性愚蠢弯矩的大年夜幅度削减。

When a beam has a more general loading than that of equal and opposite end moments, the in-plane bending moment varies along the beam, and so when yielding occurs its distribution also varies. Because of this the beam acts as if non-uniform, and the torsion equilibrium equation becomes more com-plicated. Nevertheless, numerical solutions have been obtained for some hot- rolled beams with a number of different loading arrangements [22, 23], and some of these (for unequal end moments M and mM) are shown in Fig. 6.11, together with approximate solutions given by

in which Mob is given by equations 6.4 and 6.5.

当梁比遭受等大年夜异号弯矩的梁遭受更多的荷载时，其平面内曲折弯矩沿梁长变更，当屈从产生时，弯矩分派也产生变更。因为这梁如同不平均感化，同时扭转均衡公式也加倍复杂。然而，我们差不多获得了遭受不合荷载情形的热轧型钢梁的数学解答。个中的一些（遭受不等弯矩M和mM）如图6.11所示，近似解答公式为：

个中Mob在公式6.4和6.5中给出。

In this equation, the effects of the bending moment distribution are included in both the elastic buckling resistance MobmMyzthrough the use of the end moment ratio m in the moment modification factor m and also through the direct use of min equation 6.22.

This latter use causes the inelastic buckling moments M1to approach the elastic buckling moment Mobas the end moment ratio increases towards m1.

在那个公式中，梁曲折弯矩分派的阻碍经由过程梁端弯矩比率m在弹性愚蠢抗击弯矩

MobmMyz中的m的应用以及公式6.22中m的直截了当感化来表示出来。后一应用导致了非弹性愚蠢弯矩M1接近弹性愚蠢弯矩Mob，梁端弯矩比率接近m1。

The most severe case is that of equal and opposite end moments (m1), for which yielding is constant along the beam so that the resistance to lateral buckling is reduced everywhere. Less severe cases are those of beams with unequal end moments M andmMwithm0, where yielding is confined to short regions near the supports, for which the reductions in the section properties are comparatively unimportant. The least severe case is that of equal end moments that bend the beam in double curvature (m1), for which the moment gradient is steepest and the regions of yielding are most limited.

最大年夜的阻碍确实是等大年夜异号弯矩(m1)使屈从沿全梁产生，这导致抗击侧向愚蠢的才能减小。其次是遭受不等弯矩M和mM（m0）的梁，离支座专

门近的范畴内也会屈从，导致截面机能的减小变得不重要。别的遭受相等端弯矩

(m1)的梁会双倍曲折，这种情形下弯矩的斜率最陡，屈从区域也最小。

The range of modified slenderness

MP/Mob for which a beam can reach the full

plastic moment Mp depends very much on the loading arrangement. An approximate expression for the limit of this range for beams with end moments M and can be obtained from equation 6.23 as

达到全塑性弯矩Mp的梁的修改长细比MP/Mob的范畴专门大年夜程度上取决于荷载的分布情形。遭受不等弯矩M和mM的梁的这种范畴限制的近似表述公式如公式6.23：

In the case of a simply supported beam with an unbraced central concen-trated load, yielding is confined to a small central portion of the beam, so that any reductions in the section properties are limited to this region. Inelastic buckling can be approximated by using equation 6.22 withm0.7,m1.35.

关于遭受跨中集中荷载的简支梁，只在梁的中间一小部分区域内产生屈从。因此只在这一小区域内截面机能降低。非弹性愚蠢近似可用公式6.22运算，个中

m0.7,m1.35。

6.4 Real beams

6.4.1 BEHAVIOUR OF REAL BEAMS

Real beams differ from the ideal beams analysed in section 6.2.1 in much the same way as do real compression members (see section 3.4.1). Thus any small 'imperfections such as initial curvature, twist, eccentricity of load, or hori-zontal load components cause the beam

to behave as if it had an equivalent initial curvature and twist (see section 6.2.2), as shown by curve A in Fig. 6.12. On the other hand, imperfections such as residual stresses or variations in material properties cause the beam to behave as shown by curve B in Fig. 6.12. The behaviour of real beams having both types of imperfection is indicated by curve C in Fig. 6.12, which shows a transition from the elastic behaviour of a beam with curvature and twist to the inelastic post-buckling behaviour of a beam with residual stresses.

6.4 实际的梁

6.4.1 实际梁的机能

在6.2.1节平分析了，就跟实际紧缩构件（见3.4.1节）一样，实际梁在专门多方面跟幻想梁不合。任何缺点比如初始曲折，初始扭转，专门荷载，或荷载的水等分力都邑使梁表示的如同存在初曲折和初扭曲（见6.2.2节），如图6.12中曲线A所示。另一方面，如残存应力或材料机能的变更又会使梁的表示如图6.12中曲线B所示。实际梁的表示受表现从存在曲折和扭转的梁的弹性性质到存在残存应力的梁的非弹性愚蠢性质之间变更过程的图6.12中曲线C所注解的梁的各类缺点的阻碍。

6.4.2.DESIGN RULES

6.4.2.1 Simply supported beams in uniform bending

It is possible to develop a refined analysis of the behaviour of real beams which includes the effects of all types of imperfection. However, the use of such an analysis is unwarranted because the magnitudes of the imperfections are uncertain. Instead, design rules are often based on a simple analysis for one type of equivalent imperfection which allows approximately for all im-perfections, or on approximations of experimental results such as those shown in Fig. 6.13. 6.4.2 设计方法

6.4.2.1平均曲折的简支梁

对实际梁的特点开展推敲所出缺点的精确分析是可能的。然而这种分析方法的应用是没有依照的，因为初始缺点的大年夜小不确信。因此设计方法平日是基于对一种能够许可所出缺点的近似等效缺点的简单分析，或者是基于如图6.13所示实验成果的近似。

The AS4100 uses the simple semi-empirical equation

in

which

to relate the nominal moment capacity Mb of hot-rolled and welded beams to the nominal

section capacity MS (see section 4.7.2) and the elastic buckling moment mMyz, and this is shown in Figs 6.13 and 6.14 for beams with MSMP. AS4100采取简单的半体会公式：

个中：

来把

热轧型钢梁和焊接组合梁的名义弯矩承载力Mb，名义截面承载力MS（见4.7.2节）和弹性愚蠢弯矩mMyz接洽起来。关于MSMP的梁来说，这在图6.13和图6.14中获得表示。

For very slender beams with high values of modified slenderness

(MS/Myz), the

nominal uniform bending moment capacity Mb shown in Fig. 6.13 approaches 0.9 times the elastic buckling moment Myz, while for stocky beams the moment capacity

Mbreaches the nominal section capacity MS, and so is governed by yielding or local buckling, as discussed in section 4.7.2. For beams of intermediate slenderness, equation 6.24 provides a transition be-tween these limits, which is close to the lower bound of the experimental results shown in Fig. 6.13,

关于进行了长细比(MS/Myz)修改的细长梁来说，图6.13中的名义平均曲折弯矩承载才能Mb大年夜约是弹性愚蠢弯矩Myz的0.9倍，而对短梁来说，名义弯矩承载力

Mb能达到名义截面承载力MS，因此受屈从或局部愚蠢（掉稳）操纵，如4.7.2.中评论辩论。关于中心细长的梁，公式6.24供给了这些范畴的转换，接近图6.13中显示的较低范畴的实验成果。 6.4.2.2 Unequal end moments

For beams with unequal end moments M and mM, equations 6.5 and 6.24 give

nominal moment capacities which increase with the end moment ratio m, as shown in Fig. 6.14. It should be noted that the moment modification factor m is used only once in equation 6.24, and that the elastic buckling moment M0 used in equation 6.24 is for uniform bending（m1）.Also shown in Fig. 6.14 are the approximate inelastic buckling resistances given by equation 6.22. It can be seen that the moment capacities are reduced below the inelastic buckling resistances, thus providing allowances for the effects of geometrical imperfections on the strengths of real beams. 6.4.2.2不相等的梁端弯矩

关于遭受不等弯矩M和mM的梁，公式6.5和6.24给出了跟着梁端弯矩比例m而增长的名义弯矩承载力的运算方法，如图6.14所示。我们应当留意到弯矩修改系数m仅应用于公式6.24，公式6.24中的弹性愚蠢弯矩M0仅针关于平均曲折（m1）。图6.24中还表现了公式6.22所得的非弹性愚蠢抗击力的近似值。我们还看到弯矩承载力在非弹性愚蠢抗击力之下，从而能够许可实际梁的几何缺点的阻碍。

6.4.2.3 Other moment distributions

The nominal moment capacities Mbfor beams with other bending moment distributions are given by equation 6.24 when the appropriate values of the moment

modification factor mobtained from Fig. 6.7 or equation 6.13 are substituted. 6.4.2.3其它弯矩分派情势

遭受其它情势弯矩的梁的名义弯矩承载力Mb由公式6.24给出，弯矩修改系数m能够由图6.7近似得出，或由公式6.13得出。 6.4.2.4 Top flange loading on a beam

For a beam supported at both ends with top flange loading which moves laterally with the flange (see section 6.2.1.3 for example), AS4100 requires the length L used in the calculation of the moment capacity to be replaced by

in which K11.4. 6.4.2.4梁的上翼缘遭受荷载

关于两端支撑，上翼缘荷载跟着翼缘侧向移动的梁（见6.2.1.3中例子），AS4100要求用于估算弯矩承载力的长度L用下述公式代替：

个中K11.4。 6.4.2.5 Cantilevers

Cantilevers which are free to deflect laterally and twist at the unsupported end are treated by AS4100 as equivalent beams, with values of m0.25for uniform bending, 1.25 for end load, and 2.25 for uniformly distributed load. For a cantilever with top flange loading which moves laterally with the flange, the length Le used in the calculation of the moment capacity is increased to 2L. 6.4.2.5悬臂梁

自由端能够自由侧向偏移和扭转的悬臂梁被AS4100进行等效处理后再进行运算，关于平均曲折m0.25，端部集中荷载m1.25，均布荷载m2.25，关于上翼缘荷载跟着翼缘侧向移动的悬臂梁，用于估算弯矩承载力的长度Le为2L。