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1、7 Rigid-Frame StructuresA rigid-frame high-rise structure typically comprises parallel or orthogonally arranged bents consisting of columns and girders with moment resistant joints. Resistance to horizontal loading is provided by the bending resistance of the columns, girders, and joints. The contin
2、uity of the frame also contributes to resisting gravity loading, by reducing the moments in the girders.The advantages of a rigid frame are the simplicity and convenience of its rectangular form.Its unobstructed arrangement, clear of bracing members and structural walls, allows freedom internally fo
3、r the layout and externally for the fenestration. Rigid frames are considered economical for buildings of up to about 25 stories, above which their drift resistance is costly to control. If, however, a rigid frame is combined with shear walls or cores, the resulting structure is very much stiffer so
4、 that its height potential may extend up to 50 stories or more. A flat plate structure is very similar to a rigid frame, but with slabs replacing the girders As with a rigid frame, horizontal and vertical loadings are resisted in a flat plate structure by the flexural continuity between the vertical
5、 and horizontal components. As highly redundant structures, rigid frames are designed initially on the basis of approximate analyses, after which more rigorous analyses and checks can be made. The procedure may typically include the following stages:1. Estimation of gravity load forces in girders an
6、d columns by approximate method.2. Preliminary estimate of member sizes based on gravity load forces with arbitrary increase in sizes to allow for horizontal loading.3. Approximate allocation of horizontal loading to bents and preliminary analysis of member forces in bents.4. Check on drift and adju
7、stment of member sizes if necessary.5. Check on strength of members for worst combination of gravity and horizontal loading, and adjustment of member sizes if necessary.6. Computer analysis of total structure for more accurate check on member strengths and drift, with further adjustment of sizes whe
8、re required. This stage may include the second-order P-Delta effects of gravity loading on the member forces and drift.7. Detailed design of members and connections.This chapter considers methods of analysis for the deflections and forces for both gravity and horizontal loading. The methods are incl
9、uded in roughly the order of the design procedure, with approximate methods initially and computer techniques later. Stability analyses of rigid frames are discussed in Chapter 16.7.1 RIGID FRAME BEHAVIORThe horizontal stiffness of a rigid frame is governed mainly by the bending resistance of the gi
10、rders, the columns, and their connections, and, in a tall frame, by the axial rigidity of the columns. The accumulated horizontal shear above any story of a rigid frame is resisted by shear in the columns of that story (Fig. 7.1). The shear causes the story-height columns to bend in double curvature
11、 with points of contraflexure at approximately mid-story-height levels. The moments applied to a joint from the columns above and below are resisted by the attached girders, which also bend in double curvature, with points of contraflexure at approximately mid-span. These deformations of the columns
12、 and girders allow racking of the frame and horizontal deflection in each story. The overall deflected shape of a rigid frame structure due to racking has a shear configuration with concavity upwind, a maximum inclination near the base, and a minimum inclination at the top, as shown in Fig. 7.1.The
13、overall moment of the external horizontal load is resisted in each story level by the couple resulting from the axial tensile and compressive forces in the columns on opposite sides of the structure (Fig. 7.2). The extension and shortening of the columns cause overall bending and associated horizont
14、al displacements of the structure. Because of the cumulative rotation up the height, the story drift due to overall bending increases with height, while that due to racking tends to decrease. Consequently the contribution to story drift from overall bending may, in. the uppermost stories, exceed tha
15、t from racking. The contribution of overall bending to the total drift, however, will usually not exceed 10% of that of racking, except in very tall, slender, rigid frames. Therefore the overall deflected shape of a high-rise rigid frame usually has a shear configuration.The response of a rigid fram
16、e to gravity loading differs from a simply connected frame in the continuous behavior of the girders. Negative moments are induced adjacent to the columns, and positive moments of usually lesser magnitude occur in the mid-span regions. The continuity also causes the maximum girder moments to be sens
17、itive to the pattern of live loading. This must be considered when estimating the worst moment conditions. For example, the gravity load maximum hogging moment adjacent to an edge column occurs when live load acts only on the edge span and alternate other spans, as for A in Fig. 7.3a. The maximum ho
18、gging moments adjacent to an interior column are caused, however, when live load acts only on the spans adjacent to the column, as for B in Fig. 7.3b. The maximum mid-span sagging moment occurs when live load acts on the span under consideration, and alternate other spans, as for spans AB and CD in
19、Fig. 7.3a.The dependence of a rigid frame on the moment capacity of the columns for resisting horizontal loading usually causes the columns of a rigid frame to be larger than those of the corresponding fully braced simply connected frame. On the other hand, while girders in braced frames are designe
20、d for their mid-span sagging moment, girders in rigid frames are designed for the end-of-span resultant hogging moments, which may be of lesser value. Consequently, girders in a rigid frame may be smaller than in the corresponding braced frame. Such reductions in size allow economy through the lower
21、 cost of the girders and possible reductions in story heights. These benefits may be offset, however, by the higher cost of the more complex rigid connections.7.2 APPROXIMATE DETERMINATION OF MEMBER FORCES CAUSED BY GRAVITY LOADSIMGA rigid frame is a highly redundant structure; consequently, an accu
22、rate analysis can be made only after the member sizes are assigned. Initially, therefore, member sizes are decided on the basis of approximate forces estimated either by conservative formulas or by simplified methods of analysis that are independent of member properties. Two approaches for estimatin
23、g girder forces due to gravity loading are given here.7.2.1 Girder ForcesCode Recommended ValuesIn rigid frames with two or more spans in which the longer of any two adjacent spans does not exceed the shorter by more than 20 %, and where the uniformly distributed design live load does not exceed thr
24、ee times the dead load, the girder moment and shears may be estimated from Table 7.1. This summarizes the recommendations given in the Uniform Building Code 7.1. In other cases a conventional moment distribution or two-cycle moment distribution analysis should be made for a line of girders at a floo
25、r level.7.2.2 Two-Cycle Moment Distribution 7.2.This is a concise form of moment distribution for estimating girder moments in a continuous multibay span. It is more accurate than the formulas in Table 7.1, especially for cases of unequal spans and unequal loading in different spans.The following is
26、 assumed for the analysis:1. A counterclockwise restraining moment on the end of a girder is positive and a clockwise moment is negative.2. The ends of the columns at the floors above and below the considered girder are fixed.3. In the absence of known member sizes, distribution factors at each join
27、t are taken equal to 1 /n, where n is the number of members framing into the joint in the plane of the frame.Two-Cycle Moment DistributionWorked Example. The method is demonstrated by a worked example. In Fig, 7.4, a four-span girder AE from a rigid-frame bent is shown with its loading. The fixed-en
28、d moments in each span are calculated for dead loading and total loading using the formulas given in Fig, 7.5. The moments are summarized in Table 7.2.The purpose of the moment distribution is to estimate for each support the maximum girder moments that can occur as a result of dead loading and patt
29、ern live loading. A different load combination must be considered for the maximum moment at each support, and a distribution made for each combination. The five distributions are presented separately in Table 7.3, and in a combined form in Table 7.4. Distributions a in Table 7.3 are for the exterior
30、 supports A and E. For the maximum hogging moment at A, total loading is applied to span AB with dead loading only on BC. The fixed-end moments are written in rows 1 and 2. In this distribution only .the resulting moment at A is of interest. For the first cycle, joint B is balanced with a correcting
31、 moment of - (-867 + 315)/4 = - U/4 assigned to MBA where U is the unbalanced moment. This is not recorded, but half of it, ( - U/4)/2, is carried over to MAB. This is recorded in row 3 and then added to the fixed-end moment and the result recorded in row 4.The second cycle involves the release and
32、balance of joint A. The unbalanced moment of 936 is balanced by adding -U/3 = -936/3 = -312 to MBA (row 5), implicitly adding the same moment to the two column ends at A. This completes the second cycle of the distribution. The resulting maximum moment at A is then given by the addition of rows 4 an
33、d 5, 936 - 312 = 624. The distribution for the maximum moment at E follows a similar procedure.Distribution b in Table 7.3 is for the maximum moment at B. The most severe loading pattern for this is with total loading on spans AB and BC and dead load only on CD. The operations are similar to those i
34、n Distribution a, except that the T first cycle involves balancing the two adjacent joints A and C while recording only their carryover moments to B. In the second cycle, B is balanced by adding - (-1012 + 782)/4 = 58 to each side of B. The addition of rows 4 and 5 then gives the maximum hogging mom
35、ents at B. Distributions c and d, for the moments at joints C and D, follow patterns similar to Distribution b.The complete set of operations can be combined as in Table 7.4 by initially recording at each joint the fixed-end moments for both dead and total loading. Then the joint, or joints, adjacen
36、t to the one under consideration are balanced for the appropriate combination of loading, and carryover moments assigned .to the considered joint and recorded. The joint is then balanced to complete the distribution for that support.Maximum Mid-Span Moments. The most severe loading condition for a m
37、aximum mid-span sagging moment is when the considered span and alternate other spans and total loading. A concise method of obtaining these values may be included in the combined two-cycle distribution, as shown in Table 7.5. Adopting the convention that sagging moments at mid-span are positive, a m
38、id-span total; loading moment is calculated for the fixed-end condition of each span and entered in the mid-span column of row 2. These mid-span moments must now be corrected to allow for rotation of the joints. This is achieved by multiplying the carryover moment, row 3, at the left-hand end of the
39、 span by (1 + 0.5 D.F. )/2, and the carryover moment at the right-hand end by -(1 + 0.5 D.F.)/2, where D.F. is the appropriate distribution factor, and recording the results in the middle column. For example, the carryover to the mid-span of AB from A = (1 + 0.5/3)/2 x 69 = 40 and from B = -(1+ 0.5/
40、4)/2 x (-145) = 82. These correction moments are then added to the fixed-end mid-span moment to give the maximum mid-span sagging moment, that is, 733 + 40 + 82 = 855.7.2.3 Column ForcesThe gravity load axial force in a column is estimated from the accumulated tributary dead and live floor loading a
41、bove that level, with reductions in live loading as permitted by the local Code of Practice. The gravity load maximum column moment is estimated by taking the maximum difference of the end moments in the connected girders and allocating it equally between the column ends just above and below the joi
42、nt. To this should be added any unbalanced moment due to eccentricity of the girder connections from the centroid of the column, also allocated equally between the column ends above and below the joint.第七章框架结构 高层框架结构一般由平行或正交布置的梁柱结构组成，梁柱结构是由带有能承担弯矩作用节点的梁、柱组成。具有抗弯能力的梁、柱和节点共同作用抵抗水平荷载。连续框架可降低梁的跨中弯矩而有利于抵
43、抗重力荷载。 框架结构有简捷和便于采用矩形体系的优点。由于这种布置形式没有斜支撑和结构墙体，因此，没有不便利之处，内部可以自由布置，外部可以自由设计门、窗。框架结构对于25层以内的建筑是经济的，超过25层由于要限制其位移而花费的代价高，显得很不经济。如果框架与剪力墙及芯筒相结合，刚度能够大幅度提高，可以建造50层以上的建筑。板柱结构与框架结构非常相似，不同之处仅是用板代替了梁。和框架结构一样，板柱结构是通过其水平和竖向构件之间的连续抗弯作用来抵抗水平和竖向荷载。 对于高次超静定框架结构，应根据近似分析进行初步设计，随后进行精确分析和校核。分析过程一般包括以下几步: 1按近似方法确定梁和柱所受重
44、力荷载; 2初步确定在重力荷载作用下构件的截面尺寸，考虑水平荷载的作用进行构件截面尺寸的任意调整; 3将水平荷载分配到各梁柱结构上，对这些结构构件的内力进行初步分析; 4检验位移并对构件截面尺寸做必要的调整; 5按最不利的重力荷载和水平荷载组合检验构件强度，做必要的构件截面尺寸调整; 6为了更精确地验算构件强度和位移，利用计算机对结构进行整体分析，需要时则近一步调整构件截面尺寸。这一阶段中应包括考虑重力荷载对构件内力和位移产生的一二阶效应; 7构件和节点的详细设计。本章讨论在重力和水平荷载作用下结构的变形和内力分析方法。这些方法基本上按照设计过程中的次序介绍，首先是近似法，然后介绍计算机分析技
45、术。框架结构的稳定性分析将在第十六章中讨论。7.1框架结构的性能 框架结构的侧向刚度主要取决于梁、柱及节点的抗弯能力，在较高的框架中主要取决于柱子的轴向刚度。作用于框架任一层间的水平集中剪力由该层柱子的抗剪能力抵抗(图7. 1)。剪力使框架结构每层的柱产生双曲率弯曲，其反弯点大约在层高的中间部位。上、下柱引起的作用于节点处的弯矩由相邻梁承担，该梁、柱的变形引起框架的整体变形，使各层间产生水平位移。在水平推力作用下结构的整体变形和剪力图如图7. 1所示，其凹面朝向风荷载作用方向，最大倾角在基底附近，最小倾角在顶端。外部水平荷载产生的总弯矩由各层间两个边柱中的轴向拉、压力组成的力矩抵抗(图7.2
46、)。柱子的伸、缩引起结构的整体弯曲变形，并产生相应的水平位移。因为转角沿建筑高度累加，所以整体弯曲变形引起的层间位移随高度增加而增加，而剪切变形引起的层间位移随高度的增加而减小。其结果在建筑的最顶部整体弯曲对层间位移的贡献会大大超过剪切变形对层间位移的贡献。但是，整体弯曲变形对总位移的贡献与剪切变形对总位移的贡献之比不会超过10，除非在极高或细长的框架中。因此，高层框架结构变形型式为剪切型。从梁的连接受力性能来看，高层建筑采用的刚性节点连续的框架不同于一般简单连接的普通框架。梁在柱边附近产生负弯矩，跨中正弯矩值常常很小。这种连续性能使梁中最大弯矩对活荷载的作用方式非常敏感。如果能够估计出产生最
47、不利弯矩的因素，则必须加以认真的考虑。例如，重力荷载作用下梁在边柱附近产生的最大负弯矩只会在活荷载作用于边跨和相间跨时才能发生，如图7.3a中的A点。而梁在内柱附近产生的最大负弯矩只会在活荷载作用于相邻跨时才能发生，如图7.3a中的B点。当活荷载作用于本跨和相间跨时，梁的跨中正弯矩最大，如图7. 3a中的AB和CD跨。框架的尺寸取决于柱子在水平荷载作用下的抗弯强度，这往往会使框架柱的截面尺寸大于相应全对角支撑简单连接框架的柱截面尺寸。另外，框架支撑结构中的梁被设计为只具有跨中正弯矩，而框架结构的梁则被设计为端部为负弯矩和跨中为正弯矩，跨中弯矩值较小。因此，框架结构中梁的截面尺寸会小于相应的框架
48、支撑结构中梁的截面尺寸。梁截面的减小将会降低其造价，有时可以降低层高，经济效益明显。但是，由于刚性节点的处理相当复杂，代价较高，使上述经济优势被削弱。7.2重力荷载作用下构件内力的近似计算框架结构是多次超静定结构，因此，只有在确定了构件截面尺寸后才能进行精确分析。所以，在初步设计阶段，可根据传统的公式和不考虑构件特征值的简化分析法近似确定构件中的内力，以此为基础确定构件的截面尺寸。下面将讨论在重力荷载作用下构件内力计算的两种方法。 7.2.1梁的内力规范推荐值对于两跨以上的框架结构，当任何相邻两跨中的长跨不超过短跨的20%跨度，同时设计均布活荷载不超过3倍的恒载时，梁的弯矩和剪力可以按表7.1
49、确定。表中各数值是根据统一建筑规范【7.1】中的推荐值给出。对于其它情况，可按照楼面连续梁采月传统弯矩分配法或两次循环弯矩分配法进行分析确定。7.2.2弯矩分配法【7.2】 弯矩分配法用于计算多跨连续梁的弯矩是非常便利的形式。该方法的计算结果比表7.1中的推荐公式计算结果更精确，特别是对于不等跨和荷载变化较大的情况。弯矩分配法的分析假定如下: 1梁端约束弯矩以反时针方向为正，顺时针方向为负; 2被分析的梁与上、卞柱的连接为固接;3当构件尺寸尚未确定时，每个节点的分配系数取1/n，n是框架平面内连接在各节点上的构件总数。 两次循环弯矩分配实例. 现以一个实例具体说明两次循环弯矩分配的过程。在图7.4中表示出一个取自框架单榻结构的中跨连续梁AE和作于其上的荷载。恒载和全部荷载作用下每跨的固端弯矩采用图7.5中的公式计算。这些弯矩
