EUROPEAN CODE REQUIREMENTS FOR DESIGN OF CONCRETE FLOORS, INCLUDING POST-TENSIONING 1

Yur Partner in Structural Cncrete Design TN175_EC2_implementatin_040409 EUROPEAN CODE REQUIREMENTS FOR DESIGN OF CONCRETE FLOORS, INCLUDING POST-TENSIONING 1 This Technical Nte details the requirements f the Eurpean Cde EC2 (ENV 1992-1-1:2004) fr the design f cncrete flr systems, including pst-tensining, and the implementatin f these requirements in the Builder Platfrm prgrams. The implementatin fllws the EC2 s prcedure f calculating a Demand, referred t as design value fr each design sectin, and a Resistance, fr the same sectin, referred t as design capacity. Design value and design capacity are generic terms that apply t displacements as well as actins. Fr each lading cnditin, r instance defined in EC2, the design is achieved by making the resistance exceed the assciated demand Design Value. Where necessary, reinfrcement is added t meet this cnditin. The implementatin is brken dwn int the fllwing steps: Serviceability limit state (SLS) Check fr cmputed stresses in cncrete Check fr cracking and crack reinfrcement Minimum reinfrcement Based n crack width Based n gemetry Deflectin check Strength limit state (ULS) Bending f sectin With r withut prestressing With r withut axial lading Punching shear (tw-way shear) Beam shear (ne-way shear) Initial cnditin (transfer f prestressing) (ILS) Check fr cmputed stresses in cncrete Prvide rebar In the fllwing, the values in square brackets [ ] are defaults f the prgram. They can be changed by the user. MATERIAL AND MATERIAL FACTORS Cncrete Cylinder strength at 28 days, as specified by the user f ck characteristic cmpressive cylinder strength at 28 days; 1 Cpyright 2006 supprt@adaptsft.cm www.adaptsft.cm ADAPT Crpratin, Redwd City, Califrnia, USA, Tel: (650) 306-2400 Fax (650) 306 2401 ADAPT Internatinal Pvt. Ltd, Klkata, India, Tel: 91 33 302 86580 Fax: 91 33 224 67281

Bilinear stress/strain diagram with the hrizntal branch at f cd ; maximum strain at 0.0035; strain at limit f prprtinality 0.00175 σ f cd 0.00175 0.0035 ε Mdulus f elasticity f cncrete is autmatically calculated and displayed by the prgram using f ck, and the relatinship (2.1-15) 2 f the cde given belw. User is given the ptin t verride the cde value and specify a user defined substitute. E ci E c [(f ck +Δf)/f cm ] 1/3 where, E ci E c f ck f f cm mdulus f elasticity at 28 days 2.15 *10 4 MPa characteristic cylinder strength at 28 days 8 MPa 10 MPa Nnprestressed Steel Bilinear stress/strain diagram with the hrizntal branch at f yd f yk / γ s Mdulus f elasticity is user defined [200000 MPa] N limit n tensile strain is impsed σ f yd 200000 MPa f yd /E s ε Prestressing Steel Bilinear stress/strain diagram with the hrizntal branch at (0.90f pk )/ γ s Mdulus f elasticity is user defined [190000 MPa] N limit n tensile strain is impsed 2 CEB-FIP MODEL CODE 1990 2

σ 0.9f pk /γ s 190000 MPa f pd /E s ε Material Factrs Cncrete γ s 1.50 Nnprestressed steel γ s 1.15 Prestressing steel γ s 1.15 LOAD COMBINATIONS The prgram autmatically generates the fllwing lad cmbinatins and perfrms the assciated design checks. The default lad cmbinatins f the prgram can be edited by the user. Als, users can define additinal lad cmbinatins. Service (quasi-permanent) Service (frequent) Strength cnditin Initial (transfer) In additin t the abve, the prgram has a N cde check ptin fr cmbinatins, when a user is interested in the respnse f the flr system t a user defined lading, as ppsed t perfrming a cde check. N cde check The parts and factrs f the prgram s autmatically generated lad cases and lad cmbinatins are listed belw. Except fr the initial (transfer) cnditin, which is nt explicitly defined in the cde, the remainder f the cmbinatins fllws EC2 stipulatins. Service (quasi-permanent) 1.00 x Selfweight + 1.00 x Dead lad + 0.30 x Live lad + 1.00 x Prestressing Service (frequent) 1.00 x Selfweight + 1.00 x Dead lad + 0.50 x Live lad + 1.00 x Prestressing Strength 1.35 x Selfweight + 1.35 x Dead lad + 1.50 x Live lad + 1.00 x Hyperstatic Initial (transfer) 1.00 x Selfweight + 1.15 x Prestressing The factrs f the initial cnditin are based upn the cmmn practice f engineers in the USA. 3

DESIGN FOR FLEXURE WITH OR WITHOUT AXIAL LOAD Serviceability Check Fr frequent lad cnditin the cde stipulated stress limitatins explained belw are used as default. Hwever, user can edit the default values. Cncrete Maximum cmpressive stress 3 0.60 f ck. If calculated stress at any lcatin exceeds the allwable, the prgram identifies the lcatin graphically n the screen and ntes it in its tabular reprts. The maximum allwable hypthetical tensile stress 4 used is (fct,eff). Where calculated values exceed this threshld, the prgram prvided reinfrcement t cntrl cracking. Nnprestressed Reinfrcement The maximum allwable stress (0.80 f yk ) given in the cde is used. If the calculated stress exceeds the allwable value, the prgram autmatically increases the area f steel t lwer the calculated stress t the cde specified limit. Prestressing steel The maximum allwable stress under service cnditin (0.75 f yk ) given in the cde is used. If this value exceeds, the prgram reprt it t the user n the cmputer screen. Stress limitatins used fr the quasi-permanent lad cmbinatin are as fllws: Cncrete Maximum cmpressive stress 0.45 f ck. If stress at any lcatin exceeds, the prgram displays that lcatin with a change in clr (r brken lines fr black and white display), alng with a nte n the text utput. The maximum allwable hypthetical tensile stress used is (fct,eff). Where calculated values exceed this threshld, the prgram prvides mre reinfrcement t limit the crack width t the cde specified limit and t cntrl cracking. Nnprestressed Reinfrcement Nne required n check made Prestressing steel Nne required - n check made Cracking and reinfrcement fr crack cntrl The minimum reinfrcement fr crack cntrl ( A smin ) is based n sectin 7.3.2 f the cde. The fllwing values are used in the evaluatin f A smin. A smin where, k k f A σs c ct,eff ct 3 EN 1992-1-1:2004(E), Sectin 7.2(2) 4 EN 1992-1-1:2004(E), Sectin 7.3.2(4) 4

k fr members with least dimensin < 300 mm k 1.0 fr members with least dimensin > 800 mm k 0.65 fr ther members, linear interplatin is used. is determined based n the maximum fiber stresses as fllws: k c Fr pure tensin k c 1.0 Fr bending r bending cmbined with axial frces: Technical Nte - Fr rectangular sectins and webs f bx sectins and T-sectins: σ c k c 0.4 1- k(h/h*)f 1 ct,eff - Fr flanges f bx sectins and T-sectins: F Af where NED σc ; average precmpressin bh k c 0.9 cr 0.5 ct ct,eff N ED Axial frce at the serviceability limit state. h* h fr h <1.0 m 1.0m fr h 1.0 m k 1 1.5 if N ED is a cmpressive frce 2h*/3h if N ED is a tensile frce F cr, but nt greater than 1 Abslute value f the tensile frce within the flange due t the cracking mment calculated with f ct,eff. f ct,eff tensile strength f cncrete at time f crack frmatin, f ctm, but nt less than 3 MPa mean axial tensile strength accrding t Table 3.1 f the cde f ctm Minimum Overall Reinfrcement Each design sectin is checked t satisfy the minimum verall reinfrcement (A s ), using Sectin 9.2.1.1 f the cde. A s > (0.26 b t d f ctm / f yk ), but nt less than 0.0013 b t d where d depth t the centrid f the mild steel. If n mild steel is required, the cver specified by the user and the user s chice f reinfrcement is used t calculate d b t mean width f the tensin zne. f pk is used in lieu f f yk, when sectin is prestressed. If bth prestressing and nnprestressed steel are present, weighted average f their characteristic strengths is used. 5

A A+ A s s ps f f pk yk Crack Width Limitatin Sectins are assumed cracked, if the flexural tensile stress exceeds f ct,eff (7.1 (2)). Fr crack width calculatins, f ct,eff may be assumed t be equal t f ctm. Calculatin is based n sectin 7.3.4 5. Design crack width, wk sr, max ( εsm ε cm) Where, s r,max ε sm ε cm maximum crack spacing mean strain in the reinfrcement under the relevant cmbinatin f lads, including the effect f impsed defrmatins and taking int accunt the effects f tensin stiffening. Only the additinal tensile strain beynd the state f zer strain f cncrete at the same level is cnsidered. mean strain in the cncrete between cracks fct, eff σ s kt ( 1+αρ e p,eff) ρp, eff σ εsm ε cm 0.6 Es E Where, σ s s s the stress in the tensin reinfrcement calculated n the basis f a cracked sectin [ cnservatively assumed f yk ]; α e E s /E cm ; ; E s mdulus f elasticity f steel; E cm mdulus f elasticity f cncrete (secant mdulus) [Eci] ; ρ p,eff (A s +ξ 2 1 A p )/A c,eff ; A s area f nnprestressed reinfrcement in tensin zne; A p area f tendns within A c,eff ; A c,eff effective area f cncrete in tensin surrunding the reinfrcement r prestressing tendns f depth h c,ef [h c,ef * (bar spacing)] f ct,eff mean value f the tensile strength f cncrete [f ctm ]; f ctm mean tensile strength f cncrete f ctm 0.30xf ck (2/3) fr f ck <50 MPa (Table 3.1), but nt less than 3 MPa; h c,ef lesser f 2.5(h-d), (h-x)/3 r h/2 h depth f the member x depth f neutral axis frm the cmpressin fiber The fllwing figure reprduced frm EC2 explains the preceding parameters graphically 5 EN 1992-1-1:2004(E), Sectin 7.2(2). 6

ξ 1 ξ φ ξ * ; adjusted rati f bnd strength taking int accunt the φ s p different diameters f prestressing and nnprestressed steel; rati f bnd strength f prestressing and nn-prestressed reinfrcement steel accrding t Table 6.2 f EC2, as reprduced belw: Fr strands: ξ 0.5 fr f ck < 50; Fr fr bars and wires: ξ 0.3 fr f ck < 50; and 0.25 fr f ck > 70 MPa and 0.15 fr f ck > 70 MPa Interplate fr intermediate values. Φ s Φ p largest diameter f nn-prestressed steel used; diameter, r equivalent diameter f prestressing steel; k t factr dependent n the duratin f the lad 0.6 fr shrt term lading 7

0.4 fr lng term lading s r,max 1.3(h-x) Using these parameters, the prgram calculates the design crack width (w k ) f each design sectin. If the calculated value exceeds the allwable, reinfrcement is added t that sectin, in rder t reduce the crack width t within the allwable value given belw. The allwable crack width depends n the Expsure classificatin. Crack width fr nnprestressed cncrete Expsure 2-4 Width is limited t 0.3 mm. Crack width fr prestressed cncrete Expsure 1-2 Width is limited t 0.2 mm fr frequent lad cmbinatin. Width is limited t 0.3 mm fr quasi-permanent lad cmbinatin. The prgram uses the abve values fr allwable crack width fr prestressed (gruted and unbnded systems) and nnprestressed structures independent f the expsure classes. Strength Check in Bending Plane sectins remain plane. Strain cmpatibility is used t determine the frces n a sectin. Maximum cncrete strain in cmpressin is limited t 0.0035. Maximum allwable value fr the neutral axis x is determined based n cncrete strength f the sectin f ck. Fr f ck < 35 MPa x/d < 0.45 Fr f ck > 35 MPa x/d < 0.35 Where necessary, cmpressin reinfrcement is added t enfrce the abve requirement. If a sectin is made up f mre than ne cncrete material, the entire sectin is designed using the cncrete prperties f lwest strength in that sectin Fr prestressed and nnprestressed reinfrcement that crss a design sectin at an angle ther than 90 degrees, the change in strain f the reinfrcement due t flexing f the design sectin abut its wn axis cnsidered t cntribute t the design capacity is calculated frm the fllwing relatinship: Cntributry change f strain in reinfrcement (strain nrmal t the design sectin at lcatin f rebar)*cs 2 θ where θ is the angle between the nrmal t the design sectin and the reinfrcement. The strain in each reinfrcement is calculated separately, based n the lcatin reinfrcement n the crss-sectin and the angle it makes with the nrmal t the design sectin. Additinal cnsideratins fr strain in prestressing steel If bnded, strain is calculated using the stress-strain curve and the angle f strand with the design sectin If unbnded, the strain is calculated using the same prcedure as bnded tendns, but the calculated strain is adjusted as fllws: The stress increase is reduced by the factr (L s /L T ) 8

The stress increase is limited t (L s /L T )100 MPa Stress in nnprestressed steel is based n stress-strain relatinship assumed 6 Rectangular cncrete blck is used with maximum stress equal t ηf cd η1 fr f ck 50MPa, η1-(f ck -50)/200 fr 50 < f ck 90MPa Fr flanged sectins, the fllwing prcedure is adpted: Technical Nte If x is within the flange, the sectin is treated as a rectangle If x exceeds the flange thickness, unifrm cmpressin is assumed ver the flange. The stem is treated as a rectangular sectin One-Way Shear Check The design is based n the fllwing (sectin 6.2 f the cde): where, V Sd < V Rd V Sd design value; and V Rd design resistance (capacity). Three resistance values are calculated and checked against the design values. These are: V Rd1 design shear resistance withut shear reinfrcement V Rd2 maximum design shear resistance that can be carried withut crushing f ntinal cncrete cmpressive struts V Rd3 design shear resistance f sectin with shear reinfrcement Cde frmulas are used t calculate the abve values. In using the cde frmulas, the fllwing cnsideratins are bserved. Punching Shear v sd V Sd / b w d Fr b w the smallest width f the sectin is used Shear due t the vertical cmpnent f prestressing tendns is ignred Lss f crss-sectinal area due t pst-tensining ducts is ignred Cde prvisin fr increase in shear capacity fr sectins that are clse t a supprt is nt implemented All available tensin reinfrcement is included in the calculatin f A s1 Curtailment lengths fr lngitudinal bars are nt calculated The Stirrup spacing are calculated based n the required shear reinfrcement and is limited t minimum f (0.75(d+ctα), 300mm), where α is the angle between the shear reinfrcement and lngitudinal axis f the beam/slab. Categrizatin f clumns: N criterin is mentined in EC2 regarding categrizatins f clumns fr punching shear check. The prgram uses ACI-318 criteria as detailed belw. Based n the gemetry f the flr slab at the vicinity f a clumn, each clumn is categrized int t ne f the fllwing ptins: 6 EN 1992-1-1:2004(E), Eq 3.21 and 3.22 9

1. Interir clumn Each face f the clumn is at least fur times the slab thickness away frm a slab edge 2. Edge clumn One side f the clumn nrmal t the axis f the mment is less than fur times the slab thickness away frm the slab edge 3. Crner clumn Tw adjacent sides f the clumn are less than fur times the slab thickness frm slab edges parallel t each 4. End clumn One side f the clumn parallel t the axis f the mment is less than fur times the slab thickness frm a slab edge In cases 2, 3 and 4, clumn is assumed t be at the edge f the slab. The verhang f the slab beynd the face f the clumn is nt included in the calculatins. Hence, the analysis perfrmed is smewhat cnservative. Design Stress Several critical perimeters arund each clumn are cnsidered. Fr each critical perimeter, a representative design shear stress (v u ) is calculated, using a cmbinatin f the direct shear and mment: v u β V A u where V u is the abslute value f the direct shear and β is an amplificatin factr f shear t take int accunt the effect f mments and eccentricities. β 1.15 fr interir clumns β 1.40 fr edge and end clumns β 1.50 fr crner clumns Fr a clumn with dimensins a and b, r drp cap with dimensins a and b, and a critical sectin which is at distance c frm the face f clumn r drp cap, A is given by: 1. Interir clumn: A 2(a+ b) + 2π c 2. Edge clumn: (a is parallel t the axis f mment) A 2a+ b+ π c 3. Crner Clumn: π A a+ b+ c 2 10

4. End clumn: (a is parallel t the axis f mment) A a+ 2b+ π c Allwable stress: Allwable stress f a sectin is calculated based n the fllwing equatin: v v + k σ Rd,c min 1 cp where, 3/2 1/2 vmin 0.035k fck k 1 0.1 k 1+ (200/d) 1/2 NED σcp Ac 2.0, d in mm σ cp Nrmal cncrete stress in the critical sectin (MPa, psitive if cmpressin) N ED Lngitudinal frce acrss the cntrl sectin (in N). A c Area f the cncrete Critical sectins The critical sectins fr stress check are: (1) at the face f clumn; (2) at 2d frm the face f the clumn, where d is the effective depth f the slab; and (3) additinal sectins at 0.75d intervals, where required. If a drp cap is present, depending n the size f the cap, the stresses are checked bth within the drp cap, and beynd the perimeter f the drp cap. Hwever, if the drp cap is t small t be effective t resist punching shear, the stress check is perfrmed utside the perimeter f the drp cap nly. Fr a drp cap t be cnsidered effective, its hrizntal extensin frm the face f the clumn (l H ) shuld exceed twice the prjectin f the cap belw the slab sffit (2h H ) 7. The first critical sectin is at r cnt 8 frm the center f the clumn, and the subsequent sectins are at 0.75d intervals. Generally, critical sectins bth within the drp cap and beynd it will be checked. Stress check: Stresses are calculated at the critical sectins and cmpared against the allwable values: If vu < vrd,c n punching shear reinfrcement is required if v u > v max at the face f the clumn, punching stress is excessive; the sectin shuld be revised. The prgram displays graphically and reprts in table the lcatins that need revisin. 7 EN 1992-1-1:2004, Sectin 6.4.2 (8) 8 EN 1992-1-1:2004, Eqn 6.34-6.37 11

where v max maximum allwable shear stress fr a perimeter f the clumn is v max 0.5vf cd where v 0.6 1- (f ck / 250) Flw Chart f TR43 Reprt fr Pst-Tensining Design If v u > v Rd,c prvide punching shear reinfrcement Stress check is perfrmed until n shear reinfrcement is needed. Where drp caps exist, stresses are checked within the drp cap until the design stress is less than permissible, then in a similar manner the stresses are checked utside the drp cap. Shear reinfrcement: Where needed, shear reinfrcement is prvided accrding t the fllwing: A A s s,min ( vu 0.75vRd,c) u d sr 1.5 d f sin(α) ywd,ef 0.08 fck sr st csα f ( 1.5sinα + ) yk where, fywd,ef 250 + 0.25d fywd f ywd design strength f punching shear reinfrcement s r spacing f shear links in the radial directin s t spacing f shear links in the tangential directin u perimeter f the critical sectin α the angle f shear reinfrcement with the plane f slab d effective depth Arrangement f shear reinfrcements: Shear reinfrcement can be in the frm f shear studs r shear stirrups (links). In case f shear links, the number f shear links ( N shear_links ) in a critical sectin and distance between the links ( Dist shear_links ) are given by: N shear_links A A s shear_link u Distshear_links Nshear_links where, A shear-link area f the single shear link The calculated distance will be cmpared with the maximum allwable by the cde and will be adjusted accrdingly. 12

If shear studs are used, the number f shear studs per rail ( N shear_studs ) and the distance between the studs ( Dist shear_studs ) are given by: As Nshear_studs Ashear_stud Nrails s Distshear_studs N shear_studs where, A shear-stud area f the shear stud s spacing between the critical sectins. Shear reinfrcement is prvided in three layers (perimeters) frm the face f supprt including the first critical sectin, i.e., within the distance 2d frm the face f supprt. Initial Cnditin (Transfer f prestressing) Stress limitatins used fr the initial lad cmbinatin are as fllws: Cncrete Maximum cmpressive stress 9 0.60 f ci. Since the EC2 cde is nt specific abut the allwable tensile stress limit, prgram uses the ACI-05 clde values (0.25 fci) as a default. But the user has the ptin t verride thse values. Where calculated values exceed this threshld, the prgram prvided reinfrcement t cntrl cracking. Reinfrcement Reinfrcement will be prvided fr initial cnditin if tensile stress exceeds allwable stress. Rebar is prvided based n ACI cde and will be placed n tensin side: AsT/(0.5Fy) Where: As: Area f reinfrcement T: ttal tensile frce n tensin blck Fy: Yield Stress f the steel but nt mre that 60 ksi NOTATION As f cd area f the reinfrcement; design value f cncrete cylinder cmpressin strength; 9 EN 1992-1-1:2004(E), Sectin 5.10.2.2(5) 13

f ck f ct,eff f ctm f pk f yk characteristic cmpressive cylinder strength at 28 days; mean value f the tensile strength f cncrete; mean tensile strength f cncrete f ctm 0.30xf ck (2/3) fr f ck <50 MPa (Table 3.1), but nt less than 3 MPa; characteristic tensile strength f prestressing steel [1860 MPa]; characteristic yield strength f steel, [460 MPa]; fywd, ef effective design strength f the punching shear reinfrcement; k 1 k 2 L s L T r cnt s v sd V Sd V Rd x w k a cefficient that takes accunt f the bnd prperties f the bars; a cefficient that takes accunt f the frm f the strain distributin; span length f tendn; ttal length f tendn; distance f critical sectin frm the center f the clumn, if drp cap exists; spacing between successive critical sectins; design shear stress; design shear frce; design shear resistance; depth f neutral axis; and design (cmputed) crack width. 14