NCMA TEK National Concrete Masonry Association an information series from the national athority on concrete masonry technology STRENGTH DESIGN OF CONCRETE MASONRY WALLS FOR AXIAL LOAD & FLEXURE TEK 14-11B Strctral (3) Keywords: axial strength, design aids, design example, interaction diagrams, loadbearing walls, load combinations, strength design, flexral strength, reinforced concrete masonry, strctral design INTRODUCTION The strctral design of bildings reqires a variety of loads to be acconted for: dead and live loads, those from wind, earthqake, lateral soil pressre, lateral flid pressre as well as forces indced by temperatre changes, creep, shrinkage and differential movements. Becase most loads can act simltaneosly with another, the designer mst consider how these varios loads interact on the wall. For example, a concentrically applied compressive axial load can offset tension de to lateral load, effectively increasing flexral capacity. Bilding codes dictate which load combinations mst be considered, and reqire that the strctre be designed to resist all possible combinations. The design aids in this TEK cover combined axial compression or axial tension and flexre, as determined sing the strength design provisions of Bilding Code Reqirements for Masonry Strctres (ref. 3). For concrete masonry walls, these design provisions are otlined in TEK 14-4A, Strength Design of Concrete Masonry (ref. 1). Axial load-bending moment interaction diagrams accont for the interaction between moment and axial load on the design capacity of a wall. This TEK shows the portion of the interaction diagram that applies to the majority of wall designs. Althogh negative moments are not shown, the figres may be sed for these conditions, since reinforcement in the center of the wall will provide eqal strength nder either a positive or negative moment of the same magnitde. Conditions otside of this area may be determined sing Concrete Masonry Wall Design Software or Concrete Masonry Design Tables (refs. 4, 5). The reader is referred to Loadbearing Concrete Masonry Wall Design (ref. ) for a fll discssion of interaction diagrams. Figres 1 throgh 8 apply to flly or partially groted reinforced concrete masonry walls with a specified compressive strength f' m of 1,5 psi (1.34 MPa), and a maximm wall height of ft (6.1 m), Grade 6 (414 MPa) vertical reinforcement, with reinforcing bars positioned in the center of the wall and reinforcing bar spacing s from 8 in. to 1 in. ( 3 to 3,48 mm). The following discssion applies to simply spported walls and is limited to niform lateral loads. Other spport and loading conditions shold comply with applicable engineering procedres. Each figre applies to one specific wall thickness and one reinforcing bar size. In strength design, two different deflections are calclated; one for service level loads (δ s ) and another for factored loads (δ ). For a niformly loaded simply spported wall, the reslting bending moment is as follows: M x = W x h /8 + P xf (e/) + P x δ x (Eqn. 1) In the above eqation, notations with "x" are replaced with factored or service level vales as appropriate. The first term on the right side of Eqation 1 represents the maximm moment of a niform load at the mid-height of the wall (normally wind or earthqake loads). The second term represents the moment indced by eccentrically applied floor or roof loads. The third term is the P-delta effect, which is the moment indced by vertical axial loads and lateral deflection of the wall. DESIGN EXAMPLE An 8-in. (3-mm) thick, ft (6.1 m) high reinforced simply spported concrete masonry wall (115 pcf (1,84 kg/m 3 )) is to be designed to resist wind load as well as eccentrically applied axial live and dead loads as depicted in Figre 9. The designer mst determine the reinforcement size spaced at 4 in. (61 mm) reqired to resist the applied loads, listed below. D = 5 lb/ft (7.6 kn/m), at e =.75 in. (19 mm) L = 5 lb/ft (3.6 kn/m), at e =.75 in. (19 mm) W = psf (1. kpa) The wall weight at midheight for 115 pcf (1,84 kg/m 3 ) nit concrete density is 49 lb/ft (39 kg/m ) (ref. 7, Table 1). P w = (49 lb/ft )(1 ft) = 49 lb/ft (7. kn/m) TEK 14-11B 3 National Concrete Masonry Association (replaces TEK 14-11A)
5, 4, 3,, 1, -1, 1, s = 1, 3, 4, 5, 6, Figre 1 8-Inch (3-mm) Concrete Masonry Wall With No. 4 (M # 13) Reinforcing Bars 5, 4, 3,, 1, s = 1-1, 1,, 3, 4, 5, 6, Figre 8-Inch (3-mm) Concrete Masonry Wall With No. 5 (M # 16) Reinforcing Bars 5, 4, 3,, 1, -1, 1, s = 1, 3, 4, 5, 6, Figre 3 8-Inch (3-mm) Concrete Masonry Wall With No. 6 (M # 19) Reinforcing Bars
1, 8, 6, 4,, -,, s = 1 4, 6, 8, Figre 4 1-Inch (54-mm) Concrete Masonry Wall With No. 4 (M # 13) Reinforcing Bars 1, 8, 6, 4,, s = 1 -,, 4, 6, 8, Figre 5 1-Inch (54-mm) Concrete Masonry Wall With No. 5 (M # 16) Reinforcing Bars The applicable load combination (ref. 6) for this example is: 1.D + 1.6W + f 1 L +.5L r (Eqn. ) Dring design, all load combinations shold be checked. For brevity, only the combination above will be evalated here. First determine the cracking moment M cr : M cr = S n f r = 9,199 lb-in./ft (3,41 m. N/m), where S n = 93. in. 3 /ft (5.1 x 1 6 mm 3 /m) (ref. 8, Table 1) f r = 98.7 psi (.68 MPa) (ref. 1, Table 1 interpolated for grot at 4 in. (61 mm) o.c.) To check service level load deflection and moment, the following analysis is performed in an iterative process. First iteration, δ s = M ser1 = () (1)/8 + (5 + 5)(.75/) + (5 + 5 + 49)() = 1,89 in.-lb/ft (4,555 m. N/m) (from Eqn. 1) Since M cr < M ser1, therefore analyze as a cracked section. 5M crh 5( M ser M cr ) h δ s1 = + (1) 48E I 48E I (Eqn. 3) m g m cr where: E m = 9f' m = 1,35, psi (9,38 MPa) I g = 369.4 in. 4 /ft(54 x 1 6 mm 4 /m) (ref. 8, Table 1) I cr = 1. in. 4 /ft(54 x 1 6 mm 4 /m) (Table 1) δ s 1 5(9,199)(4) 5(1,89 9,199)(4) = + 48(1,35,)(369.4) 48(1,35,)(1.) =.76 in. (19 mm) Second iteration, δ s =.76 in. (19 mm) M ser = 1,89 + (5 + 5 + 49)(.76) = 13,47 in.-lb/ft (4,91 m. N/m) δ s =.97 in. (5 mm) Third iteration, M ser = 13,511 in.-lb/ft (5,8 m. N/m), δ s3 = 1. in. (6 mm). Becase δ s3 is within 5% of δ s, then δ s = δ s3. Check δ s against the maximm service load deflection: δ s <.7h =.7(4) = 1.68 in. (43 mm) > 1. in. (6 mm), OK. If M ser < M cr, instead of sing Eqation for deflection, we wold have sed:
1, 8, 4, s = 1 -,, 4, 6, 8, 1, 1, 14, Figre 6 1-Inch (35-mm) Concrete Masonry Wall With No. 4 (M # 13) Reinforcing Bars 1, 8, 4, s = 1 -,, 4, 6, 8, 1, 1, 14, Figre 7 1-Inch (35-mm) Concrete Masonry Wall With No. 5 (M # 16) Reinforcing Bars 1, 8, 4, -,, s = 1 4, 6, 8, 1, 1, 14, Figre 8 1-Inch (35-mm) Concrete Masonry Wall With No. 6 (M # 19) Reinforcing Bars
Table 1 Cracked Moment of Inertia, I cr, in. 4 /ft a Bar size, Spacing of reinforcement, in. (mm) No. (M #) 8 (3) 16 (46) 4 (61) 3 (813) 4 (1,16) 48 (1,19) 7 (1,89) 96 (,438) 1 (3,48) 8-inch (3-mm) wall thickness: 4 (13) 47.9 8.9 1. 16.6 13.7 11.8 8.5 6.38 5.1 5 (16) 63.8 4. 9.6 3.7 19.8 17. 1.1 9.4 7.74 6 (19) 78.5 51. 38.5 31.1 6..7 16.3 1.8 1.5 1-inch (54-mm) wall thickness: 4 (13) 81.8 48.5 34.9 7.4.6 19.3 13.5 1.4 8.47 5 (16) 11.5 67.9 49.7 39.5 3.9 8. 19.9 15.4 1.6 1-inch (35-mm) wall thickness: 4 (13) 15.7 73.4 5.5 41.1 33.8 8.8. 15.4 1.5 5 (16) 171.6 13.7 75.4 59.6 49.4 4.3 9.7 3. 18.8 6 (19) 16.1 134.3 99.4 79.3 66. 56.9 4.3 31.4 5.7 a Intermediate spacings may be interpolated. 5M h ser δ s = 48EmI (Eqn. 4) g To determine deflection and moment de to factored loads, an identical calclation is performed as for service loads with the exception that factored loads are sed in Eqations 1 and 3 or Eqations 1 and 4. First iteration, δ =, sing Eqation 1: lateral = 1.6[()() (1)/8] = 19, roof & floor = 1.(5)(.75/) +.5(5)(.75/) = 81 P-delta = [1.(5 + 49) +.5(5)] = M 1 = lateral + roof & floor + P-delta = 19,481lb-in./ft (7,1 m. N/m) From Eqation 3, sing M 1 instead of M ser, δ 1 =.9 in. (58 mm). Second iteration, M =,543 lb-in./ft (8,356 m. N/m), δ =.94 in. (75 mm). Third iteration, M 3 = 3,41 lb-in./ft (8,678 m. N/m), δ 3 = 3.1 in. (79 mm). Forth iteration, M 4 = 3,65 lb-in./ft (8,767 m. N/m), δ 4 = 3.17 in. (81 mm). δ 4 is within 5% of δ 3. Therefore, M = M 4 = 3,65 lb-in./ft = 1,971 lb-ft/ft (8,767 m. N/m). P = 1.(5 + 49) +.5(5) = 1,337 lb/ft ( kn/m) To determine the reqired reinforcement size and spacing to resist these loads, P and M are plotted on the appropriate interaction diagram ntil a satisfactory design is fond. If the axial load is sed to offset stresses de to bending, only the nfactored dead load shold be considered. Figre 1 shows that No. 4 bars at 4 in. (M #13 at 61 mm) on center is adeqate. If a larger bar spacing is desired, No. 5 at 3 in. (M #16 at 813 mm) or No. 6 at 48 in. (M #19 at 119 mm) also appear to meet the design reqirements (see Figres and 3, respectively). However, the design procedre shold be repeated and verified with the new grot spacings and associated properties. Althogh above grade wall design is seldom governed by ot-of-plane shear, the shear capacity shold be checked. NOMENCLATURE D dead load, lb/ft (kn/m) E m modls of elasticity of masonry in compression, psi (MPa) e eccentricity of axial load - measred from centroid of wall, in. (mm) f m specified masonry compressive strength, psi (MPa) f r modls of rptre, psi (MPa) f 1 factor for floor load: = 1. for floors in places of pblic assembly, for live loads in excess of 1 psf (4.8 kpa) and for parking garage live loads; =.5 otherwise h height of wall, in. (mm) I cr moment of inertia of cracked cross-sectional area of a member, in. 4 /ft(mm 4 /m) I g moment of inertia of gross cross-sectional area of a member, taken here as eqal to I avg, in. 4 /ft (mm 4 /m) L live load, lb/ft (kn/m) L r roof live load, lb/ft (kn/m) M cr nominal cracking moment strength, in.-lb/ft (kn. m/m) M ser service moment at midheight of a member, inclding P- delta effects, in.-lb/ft (kn. m/m) M factored moment, in.-lb/ft or ft-lb/ft (kn. m/m) P factored axial load, lb/ft (kn/m) P f factored load from tribtary floor or roof areas, lb/ft (kn/ m) P w load de to wall weight, lb/ft (kn/m) S n section modls of the net cross-sectional area of a member, in. 3 /ft(mm 3 /m) s spacing of vertical reinforcement, in. (mm) W wind load, psf (kn/m ) δ s horizontal deflection at midheight nder service loads, in. (mm) deflection de to factored loads, in. (mm) δ
W = psf (1. kpa) (sction) P (dead & live) e = 3 4 in. (19 mm) C L ft (6.1 m) REFERENCES 1. Strength Design of Concrete Masonry, TEK 14-4A. National Concrete Masonry Association,.. Loadbearing Concrete Masonry Wall Design, TEK 14-5A. National Concrete Masonry Association,. 3. Bilding Code Reqirements for Masonry Strctres, ACI 53-/ASCE 5-/TMS 4-. Reported by the Masonry Strctres Joint Committee,. 4. Concrete Masonry Wall Design Software, CMS-1. National Concrete Masonry Association,. 5. Concrete Masonry Design Tables, TR 11A. National Concrete Masonry Association,. 6. Minimm Design Loads for Bildings and Other Strctres, ASCE 7-. American Society of Civil Engineers,. 7. Concrete Masonry Wall Weights, TEK 14-13A. National Concrete Masonry Association,. 8. Section Properties of Concrete Masonry Walls, TEK 14-1. National Concrete Masonry Association, 1993. Figre 9 Wall Section for Loadbearing Wall Design Example METRIC CONVERSIONS To convert: To metric nits: Mltiply English nits by: ft m.348 lb-ft/ft m. N/m 4.448 lb-in/ft m. N/m.3769 in. mm 5.4 in. 4 /ft mm 4 /m 1,366, lb/ft kn/m.145939 psi MPa.689476 NATIONAL CONCRETE MASONRY ASSOCIATION To order a complete TEK Manal or TEK Index, 1375 Snrise Valley Drive, Herndon, Virginia 171 contact NCMA Pblications (73) 713-19 www.ncma.org