Reinforced Concrete Design

Transcription

1 Reinored Conrete Design Notation: a = depth o the eetive ompression blok in a onrete beam A = name or area A g = gross area, equal to the total area ignoring an reinorement A s = area o steel reinorement in onrete beam design A s = area o steel ompression reinorement in onrete beam design A st = area o steel reinorement in onrete olumn design A v = area o onrete shear stirrup reinorement ACI = Amerian Conrete Institute b = width, oten ross-setional b E = eetive width o the lange o a onrete T beam ross setion b = width o the lange b w = width o the stem (web) o a onrete T beam ross setion = distane rom the top to the neutral axis o a onrete beam (see x) = shorthand or lear over C = name or entroid = name or a ompression ore C = ompressive ore in the ompression steel in a doubl reinored onrete beam C s = ompressive ore in the onrete o a doubl reinored onrete beam d = eetive depth rom the top o a reinored onrete beam to the entroid o the tensile steel d = eetive depth rom the top o a reinored onrete beam to the entroid o the ompression steel d b = bar diameter o a reinoring bar D = shorthand or dead load DL E E = shorthand or dead load = modulus o elastiit or Young s modulus = shorthand or earthquake load = modulus o elastiit o onrete 201 E s = modulus o elastiit o steel = smbol or stress = ompressive stress = onrete design ompressive stress s = stress in the steel reinorement or onrete design s = ompressive stress in the ompression reinorement or onrete beam design = ield stress or strength t = ield stress or strength o transverse reinorement F = shorthand or luid load G = relative stiness o olumns to beams in a rigid onnetion, as is Ψ h = ross-setion depth H = shorthand or lateral pressure load h = depth o a lange in a T setion I transormed = moment o inertia o a multimaterial setion transormed to one material k = eetive length ator or olumns = length o beam in rigid joint l b l l d l dh l n L L r LL M n M u n = length o olumn in rigid joint = development length or reinoring steel = development length or hooks = lear span rom ae o support to ae o support in onrete design = name or length or span length, as is l = shorthand or live load = shorthand or live roo load = shorthand or live load = nominal lexure strength with the steel reinorement at the ield stress and onrete at the onrete design strength or reinored onrete beam design = maximum moment rom atored loads or LRFD beam design = modulus o elastiit transormation oeiient or steel to onrete

2 n.a. = shorthand or neutral axis (N.A.) ph = hemial alkalinit P = name or load or axial ore vetor P o = maximum axial ore with no onurrent bending moment in a reinored onrete olumn P n = nominal olumn load apait in onrete design P u = atored olumn load alulated rom load ators in onrete design R = shorthand or rain or ie load = radius o urvature in beam deletion relationships (see ρ) R n = onrete beam design ratio = M u /bd 2 s = spaing o stirrups in reinored onrete beams S = shorthand or snow load t = name or thikness T = name or a tension ore = shorthand or thermal load U = atored design value V = shear ore apait in onrete V s = shear ore apait in steel shear stirrups V u = shear at a distane o d awa rom the ae o support or reinored onrete beam design w = unit weight o onrete w DL = load per unit length on a beam rom dead load w LL = load per unit length on a beam rom live load w sel wt = name or distributed load rom sel weight o member w u = load per unit length on a beam rom load ators W x β 1 ε ε t ε λ φ φ γ ρ ρ balaned υ = shorthand or wind load = horizontal distane = distane rom the top to the neutral axis o a onrete beam (see ) = vertial distane = oeiient or determining stress blok height, a, based on onrete strength, = elasti beam deletion = strain = strain in the steel = strain at the ield stress = modiiation ator or lightweight onrete = resistane ator = resistane ator or ompression = densit or unit weight = radius o urvature in beam deletion relationships (see R) = reinorement ratio in onrete beam design = A s /bd = balaned reinorement ratio in onrete beam design = shear strength in onrete design Reinored Conrete Design Strutural design standards or reinored onrete are established b the Building Code and Commentar (ACI ) published b the Amerian Conrete Institute International, and uses strength design (also known as limit state design). = onrete ompressive design strength at 28 das (units o psi when used in equations) Materials Deormed reinoring bars ome in grades 40, 60 & 75 (or 40 ksi, 60 ksi and 75 ksi ield strengths). Sizes are given as # o 1/8 up 202

3 to #8 bars. For #9 and larger, the number is a nominal size (while the atual size is larger). Reinored onrete is a omposite material rom a mixture o ement, oarse aggregate, ine aggregate and water, and the average densit is onsidered to be 150 lb/t 3. It has the properties that it will reep (deormation with long term load) and shrink (a result o hdration) that must be onsidered. Plane setions o omposite materials an still be assumed to be plane (strain is linear), but the stress distribution is not the same in both materials beause the modulus o elastiit is dierent. (=E ε) = E1ε = 2 = E2ε = where R (or ρ) is the radius o urvature In order to determine the stress, we an deine n as the ratio o the elasti moduli: n is used to transorm the width o the seond material suh that it sees the equivalent element stress. E R E R E n = E 2 1 Transormed Setion and I In order to determine stresses in all tpes o material in the beam, we transorm the materials into a single material, and alulate the loation o the neutral axis and modulus o inertia or that material. ex: When material 1 above is onrete and material 2 is steel: to transorm steel into onrete E2 n = = E 1 E E steel onrete to ind the neutral axis o the equivalent onrete member we transorm the width o the steel b multipling b n to ind the moment o inertia o the equivalent onrete member, I transormed, use the new geometr resulting rom transorming the width o the steel onrete stress: steel stress: onrete steel = = I I M transorme d Mn transorme d 203

4 Reinored Conrete Beam Members Strength Design or Beams Strength design method is similar to LRFD. There is a nominal strength that is redued b a ator φ whih must exeed the atored design stress. For beams, the onrete onl works in ompression over a retangular stress blok above the n.a. rom elasti alulation, and the steel is exposed and reahes the ield stress, For stress analsis in reinored onrete beams the steel is transormed to onrete an onrete in tension is assumed to be raked and to have no strength the steel an be in tension, and is plaed in the bottom o a beam that has positive bending moment 204

5 The neutral axis is where there is no stress and no strain. The onrete above the n.a. is in ompression. The onrete below the n.a. (shown as x, but also sometimes named ) is onsidered ineetive. The steel below the n.a. is in tension. (Shown as x, but also sometimes named.) Beause the n.a. is deined b the moment areas, we an solve or x knowing that d is the distane rom the top o the onrete setion to the entroid o the steel: x bx nas( d x ) = 0 2 x an be solved or when the equation is rearranged into the generi ormat with a, b & in the 2 binomial equation: ax + bx + = 0 b b ± x = b 2 4a 2a T-setions I the n.a. is above the bottom o a lange in a T setion, x is ound as or a retangular setion. h b w h I the n.a. is below the bottom o a lange in a T setion, x is ound b inluding the lange and the stem o the web (b w ) in the moment area alulation: ( x h ) h b h x ( x h ) ( ) bw nas d x = 2 b w Load Combinations - (Alternative values allowed) 1.4D 1.2D + 1.6L + 0.5(L r or S or R) 1.2D + 1.6(L r or S or R) + (1.0L or 0.5W) 1.2D + 1.0W + 1.0L + 0.5(L r or S or R) 1.2D + 1.0E + 1.0L + 0.2S 0.9D + 1.0W 00.9D + 1.0E Internal Equilibrium C = ompression in onrete = stress x area = 0.85 ba T = tension in steel = stress x area = A s h A s b C or x a= d β 1 x n.a. T atual stress 0.85 a/2 T Whitne stress blok C 205

6 C = T and Mn = T(d-a/2) where = onrete ompression strength 4000 a = height o stress blok β = (0.05) 0.65 β 1 = ator based on 1000 x or = loation to the neutral axis b = width o stress blok = steel ield strength A s = area o steel reinorement d = eetive depth o setion (depth to n.a. o reinorement) As With C=T, As = 0.85 ba so a an be determined with a = = β b Criteria or Beam Design For lexure design: Mu φmn φ = 0.9 or lexure (when the setion is tension ontrolled) so, Mu an be set =φmn =φt(d-a/2) = φ As (d-a/2) Reinorement Ratio The amount o steel reinorement is limited. Too muh reinorement, or over-reinored will not allow the steel to ield beore the onrete rushes and there is a sudden ailure. A beam with the proper amount o steel to allow it to ield at ailure is said to be under reinored. As The reinorement ratio is just a ration: ρ = (or p) and must be less than a value bd determined with a onrete strain o and tensile strain o (minimum). When the strain in the reinorement is or greater, the setion is tension ontrolled. (For smaller strains the resistane ator redues to 0.65 beause the stress is less than the ield stress in the steel.) Previous odes limited the amount to 0.75ρ balaned where ρ balaned was determined rom the amount o steel that would make the onrete start to rush at the exat same time that the steel would ield based on strain (ε ) o d The strain in tension an be determined rom εt = (0.003). At ield, ε =. E The resistane ator expressions or transition and ompression ontrolled setions are: s 0.15 φ = ( εt ε ) (0.005 ε ) 0.25 φ = ( εt ε ) (0.005 ε ) or spiral members (not less than 0.75) or other members (not less than 0.65) 206

7 Flexure Design o Reinorement One method is to wisel estimate a height o the stress blok, a, and solve or A s, and alulate a new value or a using M u. 1. guess a (less than n.a.). ba 2. As = solve or a rom Mu = φas (d-a/2) : M u a = 2 d φas 4. repeat rom 2. until a ound rom step 3 mathes a used in step 2. Design Chart Method: M 1. alulate R = n n 2 bd 2. ind urve or and to get ρ 3. alulate A s and a A s = ρbd and As a = 0.85 b An method an simpli the size o d using h = 1.1d rom Reinored Conrete, 7th, Wang, Salmon, Pinheira, Wile & Sons, 2007 Maximum Reinorement Based on the limiting strain o in the steel, x(or ) = 0.375d so β 1 a = ( d ) to ind A s-max (β 1 is shown in the table above) Minimum Reinorement Minimum reinorement is provided even i the onrete an resist the tension. This is a means to ontrol raking. Minimum required: As = 3 ( bwd ) 207 (tensile strain o 0.004) but not less than: 200 As = ( bwd ) where is in psi. This an be translated to ρ = min but not less than

8 Compression Reinorement I a setion is doubl reinored, it means there is steel in the beam seeing ompression. The ore in the ompression steel at ield is equal to stress x area, C s = A F. The total ompression that balanes the tension is now: T = C + C s. And the moment taken about the entroid o the ompression stress is Mn = T(d-a/2)+C s (a-d ) where A s is the area o ompression reinorement, and d is the eetive depth to the entroid o the ompression reinorement T-setions (pan joists) T beams have an eetive width, b E, that sees ompression stress in a wide lange beam or joist in a slab sstem. For interior T-setions, b E is the smallest o L/4, b w + 16t, or enter to enter o beams For exterior T-setions, b E is the smallest o b w + L/12, b w + 6t, or b w + ½(lear distane to next beam) When the web is in tension the minimum reinorement required is the same as or retangular setions with the web width (b w ) in plae o b. M n =C w (d-a/2)+c (d-h /2) (h is height o lange or t) When the lange is in tension (negative bending), the minimum reinorement required is the greater value o or where is in psi, b w is the beam width, and b is the eetive lange width Lightweight Conrete Lightweight onrete has strength properties that are dierent rom normalweight onretes, and a modiiation ator, λ, must be multiplied to the strength value o. or onrete or some speiiations (ex. shear). Depending on the aggregate and the lightweight onrete, the value o λ ranges rom 075 to 0.85, 0.85, or 0.85 to 1.0. λ is 1.0 or normalweight onrete. 208

9 Cover or Reinorement Cover o onrete over/under the reinorement must be provided to protet the steel rom orrosion. For indoor exposure, 3/4 inh is required or slabs, 1.5 inh is tpial or beams, and or onrete ast against soil, 3 inhes is tpial. Bar Spaing Minimum bar spaings are speiied to allow proper onsolidation o onrete around the reinorement. Slabs One wa slabs an be designed as one unit -wide beams. Beause the are thin, ontrol o deletions is important, and minimum depths are speiied, as is minimum reinorement or shrinkage and rak ontrol when not in lexure. Reinorement is ommonl small diameter bars and welded wire abri. Maximum spaing between bars is also speiied or shrinkage and rak ontrol as ive times the slab thikness not exeeding 18. For required lexure reinorement spaing the limit is three times the slab thikness not exeeding 18. Shrinkage and temperature reinorement (and minimum or lexure reinorement): Minimum or slabs with grade 40 or 50 bars: ρ = = or As-min = 0.002bt bt Minimum or slabs with grade 60 bars: ρ = = or As-min = bt bt A s A s Shear Behavior Horizontal shear stresses our along with bending stresses to ause tensile stresses where the onrete raks. Vertial reinorement is required to bridge the raks whih are alled shear stirrups. The maximum shear or design, V u is the value at a distane o d rom the ae o the support. 209

10 Nominal Shear Strength The shear ore that an be resisted is the shear stress ross setion area: V = ν The shear stress or beams (one wa) υ = 2λ so φv = φ2λ bwd where b w = the beam width or the minimum width o the stem. φ = 0.75 or shear λ = modiiation ator or lightweight onrete One-wa joists are allowed an inrease o 10% V i the joists are losel spaed. Stirrups are neessar or strength (as well as rak ontrol): V where A v = area o all vertial legs o stirrup s = spaing o stirrups d = eetive depth s Av td = bwd s 8 (max) b w d For shear design: V φv + φv φ = 0.75 or shear u s Spaing Requirements φv Stirrups are required when V u is greater than. A minimum is required beause shear ailure 2 o a beam without stirrups is sudden and brittle and beause the loads an var with respet to the design values. greater o and smaller o and NOTE: setion numbers are pre ACI Eonomial spaing o stirrups is onsidered to be greater than d/4. Common spaings o d/4, d/3 and d/2 are used to determine the values o φv s at whih the spaings an be inreased. φ Av td φ Vs = s 210

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12 Development Length in Compression db ld = d 50λ Hook Bends and Extensions The minimum hook length is l dh db = 50λ b or 8 in. minimum but not less than the larger o 8d b and 6 in Modulus o Elastiit & Deletion E or deletion alulations an be used with the transormed setion modulus in the elasti range. Ater that, the raked setion modulus is alulated and E is adjusted. Code values: E = 57, 000 (normal weight) E = w , w = 90 lb/t lb/t 3 Deletions o beams and one-wa slabs need not be omputed i the overall member thikness meets the minimum speiied b the ode, and are shown in (see Slabs). The span lengths or ontinuous beams or slabs is taken as the lear span, l n. Criteria or Flat Slab & Plate Sstem Design Sstems with slabs and supporting beams, joists or olumns tpiall have multiple bas. The horizontal elements an at as one-wa or two-wa sstems. Most oten the lexure resisting elements are ontinuous, having positive and negative bending moments. These moment and shear values an be ound using beam tables, or rom ode speiied approximate design ators. Flat slab two-wa sstems have drop panels (or shear), while lat plates do not. Two wa shear at olumns is resisted b the thikness o the slab at a perimeter o d/2 awa rom the ae o the support b the shear stress ross setion area: V = ν b d o 212

13 The shear stress (two wa) υ = 4λ so φv = φ4λ bod where b o = perimeter length. φ = 0.75 or shear λ = modiiation ator or lightweight onrete Criteria or Column Design (Amerian Conrete Institute) ACI Code and Commentar: P u φ P n where P u is a atored load φ is a resistane ator P n is the nominal load apait (strength) Load ombinations, ex: 1.4D (D is dead load) 1.2D + 1.6L (L is live load) 1.2D + 1.6L r + 0.5W (W is wind load) 0.90D + 1.0W For ompression, φ = 0.75 and P n = 0.85P o or spirall reinored, φ = 0.65 P n = 0.8P o or tied olumns where P o = ( Ag Ast ) + Ast and P o is the name o the maximum axial ore with no onurrent bending moment. Columns whih have reinorement ratios, A ρ = g st A g, in the range o 1% to 2% will usuall be the most eonomial, with 1% as a minimum and 8% as a maximum b ode. Bars are smmetriall plaed, tpiall. Columns with Bending (Beam-Columns) Conrete olumns rarel see onl axial ore and must be designed or the ombined eets o axial load and bending moment. The interation diagram shows the redution in axial load a olumn an arr with a bending moment. Design aids ommonl present the interation diagrams in the orm o load vs. equivalent eentriit or standard olumn sizes and bars used. 213

14 Eentri Design The strength interation diagram is dependent upon the strain developed in the steel reinorement. I the strain in the steel is less than the ield stress, the setion is said to be ompression ontrolled. Below the transition zone, where the steel starts to ield, and when the net tensile strain in the reinorement exeeds the setion is said to be tension ontrolled. This is a dutile ondition and is preerred. Rigid Frames Monolithiall ast rames with beams and olumn elements will have members with shear, bending and axial loads. Beause the joints an rotate, the eetive length must be determined rom methods like that presented in the handout on Rigid Frames. The harts or evaluating k or non-swa and swa rames an be ound in the ACI ode. Frame Columns Beause joints an rotate in rames, the eetive length o the olumn in a rame is harder to determine. The stiness (EI/L) o eah member in a joint determines how rigid or lexible it is. To ind k, the relative stiness, G or Ψ, must be ound or both ends, plotted on the alignment harts, and onneted b a line or braed and unbraed ames. Σ EI l G =Ψ = Σ EI l b where E = modulus o elastiit or a member I = moment o inertia o or a member l = length o the olumn rom enter to enter l b = length o the beam rom enter to enter For pinned onnetions we tpiall use a value o 10 or Ψ. For ixed onnetions we tpiall use a value o 1 or Ψ. 214

15 Braed non-swa rame Unbraed swa rame 215

16 Fatored Moment Resistane o Conrete Beams, φm n (k-t) with = 4 ksi, = 60 ksi a Approximate Values or a/d Approximate Values or ρ b x d (in) x 14 2 #6 2 #8 3 # x 18 3 #5 2 #9 3 # x 22 2 #7 3 #8 (3 #10) x 16 2 #7 3 #8 4 # x 20 2 #8 3 #9 4 # x 24 2 #8 3 #9 (4 #10) x 20 3 #7 4 #8 5 # x 25 3 #8 4 #9 4 # x 30 3 #8 5 #9 (5 #11) x 24 3 #8 5 #9 6 # x 30 3 #9 6 #9 (6 #11) x 36 3 #10 6 #10 (7 #11) x 30 3 # 10 7 # 9 6 # x 35 4 #9 5 #11 (7 #11) x 40 6 #8 6 #11 (9 #11) x 32 6 #8 7 #10 (8 #11) x 40 6 #9 7 #11 (10 #11) x 48 5 #10 (8 #11) (13 #11) a Table ields values o atored moment resistane in kip-t with reinorement indiated. Reinorement hoies shown in parentheses require greater width o beam or use o two stak laers o bars. (Adapted and orreted rom Simpliied Engineering or Arhitets and Builders, 11 th ed, Ambrose and Tripen,

17 Column Interation Diagrams 217

18 Column Interation Diagrams 218

19 Beam / One-Wa Slab Design Flow Chart Collet data: L, ω, γ, llimits, hmin; ind beam harts or load ases and atual equations (sel weight = area x densit) Collet data: load ators,, Find V s & Mu rom onstruting diagrams or using beam hart ormulas with the atored loads (Vu-max is at d awa rom ae o support) Assume b & d (based on hmin or slabs) Determine Mn required b Mu/ φ, hoose method Chart (Rn vs ρ) Selet ρmin ρ ρmax Find Rn o hart with, and selet ρmin ρ ρmax Choose b & d ombination based on Rn and hmin (slabs), estimate h with 1 bars (#8) Calulate As = ρbd Selet bar size and spaing to it width or 12 in strip o slab and not exeed limits or rak ontrol Inrease h, ind d* Find new d / adjust h; Is ρmin ρ ρmax? NO YES Calulate a, φmn Inrease h, ind d Is Mu φmn? NO Yes (on to shear reinorement or beams) 219

20 Beam / One-Wa Slab Design Flow Chart - ontinued Beam, Adequate or Flexure Determine shear apait o plain onrete based on, b & d, φv Is Vu (at d or beams) φv? NO Beam? NO YES NO Beam? Determine φvs = (Vu - φv ) Inrease h and re-evaluate lexure (As and φmn o previous page)* YES Is φvs φ? NO Is Vu < ½ φv? NO (4φV) YES YES Determine s & Av Find where V = φv and provide minimum Av and hange s Find where V = ½ φv and provide stirrups just past that point Yes (DONE) 220