BEAM DESIGN In order to be able to design beams, we need both moments and shears. 1. Moment a) From diret design method or equivalent frame method b) From loads applied diretly to beams inluding beam weight or max M bm = W bm l n 2 /10 ACI 8.3.3 2. Shear what to use for loads? ACI 13.6.8 (see next page) 1
2
Figure 2. Shear Perimeter in Slabs with Beams. (MaGregor 1997). 3
Without Beams, Flat Plates and Flat Slabs Tests of flat plate strutures indiate that in most pratial ases, the apaity is governed by shear. Two types of shear may be ritial in the design of the flat slabs: 1) Beam type shear leading to diagonal tension failure. (long narrow slabs). A potential diagonal rak extends in a plane aross the entire width l 2 of the slab. d Critial setion a distane "d" from the fae of olumn or apital. V u φv or else shear reinforement is required V = 2 f l d Eq. 11-3 of ACI 2 V = (1.9 f + 2500 V / M ) l d 3.5 f l d Eq. 11-5 of ACI u u 2 2 4
1.(a). Exterior Columns When design is arried out using the Diret Design Method, ACI Se. 13.6.3.6 speifies that the moment that is transferred from a slab to an edge olumn is 0.3M 0. This moment is used to ompute the shear stresses due to moment transfer to the edge olumn, as shown in Se. 13-8. Although the ACI Code does not speifially state, this moment an be assumed to be about the entroid of the shear perimeter. The exterior negative moment from the Diret Design Method alulation is divided between the olumns above and below the slab in proportion to the olumn stiffnesses, 4EI/l. The resulting olumn moments are used in the design of the olumns. 2.(b). Interior Columns At interior olumns, the moment transfer alulations and the total moment used in the design of the olumns above and below the floor are based on an unbalaned moment resulting from an uneven distribution of live load. The unbalaned moment is omputed assuming that the longer span adjaent to the olumn is loaded with the fatored dead load and half the fatored live load, while the shorter span arries only the fatored dead load. The total unbalaned negative moment at the joint is thus M 2 2 ( w + 0.5w ) l l w l l = 0.65 8 8 d l 2 n d 2 n where w d and w l refer to the fatored dead and live loads on the longer span and w d, l 2, and l n refer to the shorter span adjaent to the olumn. The fator 0.65 is the fration of the stati moment assigned to the negative moment at an interior support. The fators 0.65 and 1/8 ombine to give 0.081. A portion of the unbalaned moment is distributed to the slabs, and the rest goes to the olumns. Sine slab stiffnesses have not been alulated, it is assumed that most of the moment is transferred to the olumns, giving 2 M ( ) ( ) 2 ol = 0.07 wd + 0.5wl l2ln w d l 2 l n (ACI Eq. 13-4) The moment, M ol, is used to design the slab-to-olumn joint. It is distributed between the olumns above and below the joint in the ratio of their stiffnesses to determine the moments used to design the olumns. 5
2. SHEAR DESIGN IN FLAT PLATES AND FLAT SLABS When two-way slabs are supported diretly by olumns, as in flat slabs and flat plates, or when slabs arry onentrated loads, as in footings, shear near the olumns is of ritial importane. Tests of flat plate strutures indiate that, in most pratial ases, the apaity is governed by shear. a. Slabs without Speial Shear Reinforement Two kinds of shear may be ritial in the design of flat slabs, flat plates, or footings. The first is the familiar beam-type shear leading to diagonal tension failure. Appliable partiularly to long narrow slabs or footings, this analysis onsiders the slab to at as a wide beam, spanning between supports provided by the perpendiular olumn strips. A potential diagonal rak extends in a plane aross the entire width 12 of the slab. The ritial setion is taken a distane d from the fae of the olumn or apital. As for beams, the design shear strength (φv, must be at least equal to the required strength V u at fatored loads. The nominal shear strength V should be alulated by V = 2 f b d w with bw = l2 in this ase. Alternatively, failure may our by punhing shear, with the potential diagonal rak following the surfae of a trunated one or pyramid around the olumn, apital, or drop panel, as shown in Fig. 13.14a. The failure surfae extends from the bottom of the slab, at the support, diagonally upward to the top surfae. The angle of inlination with the horizontal, θ (see Figure below), depends upon the nature and amount of reinforement in the slab. It may range between about 20 and 45. The ritial setion for shear is taken perpendiular to the plane of the slab and a distane d/2 from the periphery of the support, as shown. The shear fore V u to be resisted an be alulated as the total fatored load on the area bounded by panel enterlines around the olumn less the load applied within the area defined by the ritial shear perimeter, unless signifiant moments must be transferred from the slab to the olumn (see Se. 3). Figure 1. Failure surfae defined by punhing shear (Nilson s Book) 6
At suh a setion, in addition to the shearing stresses and horizontal ompressive stresses due to negative bending moment, vertial or somewhat inlined ompressive stress is present, owing to the reation of the olumn. The simultaneous presene of vertial and horizontal ompression inreases the shear strength of the onrete. For slabs supported by olumns having a ratio of long to short sides not greater than 2, tests indiate that the nominal shear strength may be taken equal to V = 4 f b d ACI Eq. (11-35) 0 aording to ACI Code 11.12.2, where b o = the perimeter along the ritial setion. However, for slabs supported by very retangular olumns, the shear strength predited by Equation (11-35) has been found to be unonservative. The value of V approahes 2 f bd 0 as β, the ratio of long to short sides of the olumn, beomes very large. Based on this, ACI Code 11.12.2 states further that V in punhing shear shall not be taken greater than V 4 = 2 + β f b d 0 ACI Eq. (11-33) Further tests have shown that the shear strength V dereases as the ratio of ritial perimeter to slab depth, bold, inreases. Aordingly, ACI Code 11.12.2 states that V in punhing shear must not be taken greater than V α sd = 2 + b0 f b d 0 ACI Eq. (11-34) where a s is 40 for interior olumns, 30 for edge olumns, and 20 for orner olumns, i.e., olumns having ritial setions with 4, 3, or 2 sides, respetively. The shear design strength is the smallest of three equations given above. 7
Figure 2. L1 L L 1 1 L 1 L 2 L 2 L 2 L 2 L 2 8
Figure 3. Failure surfae defined by punhing shear (KaGregor s Book) 9
3. TRANSFER OF MOMENTS AT COLUMNS The analysis for punhing shear in flat plates and flat slabs presented in Se. 2 assumed that the shear fore V u was resisted by shearing stresses uniformly distributed around the perimeter b o of the ritial setion, a distane d/2 from the fae of the supporting olumn. The nominal shear strength V was given by Eqs. (11-33, 11-34, 11-35). If signifiant moments are to be transferred from the slab to the olumns, as would result from unbalaned gravity loads on either side of a olumn or from horizontal loading due to wind or seismi effets, the shear stress on the ritial setion is no longer uniformly distributed. The situation an be modeled as shown in Figure 4. Here V u represents the total vertial reation to be transferred to the olumn, and M,, represents the unbalaned moment to be transferred, both at fatored loads. The vertial fore V u auses shear stress distributed more or less uniformly around the perimeter of the ritial setion as assumed earlier, represented by the inner pair of vertial arrows, ating downward. The unbalaned moment M u auses additional loading on the joint, represented by the outer pair of vertial arrows, whih add to the shear stresses otherwise present on the right side, in the sketh, and subtrat on the left side. Tests indiate that for square olumns about 60 perent of the unbalaned moment is transferred by flexure (fores T and C in Figure 4) and about 40 perent by shear stresses on the faes of the ritial setion. For retangular olumns, it is reasonable to suppose that the portion transferred by flexure inreases as the width of the ritial setion that resists the moment inreases, i.e., as 2 + d beomes larger relative to l + d in Figure 4. Aording to ACI Code 13.5.3, the moment onsidered to be transferred by flexure is where M = γ M ACI Eq. (13-1) ub f u 1 γ f = 2 1 + b / b 3 1 2 M u ACI Eq. (13-1) while that assumed to be transferred by shear, by ACI Code 11.12.6, is M uv 1 = 1 2 1 + b / b 3 1 2 M u ACI Eq. (13-) Please note: 10
Where: b = + d, b = + d for Interior Column 1 1 2 2 d b1 = 1+, b2 = 2 + d 2 for Edge Column d d b1 = 1+, b2 = 2 + 2 2 for Corner Column It is seen that for a square olumn these equations indiate that 60 perent of the unbalaned moment is transferred by flexure and 40 perent by shear, in aordane with the available data (see R11.12.6.1 page 184 of ACI). If 2 is very large relative to 1, nearly all the moment is transferred by flexure. 11
Figure 4. 12
Figure 5. 13
ACI 13.5.3. The moment M ub an be aommodated by onentrating a suitable fration of the slab olumn-strip reinforement near the olumn. Aording to ACI Code 13.5.3, this steel must be plaed within a width between lines 1.5h on eah side of the olumn or apital, where h is the total thikness of the slab or drop panel. Figure 6. Shear Stress Distribution Around Column Edges Transfer Nominal Moment Strength M n. 14
Figure 7. Shear Stress Distribution Around Column Edges: (a) Interior Column, (b) End Column (See ACI Page 186, Figure R11.12.6.2) The moment M uv together with the vertial reation delivered to the olumn, auses shear stresses assumed to vary linearly with distane from the entroid of the ritial setion, as indiated for an interior olumn by Figure 4. The stresses an be alulated from v l V A = u uv l ACI R11.12.6.2 Page 184 M J v r V A = u + uv r ACI R11.12.6.2 Page 185 M J where 15
A = area of ritial setion = 2d ( + d) + ( + d) 1 2 l, r = distanes from entroid of ritial setion to left and right fae of setion respetively J = property of ritial setion analogous to polar moment of inertia The quantity J is to be alulated from 3 3 2 d ( 1+ d) 2( 1+ dd ) 1+ d J = + + 2 d( 2 + d) 12 12 12 2 ACI Note the impliation, in the use of the parameter J in the form of a polar moment of inertia, that shear stresses indiated on the near and far faes of the ritial setion in Figure 4 have horizontal as well as vertial omponents. Aording to ACI Code 11.12.6, the maximum shear stress alulated by Eq. (13-) must not exeed φ vn. For slabs without shear reinforement, φvn = φv / b0d, where V is the smallest value given by Eqs. (13- ), (13-), or (13-). For slabs with shear reinforement other than shearheads, φ v = φ (V + V S )/b o d. n ACI Code 13.5.3.3 permits some inrease in the amount of unbalaned moment assumed to be transferred by flexure, with a orresponding derease in the amount transferred by shear, provided that a speified redution is made in the allowable shear apaity at that support. Equations similar to those above an be derived for the edge olumns shown in Figures 4 and 5 for a orner olumn. Note that although the entroidal distanes l and r are equal for the interior olumn, this is not true for the edge olumn of Fig. 4 or for a orner olumn. Aording to ACI Code 13.6.3.6, when the diret design method is used, the moment to be transferred between slab and an edge olumn by shear is to be taken equal to 0.30M o, where M o is found from Eq. (13.1). This is intended to ompensate for assigning a high proportion of the stati moment to the positive and interior negative moment regions aording to Table 13.1, and to ensure that adequate shear strength is provided between slab and edge olumn, where unbalaned moment is high and the ritial setion width is redued. The appliation of moment to a olumn from a slab or beam introdues shear to the olumn also, as is lear from Fig. 13.24a. This shear must be onsidered in the design of lateral olumn reinforement. As was pointed out in Se. 13.6, most flat plate strutures, if they are overloaded, fail in the region lose to the olumn, where large shear and bending fores must be 13.8 16
Example Column Moments Use Equivalent Frame Method or get from analysis. Exterior Column ACI 13.6.3. Table must be used The olumn supporting an edge beam must provide resisting moment equal to the applied from the edge of slab. Interior Column ACI Eq. 13-4 page 228 (Set. 13.6.9.2): 2 ( ) ( ) 2 M = 0.07 w + 0.5w l l w l l ol d l 2 n d 2 n (ACI Eq. 13-4) w d w + 0.5w d l l l n n 17
4. Openings in Slabs Almost invariably, flab systems must inlude openings. These may be of substantial size, as required by stairways and elevator shafts, or they may be of smaller dimensions, suh as those needed to aommodate heating, plumbing, and ventilating risers; floor and roof drains; and aess hathes. Relatively small openings usually are not detrimental in beam-supported slabs. As a general rule, the equivalent of the interrupted reinforement should be added at the fides of the opening. Additional diagonal bars should be inluded at the orners to ontrol the raking that will almost inevitably our there. The importane of small openings in slabs supported diretly by olumns (flat slabs and flat plates) depends upon the loation of the opening with respet to the olumns. From a strutural point of view, they are best loated away from the olumns, preferably in the area ommon to the flab middle strips. Unfortunately, arhitetural and funtional onsiderations usually ause them to be loated lose to the olumns. In this afe, the redution in effetive shear perimeter if the major onern, beause suh floors are usually shear-ritial. Aording to ACI Code 11.12.5, if the opening if lose to the olumn (within 10 flab thiknesses or within the olumn strips), then that part of b o inluded within the radial lines projeting from the opening to the entroid of the olumn should be onsidered ineffetive. If shearheads (fee Se. 13.6d) are used under suh irumstanes, the redution in width of the ritial setion if found in the fame way, exept that only one-half the perimeter inluded within the radial lines need be deduted. With regard to flexural requirements, the total amount of steel required by alulation must be provided regardless of openings. Any steel interrupted by holes should be mathed with an equivalent amount of supplementary reinforement on either fide, properly lapped to transfer stress by bond. Conrete ompression area to provide the required strength must be maintained; usually this would be restritive only near the olumns. Aording to ACI Code 13.4.2, openings of any size may be loated in the area ommon to interseting middle strips. In the area ommon to interseting olumn strips, not more than one-eighth of the width of the olumn strip in either span an be interrupted by openings. In the area ommon to one middle strip and one olumn strip, not more than one-quarter of the reinforement in either strip may be interrupted by the opening. ACI Code 13.4.1 permits openings of any size if it an be shown by analysis that the strength of the flab if at least equal to that required and that all servieability onditions, i.e., raking and defletion limits, are met. The strip method of analysis and design for openings in slabs, by whih speially reinfored integral beams, or strong bands, of depth equal to the flab depth are used to frame the openings, will be desribed in detail in Chapter 15. Very large openings should preferably be framed by beams or slab bands on inreased depth to restore the ontinuity of the slab. The beams must be designed to arry a portion of the floor load, in addition to slab. The beams must be designed to arry a portion of the floor load, in addition to loads applied diretly by partition walls, elevator support beams, or stair slabs. 18