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 insizes to allow for horizontal loading.3. Approximate allocation of horizontal loading to bents and preliminary analysis of memberforces 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, andadjustment 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 BEHA VIORThe 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 GRA VITY 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 Forces—Code 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 Table7.1. 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 clockwisemoment 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 Distribution—Worked 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 hoggingmoment 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 M BA where U is the unbalanced moment. This is not recorded, but half of it, ( - U/4)/2, is carried over to M AB. 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 M BA (row 5), implicitly adding the same moment tothe 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。