Two-Way Concrete Floor Slab with Beams Design and Detailing (CSA A )

Transcription

1 Two-Way Conrete Floor Slab with Beams Design and Detailing (CSA A.-14)

2 Two-Way Conrete Floor Slab with Beams Design and Detailing (CSA A.-14) Design the slab system shown in Figure 1 for an intermediate floor where the story height =.7 m, olumn rosssetional dimensions = 450 mm 450 mm, edge beam dimensions = 50 mm 700 mm, interior beam dimensions = 50 mm x 500 mm, and unfatored live load = 4.8 kn/m. The lateral loads are resisted by shear walls. Normal weight onrete with ultimate strength (f = 5 MPa) is used for all members, respetively. And reinforement with F y = 400 MPa is used. Use the Elasti Frame Method (EFM) and ompare the results with spslab model results. Figure 1 Two-Way Slab with Beams Spanning between all Supports Version: Ot-0-018

3 Contents 1. Preliminary Slab Thikness Sizing Two-Way Slab Analysis and Design Using Elasti Frame Method (EFM) Elasti frame method limitations Frame members of elasti frame Elasti frame analysis Design moments Distribution of design moments Flexural reinforement requirements Column design moments.... Design of Interior, Edge, and Corner Columns Two-Way Slab Shear Strength One-Way (Beam ation) Shear Strength Two-Way (Punhing) Shear Strength Two-Way Slab Defletion Control (Servieability Requirements) Immediate (Instantaneous) Defletions Time-Dependent (Long-Term) Defletions (Δ lt) spslab Software Program Model Solution Summary and Comparison of Design Results Conlusions & Observations Version: Ot-0-018

4 Code Design of Conrete Strutures (CSA A.-14) and Explanatory Notes on CSA Group standard A.-14 Design of Conrete Strutures Referenes CAC Conrete Design Handbook, 4 th Edition, Cement Assoiation of Canada Notes on ACI Building Code Requirements for Strutural Conrete, Twelfth Edition, 01 Portland Cement Assoiation. Design Data Floor-to-Floor Height =.7 m (provided by arhitetural drawings) Columns = mm Interior beams = mm Edge beams = mm w = 4 kn/m f = 5 MPa f y = 400 MPa Live load, L o = 4.8 kn/m Solution 1. Preliminary Slab Thikness Sizing Control of defletions. CSA A. (1..5) In lieu of detailed alulation for defletions, CSA A. Code gives minimum thikness for two-way slab with beams between all supports on all sides in Clause Ratio of moment of inertia of beam setion to moment of inertia of a slab (α) is omputed as follows: Ib CSA A. (1..5) I s The moment of inertia for the effetive beam and slab setions an be alulated as follows: I b bwh hs h CSA A. (Eq. 1.4) The preliminary thikness of 155 mm is assumed and it will be heked in next steps. Edge Beams: The effetive beam and slab setions for the omputation of stiffness ratio for edge beam is alulated as follows: 1

5 For North-South Edge Beams: I b I s mm , mm For East-West Edge Beams:. I b mm , I s mm Interior Beams: For North-South Interior Beams: I b mm For East-West Interior Beams: I b mm The average of α for the beams on four sides of exterior and interior panels are alulated as: ( ) For exterior panels: m 5. 4 (.68.1) For interior panels: m.40 4 α m shall not be taken greater than.0, then α m =.0 for both exterior and interior panels.

6 The minimum slab thikness is given by: h min f y ln 0.6 1, m CSA A.-14 (1..5) Where: l lear span in the long diretion measured fae to fae of olumns 6.05 m 6050 mm n lear span in the long diretion lear span in the short diretion h min 400 6, , The assumed thikness is more than the h min. Use 155 mm slab thikness.

7 . Two-Way Slab Analysis and Design Using Elasti Frame Method (EFM) EFM (as known as Equivalent Frame Method in the ACI 18) is the most omprehensive and detailed proedure provided by the CSA A. for the analysis and design of two-way slab systems where these systems may, for purposes of analysis, be onsidered a series of plane frames ating longitudinally and transversely through the building. Eah frame shall be omposed of equivalent line members interseting at member enterlines, shall follow a olumn line, and shall inlude the portion of slab bounded laterally by the enterline of the panel on eah side. CSA A.-14 ( ) Probably the most frequently used method to determine design moments in regular two-way slab systems is to onsider the slab as a series of two-dimensional frames that are analyzed elastially. When using this analogy, it is essential that stiffness properties of the elements of the frame be seleted to properly represent the behavior of the three-dimensional slab system. In a typial frame analysis it is assumed that at a beam-olumn onnetion all members meeting at the joint undergo the same rotation. For uniform gravity loading this redued restraint is aounted for by reduing the effetive stiffness of the olumn by either Clause 1.8. or Clause CSA A.-14 (N.1.8) Eah floor and roof slab with attahed olumns may be analyzed separately, with the far ends of the olumns onsidered fixed. CSA A.-14 (1.8.1.) The moment of inertia of olumn and slab-beam elements at any ross-setion outside of joints or olumn apitals shall be based on the gross area of onrete at that setion. CSA A.-14 (1.8..5) An equivalent olumn shall be assumed to onsist of the atual olumns above and below the slab- beam plus an attahed torsional member transverse to the diretion of the span for whih moments are being determined. CSA A.-14 (1.8..5).1. Elasti frame method limitations In EFM, live load shall be arranged in aordane with whih requires slab systems to be analyzed and designed for the most demanding set of fores established by investigating the effets of live load plaed in various ritial patterns. CSA A.-14 (1.8.4) Complete analysis must inlude representative interior and exterior elasti frames in both the longitudinal and transverse diretions of the floor. CSA A.-14 ( ) Panels shall be retangular, with a ratio of longer to shorter panel dimensions, measured enter-to-enter of supports, not to exeed. CSA A.-14 (.1a) For slab systems with beams between supports, the relative effetive stiffness of beams in the two diretions is not less than 0. or greater than. CSA A.-14 (.1b) Column offsets are not greater than 0% of the span (in the diretion of offset) from either axis between enterlines of suessive olumns. CSA A.-14 (.1) The reinforement is plaed in an orthogonal grid. CSA A.-14 (.1d) 4

8 .. Frame members of elasti frame Determine moment distribution fators and fixed-end moments for the elasti frame members. The moment distribution proedure will be used to analyze the elasti frame. Stiffness fators k, arry over fators COF, and fixed-end moment fators FEM for the slab-beams and olumn members are determined using the design aids tables at Appendix 0A of PCA Notes on ACI These alulations are shown below. a. Flexural stiffness of slab-beams at both ends, K sb. N1 450 N , ,500 6,500 1 For stiffness fators, k k 4.15 PCA Notes on ACI (Table A1) F1 F NF FN E Isb E Isb Thus, K k 4.15 PCA Notes on ACI (Table A1) sb NF 1 1 Where I sb is the moment of inertia of slab-beam setion shown in Figure and an be omputed with the aid of Figure as follows: bh w Isb C t K sb mm 1 1 E , E N.m 9 4 Figure Cross-Setion of Slab-Beam Carry-over fator COF = PCA Notes on ACI (Table A1) Fixed-end moment FEM w u PCA Notes on ACI (Table A1) 1 Figure Coeffiient C t for Gross Moment of Inertia of Flanged Setions 5

9 b. Flexural stiffness of olumn members at both ends, K. Referring to Table A7, Appendix 0A: For Interior Columns: t a / 4.5 mm, t 77.5 mm b ta H H.7 m 700 mm, H mm, 5.45, 1.16 t H Thus, k 6.55 and k 4.91 by interpolation., top, bottom 4 4 (450) I.4 10 mm m, 700 mm k E I 9 4 K PCA Notes on ACI (Table A7) K K, top, bottom E, E N.m E, E N.m b For Exterior Columns: t a / 6.5 mm, t 77.5 mm b ta H H.7 m, 700 mm, H, , 000 mm, 8.0, 1. t H Thus, k 8.45 and k 5.47 by interpolation., top, bottom 4 4 (450) I.4 10 mm ft,700 mm k E I 9 4 K PCA Notes on ACI (Table A7) K K, top, bottom E, E N.m E E,700 b 6

10 . Torsional stiffness of torsional members, K t. 9EsC Kt CSA A.-14 (1.8..8) [ t (1 ) ] t For Interior Columns: 9 9E Kt 5, 500(0.918) Where: ,500 t E N.m x x y C 10.6 CSA A.-14 (1.8..9) y x 1 = 50 mm x = 155 mm x 1 = 50 mm x = 150 mm y 1 = 45 mm y = 1,040 mm y 1 = 500 mm y = 45 mm C 1 = C = C 1 = C = C = = mm 4 C = = mm 4 Figure 4 Attahed Torsional Member at Interior Column 7

11 For Exterior Columns: 9 9E Kt 5, 500(0.918) Where: ,500 t E N.m x x y C 10.6 CSA A.-14 (1.8..9) y x 1 = 50 mm x = 155 mm x 1 = 50 mm x = 155 mm y 1 = 545 mm y = 895 mm y 1 =700 mm y = 545 mm C 1 = C = C 1 = C = C = = mm 4 C = = mm 4 Figure 5 Attahed Torsional Member at Exterior Column 8

12 d. Inreased torsional stiffness due to parallel beams, K ta. For Interior Columns: Figure 6 Slab-Beam in the Diretion of Analysis K ta Where: I s K I E t sb 9 Is.010 l h 6, mm 1 1 For Exterior Columns: E N.m K ta K I E t sb 9 Is E N.m e. Equivalent olumn stiffness K e. K e K K K ta K ta Where K ta is for two torsional members one on eah side of the olumn, and K is for the upper and lower olumns at the slabbeam joint of an intermediate floor. For Interior Columns: K e 4 ( E E )( E ) 4 E E E E ( ) ( ) Figure 7 Equivalent Column Stiffness 9

13 For Exterior Columns: K e 4 ( E E )( E ) 4 E E E E ( ) ( ) f. Slab-beam joint distribution fators, DF. At exterior joint, DF E 4 ( E E) 0.87 At interior joint, DF E ( E E) COF for slab-beam = Figure 8 Slab and Column Stiffness.. Elasti frame analysis Determine negative and positive moments for the slab-beams using the moment distribution method. With an unfatored live-to-dead load ratio: L D (4155 / 1000) 4 The frame will be analyzed for five loading onditions with pattern loading and partial live load as allowed by CSA A.-14 (1.8.4). a. Fatored load and Fixed-End Moments (FEM s). Fatored dead load w 1.5( ) 5.1 kn/m df Where (0.446 kn/m = (0.45 x 0.5) 4 / 6.5 is the weight of beam stem per foot divided by l ) Fatored live load w 1.5(4.8) 7. kn/m Lf Fatored load w w w 1.41 kn/m f Df Lf 1 FEM's for slab-beam mnf wf PCA Notes on ACI (Table A1) FEM due to wdf wlf ( ) kn.m 4 FEM due to wdf wlf ( ) kn.m FEM due to wdf ( ) kn.m 10

14 b. Moment distribution. Moment distribution for the five loading onditions is shown in Table 1 (The unit for moment values is kn.m). Counter-lokwise rotational moments ating on member ends are taken as positive. Positive span moments are determined from the following equation: M u( midspan) ( MuL MuR ) Mo Where M o is the moment at the midspan for a simple beam. When the end moments are not equal, the maximum moment in the span does not our at the midspan, but its value is lose to that midspan for this example. Positive moment in span 1- for loading (1): M f 5.5 (11.1.8) ( ) 1.0 kn.m 8 Positive moment span - for loading (1): M f 5.5 ( ) ( ) 91.5 kn.m 8 Table 1 Moment Distribution for Partial Frame (Transverse Diretion) Joint 1 4 Member DF COF Loading (1) All spans loaded with full fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist CO Dist M Midspan M

15 Loading () First and third spans loaded with /4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Loading () Center span loaded with /4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Loading (4) First span loaded with /4 fatored live load and beam-slab assumed fixed at support two spans away FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M 1

16 Loading (5) First and seond spans loaded with /4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Max M Max M Design moments Positive and negative fatored moments for the slab system in the diretion of analysis are plotted in Figure 9. The negative moments used for design are taken at the faes of supports (retangle setion or equivalent retangle for irular or polygon setions) but not at distanes greater than from the enters of supports. CSA A.-14 ( ) 450 mm < ,500 = 96.5 mm (use fae of support loation) 1

17 Figure 9 Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Fatored Live Load exept as Noted).5. Distribution of design moments Chek Appliability of Diret Design Method: 1. There shall be a minimum of three ontinuous spans in eah diretion ( spans) CSA A.-14 (1.9.1.). Suessive span lengths entre-to-entre of supports in eah diretion shall not differ by more than onethird of the longer span (span lengths are equal) CSA A.-14 (1.9.1.). All loads shall be due to gravity only and uniformly distributed over an entire panel (Loads are uniformly distributed over the entire panel) CSA A.-14 ( ) 4. The fatored live load shall not exeed twie the fatored dead load (Fatored live-to-dead load ratio of 1.8 <.0) CSA A.-14 ( ) 5. For slabs with beams between supports, the relative effetive stiffness of beams in the two diretions ( l / l ) is not less than 0. or greater than 5.0. CSA A.-14 ( ) , l 5.5 m 5,500 mm , l 5.5m 5,500 mm 1 l.1 6, l , 500 O.K. Sine all the riteria are met, Diret Design Method an be utilized. 14

18 b. Distribute fatored moments to olumn and middle strips: The negative and positive fatored moments at ritial setions may be distributed to the olumn strip and the two half-middle strips of the slab-beam aording to the Diret Design Method (DDM) in 1.9, provided that limitations in is satisfied. CSA A..-14 (1.) Beams shall be reinfored to resist the following fration of the positive or interior negative fatored moments determined by analysis or determined as speified in Clause CSA A..-14 (1.1..1) Portion of design moment resisted by beam: End Span Interior Span 1 l l Fatored moments at ritial setions are summarized in Table. Exterior Negative Fatored Moments (kn.m) Table - Lateral distribution of fatored moments Beam Strip Perent Beam Strip Moment (kn.m) Column Strip Column Strip Perent Column Strip Moment (kn.m) Moments in Two Half-Middle Strips* (kn.m) Positive Interior Negative Negative Positive *That portion of the fatored moment not resisted by the olumn strip is assigned to the two half-middle strips.6. Flexural reinforement requirements a. Determine flexural reinforement required for strip moments The flexural reinforement alulation for the olumn strip of end span interior negative loation is provided below: M 1.55 kn.m f Column strip width, b = (5,500 /) - 50 =,400 mm Use d avg = 17 mm In this example, jd is assumed equal to 0.98d. The assumption will be verified one the area of steel in finalized. Assume jd 0.98 d 447. mm Column strip width, b = (5,500 /) - 50 =,400 mm Middle strip width, b 6,500, , 750 mm 15

19 6 M f As 07.5 mm f jd s y CSA A.-14 (10.1.7) ' f Af s s y Realulate ' a' for the atual As 07.5 mm a 15.6 mm 1 f ' b , 400 a mm The tension reinforement in flexural members shall not be assumed to reah yield unless: d fy jd d a 0.98d CSA A.-14 (10.5.) As,min mm > 07.5 mm CSA A.-14 (7.8.1) A s 774 mm Maximum spaing: CSA A.-14 (1.10.4) - Negative reinforement in the band defined by b b: 1.5hs.5 mm 50 mm - Remaining negative moment reinforement: hs 465 mm 500 mm Provide 6 15M bars with A s = 00 mm and s =,400/6 = 400 mm s max The flexural reinforement alulation for the beam strip of end span interior negative loation is provided below: M kn.m f Beam strip width, b = 50 mm Use d = 468 mm jd is assumed equal to 0.948d. The assumption will be verified one the area of steel in finalized. Assume jd d 44.6 mm 6 M f As 66.6 mm f jd s y CSA A.-14 (10.1.7) ' f CSA A.-14 (10.1.7) ' f Af s s y Realulate ' a' for the atual As 66.6 mm a mm 1 f ' b a mm The tension reinforement in flexural members shall not be assumed to reah yield unless: d fy CSA A.-14 (10.5.) 16

20 jd d a 0.948d ' 0. f 0. 5 As,min bt h mm CSA A.-14 ( ) f mm A s y Provide 5M bars with A s = 500 mm Beam Strip Column Strip Middle Strip Beam Strip Column Strip Middle Strip All the values on Table are alulated based on the proedure outlined above. Span Loation Table - Required Slab Reinforement for Flexure [Elasti Frame Method (EFM)] Mf (kn.m) b * (mm) d ** (mm) As Req d for flexure (mm ) Min As (mm ) Reinforement Provided As Prov. for flexure (mm ) End Span Exterior Negative M 1,000 Positive M 1,000 Interior Negative M 1,000 Exterior Negative 0.00, M 1,00 Positive 1.47, M 1,00 Interior Negative 1.55, M 1,00 Exterior Negative 0.00, , M 1,800 Positive.55, , M 1,800 Interior Negative 49.9, , M 1,800 Interior Span Positive M 1,000 Positive 17.10, M 1,00 Positive 6.71, , M 1,800 * Column strip width, b = (5,500/) - 50 =,400 mm * Middle strip width, b = 6,500-, =,750 mm * Beam strip width, b = 50 mm ** Use average d = = 17 mm for Column and Middle strips ** Use average d = = 457 mm for Beam strip Positive moment regions ** Use average d = = 468 mm for Beam strip Negative moment regions Min. As = 0.00 b h = 0.1 b for Column and Middle strips CSA A.-14 (7.8.1) Min. As = (0.(f')^0.5/fy*b*d for Beam strip CSA A.-14 ( ) b. Calulate additional slab reinforement at olumns for moment transfer between slab and olumn by flexure 17

21 Portion of the unbalaned moment transferred by flexure is γ f M f Where: 1 f CSA A.-14 (1.10.) 1 ( / ) b / b 1 b1 Width width of the ritial setion for shear measured in the diretion of the span for whih moments are determined aording to CSA A.-14, lause 1 (see Figure 10). b Width of the ritial setion for shear measured in the diretion perpendiular to b1 aording to CSA A.-14, lause 1 (see Figure 10). b b = Effetive slab width = hs CSA A.-14 (.) For Exterior Column: d 17 b mm, b d mm, bb h 450 (155) 915 mm 1 f ( / ) 51.5 / 577 f M, kn.m f net Figure 10 Critial Shear Perimeters for Columns 0.81 f ' b M A d d b f f, net s, req ' d ' s f y 0.81 f bb A s, req ' d , 507 mm As,min mm <,507 mm CSA A.-14 (7.8.1) A s, req ' d,507 mm 18

22 A ( A ) ( A ) A s, provided s, provided ( beam) s, provided ( b b b beam ) mm < A, 507 mm,400 s, provided s, req ' d Additional slab reinforement at the exterior olumn is required. A ', mm req d add Use 1-15M A 1 00,400 in. > A,4.5 mm provided, add req ' d, add Table 4 - Additional Slab Reinforement at olumns for moment transfer between slab and olumn [Elasti Frame Method (EFM)] Span Loation Effetive slab width, b b (mm) d (mm) γ f M u * (kn.m) γ f M u (kn.m) A s req d within b b (mm ) A s prov. for flexure within b b (mm ) Add l Reinf. Column Strip Exterior Negative Interior Negative End Span ,507 1,8 1-15M ,0 1,8 - *M f is taken at the enterline of the support in Elasti Frame Method solution. b. Determine transverse reinforement required for beam strip shear The transverse reinforement alulation for the beam strip of end span exterior loation is provided below. Figure 11 Shear at ritial setions for the end span (at distane d v from the fae of the olumn) dv Max (0.9 d, 0.7 h) Max ( , ) mm CSA A.-14 (.) The required shear at a distane d from the fae of the supporting olumn V u_d= 15 kn (Figure 11). Vr,max / kn setion is adequate CSA A.-14 (11..) V f b d CSA A.-14 (Eq. 11.5) ' w v 19

23 V / 1, kn<15 kn Stirrups are required. Distane from the olumn fae beyond whih minimum reinforement is required: V V V ACI ( ) s f _ d V kn s Av Vf V mm / mm s f d ot ot 5 Where Av s req min 5 yt 0.06 f bw f yt v ' CSA A.-14 ( ) CSA A.-14 (11..6.) CSA A.-14 (11..8.) A v s min mm /mm 400 s req Av mm Av 0.6 s req Chek whether the required spaing based on the shear demand meets the spaing limits for shear reinforement per CSA A.-14 (11..8). ' 0.15 fbwdv 9.7 Vf CSA A.-14 (11..8.) Therefore, maximum stirrup spaing shall be the smallest of 0.7d v and 600 mm. CSA A.-14 ( ) 0.7d v mm smax lesser of lesser of lesser of 88 mm 600 mm 600 mm 600 mm Sine s s use s req ' d max max Selet s provided = 80 mm 10M stirrups with first stirrup loated at distane 140 mm from the olumn fae. The distane where the shear is zero is alulated as follows: l 5.5 x VuL, 0..5 m,50 mm V V f, L f, R The distane at whih no shear reinforement is required is alulated as follows: x.5 x1 x V m 1,480 mm V 0. f s 1 provided x 1,480 1 # of stirrups use 6 stirrups s 80 provided 0

24 All the values on Table 5 are alulated based on the proedure outlined above. Span Loation Table 5 - Required Beam Reinforement for Shear Av,min/s mm /mm Av,req'd/s mm /mm End Span sreq'd mm smax mm Reinforement Provided Exterior mm Interior mm Interior Span Interior mm 1

25 .7. Column design moments The unbalaned moment from the slab-beams at the supports of the frame are distributed to the atual olumns above and below the slab-beam in proportion to the relative stiffness of the atual olumns. Referring to Fig. 9, the unbalaned moment at joints 1 and are: Joint 1 = kn.m Joint = = kn.m The stiffness and arry-over fators of the atual olumns and the distribution of the unbalaned moments to the exterior and interior olumns are shown in Fig 1. Figure 1 - Column Moments (Unbalaned Moments from Slab-Beam)

26 In summary: Design moment in exterior olumn = kn.m Design moment in interior olumn = 5.40 kn.m The moments determined above are ombined with the fatored axial loads (for eah story) and fatored moments in the transverse diretion for design of olumn setions. A detailed analysis to obtain the moment values at the fae of interior, exterior, and orner olumns from the unbalaned moment values an be found in the Two-Way Flat Plate Conrete Floor Slab Design example.. Design of Interior, Edge, and Corner Columns The design of interior, edge, and orner olumns is explained in the Two-Way Flat Plate Conrete Floor Slab Design example. 4. Two-Way Slab Shear Strength Shear strength of the slab in the viinity of olumns/supports inludes an evaluation of one-way shear (beam ation) and two-way shear (punhing) in aordane with CSA A.-14 lause One-Way (Beam ation) Shear Strength One-way shear is ritial at a distane d v from the fae of the olumn. Figure 1 shows the V f at the ritial setions around eah olumn. Sine there is no shear reinforement, the design shear apaity of the setion equals to the design shear apaity of the onrete: V V V V V, ( V V 0) CSA A.-14 (Eq. 11.4) r s p Where: s p V f b d CSA A.-14 (Eq. 11.5) ' w v 1 for normal weight onrete 0.1 for slabs with overall thikness not greater than 50 mm CSA A.-14 (11..6.) dv Max (0.9 d,0.7 h) Max (0.917, ) 114 mm CSA A.-14 (.) ' avg f 5 MPa 8 MPa CSA A.-14 (11..4) 114 V , kn > Vf 1000 Beause Vr V f at all the ritial setions, the slab has adequate one-way shear strength.

27 Figure 1 One-way shear at ritial setions (at distane d v from the fae of the supporting olumn) 4.. Two-Way (Punhing) Shear Strength Two-way shear is ritial on a retangular setion loated at d slab/ away from the fae of the olumn. The fatored shear fore V f in the ritial setion is alulated as the reation at the entroid of the ritial setion minus the self-weight and any superimposed surfae dead and live load ating within the ritial setion. The fatored unbalaned moment used for shear transfer, M unb, is alulated as the sum of the joint moments to the left and right. Moment of the vertial reation with respet to the entroid of the ritial setion is also taken into aount. For the exterior olumn: Vf kn / Munb ft-kip 1 For the exterior olumn in Figure 14, the loation of the entroidal axis z-z is: AB moment of area of the sides about AB area of the sides AB (50 67 ( / ) ((514 50) 17 (514 50) / ) 0.4 mm (50 67 (514 50) 17) ( ) 17 Figure 14 Critial setion of exterior support of interior frame 4

28 A ( (514 50)) 17 (577 50) mm The polar moment J of the shear perimeter is: 5 J bbeam, Ext dbeam, Ext dbeam, Extbbeam, Ext bbeam, Ext bbeam, Ext dbeam, Ext b1 bbeam, Ext AB 1 1 b b d d b b b b beam, Ext slab, Ext slab 1 beam, Ext 1 beam, Ext b1 bbeam, Ext dslab AB b d b b d beam, Int beam. Int beam, Int slab AB J J mm v f 10 4 γ 1 γ CSA A.-14 (Eq. 1.8) The length of the ritial perimeter for the exterior olumn: bo ( / ) (450 17) 1604 mm V f v f = + b o d γvm J unb e CSA A.-14 (Eq.1.9) v MPa f 5 10 The fatored resisting shear stress, V r shall be the smallest of: CSA A.-14 (1..4.1) a) b) ) ' v r= v f MPa 1 d 17 s ' v r= v 0.19 f MPa bo 1604 ' v r= v 0.8 f MPa In this example, sine the d avg = mm around the joint for two-way shear, exeeds 00 mm, therefore the value of v obtained above shall be multiplied by 100/(1000+d). CSA A.-14 (1..4.) 5

29 v =1.115 MPa (1000 d) ( ) Sine vr vf at the ritial setion, the slab has adequate two-way shear strength at this joint. For the interior olumn: Vf kn 6 10 Munb (0) 19.0 kn.m For the interior olumn in Figure 15, the loation of the entroidal axis z-z is: b 577 1, Int AB 88.5 mm A 4 (50 47 (577 50) 17) mm 5 The polar moment J of the shear perimeter is: Figure 15 Critial setion of interior support of interior frame J 1 1 bbeam, Int dbeam, Int dbeam, Intbbeam, Int bbeam, Int b1 bbeam, Int bbeam, Int dbeam, Int AB b1 bbeam, Int b1 bbeam, Int d d slab, Int slab b1 bbeam, Int b1 bbeam, Int dslab AB 1 1 b d b b d beam, Int beam. Int beam, Int slab AB J

30 10 4 J mm γ 1 γ ACI (Eq ) v f The length of the ritial perimeter for the exterior olumn: bo 4 (450 17),08 mm V f v f = + b o d γvm J unb e CSA A.-14 (Eq.1.9) v 4581, , MPa f 5 10 The fatored resisting shear stress, V r shall be the smallest of: CSA A.-14 (1..4.1) a) b) ) ' v r= v f MPa 1 d 417 s ' v r= v 0.19 f MPa bo,08 ' v r= v 0.8 f MPa In this example, sine the d avg = 6. mm around the joint for two-way shear, exeeds 00 mm, therefore the value of v obtained above shall be multiplied by 100/(1000+d). CSA A.-14 (1..4.) v =1.01 MPa (1000 d) ( ) Sine vr vf at the ritial setion, the slab has adequate two-way shear strength at this joint. 5. Two-Way Slab Defletion Control (Servieability Requirements) Sine the slab thikness was seleted based on the minimum slab thikness equations in CSA A.-14, the defletion alulations are not required. However, the alulations of immediate and time-dependent defletions are overed in this setion for illustration and omparison with spslab model results Immediate (Instantaneous) Defletions The alulation of defletions for two-way slabs is hallenging even if linear elasti behavior an be assumed. Elasti analysis for three servie load levels (D, D + L sustained, D+L Full) is used to obtain immediate defletions of the two-way slab in this example. However, other proedures may be used if they result in preditions of defletion in reasonable agreement with the results of omprehensive tests. ACI (4..) The effetive moment of inertia (I e) is used to aount for the raking effet on the flexural stiffness of the slab. I e for unraked setion (M r > M a) is equal to I g. When the setion is raked (M r < M a), then the following equation should be used: 7

31 M I I I I I r e r g r g M a Where: CSA A.-14 (Eq.9.1) M a = Maximum moment in member due to servie loads at stage defletion is alulated. The values of the maximum moments for the three servie load levels are alulated from strutural analysis as shown previously in this doument. These moments are shown in Figure 16. Figure 16 Maximum Moments for the Three Servie Load Levels For positive moment (midspan) setion of the exterior span: M Craking moment. M r fi 9.00 / ( ) r g 6 r kn.m CSA A.-14 (Eq.9.) Yt

32 f r should be taken as half of Eq.8. CSA A.-14 (9.8..) f r = Modulus of rapture of onrete. f r 0.6 f ' MPa CSA A.-14 (Eq.8.) I g = Moment of inertia of the gross unraked onrete setion I g mm for T-setion (see Figure 1) y t = Distane from entroidal axis of gross setion, negleting reinforement, to tension fae, in. y mm (see Figure 17) t I r Figure 17 I g alulations for slab setion near support = Moment of inertia of the raked setion transformed to onrete. CAC Conrete Design Handbook 4 th Edition (5..) As alulated previously, the positive reinforement for the end span frame strip is 15 15M bars loated at 0 mm along the slab setion from the bottom of the slab and 5M bars loated at 0 mm along the beam setion from the bottom of the beam. Three of the slab setion bars are not ontinuous and will be exluded from the alulation of I r. Figure 18 shows all the parameters needed to alulate the moment of inertia of the raked setion transformed to onrete at midspan. Figure 18 Craked Transformed Setion (positive moment setion) E s = Modulus of elastiity of slab onrete. E s ',447 (,00 f 6,900) (,00 5 6,900) 5,684 MPa,00,00 CSA A.-14(8.6..) Es 00, 000 n 7.79 CAC Conrete Design Handbook 4 th Edition (Table 6.a) E 5, 684 s 9

33 b 6,500 a,50 mm b n A, n A, , mm s beam s slab 6 s, beam s, beam s, slab s, slab 1 n A d n A d mm 6 b b 4a 6, , , kd 8.84 mm a, 50 b( kd) I na d kd na d kd r s, slab ( slab ) s, beam ( beam ) I r 6, 500 (8.84) mm 9 4 For negative moment setion (near the interior support of the end span): The negative reinforement for the end span frame strip near the interior support is 7 #4 bars loated at 1.0 in. along the setion from the top of the slab. M r 9 fi r g.00 / ( ) kn.m CSA A.-14 (Eq.9.) Y 50 t f r 0.6 f ' MPa CSA A.-14 (Eq.8.) I mm g 9 4 y 50 mm t Figure 19 I g alulations for slab setion near support E s ',447 (,00 f 6,900) (,00 5 6,900) 5,684 MPa,00,00 CSA A.-14(8.6..) 0

34 Es 00, 000 n 7.79 CAC Conrete Design Handbook 4 th Edition (Table 6.a) E 5, 684 s bbeam 50 B mm na s, total 1 CAC Conrete Design Handbook 4 th Edition (Table 6.a) db kd 1 mm B 468 CAC Conrete Design Handbook 4 th Edition (Table 6.a) b ( kd) I na d kd beam r s, total ( ) CAC Conrete Design Handbook 4 th Edition (Table 6.a) I r 50 (1) mm 9 4 Figure 0 Craked Transformed Setion (interior negative moment setion for end span) The effetive moment of inertia proedure desribed in the Code is onsidered suffiiently aurate to estimate defletions. The effetive moment of inertia, I e, was developed to provide a transition between the upper and lower bounds of I g and I r as a funtion of the ratio M r/m a. For onventionally reinfored (nonprestressed) members, the effetive moment of inertia, I e, shall be alulated by by Eq. (9.1) in CSA A.-14 unless obtained by a more omprehensive analysis. For ontinuous prismati members, the effetive moment of inertia may be taken as the weighted average of the values obtained from Eq. (9.1) in CSA A.-14 for the ritial positive and negative moment setions. CSA A.-14(9.8..4) For the exterior span (span with one end ontinuous) with servie load level (D+LL full): M r Ie Ir Ig Ir, M r 1.88 kn.m < M a =179.9 kn.m M a ACI (4...5a) Where I e - is the effetive moment of inertia for the ritial negative moment setion (near the support) I e mm

35 For positive moment setion (midspan): M r Ie Ir Ig Ir, M r 7.7 kn.m < M a = 9.07 kn.m M a + Where I e is the effetive moment of inertia for the ritial positive moment setion (midpan) I e mm Where I e + is the effetive moment of inertia for the ritial positive moment setion (midspan). Sine midspan stiffness (inluding the effet of raking) has a dominant effet on defletions, midspan setion is heavily represented in alulation of I e and this is onsidered satisfatory in approximate defletion alulations. The averaged effetive moment of inertia (I e,avg) is given by: I, 0.85 I 0.15 I for end span CSA A.-14 (9.8..4) e avg e e I, mm e avg Where: I e I e = The effetive moment of inertia for the ritial negative moment setion near the support. = The effetive moment of inertia for the ritial positive moment setion (midspan). For the interior span (span with both ends ontinuous) with servie load level (D+LL full): M r Ie Ir Ig Ir, M r 1.88 kn.m < M a =16.49 kn.m M a ACI (4...5a) I e mm For positive moment setion (midspan): M r Ie Ir Ig Ir, M r 7.7 kn.m < M a = kn.m M a + Where I e is the effetive moment of inertia for the ritial positive moment setion (midpan) I e mm The averaged effetive moment of inertia (I e,avg) is given by: e, avg e e, l e, r I 0.70 I 0.15 I I for interior span CSA A.-14 (9.8..4) Ie, avg mm

36 Where: I el, = The effetive moment of inertia for the ritial negative moment setion near the left support. I er, = The effetive moment of inertia for the ritial negative moment setion near the right support. Table 6 provides a summary of the required parameters and alulated values needed for defletions for exterior and interior equivalent frame. It also provides a summary of the same values for olumn strip and middle strip to failitate alulation of panel defletion. Span Ext Int zone Ig, mm 4 ( 10 9 ) Ir, mm 4 ( 10 9 ) Table 6 Averaged Effetive Moment of Inertia Calulations D M a, kn.m D + LL Sus For Frame Strip D + L full M r, kn.m D Ie, mm 4 ( 10 9 ) Ie,avg, mm 4 ( 10 9 ) D + LL Sus Left Midspan Right Left Mid Right D + L full D D + LL Sus D + L full Defletions in two-way slab systems shall be alulated taking into aount size and shape of the panel, onditions of support, and nature of restraints at the panel edges. For immediate defletions two-way slab systems the midpanel defletion is omputed as the sum of defletion at midspan of the olumn strip or olumn line in one diretion (Δ x or Δ y) and defletion at midspan of the middle strip in the orthogonal diretion (Δ mx or Δ my). Figure 1 shows the defletion omputation for a retangular panel. The average Δ for panels that have different properties in the two diretion is alulated as follows: ( x my ) ( y mx ) PCA Notes on ACI ( Eq. 8)

37 Figure 1 Defletion Computation for a retangular Panel To alulate eah term of the previous equation, the following proedure should be used. Figure shows the proedure of alulating the term Δ x. same proedure an be used to find the other terms. For exterior span - servie dead load ase: Figure Δ x alulation proedure 4 wl frame, fixed PCA Notes on ACI ( Eq. 10) 84EI Where: frame, averaged 4

38 frame, fixed = Defletion of olumn strip assuing fixed end ondition ( ) 50 w slab weight + beam weight = (6.5) 7.08 kn/m E s ', 447 (, 00 f 6, 900) (, , 900) 5, 684 MPa,00,00 CSA A.-14(8.6..) I frame,averaged = The averaged effetive moment of inertia (I e,avg) for the frame strip for servie dead load ase from Table 6 = 8. x 10 9 mm frame, fixed 0.17 mm , I frame, fixed LDF frame, fixed I g PCA Notes on ACI ( Eq. 11) Where LDF is the load distribution fator for the olumn strip. The load distribution fator for the olumn strip an be found from the following equation: LDF LDF LDF l LDF R And the load distribution fator for the middle strip an be found from the following equation: LDF m 1 LDF For the end span, LDF for exterior negative region (LDF L ), interior negative region (LDF R ), and positive region (LDF + L ) are 1.00, 0.77, and 0.77, respetively (From Table of this doument). Thus, the load distribution fator for the olumn strip for the end span is given by: LDF I,g = The gross moment of inertia (I g) for the olumn strip (for T setion) = 7.9 x 10 9 mm 4 I frame,g = The gross moment of inertia (I g) for the frame strip (for T setion) = 9.95 x 10 9 mm , fixed mm e M net, L frame L, PCA Notes on ACI ( Eq. 1) K Where: 5

39 L, = Rotation of the span left support. 7 ( Mnet, L) frame N.mm = Net frame strip negative moment of the left support. K e = effetive olumn stiffness for exterior olumn. =.05 x N.mm/rad (alulated previously) L, rad I l PCA Notes on ACI ( Eq. 14) g, L, L 8 I e frame Where: L, = Midspan defletion due to rotation of left support. I I g e frame = Gross-to-effetive moment of inertia ratio for frame strip L, mm ( M ) ( ) rad net, R frame R, 11 Ke.4510 Where R, = Rotation of the end span right support. ( M ) Net frame strip negative moment of the right support. net, R frame K e = effetive olumn stiffness for interior olumn. =.45 x N.mm/rad (alulated previously). l I mm 9, R g, R 8 Ie frame Where: 6

40 R, = Midspan delfetion due to rotation of right support. x fixed PCA Notes on ACI ( Eq. 9) x, x, R x, L mm x Following the same proedure, Δ mx an be alulated for the middle strip. This proedure is repeated for the equivalent frame in the orthogonal diretion to obtain Δ y, and Δ my for the end and middle spans for the other load levels (D+LL sus and D+LL full). Assuming square panel, Δ x = Δ y= in. and Δ mx = Δ my= 0.01 in. The average Δ for the orner panel is alulated as follows: x my y mx x my y mx in. 7

41 Table 7 - Instantaneous Defletions Column Strip Middle Strip Span LDF Δ frame-fixed, mm Δ -fixed, mm θ 1, rad D θ, rad Δθ 1, mm Δθ, mm Δ x, mm LDF Δ frame-fixed, mm Ext Int Δ m-fixed, mm θ m1, rad D θ m, rad Δθ m1, mm Δθ m, mm Δ mx, mm Span LDF Δ frame-fixed, mm Δ -fixed, mm θ 1, rad D+LL sus θ, rad Δθ 1, mm Δθ, mm Δ x, mm LDF Δ frame-fixed, mm Ext Int Δ m-fixed, mm θ m1, rad D+LL sus θ m, rad Δθ m1, mm Δθ m, mm Δ mx, mm Span LDF Δ frame-fixed, mm Δ -fixed, mm θ 1, rad D+LL full θ, rad Δθ 1, mm Δθ, mm Δ x, mm LDF Δ frame-fixed, mm Ext Int Δ m-fixed, mm θ m1, rad D+LL full θ m, rad Δθ m1, mm Δθ m, mm Δ mx, mm Span LDF LL Δ x, mm Ext Int LDF LL Δ mx, mm 8

42 5.. Time-Dependent (Long-Term) Defletions (Δlt) The additional time-dependent (long-term) defletion resulting from reep and shrinkage (Δ s) may be estimated as follows: ( ) PCA Notes on ACI ( Eq. 4) s sust Inst The total time-dependent (long-term) defletion is alulated as: ( ) ( ) (1 ) [( ) ( ) ] CSA A.-04 (N9.8..5) total lt sust Inst total Inst sust Inst Where: ( ) Immediate (instantaneous) defletion due to sustained load, in. sust Inst 1 50 ' ACI ( ) ( ) Time-dependent (long-term) total delfetion, in. total lt ( ) Total immediate (instantaneous) defletion, in. total Inst For the exterior span =, onsider the sustained load duration to be 60 months or more. ACI (Table ) ' = 0, onservatively mm s mm total lt Table 8 shows long-term defletions for the exterior and interior spans for the analysis in the x-diretion, for olumn and middle strips. 9

43 Table 8 - Long-Term Defletions Column Strip Span (Δsust)Inst, mm λδ Δs, mm (Δtotal)Inst, mm (Δtotal)lt, mm Exterior Interior Middle Strip Exterior Interior spslab Software Program Model Solution spslab program utilizes the Elasti Frame Method desribed and illustrated in details here for modeling, analysis and design of two-way onrete floor slab systems. spslab uses the exat geometry and boundary onditions provided as input to perform an elasti stiffness (matrix) analysis of the equivalent frame taking into aount the torsional stiffness of the slabs framing into the olumn. It also takes into aount the ompliations introdued by a large number of parameters suh as vertial and torsional stiffness of transverse beams, the stiffening effet of drop panels, olumn apitals, and effetive ontribution of olumns above and below the floor slab using the of equivalent olumn onept (CSA A.-14 (1.8..6)). spslab Program models the elasti frame as a design strip. The design strip is, then, separated by spslab into olumn and middle strips. The program alulates the internal fores (Shear Fore & Bending Moment), moment and shear apaity vs. demand diagrams for olumn and middle strips, instantaneous and long-term defletion results, and required flexural reinforement for olumn and middle strips. The graphial and text results will be provided from the spslab model in a future revision to this doument. 40

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66 7. Summary and Comparison of Design Results Table 9 - Comparison of Moments obtained from Hand (EFM) and spslab Solution (kn.m) Hand (EFM) spslab Exterior Span Beam Strip Column Strip Middle Strip Beam Strip Column Strip Middle Strip * negative moments are taken at the faes of supports Exterior Negative * Positive Interior Negative * Exterior Negative * Positive Interior Negative * Exterior Negative * Positive Interior Negative * Interior Span Interior Negative * Positive Interior Negative * Positive Interior Negative * Positive

67 Span Loation Exterior Negative Table 10 - Comparison of Reinforement Results Reinforement Provided for Flexure Additional Reinforement Provided for Unbalaned Moment Transfer* Total Reinforement Provided Hand spslab Hand spslab Hand spslab Exterior Span 15M 5M n/a n/a 15M 5M Beam Strip Column Strip Middle Strip Positive 15M 5M n/a n/a 15M 5M Interior Negative Exterior Negative 15M 5M M 5M 6 15M 6 15M 1 15M 1 15M 18 15M 18 15M Positive 6 15M 6 15M n/a n/a 6 15M 6 15M Interior Negative Exterior Negative 6 15M 6 15M M 6 15M 9 15M 9 15M n/a n/a 9 15M 9 15M Positive 9 15M 9 15M n/a n/a 9 15M 9 15M Interior Negative 9 15M 9 15M n/a n/a 9 15M 9 15M Interior Span Beam Strip Positive 15M 5M n/a n/a 15M 5M Column Strip Middle Strip Positive 6 15M 6 15M n/a n/a 6 15M 6 15M Positive 9 15M 9 15M n/a n/a 9 15M 9 15M Table 11 - Comparison of Beam Shear Reinforement Results Span Loation Reinforement Provided Hand spslab End Span Exterior 6 80 mm 6 81 mm Interior 6 80 mm 6 81 mm Interior Span Interior 8 80 mm 8 81 mm 64

68 Table 1 - Comparison of Two-Way (Punhing) Shear Chek Results (around Columns Faes) Support b1, mm b, mm bo, mm Vf, kn AB, mm Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior Interior Support J, mm 4 γv Munb, kn.m vu, MPa v, MPa Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior Interior Span Table 1 - Comparison of Immediate Defletion Results (mm) Column Strip D D+LLsus D+LLfull LL Hand spslab Hand spslab Hand spslab Hand spslab Exterior Interior Span Middle Strip D D+LLsus D+LLfull LL Hand spslab Hand spslab Hand spslab Hand spslab Exterior Interior Table 14 - Comparison of Time-Dependent Defletion Results Column Strip Span λδ Δs, in. Δtotal, in. Hand spslab Hand spslab Hand spslab Exterior Interior Middle Strip Span λδ Δs, in. Δtotal, in. Hand spslab Hand spslab Hand spslab Exterior Interior In all of the hand alulations illustrated above, the results are in lose or exat agreement with the automated analysis and design results obtained from the spslab model. Exerpts of spslab graphial and text output are given below for illustration. 65

69 8. Conlusions & Observations A slab system an be analyzed and designed by any proedure satisfying equilibrium and geometri ompatibility. Three established methods are widely used. The requirements for two of them are desribed in detail in CSA A..-14 Clause 1. Diret Design Method (DDM) is an approximate method and is appliable to two-way slab onrete floor systems that meet the stringent requirements of CSA A..-14 (1.9.1). In many projets, however, these requirements limit the usability of the Diret Design Method signifiantly. The Elasti Frame Method (EFM) does not have the limitations of Diret Design Method. It requires more aurate analysis methods that, depending on the size and geometry an prove to be long, tedious, and timeonsuming. StuturePoint s spslab software program solution utilizes the Elasti Frame Method to automate the proess providing onsiderable time-savings in the analysis and design of two-way slab systems as ompared to hand solutions using DDM or EFM. Finite Element Method (FEM) is another method for analyzing reinfored onrete slabs, partiularly useful for irregular slab systems with variable thiknesses, openings, and other features not permissible in DDM or EFM. Many reputable ommerial FEM analysis software pakages are available on the market today suh as spmats. Using FEM requires ritial understanding of the relationship between the atual behavior of the struture and the numerial simulation sine this method is an approximate numerial method. The method is based on several assumptions and the operator has a great deal of deisions to make while setting up the model and applying loads and boundary onditions. The results obtained from FEM models should be verified to onfirm their suitability for design and detailing of onrete strutures. The following table shows a general omparison between the DDM, EFM and FEM. This table overs general limitations, drawbaks, advantages, and ost-time effiieny of eah method where it helps the engineer in deiding whih method to use based on the projet omplexity, shedule, and budget. 66