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 continuity 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 for 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 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 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 and 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 adjustment 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 where 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 included 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 girders, 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 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 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 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 horizontal 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 that 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 frame 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 sensitive 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 hogging 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 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 designed 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 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 accurate 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 estimating 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 three 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 floor 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 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 joint 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-end 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 pattern 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 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 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 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 and 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 in 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 moments 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, adjacent 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 maximum 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 mid-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 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+ 0.5/4)/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 above 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 joint. 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.第七章框架結構 高層框架結構一般由平行或正交布置的梁柱結構組成，梁柱結構是由帶有能承擔彎矩作用節點的梁、柱組成。具有抗彎能力的梁、柱和節點共同作用抵抗水平荷載。連續框架可降低梁的跨中彎矩而有利于抵抗重力荷載。 框架結構有簡捷和便于采用矩形體系的優點。由于這種布置形式沒有斜支撐和結構墻體，因此，沒有不便利之處，內部可以自由布置，外部可以自由設計門、窗。框架結構對于25層以內的建筑是經濟的，超過25層由于要限制其位移而花費的代價高，顯得很不經濟。如果框架與剪力墻及芯筒相結合，剛度能夠大幅度提高，可以建造50層以上的建筑。板柱結構與框架結構非常相似，不同之處僅是用板代替了梁。和框架結構一樣，板柱結構是通過其水平和豎向構件之間的連續抗彎作用來抵抗水平和豎向荷載。 對于高次超靜定框架結構，應根據近似分析進行初步設計，隨后進行精確分析和校核。分析過程一般包括以下幾步: 1按近似方法確定梁和柱所受重力荷載; 2初步確定在重力荷載作用下構件的截面尺寸，考慮水平荷載的作用進行構件截面尺寸的任意調整; 3將水平荷載分配到各梁柱結構上，對這些結構構件的內力進行初步分析; 4檢驗位移并對構件截面尺寸做必要的調整; 5按最不利的重力荷載和水平荷載組合檢驗構件強度，做必要的構件截面尺寸調整; 6為了更精確地驗算構件強度和位移，利用計算機對結構進行整體分析，需要時則近一步調整構件截面尺寸。這一階段中應包括考慮重力荷載對構件內力和位移產生的一二階效應; 7構件和節點的詳細設計。本章討論在重力和水平荷載作用下結構的變形和內力分析方法。這些方法基本上按照設計過程中的次序介紹，首先是近似法，然后介紹計算機分析技術。框架結構的穩定性分析將在第十六章中討論。7.1框架結構的性能 框架結構的側向剛度主要取決于梁、柱及節點的抗彎能力，在較高的框架中主要取決于柱子的軸向剛度。作用于框架任一層間的水平集中剪力由該層柱子的抗剪能力抵抗(圖7. 1)。剪力使框架結構每層的柱產生雙曲率彎曲，其反彎點大約在層高的中間部位。上、下柱引起的作用于節點處的彎矩由相鄰梁承擔，該梁、柱的變形引起框架的整體變形，使各層間產生水平位移。在水平推力作用下結構的整體變形和剪力圖如圖7. 1所示，其凹面朝向風荷載作用方向，最大傾角在基底附近，最小傾角在頂端。外部水平荷載產生的總彎矩由各層間兩個邊柱中的軸向拉、壓力組成的力矩抵抗(圖7.2 )。柱子的伸、縮引起結構的整體彎曲變形，并產生相應的水平位移。因為轉角沿建筑高度累加，所以整體彎曲變形引起的層間位移隨高度增加而增加，而剪切變形引起的層間位移隨高度的增加而減小。其結果在建筑的最頂部整體彎曲對層間位移的貢獻會大大超過剪切變形對層間位移的貢獻。但是，整體彎曲變形對總位移的貢獻與剪切變形對總位移的貢獻之比不會超過10，除非在極高或細長的框架中。因此，高層框架結構變形型式為剪切型。從梁的連接受力性能來看，高層建筑采用的剛性節點連續的框架不同于一般簡單連接的普通框架。梁在柱邊附近產生負彎矩，跨中正彎矩值常常很小。這種連續性能使梁中最大彎矩對活荷載的作用方式非常敏感。如果能夠估計出產生最不利彎矩的因素，則必須加以認真的考慮。例如，重力荷載作用下梁在邊柱附近產生的最大負彎矩只會在活荷載作用于邊跨和相間跨時才能發生，如圖7.3a中的A點。而梁在內柱附近產生的最大負彎矩只會在活荷載作用于相鄰跨時才能發生，如圖7.3a中的B點。當活荷載作用于本跨和相間跨時，梁的跨中正彎矩最大，如圖7. 3a中的AB和CD跨。框架的尺寸取決于柱子在水平荷載作用下的抗彎強度，這往往會使框架柱的截面尺寸大于相應全對角支撐簡單連接框架的柱截面尺寸。另外，框架支撐結構中的梁被設計為只具有跨中正彎矩，而框架結構的梁則被設計為端部為負彎矩和跨中為正彎矩，跨中彎矩值較小。因此，框架結構中梁的截面尺寸會小于相應的框架支撐結構中梁的截面尺寸。梁截面的減小將會降低其造價，有時可以降低層高，經濟效益明顯。但是，由于剛性節點的處理相當復雜，代價較高，使上述經濟優勢被削弱。7.2重力荷載作用下構件內力的近似計算框架結構是多次超靜定結構，因此，只有在確定了構件截面尺寸后才能進行精確分析。所以，在初步設計階段，可根據傳統的公式和不考慮構件特征值的簡化分析法近似確定構件中的內力，以此為基礎確定構件的截面尺寸。下面將討論在重力荷載作用下構件內力計算的兩種方法。 7.2.1梁的內力規范推薦值對于兩跨以上的框架結構，當任何相鄰兩跨中的長跨不超過短跨的20%跨度，同時設計均布活荷載不超過3倍的恒載時，梁的彎矩和剪力可以按表7.1確定。表中各數值是根據統一建筑規范【7.1】中的推薦值給出。對于其它情況，可按照樓面連續梁采月傳統彎矩分配法或兩次循環彎矩分配法進行分析確定。7.2.2