极限设计的常用方法.doc

11.5.2极限设计的常用方法
一个极限设计方法，其目的在于满足三个条件（1）平衡条件（2）转动协调（3）适用性。大多数常用的极限设计方法，在开始时往往只考虑其中的一个或两个条件，而后再较核余下的条件。至于弯矩的的分布图形，如果它比弹性方法在各种荷载组合作用下所得的包络图植有最大降低时，则将认为是最经济的弯矩分布。在所发表的各种极限设计方法中，最值得注意的可能是：Baker（11.14）、（11.16）、（11.17）,Cohn(11.11)、(11.18),Sawyer(11.19)和Furlong(11.20)等方法。这四种方法扼要阐述如后：
Baker的方法
1940年以来，：Baker（11.14）、（11.16）、（11.17）推导了一个极限设计的方法。设计（方法）是建立在极限平衡要求的基础上的。对转动协调与适用性的要求作为二步骤校核之。设计是从确定极限弯矩的分布开始，这个弯矩分布是和极限荷载相平衡。这样的弯矩分布图，可由杆端能自由转动杆件承受极限荷载的简支弯矩图上，在某一个适当的位置作一条固定端弯矩的图线而得，示如图11.7。
于是各截面按此极限弯矩配筋。注意到在极限荷载下形成破坏机构，则应较核塑铰区的转动能力，以保证在极限荷载下能达到所选定的弯矩分布，同时确定使用荷载下的弯矩图形以及较核其应力值以保证构件的适用性。如发现适用性不满足和转动能力不相适应，则原来假定的极限弯矩分布须加意修正。
Cohn的方法
Cohn(11.11)、(11.18)已定出了一个基于极限平衡和适用性的方法，转动协调的要求作为第二步的校核。方法的解答是通过对各种极限荷载组合下的弹性弯矩包络图予以按比例的缩减，即乘以适当的参数xj1,这里xj是截面j达到屈服时的安全系数.此xj的值按下要求而定:在使用荷载下,内力必须和外荷保持平衡,并能形成一种或几种形式的破坏机构,从弹性包络图中所折减下来的弯矩必须是最多的.一个典型的设计是寻求在极限荷载下,既能适用使用荷载下的性能又是满足平衡条件的一组xj的最小值.于是截面就跟据这样确定下来的弯矩分布得以实现.
Sawyer的方法
Sawyer(11.19)提供了一个基于极限平衡要求又是转动协调的方法.而适用性的要求作为第二步的校核.此法是通过对一个设计作逐次逼近的调整,间接地应用转动协调的分析.设计一开始,就调整在极限荷载下各种荷载组合的弹性弯矩包络图,从而建立一个弯矩图形予以配筋.而后.对极限荷载下任一中可能的荷载组合,作既能满足平衡又是落在截面极限抗弯矩之内的任一组弯矩调整,并计算出每一塑铰区的非弹性转动度.此时,假定弯矩-曲率曲线中的屈服弯矩的0.85.然后应用弹性理论来计算由于非弹性弯曲角所引起的弯矩,则应修改配筋,即在超过极限弯矩的区域内和非弹性角过大的区域内增加配筋,直至具有适当的极限抗弯矩为止.最后,设计用弹性方法来校核,保证在使用荷载,钢筋的应力不会过大.
Furlong的方法
Furlong 的极限设计方法(11.20),它包括了无侧移结构确定的极限弯矩. Furlong分析了不同跨数的连续梁在各种不利荷载作用与组合情况,从而得出了能满足极限平衡与适用性要求(受拉筋在使用荷载下会屈服)的设计弯矩可能分布图形.然后分析这种极限弯矩分布状态所造成的塑性转角,以便决定曲率延性的要求,将求得的(可能的)设计弯矩的分布列成表格,并用一个简单的公式来给出曲率延性的要求,因此可得到一个方便的设计方法.若设计一个梁,其截面的配筋使在每一跨内的各截面抵抗弯矩和其支承的极限荷载相平衡:而极限抗弯弯矩应等于或大于MF与表11.1中相应系数的乘积,此于,MF=极限荷载下简支梁的最大弯矩.同样,截面还须符合下式:

Φu/φy≥1+0.25ln/d
于此, Φu=极限曲率； φy=开始屈服时曲率；ln=净跨；d=截面的有效高度。
表11.1支承端不同约束梁的弯矩系数表（11.20）
支承端约束情况
弯矩约束
仅跨中有一集中载的梁
其他荷载情况的梁
两端约束的梁
一端约束的梁
负弯矩
正弯矩
负弯矩
正弯矩
0.37
0.42
0.56
0.50
0.50
0.33
0.75
0.46
图11.10，示出了连续梁的一个中间跨在均布荷载Wu下，确定其弯矩的一个例子。截面设计是应用了相似于图6.9与6.10的图表来满足公式11.18的要求。很显然，Furlong的方法是一种简单而直接的方法。应用于所规定的极限弯矩，设计者可以避免用反复试凑的繁琐工作，且不需进行塑铰转动能力和适应性的校核。

11.5.2 Available Limit Design Methods
A limit design procedure aims to satisfy three conditions(1)limit equilibrium,(2)rotational compatibility,and (3)serviceability. Most of the available limit design methods consider one or two of these conditions initially, the remaining condition or conditions being the object of a sudsequent check.The most economical distribution of moments which results in the greatest moment reduction when compared with the elastic envelope moments obtainedfrom the various design loading combinations. Of the limit design methods that the most attention are those due to Baker,11.14,11.16,11.17,;Cohn,11.11,11.18:Sawyer,11.19 and Furlong,11.20 .These four methods are briefly described below.
Baker＇s Method
Baker 11.14,11.16,11.17 has been developing a method of limit design since the 1940s .The design is based on the requirements of limit equilibrium .The requirements of rotational compatibility and serviceability are checked as sudsequent steps . then design is commenced by determining a distribution of ultimate bending moments which is in equilibrium with the ultimate loads. This may be obtained by drawing the free bending moments diagram for the members supporting the ultimate loads when the ends are free of rotational restraint , and drawing the fixing moment line at some convenient position, as in Fig 11.7.The sections are reinforced for those ultimate moments. Note that a collapse mechanism has developed at the ultimate load. The rotation capacity of the plastic hinge regions is then checked to ensure that the chosen distribution of bending moments can be developed at the ultimate load , and the pattern of moments at the service load is determined and the stresses checked to ensure that the members are serviceable. The assumed distribution of ultimate moments may need to be modified if inadequate rotation capacity or unsatisfactory serviceability is found.
Cohn＇s Method
Cohn11.11,11.18 has developed a method based on the requirements of limit equilibrium and serviceability . The requirement of rotational compatibility is checked as a subsequent step .The solution is obtained by scaling down the elastic envelop moments, obtained from the various ultimate load combinations, by multiplying by appropriate parameters xj≤1，where xj is the yield safety parameter for section j. The value of xj is set by the following requirements : at the service load ,the internal forces must be in equilibrium with the external loads and one or more collapse mechanisms must form :and the overall moment reductions form the elastic envelop must be a maximum. A typical design seeks the minimum value for xj consistent with acceptable service load behavior and the equilibrium conditions at the ultimate load .The sections are designed on the basis of the determined distribution of bending moments ,and the
plastic hinge regions are checked to ensure that they have sufficient rotation capacity to develop the assumed moment distribution at the ultimate load.
Sawyer＇s Method
Sawyer11.19 has presented an approach based on the requirements of limit equilibrium and rotational compatibility .The requirement of serviceability is checked by a subsequent step . The method uses a rotational compatibility analysis indirectly by adjusting a given design by successive approximations. The design is commenced by adjusting the elastic envelop moments obtained from the various design loading combinations at ultimate load , toestablish a bending moment pattern for which reinforcement is provided . Foe eah possible loading combination at ultimate load ,using any set of adjusted moments that satisfies static equilibrium and falls within the ultimate resisting moments of the sections , the inelastic rotation at each plastic region is calculated . A moment-curvature curve with a yield moment of 0.85 of the ultimate moment may be assumed .Elastic theory is then used to calculate the moments resulting from these inelastic bending angles and the external loading imposed on the structure . If the calculated moments exceed the ultimate resisting moments of the sections , the reinforcement is revised by adding reinforcement to regions in which the ultimate moment is exceed or to regions in which the inelastic angle developed is excessive. The moment introduced by the inelastic angles and the eternal loading are recalculated , and the reinforcement is adjusted , until the adequacy of the ultimate moments of resistance has been demonstrated . The design is then checked by elastic theory to ensure that the stesses at the steel stresses at the service load are not excessive.
Furlong’s Method
The limit design method of Furlong involves assigned ultimate moments for structures braced against sway. The worst cases of different types and arrangements of loading on various arrangements of spans were analyzed by Furlong to determine the possible patterns of design moments in continuous beams that would satisfy the requirements of serviceability (tension steel not to yield at the service loads) and limit equilibrium. Then the plastic rotations resulting from these distributions of ultimate moments were analyzed to determine the curvature ductility requirements. The possible distributions of design moments so found were tabulated, and a simple equation was given for the curvature ductility requirements. A convenient design approach results. To design a beam, the sections are reinforced so that in each span the ultimate moments of resistance are in equilibrium with the ultimate load to the supported and the ultimate moments of resistance are equal to or greater than the product of MF and the appropriate coefficient given in Table 11.1, where MF=maximum bending moment in the span due to the ultimate loads when the ends are free of rotational restraint. The sections are also proportioned so that

where =ultimate curvature, =curvature at first yield, =clear span and =effective depth of section.

Table 11.1 Beam Moment Coefficients for Various End Restraints
End Restraint
Type of Moment
Beams Loaded Only by One Force at Midspan
All Other Beams
Span with two
Negative moment
0.37
0.50
ends restrained
Positive moment
0.42
0.33
Span with one
Negative moment
0.56
0.75
end restrained
Positive moment
0.50
0.46

An example of the ultimate resisting moments for an interjor span of a continuous beam carrying a uniform load WU per unit length is represented in Fig.11.10. The sections would be proportioned to satisfy Eq. 11.18 using charts similar to Figs.69 and 6.10. It is apparent that Furlong' method gives a simple direct design approach. The use of assigned ultimate moments means that the designer avoids the complexities of trial-and-error solutions and does not have to check for plastic rotation capacity and serviceability.