Two-Way Joist Concrete Slab Floor (Waffle Slab) System Analysis and Design

Transcription

1 Two-Way Joist Conrete Slab Floor (Waffle Slab) System Analysis and Design Version: May

2 Two-Way Joist Conrete Slab Floor (Waffle Slab) System Analysis and Design Design the onrete floor slab system shown below for an intermediate floor with partition weight of 50 psf, and unfatored live load of 100 psf. The lateral loads are independently resisted by shear walls. A flat plate system will be onsidered first to illustrate the impat longer spans and heavier applied loads. A waffle slab system will be investigated sine it is eonomial for longer spans with heavy loads. The dome voids redue the dead load and eletrial fixtures an be fixed in the voids. Waffle system provides an attrative eiling that an be left exposed when possible produing savings in arhitetural finishes. The Equivalent Frame Method (EFM) shown in ACI 318 is used in this example. The hand solution from EFM is also used for a detailed omparison with the model results of spslab engineering software program from StruturePoint. Figure 1 - Two-Way Flat Conrete Floor System Version: May

3 Contents 1. Preliminary Member Sizing Flexural Analysis and Design Equivalent Frame Method (EFM) Limitations for use of equivalent frame method Frame members of equivalent frame Equivalent frame analysis Fatored moments used for Design Fatored moments in slab-beam strip Flexural reinforement requirements Fatored moments in olumns Design of Columns by spcolumn Determination of fatored loads Moment Interation Diagram Shear Strength One-Way (Beam ation) Shear Strength At distane d from the supporting olumn At the fae of the drop panel Two-Way (Punhing) Shear Strength Around the olumns faes Around drop panels Servieability Requirements (Defletion Chek) Immediate (Instantaneous) Defletions Time-Dependent (Long-Term) Defletions (Δ lt) spslab Software Program Model Solution Summary and Comparison of Design Results Conlusions & Observations Version: May

4 Code Building Code Requirements for Strutural Conrete (ACI ) and Commentary (ACI 318R-14) Referene Conrete Floor Systems (Guide to Estimating and Eonomizing), Seond Edition, 00 David A. Fanella, Portland Cement Assoiation. PCA Notes on ACI Building Code Requirements for Strutural Conrete, Twelfth Edition, 013, Portland Cement Assoiation. Simplified Design of Reinfored Conrete Buildings, Fourth Edition, 011 Mahmoud E. Kamara and Lawrene C. Novak Control of Defletion in Conrete Strutures (ACI 435R-95), Amerian Conrete Institute Reinfored Conrete Design...Hassoun, MGraw Hill Design Data Story Height = 13 ft (provided by arhitetural drawings) Superimposed Dead Load, SDL = 50 psf for Frame walls, hollow onrete masonry unit wythe, 1 in. thik, 15 pf unit density, with no grout ASCE/SEI 7-10 (Table C3-1) Live Load, LL = 100 psf for Rereational uses Gymnasiums ASCE/SEI 7-10 (Table 4-1) f = 5000 psi (for slab) f = 6000 psi (for olumns) f y = 60,000 psi Solution 1. Preliminary Member Sizing Preliminary Flat Plate (without Joists) a. Slab minimum thikness Defletion ACI ( ) In lieu of detailed alulation for defletions, ACI 318 minimum slab thikness for two-way onstrution without interior beams is given in Table For flat plate slab system, the minimum slab thikness per ACI are: ln 376 Exterior Panels: hs 1.53 in ACI (Table ) But not less than 5 in. ACI ( (a)) ln 376 Interior Panels: hs in ACI (Table ) But not less than 5 in. ACI ( (a)) Where l n = length of lear span in the long diretion = 33 x 1 0 = 376 in. Use 13 in. slab for all panels (self-weight = 150 pf x 13 in. /1 = 16.5 psf) 1

5 b. Slab shear strength one way shear Evaluate the average effetive depth (Figure ): db 0.75 dl hs lear db in. db 0.75 dt hs lear in. d avg dl dt in. Where: lear = 3/4 in. for # 6 steel bar ACI (Table ) d b = 0.75 in. for # 6 steel bar Figure - Two-Way Flat Conrete Floor System Fatored dead load, qdu 1. ( ) 55 psf Fatored live load, qlu psf ACI (5.3.1) Total fatored load, qu psf Chek the adequay of slab thikness for beam ation (one-way shear) ACI (.5) at an interior olumn: Consider a 1-in. wide strip. The ritial setion for one-way shear is loated at a distane d, from the fae of support (see Figure 3): Tributary are for one-way shear is A Tributary Vu qu ATributary kips ft V f b d ACI (Eq ) ' w Where 1 for normal weight onrete

6 11.51 V kips V 1000 Slab thikness of 13 in. is adequate for one-way shear. u. Slab shear strength two-way shear Chek the adequay of slab thikness for punhing shear (two-way shear) at an interior olumn (Figure 4): Tributary area for two-way shear is A Tributary (33 33) 108 ft 1 Vu qu ATributary kips V ACI (Table.6.5.(a)) ' 4 f bwd (For square interior olumn) V kips 1000 V kips V u Slab thikness of 13 in. is not adequate for two-way shear. This is expeted as the self-weight an applied loads are very hallenging for a flat plate system. Figure 3 Critial Setion for One-Way Shear Figure 4 Critial Setion for Two-Way Shear In this ase, four options an be onsidered: 1) inrease the slab thikness further, ) use headed shear reinforement in the slab, 3) apply drop panels at olumns, or 4) use two-way joist slab system. In this example, the latter option will be used to ahieve better understanding for the design of two-way joist slab often alled twoway ribbed slab or waffle slab. Chek the appliable joist dimensional limitations as follows: 1) Width of ribs shall be at least 4 in. at any loation along the depth. ACI (9.8.1.) Use ribs with 6 in. width. 3

7 ) Overall depth of ribs shall not exeed 3.5 times the minimum width. ACI ( ) 3.5 x 6 in. = 1 in. Use ribs with 14 in. depth. 3) Clear spaing between ribs shall not exeed 30 in. ACI ( ) Use 30 in. lear spaing. 4) Slab thikness (with removable forms) shall be at least the greater of: ACI ( ) a) 1/1 lear distane between ribs = 1/1 x 30 =.5 in. b) in. Use a slab thikness of 3 in. >.5 in. Figure 5 Joists Dimensions In waffle slabs a drop panel is automatially invoked to guarantee adequate two-way (punhing) shear resistane at olumn supports. This is evident from the flat plate hek onduted using 13 in. indiating insuffiient punhing shear apaity above. Chek the drop panel dimensional limitations as follows: 1) The drop panel shall projet below the slab at least one-fourth of the adjaent slab thikness. ACI (8..4(a)) Sine the slab thikness (h MI alulated in page 7 of this doument) is 1 in., the thikness of the drop panel should be at least: hdp, min 0.5 h in. MI Drop panel depth are also ontrolled by the rib depth (both at the same level).for nominal lumber size (x), h dp = h rib = 14 in. > h dp, min = 3 in. The total thikness inluding the atual slab and the drop panel thikness (h) = h s + h dp = = 17 in. ) The drop panel shall extend in eah diretion from the enterline of support a distane not less than onesixth the span length measured from enter-to-enter of supports in that diretion. ACI (8..4(b)) 4

8 L 1,dp_min L1 L ft L,dp_min L L ft Use L = L = 1 ft > L L 11 ft 1,dp,dp 1,dp_min,dp_min Based on the previous disussion, Figure 6 shows the dimensions of the seleted two-way joist system. 5

9 Figure 6 Two-Way Joist (Waffle) Slab 6

10 Preliminary Two-Way Joist Slab (Waffle Slab) For slabs with hanges in thikness and subjeted to bending in two diretions, it is neessary to hek shear at multiple setions as defined in the ACI The ritial setions shall be loated with respet to: 1) Edges or orners of olumns. ACI (.6.4.1(a)) ) Changes in slab thikness, suh as edges of drop panels. ACI (.6.4.1(b)) a. Slab minimum thikness Defletion ACI ( ) In lieu of detailed alulation for defletions, ACI 318 Code gives minimum slab thikness for two-way onstrution without interior beams in Table For this slab system, the minimum slab thiknesses per ACI are: ln 376 Exterior Panels: hs 11.4 in ACI (Table ) But not less than 4 in. ACI ( (b)) ln 376 Interior Panels: hs 10.4 in ACI (Table ) But not less than 4 in. ACI ( (b)) Where l n = length of lear span in the long diretion = 33 x 1 0 = 376 in. For the purposes of analysis and design, the ribbed slab will be replaed with a solid slab of equivalent moment of inertia, weight, punhing shear apaity, and one-way shear apaity. The equivalent thikness based on moment of inertia is used to find slab stiffness onsidering the ribs in the diretion of the analysis only. The ribs spanning in the transverse diretion are not onsidered in the stiffness omputations. This thikness, h MI, is given by: h MI 1/3 1/3 1 I rib in. brib 36 spslab Software Manual (Eq. -11) Where: I rib b rib = Moment of inertia of one joist setion between enterlines of ribs (see Figure 7a). = The enter-to-enter distane of two ribs (lear rib spaing plus rib width) (see Figure 7a). Sine h MI = 1 in. > h min = 11.4 in., the defletion alulation an be negleted. However, the defletion alulation will be inluded in this example for omparison with the spslab software results. The drop panel depth for two-way joist (waffle) slab is set equal to the rib depth. The equivalent drop depth based on moment of inertia, d MI, is given by: dmi hmi hrib in. spslab Software Manual (Eq. -1) Where h rib = = 5 in. 7

11 Figure 7a Equivalent Thikness Based on Moment of Inertia Find system self-weight using the equivalent thikness based on the weight of individual omponents (see the following Figure). This thikness, h w, is given by: h w Vmod in. spslab Software Manual (Eq. -10) A 99 Where: mod V mod = The Volume of one joist module (the transverse joists are inluded 11 joists in the frame strip). Vmod VLongitudinal Joist VTransverse Joists VIntersetion between Joists V Longitudinal Joist ft 3 V Transverse Joists V Intersetion between Joists ft ft V ft mod A mod = The plan area of one joist module = 33 x 36/1 = 99 ft Self-weight for slab setion without drop panel = 150 pf x 8 in. /1 = psf Self-weight for drop panel = 150 pf x ( ) in. /1 = psf 8

12 Figure 7b Equivalent Thikness Based on the Weight of Individual Components b. Slab shear strength one-way shear For ritial setion at distane d from the edge of the olumn (slab setion with drop panel): Evaluate the average effetive depth: db 0.75 dl hs lear db in. db 0.75 dt hs lear in. d avg dl dt in. Where: lear = 3/4 in. for # 6 steel bar ACI (Table ) d b h s = 0.75 in. for # 6 steel bar = 17 in. = The drop depth (d MI) Fatored dead load q Du 1. ( / 1 50) 315 psf Fatored live load q Lu psf ACI (5.3.1) Total fatored load q u psf 9

13 Chek the adequay of slab thikness for beam ation (one-way shear) from the edge of the interior olumn ACI (.5) Consider a 1-in. wide strip. The ritial setion for one-way shear is loated at a distane d, from the edge of the olumn (see Figure 8) Tributary area for one-way shear is A Tributary V u qu ATributary kips ft V f ' b d ACI (Eq ) w Where 1 for normal weight onrete V kips 1000 Slab thikness is adequate for one-way shear for the first ritial setion (from the edge of the olumn). V u For ritial setion at the edge of the drop panel (slab setion without drop panel): Evaluate the average effetive depth: db 0.75 dl hs lear db in. db 0.75 dt hs lear in. d avg dl dt in. Where: lear = 3/4 in. for # 6 steel bar ACI (Table ) d b = 0.75 in. for # 6 steel bar Fatored dead load q Du 1. ( ) psf Fatored live load q Lu psf ACI (5.3.1) Total fatored load q u psf Chek the adequay of slab thikness for beam ation (one-way shear) from the edge of the interior drop panel ACI (.5) Consider a 1-in. wide strip. The ritial setion for one-way shear is loated at the fae of the solid head (see Figure 8) 10

14 Tributary area for one-way shear is A Tributary V u qu ATributary kips ft 1 V f ' b d ACI (Eq ) w Where 1 for normal weight onrete V kips V 1000 Slab thikness of 1 in. is adequate for one-way shear for the seond ritial setion (at the edge of the drop panel). u. Slab shear strength two-way shear Figure 8 Critial Setions for One-Way Shear For ritial setion at distane d/ from the edge of the olumn (slab setion with drop panel): Chek the adequay of slab thikness for punhing shear (two-way shear) at an interior olumn (Figure 9): Tributary area of two-way shear for the slab without the drop panel is: ATributary _1 (33 33) ft Tributary area of two-way shear for the slab with the drop panel is: A Tributary _ (1 1) 135 ft 1 V u qu ATributary kips 11

15 V 4 f ' b d (For square interior olumn) ACI (Table.6.5.(a)) o V kips 1000 V kips V u Slab thikness is adequate for two-way shear for the first ritial setion (from the edge of the olumn). For ritial setion at the edge of the drop panel (slab setion without drop panel): Chek the adequay of slab thikness for punhing shear (two-way shear) at an interior drop panel (Figure 9): Tributary area for two-way shear is A (33 33) ft Tributary V u qu ATributary kips V 4 f ' b d (For square interior olumn) ACI (Table.6.5.(a)) o V kips 1000 V kips V u Slab thikness of 1 in. is adequate for two-way shear for the seond ritial setion (from the edge of the drop panel). Figure 9 Critial Setions for Two-Way Shear d. Column dimensions - axial load 1

16 Chek the adequay of olumn dimensions for axial load: Tributary area for interior olumn for live load, superimposed dead load, and self-weight of the slab is A ft Tributary Tributary area for interior olumn for self-weight of additional slab thikness due to the presene of the drop panel is A ft Tributary Assuming four story building Pu nqu ATributary kips Assume 0 in. square olumn with 1 No. 11 vertial bars with design axial strength, φp n,max of Pn,max 0.80 (0.85 f ' ( Ag Ast ) f y Ast ) ACI (.4.) P n,max , 595, 0 lbs Pn,max 1,595 kips P 1,559 kips u Column dimensions of 0 in. x 0 in. are adequate for axial load.. Flexural Analysis and Design ACI 318 states that a slab system shall be designed by any proedure satisfying equilibrium and geometri ompatibility, provided that strength and servieability riteria are satisfied. Distintion of two-systems from oneway systems is given by ACI (R & R8.3.1.). ACI 318 permits the use of Diret Design Method (DDM) and Equivalent Frame Method (EFM) for the gravity load analysis of orthogonal frames and is appliable to flat plates, flat slabs, and slabs with beams. The following setions outline the solution per EFM and spslab software. For the solution per DDM, hek the flat plate example..1. Equivalent Frame Method (EFM) EFM is the most omprehensive and detailed proedure provided by the ACI 318 for the analysis and design of two-way slab systems where the struture is modeled by a series of equivalent frames (interior and exterior) on olumn lines taken longitudinally and transversely through the building. The equivalent frame onsists of three parts (for a detailed disussion of this method, refer to the flat plate design example): 1) Horizontal slab-beam strip. ) Columns or other vertial supporting members. 3) Elements of the struture (Torsional members) that provide moment transfer between the horizontal and vertial members. 13

17 .1.1. Limitations for use of equivalent frame method 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. ACI ( & 6.4.3) Complete analysis must inlude representative interior and exterior equivalent frames in both the longitudinal and transverse diretions of the floor. ACI ( ) Panels shall be retangular, with a ratio of longer to shorter panel dimensions, measured enter-to-enter of supports, not to exeed. ACI ( ).1.. Frame members of equivalent frame Determine moment distribution fators and fixed-end moments for the equivalent frame members. The moment distribution proedure will be used to analyze the equivalent 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 , (331) (331) N1 N 1 Slab thikness = h h 1 in. and drop thikness = d h in. drop thikness 5 = = slab thikness 1 MI MI MI For, stiffness fators, k k 5.54 PCA Notes on ACI (Table A1) F1 N1 NF FN Es Is Es Is Thus, K k 5.54 PCA Notes on ACI (Table A1) sb NF , 04 Ksb 396 Where, I , in.-lb s 3 3 sh 396 (1) 57, 04 in E w 33 f psi ACI ( a) s Carry-over fator COF = 0.54 PCA Notes on ACI (Table A1) Fixed-end moment, FEM = m w l PCA Notes on ACI (Table A1) n i1 NFi i 1 Uniform load fixed end moment oeffiient, m NF1 = Fixed end moment oeffiient for (b-a) = 0. when a = 0, m NF = Fixed end moment oeffiient for (b-a) = 0. when a = 0.8, m NF3 =

18 b. Flexural stiffness of olumn members at both ends, K. Referring to Table A7, Appendix 0A, For the Bottom Column: t t t a a b 3/ in., t 3/ 1.5 in b H 13 ft 156 in., H / ft H H Thus, k 6.18 and C 0.50 by interpolation., bottom AB AB 6.18E I K PCA Notes on ACI (Table A7) 13,333 K, bottom in.-lb 4 4 (0) Where I 13, 333 in E w 33 f psi ACI ( a) l = 13 ft = 156 in. For the Top Column: t t b a H H Thus, k 4.6 and C 0.67 by interpolation. BA BA 4.6E I K PCA Notes on ACI (Table A7) 13,333 K, top in.-lb. Torsional stiffness of torsional members, K t. K t 9E C s 1 3 ACI (R ) K t in.-lb (1 0 / (331)) 15

19 3 x x y Where C y 3 ACI (Eq b) C in in., 33 ft = 396 in. 3 4 d. Equivalent olumn stiffness K e. K e K e K K Kt K t ( )( 173) 10 ( ) (173) K in.-lb e 6 6 Where K t is for two torsional members one on eah side of the olumn, and K is for the upper and lower olumns at the slab-beam joint of an intermediate floor. Figure 10 Torsional Member Figure 11 Column and Edge of Slab e. Slab-beam joint distribution fators, DF. At exterior joint, 340 DF ( ) At interior joint, 340 DF ( ) COF for slab-beam =

20 Figure 1 Slab and Column Stiffness.1.3. Equivalent frame analysis Determine negative and positive moments for the slab-beams using the moment distribution method. Sine the unfatored live load does not exeed three-quarters of the unfatored dead load, design moments are assumed to our at all ritial setions with full fatored live on all spans. ACI (6.4.3.) L D (100 50) 4 a. Fatored load and Fixed-End Moments (FEM s). For slab: Fatored dead load qdu 1.(100 50) 180 psf Fatored live load qlu 1.6(100) 160 psf Fatored load q q q 340 psf For drop panels: u Du Lu Fatored dead load qdu 1.(150 5/ 1) 135 psf Fatored live load qlu 1.6(0) 0 psf Fatored load q q q 135 psf u Du Lu Fixed-end moment, FEM = m w l PCA Notes on ACI (Table A1) n i1 NFi i 1 FEM FEM ft-kips b. Moment distribution. Computations are shown in Table 1. Counterlokwise rotational moments ating on the member ends are taken as positive. Positive span moments are determined from the following equation: 17

21 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-: M u 33 ( ) 33 / 6 ( ) ( ) 33 / / 8 33 M ft-kips u 18

22 Table 1 - Moment Distribution for Equivalent Frame Joint Member DF COF FEM Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist CO Dist M, k-ft Midspan M, ft-kips.1.4. Fatored moments used for Design Positive and negative fatored moments for the slab system in the diretion of analysis are plotted in Figure 13. 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 l 1 from the enters of supports. ACI ( ) 0 in ft ft (use fae of supporting loation) 1 19

23 Figure 13 - Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Fatored Live Load) 0

24 .1.5. Fatored moments in slab-beam strip a. Chek whether the moments alulated above an take advantage of the redution permitted by ACI ( ): If the slab system analyzed using EFM within the limitations of ACI (8.10.), it is permitted by the ACI ode to redue the alulated moments obtained from EFM in suh proportion that the absolute sum of the positive and average negative design moments need not exeed the total stati moment M o given by Equation in the ACI Chek Appliability of Diret Design Method: 1. There is a minimum of three ontinuous spans in eah diretion. ACI ( ). Suessive span lengths are equal. ACI (8.10..) 3. Long-to-Short ratio is 33/33 = 1.0 <.0. ACI ( ) 4. Columns are not offset. ACI ( ) 5. Loads are gravity and uniformly distributed with servie live-to-dead ratio of 0.67 <.0 (Note: The self-weight of the drop panels is not uniformly distributed entirely along the span. However, the variation in load magnitude is small). ACI ( and 6) 6. Chek relative stiffness for slab panel. ACI ( ) Slab system is without beams and this requirement is not appliable. M o n qu (33 0 /1) ft-kips ACI (Eq ) End spans: ft-kips > M o Interior span: ft-kips > To illustrate proper proedure, the interior span fatored moments may be redued as follows: Permissible redution = /155 = Adjusted negative design moment = = ft-kips Adjusted positive design moment = = 69.6 ft-kips M o ft-kips M o ACI 318 allows the redution of the moment values based on the previous proedure. Sine the drop panels may ause gravity loads not to be uniform (Chek limitation #5 and Figure 13), the moment values obtained from EFM will be used for omparison reasons. 1

25 b. Distribute fatored moments to olumn and middle strips: After the negative and positive moments have been determined for the slab-beam strip, the ACI ode permits the distribution of the moments at ritial setions to the olumn strips, beams (if any), and middle strips in aordane with the DDM. ACI ( ) Distribution of fatored moments at ritial setions is summarized in Table. End Span Interior Span Table - Distribution of fatored moments Slab-beam Strip Column Strip Middle Strip Moment (ft-kips) Perent Moment (ft-kips) Perent Moment (ft-kips) Exterior Negative Positive Interior Negative Negative Positive 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: M ft-kips u Use d = in. (slab with drop panel where h = 17 in.) To determine the area of steel, assumptions have to be made whether the setion is tension or ompression ontrolled, and regarding the distane between the resultant ompression and tension fores along the slab setion (jd). In this example, tension-ontrolled setion will be assumed so the redution fator is equal to 0.9, and jd will be taken equal to 0.95d. The assumptions will be verified one the area of steel in finalized. Assume jd 0.95 d in. Column strip width, b 331 / 198 in. Middle strip width, b in. Mu As 1.68 in. f jd y Af s y Realulate ' a' for the atual As 1.68 in. a in f ' b a in t dt Therefore, the assumption that setion is tension-ontrolled is valid.

26 Mu As in. f ( d a / ) ( / ) y Two values of thikness must be onsidered. The slab thikness in the olumn strip is 17 in. with the drop panel and 8 in. for the equivalent slab without the drop panel based on the system weight. The weighted slab thikness, h w 1 33 / / in. As,min b h ACI (4.4.3.) w As,min in. < in. s max 5h in. lesser of lesser of 15 in. 18 in. 18 in. ACI ( ) Provide 30 - #6 bars with A s = 13.0 in. and s = 198/30 = 6.6 in. s max The flexural reinforement alulation for the olumn strip of interior span positive loation: M 18.6 ft-kips u Use d = in. (slab with rib where h = 17 in.) To determine the area of steel, assumptions have to be made whether the setion is tension or ompression ontrolled, and regarding the distane between the resultant ompression and tension fores along the slab setion (jd). In this example, tension-ontrolled setion will be assumed so the redution fator is equal to 0.9, and jd will be taken equal to 0.95d. The assumptions will be verified one the area of steel in finalized. Assume jd 0.95 d in. Column strip width, b 331 / 198 in. Middle strip width, b in. Mu As.69 in. f jd y Af s y Realulate ' a' for the atual As.69 in. a 0.19 in f ' b a in t dt Therefore, the assumption that setion is tension-ontrolled is valid. Mu As.57 in. f ( d a / ) ( / ) y As,min b heq ACI (4.4.3.) 3

27 As,min in. >.57 in. use A A.851 in. s s,min Sine olumn strip has 5 ribs provide 10 - #6 bars ( bars/ rib): A in. > A.851 in. s, provided s, required Based on the proedures outlined above, values for all span loations are given in Table 3. Column Strip Middle Strip Span Loation Table 3 - Required Slab Reinforement for Flexure [Equivalent Frame Method (EFM)] Mu (ft-kips) b (in.) d (in.) As Req d for flexure (in. ) End Span Min As (in. ) Reinforement Provided As Prov. for flexure (in. ) Exterior Negative #6 * ** 6.16 Positive (5 ribs) #7 ( bars / rib) Interior Negative # Exterior Negative #6 * ** 6.16 Positive (6 ribs) #6 ( bars / rib) Interior Negative #6 * ** 6.16 Interior Span Column Strip Positive (5 ribs) Middle Strip Positive (6 ribs) * Design governed by minimum reinforement. ** Number of bars governed by maximum allowable spaing. 10-#6 * ( bars / rib) 1-#6 * ( bars / rib) b. Calulate additional slab reinforement at olumns for moment transfer between slab and olumn by flexure The fatored slab moment resisted by the olumn (γ f M s) shall be assumed to be transferred by flexure. Conentration of reinforement over the olumn by loser spaing or additional reinforement shall be used to resist this moment. The fration of slab moment not alulated to be resisted by flexure shall be assumed to be resisted by eentriity of shear. ACI (8.4..3) Portion of the unbalaned moment transferred by flexure is γ f M s ACI ( ) Where 1 f ACI ( ) 1 ( / 3) b / b 1 b 1 = Dimension of the ritial setion b o measured in the diretion of the span for whih moments are determined in ACI 318, Chapter 8 (see Figure 14). b = Dimension of the ritial setion (see Figure 14). b o measured in the diretion perpendiular to 1 b in ACI 318, Chapter 8 b b = Effetive slab width = 3 h ACI ( ) 4

28 Figure 14 Critial Shear Perimeters for Columns For exterior support: d b 0.75 d = h - over - = in. d b1 1 0 = 7.94 in. b d = in. b in. b f 1 ( / 3) 7.94 / ft-kips f Ms Using the same proedure in.1.6.a, the required area of steel: A 4.18 in. s However, the area of steel provided to resist the flexural moment within the effetive slab width b b: 71 As, provided in

29 Then, the required additional reinforement at exterior olumn for moment transfer between slab and olumn: A, in. s additional Provide 5 - #6 additional bars with A s =.0 in. Based on the proedure outlined above, values for all supports are given in Table 4. Column Strip Table 4 - Additional Slab Reinforement required for moment transfer between slab and olumn (EFM) Span Loation M s * (ft-kips) γ f γ f M s (ft-kips) End Span Effetive slab width, b b (in.) d (in.) A s req d within b b (in. ) A s prov. For flexure within b b (in. ) Exterior Negative #6 Interior Negative *M s is taken at the enterline of the support in Equivalent Frame Method solution. Add l Reinf Fatored moments in olumns The unbalaned moment from the slab-beams at the supports of the equivalent frame are distributed to the support olumns above and below the slab-beam in proportion to the relative stiffness of the support olumns. Referring to Figure 13, the unbalaned moment at the exterior and interior joints are: Exterior Joint = ft-kips Joint = = -134 ft-kips The stiffness and arry-over fators of the atual olumns and the distribution of the unbalaned slab moments (M s) to the exterior and interior olumns are shown in Figure 14. 6

30 Figure 15 - Column Moments (Unbalaned Moments from Slab-Beam) In summary: For Top olumn: M ol,exterior= ft-kips M ol,interior = ft-kips For Bottom olumn: M ol,exterior= 4.97 ft-kips M ol,interior = 65.1 ft-kips 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. The moment values at the fae of interior, exterior, and orner olumns from the unbalaned moment values are shown in the following table. Mu kips-ft Table 5 Fatored Moments in Columns Column Loation Interior Exterior Corner Mux Muy

31 3. Design of Columns by spcolumn This setion inludes the design of interior, edge, and orner olumns using spcolumn software. The preliminary dimensions for these olumns were alulated previously in setion one. The redution of live load per ASCE 7-10 will be ignored in this example. However, the detailed proedure to alulate the redued live loads is explained in the wide-module Joist System example Determination of fatored loads Interior Column: Assume 4 story building Tributary area for interior olumn for live load, superimposed dead load, and self-weight of the slab is A ft Tributary Tributary area for interior olumn for self-weight of additional slab thikness due to the presene of the drop panel is A ft Tributary Assuming five story building Pu nqu ATributary kips M u,x = 65.1 ft-kips (see the previous Table) M u,y = 65.1 ft-kips (see the previous Table) Edge (Exterior) Column: Tributary area for exterior olumn for live load, superimposed dead load, and self-weight of the slab is A Tributary 33 0 / ft 1 Tributary area for exterior olumn for self-weight of additional slab thikness due to the presene of the drop panel is A Tributary 1 0 / 1 8 ft 1 Pu nqu ATributary kips M u,x = 4.97 ft-kips (see the previous Table) M u,y = 65.1 ft-kips (see the previous Table) Corner Column: Tributary area for orner olumn for live load, superimposed dead load, and self-weight of the slab is 8

32 A Tributary 33 0 / 33 0 / ft 1 1 Tributary area for orner olumn for self-weight of additional slab thikness due to the presene of the drop panel is A Tributary 1 0 / 1 0 / 46.7 ft 1 1 Pu nqu ATributary kips M u,x = 4.97 ft-kips (see the previous Table) M u,y = 4.97 ft-kips (see the previous Table) 9

33 3.. Moment Interation Diagram Interior Column: 30

34 31

35 Edge Column: 3

36 Corner Column: 33

37 4. 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 ACI 318 Chapter One-Way (Beam ation) Shear Strength ACI (.5) One-way shear is ritial at a distane d from the fae of the olumn as shown in Figure 3. Figures 17 and 19 show the fatored shear fores (V u) at the ritial setions around eah olumn and eah drop panel, respetively. In members without shear reinforement, the design shear apaity of the setion equals to the design shear apaity of the onrete: φv φv φv φv, ( V 0) ACI (Eq ) n s s Where: φv φλ f ' b d ACI (Eq ) w One-way shear apaity is alulated assuming the shear ross-setion area onsisting of the drop panel (if any), the ribs, and the slab portion above them, dereased by onrete over. For suh setion the equivalent shear width for single rib is alulated from the formula: d bv b spslab Software Manual (Eq. -13) 1 Where: b = rib width, in. d = distane from extreme ompression fiber to tension reinforement entroid At distane d from the supporting olumn d / in. for middle span with #6 reinforement bv in. 1 1 for normal weight onrete b L, drop nribs bv in. (See Figure 16). 34

38 Figure 16 Frame strip ross setion (at distane d from the fae of the supporting olumn) The one-way shear apaity for the ribbed slab portions shown in Figure 16 is permitted to be inreased by 10%. ACI ( ) 1.10 V V V Solid Slab Ribbed Slab V (1 1) (7 7.3) kips Beause V V at all the ritial setions, the slab has adequate one-way shear strength. u Figure 17 One-way shear at ritial setions (at distane d from the fae of the supporting olumn) At the fae of the drop panel d / in. for middle span with #6 reinforement. 35

39 15.88 bv in. 1 1 for normal weight onrete b n b in. (See Figure 17). ribs v Figure 18 Frame strip ross setion (at distane d from the fae of the supporting olumn) The one-way shear apaity for the ribbed slab portions shown in Figure 15 is permitted to be inreased by 10%. ACI ( ) V 1.10 V Ribbed Slab 5000 V (11 7.3) kips 1000 Beause V V at all the ritial setions, the slab has adequate one-way shear strength. u Figure 19 One-way shear at ritial setions (at the fae of the drop panel) 36

40 4.. Two-Way (Punhing) Shear Strength ACI (.6) Around the olumns faes Two-way shear is ritial on a retangular setion loated at d/ away from the fae of the olumn as shown in Figure 14. a. Exterior olumn: The fatored shear fore (V u) in the ritial setion is omputed 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 (d/ away from olumn fae) Vu V qu b1 b kips 144 The fatored unbalaned moment used for shear transfer, M unb, is omputed 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 / Munb M Vu b1 AB 1/ ft-kips 1 For the exterior olumn in Figure 13, the loation of the entroidal axis z-z is: AB moment of area of the sides about AB ( / ) 8.51 in. area of the sides The polar moment J of the shear perimeter is: J b d b d b1 d db1 b1 1 AB AB J J 144,09 in. 4 γ 1 γ ACI (Eq ) v f The length of the ritial perimeter for the exterior olumn: b in. o The two-way shear stress (v u) an then be alulated as: 37

41 v u V γ M b d J o u v unb AB ACI (R ) v u ( ) psi , 09 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI (Table.6.5.) v v min , , min 8.8, 44.3, psi 8.8 psi φ v psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength at this joint. b. Interior olumn: Vu V qu b1 b kips 144 M M V b / ft-kips unb u 1 AB 1 For the interior olumn in Figure 13, the loation of the entroidal axis z-z is: b AB 17.94in. The polar moment J of the shear perimeter is: J b d b d b d db 1 b AB AB J J 51,956 in. v f 4 γ 1 γ ACI (Eq ) The length of the ritial perimeter for the interior olumn: b in. o 38

42 The two-way shear stress (v u) an then be alulated as: v u V γ M b d J o u v unb AB ACI (R ) v u ( ) psi , 956 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI (Table.6.5.) v v min , , min 8.8, 44.3, psi 8.8 psi φ v psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength at this joint.. Corner olumn: In this example, interior equivalent frame strip was seleted where it only have exterior and interior supports (no orner supports are inluded in this strip). However, the two-way shear strength of orner supports usually governs. Thus, the two-way shear strength for the orner olumn in this example will be heked for illustration purposes. The analysis proedure must be repeated for the exterior equivalent frame strip to find the reation and fatored unbalaned moment used for shear transfer at the entroid of the ritial setion for the orner support Vu V qu b1 b kips / Munb M Vu b1 AB 1 / ft-kips 1 For the interior olumn in Figure 13, the loation of the entroidal axis z-z is: AB moment of area of the sides about AB ( / ) 6.99 in. area of the sides The polar moment J of the shear perimeter is: J b d b d b1 d db1 b1 1 AB AB 39

43 J J 81,483 in. 4 γ 1 γ ACI (Eq ) v f The length of the ritial perimeter for the orner olumn: b in. o The two-way shear stress (v u) an then be alulated as: v u V γ M b d J o u v unb AB ACI (R ) v u ( ) psi , 483 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI (Table.6.5.) v v min , , min 8.8, 44.3, psi 8.8 psi φ v psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength at this joint Around drop panels Two-way shear is ritial on a retangular setion loated at d/ away from the fae of the drop panel. The fatored shear fore (V u) in the ritial setion is omputed 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 (d/ away from olumn fae). Note: For simpliity, it is onservative to dedut only the self-weight of the slab and joists in the ritial setion from the shear reation in punhing shear alulations. This approah is also adopted in the spslab program for the punhing shear hek around the drop panels. a. Exterior drop panel: 40

44 Vu V qua kips 144 d that is used in the alulation of v u is given by (see Figure 0): (# of ribs within the drop panel width) hb d v spslab Software Manual (Eq. -14) the drop panel width 417 in. 7.3 in. d 3.3 in. 150 in. Figure 0 Equivalent thikness based on shear area alulation The length of the ritial perimeter for the exterior drop panel: b in. o The two-way shear stress (v u) an then be alulated as: v u Vu b d o ACI (R ) v u psi The two-way shear apaity for the ribbed slab is permitted to be inreased by 10%. ACI ( ) ' 4 ' αd s ' v min λ f, 1.10 λ f, 1.10 λ f β bo ACI (Table.6.5.) v v min , , min 311.1, 44.3, 16. psi 16. psi φ v psi 41

45 In waffle slab design where the drop panels reate a large ritial shear perimeter, the fator (b o/d) has limited ontribution and is traditionally negleted for simpliity and onservatism. This approah is adopted in this alulation and in the spslab program (spslab software manual, Eq. -46). The two-way shear apaity for the ribbed slab is permitted to be inreased by 10%. ACI ( ) v 1.10 λ f ' v psi φ v psi Sine φ v v u at the ritial setion, the slab have adequate two-way shear strength around this drop panel. b. Interior drop panel: V V q A u u V u kips 144 The length of the ritial perimeter for the interior drop panel: b in. o The two-way shear stress (v u) an then be alulated as: v u Vu b d o ACI (R ) v u psi The two-way shear apaity for the ribbed slab is permitted to be inreased by 10%. ACI ( ) ' 4 ' αd s ' v min λ f, 1.10 λ f, 1.10 λ f β bo ACI (Table.6.5.) v v min , , min 311.1, 44.3, psi psi φ v psi ' v 1.10 λ f spslab Software Manual (Eq. -46) 4

46 v psi φ v psi Sine φ v drop panel. v u at the ritial setion, the slab does not have adequate two-way shear strength around this. Corner drop panel: V V q A u u V u kips 144 The length of the ritial perimeter for the orner drop panel: b in. o The two-way shear stress (v u) an then be alulated as: v u Vu b d o ACI (R ) v u psi The two-way shear apaity for the ribbed slab is permitted to be inreased by 10%. ACI ( ) ' 4 ' αd s ' v min λ f, 1.10 λ f, 1.10 λ f β bo ACI (Table.6.5.) v v min , , min 311.1, 44.3, psi psi φ v psi ' v 1.10 λ f spslab Software Manual (Eq. -46) v psi φ v psi Sine φ v drop panel. v u at the ritial setion, the slab does not have adequate two-way shear strength around this To mitigate the defiieny in two-way shear apaity an evaluation of possible options is required: 43

47 1. Inrease the thikness of the slab system. Inreasing the dimensions of the drop panels (length and/or width) 3. Inreasing the onrete strength 4. Redution of the applied loads 5. Redution of the panel spans 6. Using less onservative punhing shear allowable (gain of 5-10%) 7. Refine the dedution of drop panel weight from the shear reation (gain of -5%) This example will be ontinued without the required modifiation disussed above to ontinue the illustration of the analysis and design proedure. 44

48 5. Servieability Requirements (Defletion Chek) Sine the slab thikness was seleted to meet the minimum slab thikness tables in ACI , the defletion alulations of immediate and time-dependent defletions are not required. They are shown below for illustration purposes and omparison with spslab software 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..3) 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: 3 3 M r M r Ie I 1 I I M M a a g r g ACI (Eq a) Where: 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

49 Figure 1 Maximum Moments for the Three Servie Load Levels For positive moment (midspan) setion: M Craking moment. r M f r r fi r g , ft-kips y t = Modulus of rapture of onrete. ACI (Eq b) f r I g 7.5 f ' psi ACI (Eq ) = Moment of inertia of the gross unraked onrete setion. See Figure. 46

50 I I # of ribs ,55 in. g g / rib 4 y t = Distane from entroidal axis of gross setion, negleting reinforement, to tension fae, in. y h y in. t rib bar Figure Equivalent gross setion for one rib - positive moment setion I = Moment of inertia of the raked setion transformed to onrete. PCA Notes on ACI (9.5..) r As alulated previously, the positive reinforement for the middle span frame strip is #6 bars loated at 1.15 in. along the setion from the bottom of the slab. Figure 3 shows all the parameters needed to alulate the moment of inertia of the raked setion transformed to onrete at midspan. Figure 3 Craked Transformed Setion - positive moment setion E s = Modulus of elastiity of slab onrete s E w 33 f psi ACI ( a) E s n 6.76 PCA Notes on ACI (Table 10-) E s b 331 B 6.05 in. na s 1 PCA Notes on ACI (Table 10-) db kd.13 in. PCA Notes on ACI (Table 10-) B

51 b( kd) I na d kd 3 3 r s ( ) PCA Notes on ACI (Table 10-) I r (.13) ,647 in. 3 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 45 #6 bars loated at 1.15 in. along the setion from the top of the slab. M f r r fi r g , ft-kips y t ACI (Eq b) 7.5 f ' psi ACI (Eq ) I g 103, 6 in. (See Figure 4) Note: A lower value of I g (60,55 in. 4 ) exluding the drop panel is onservatively adopted in alulating waffle slab defletion by the spslab software. y 9.65 in. t Figure 4 Gross setion negative moment setion s E w 33 f psi ACI ( a) E s n 6.76 PCA Notes on ACI (Table 10-) E s btotal 194 B 1.45 in. na s 1 PCA Notes on ACI (Table 10-) Where btotal in. (See Figures 4 and 5) 48

52 db kd 4.04 in. PCA Notes on ACI (Table 10-) B b ( kd) total I na ( d kd) PCA Notes on ACI (Table 10-) r 3 s I r (4.04) ,09 in. 3 4 Note: A lower value of I r (18,7 in. 4 ) exluding the drop panel is onservatively adopted in alulating waffle slab defletion by the spslab software. Figure 5 Craked transformed setion - negative moment setion 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 Eq. (4..3.5a) unless obtained by a more omprehensive analysis. I e shall be permitted to be taken as the value obtained from Eq. (4..3.5a) at midspan for simple and ontinuous spans, and at the support for antilevers. ACI (4..3.7) For ontinuous one-way slabs and beams. I e shall be permitted to be taken as the average of values obtained from Eq. (4..3.5a) for the ritial positive and negative moment setions. ACI (4..3.6) For the middle span (span with two ends ontinuous) with servie load level (D+LL full): 3 3 M M r r I I 1 I, sine M r ft-kips < M a = ft-kips e M g M r a a ACI (4..3.5a) - Where I e is the effetive moment of inertia for the ritial negative moment setion (near the support) I e 103,6 1 3,09 38,873 in