Two-Way Flat Slab (Concrete Floor with Drop Panels) System Analysis and Design

Two-Way Flat Slab (Conrete Floor with Drop Panels) System Analysis and Design

Two-Way Flat Slab (Conrete Floor with Drop Panels) System Analysis and Design Design the onrete floor slab system shown below for an intermediate floor onsidering partition weight = 0 psf, and unfatored live load = 60 psf. The lateral loads are independently resisted by shear walls. The use of flat plate system will be heked. If the use of flat plate is not adequate, the use of flat slab system with drop panels will be investigated. Flat slab onrete floor system is similar to the flat plate system. The only exeption is that the flat slab uses drop panels (thikened portions around the olumns) to inrease the nominal shear strength of the onrete at the ritial setion around the olumns. 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. Figure 1 - Two-Way Flat Conrete Floor System Version: Jan-6-018

Contents 1. Preliminary member sizing... 1. Flexural Analysis and Design... 10.1. Equivalent Frame Method (EFM)... 10.1.1. Limitations for use of equivalent frame method... 11.1.. Frame members of equivalent frame... 11.1.3. Equivalent frame analysis... 14.1.4. Fatored moments used for Design... 16.1.5. Fatored moments in slab-beam strip... 18.1.6. Flexural reinforement requirements... 19.1.7. Fatored moments in olumns... 3. Design of Columns by spcolumn... 4 3.1. Determination of fatored loads... 4 3.. Moment Interation Diagram... 6 4. Shear Strength... 9 4.1. One-Way (Beam ation) Shear Strength... 9 4.1.1. At distane d from the supporting olumn... 9 4.1.. At the fae of the drop panel... 30 4.. Two-Way (Punhing) Shear Strength... 31 4..1. Around the olumns faes... 31 4... Around drop panels... 34 5. Servieability Requirements (Defletion Chek)... 37 5.1. Immediate (Instantaneous) Defletions... 37 5.. Time-Dependent (Long-Term) Defletions (Δ lt)... 49 6. spslab Software Program Model Solution... 50 7. Summary and Comparison of Design Results... 71 8. Conlusions & Observations... 74 8.1. One-Way Shear Distribution to Slab Strips... 74 8.. Two-Way Conrete Slab Analysis Methods... 77 Version: Jan-6-018

Code Building Code Requirements for Strutural Conrete (ACI 318-14) and Commentary (ACI 318R-14) Referene Conrete Floor Systems (Guide to Estimating and Eonomizing), Seond Edition, 00 David A. Fanella Notes on ACI 318-11 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) Design Data Story Height = 13 ft (provided by arhitetural drawings) Superimposed Dead Load, SDL =0 psf for framed partitions, wood studs, x, plastered sides ASCE/SEI 7-10 (Table C3-1) Live Load, LL = 60 psf ASCE/SEI 7-10 (Table 4-1) 50 psf is onsidered by inspetion of Table 4-1 for Offie Buildings Offies (/3 of the floor area) 80 psf is onsidered by inspetion of Table 4-1 for Offie Buildings Corridors (1/3 of the floor area) LL = /3 x 50 + 1/3 x 80 = 60 psf f = 5000 psi (for slab) f = 6000 psi (for olumns) f y = 60,000 psi Solution 1. Preliminary member sizing For Flat Plate (without Drop Panels) a. Slab minimum thikness Defletion ACI 318-14 (8.3.1.1) In lieu of detailed alulation for defletions, ACI 318 Code gives minimum slab thikness for two-way onstrution without interior beams in Table 8.3.1.1. For this flat plate slab systems the minimum slab thiknesses per ACI 318-14 are: ln 340 Exterior Panels: hs 11.33 in. ACI 318-14 (Table 8.3.1.1) 30 30 But not less than 5 in. ACI 318-14 (8.3.1.1(a)) ln 340 Interior Panels: hs 10.3 in. ACI 318-14 (Table 8.3.1.1) 33 33 But not less than 5 in. ACI 318-14 (8.3.1.1(a)) 1

Where l n = length of lear span in the long diretion = 30 x 1 0 = 340 in. Try 11 in. slab for all panels (self-weight = 150 pf x 11 in. /1 = 137.5 psf) b. Slab shear strength one way shear At a preliminary hek level, the use of average effetive depth would be suffiient. However, after determining the final depth of the slab, the exat effetive depth will be used in flexural, shear and defletion alulations. Evaluate the average effetive depth (Figure ): db 0.75 dl hs lear db 11 0.75 0.75 9.13 in. db 0.75 dt hs lear 11 0.75 9.88 in. d avg dl dt 9.13 9.88 9.51 in. Where: lear = 3/4 in. for # 6 steel bar ACI 318-14 (Table 0.6.1.3.1) d b = 0.75 in. for # 6 steel bar Figure - Two-Way Flat Conrete Floor System Fatored dead load, qdu 1. (137.5 0) 189 psf Fatored live load, qlu 1.6 60 96 psf ACI 318-14 (5.3.1) Total fatored load, qu 189 96 85 psf Chek the adequay of slab thikness for beam ation (one-way shear) ACI 318-14 (.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 0.85 13.37 3.81 kips 30 0 9.51 1 13.37 ft 1 1 1

V f b d ACI 318-14 (Eq..5.5.1) ' w Where λ = 1 for normal weight onrete, more information an be found in Conrete Type Classifiation Based on Unit Density tehnial artile. 9.51 V 0.75 1.0 5000 1 1.09 kips V 1000 Slab thikness of 11 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): 0 9.51 Tributary area for two-way shear is A Tributary (30 30) 894 ft 1 Vu qu ATributary 0.85 894 54.78 kips V ACI 318-14 (Table.6.5.(a)) ' 4 f bwd (For square interior olumn) 9.51 V 4 5000 40 9.51 317 kips 1000 V 0.75 317 37.8 kips V u Slab thikness of 11 in. is not adequate for two-way shear. It is good to mention that the fatored shear (V u) used in the preliminary hek does not inlude the effet of the unbalaned moment at supports. Inluding this effet will lead to an inrease of V u value as shown later in setion 4.. Figure 3 Critial Setion for One-Way Shear Figure 4 Critial Setion for Two-Way Shear In this ase, four options ould be used: 1) to inrease the slab thikness, ) to inrease olumns ross setional dimensions or ut the spaing between olumns (reduing span lengths), however, this option is assumed to be not 3

permissible in this example due to arhitetural limitations, 3) to use headed shear reinforement, or 4) to use drop panels. In this example, the latter option will be used to ahieve better understanding for the design of two-way slab with drop panels often alled flat slab. 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 318-14 (8..4(a)) Sine the slab thikness (h s) is 10 in. (see page 6), the thikness of the drop panel should be at least: hdp, min 0.5 h 0.510.5 in. s Drop panel dimensions are also ontrolled by formwork onsiderations. The following Figure shows the standard lumber dimensions that are used when forming drop panels. Using other depths will unneessarily inrease formwork osts. For nominal lumber size (x), h dp = 4.5 in. > h dp, min =.5 in. The total thikness inluding the slab and the drop panel (h) = h s + h dp = 10 + 4.5 = 14.5 in. Nominal Lumber Size, in. Atual Lumber Size, in. Plyform Thikness, in. h dp, in. x 1 1/ 3/4 1/4 4x 3 1/ 3/4 4 1/4 6x 5 1/ 3/4 6 1/4 8x 7 1/4 3/4 8 Figure 5 Drop Panel Formwork Details ) 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 318-14 (8..4(b)) 1 1 1 1 L 1,dp L1 L 1 30 30 5 ft 6 6 6 6 1 1 1 1 L,dp L L 30 30 5 ft 6 6 6 6 Based on the previous disussion, Figure 6 shows the dimensions of the seleted drop panels around interior, edge (exterior), and orner olumns. 4

Figure 6 Drop Panels Dimensions 5

For Flat Slab (with Drop Panels) 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 319-14. The ritial setions shall be loated with respet to: 1) Edges or orners of olumns. ACI 318-14 (.6.4.1(a)) ) Changes in slab thikness, suh as edges of drop panels. ACI 318-14 (.6.4.1(b)) a. Slab minimum thikness Defletion ACI 318-14 (8.3.1.1) In lieu of detailed alulation for defletions, ACI 318 Code gives minimum slab thikness for two-way onstrution without interior beams in Table 8.3.1.1. For this flat plate slab systems the minimum slab thiknesses per ACI 318-14 are: ln 340 Exterior Panels: hs 10.30 in. 33 33 ACI 318-14 (Table 8.3.1.1) But not less than 4 in. ACI 318-14 (8.3.1.1(b)) ln 340 Interior Panels: hs 9.44 in. 36 36 ACI 318-14 (Table 8.3.1.1) But not less than 4 in. ACI 318-14 (8.3.1.1(b)) Where l n = length of lear span in the long diretion = 30 x 1 0 = 340 in. Try 10 in. slab for all panels Self-weight for slab setion without drop panel = 150 pf x 10 in. /1 = 15 psf Self-weight for slab setion with drop panel = 150 pf x 14.5 in. /1 = 178 psf 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 14.5 0.75 0.75 1.38 in. db 0.75 dt hs lear 14.5 0.75 13.13 in. d avg dl dt 1.38 13.13 1.75 in. Where: lear = 3/4 in. for # 6 steel bar ACI 318-14 (Table 0.6.1.3.1) d b = 0.75 in. for # 6 steel bar 6

Fatored dead load q Du 1. (178 0) 37.6 psf Fatored live load q Lu 1.6 60 96 psf ACI 318-14 (5.3.1) Total fatored load q u 37.6 96 333.6 psf Chek the adequay of slab thikness for beam ation (one-way shear) from the edge of the interior olumn ACI 318-14 (.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 7) Tributary area for one-way shear is A Tributary V u qu ATributary 0.334 13.10 4.37 kips 30 0 1.75 1 13.10 ft 1 1 1 V f ' b d ACI 318-14 (Eq..5.5.1) w Where 1 for normal weight onrete 1.75 V 0.75 1.0 5000 1 16.3 kips 1000 Slab thikness of 14.5 in. 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 10 0.75 0.75 8.13 in. db 0.75 dt hs lear 10 0.75 8.88 in. d avg dl dt 8.13 8.88 8.51 in. Where: lear = 3/4 in. for # 6 steel bar ACI 318-14 (Table 0.6.1.3.1) d b = 0.75 in. for # 6 steel bar Fatored dead load q Du 1. (15 0) 174 psf Fatored live load q Lu 1.6 60 96 psf ACI 318-14 (5.3.1) Total fatored load q u 174 96 70 psf 7

Chek the adequay of slab thikness for beam ation (one-way shear) from the edge of the interior drop panel ACI 318-14 (.5) Consider a 1-in. wide strip. The ritial setion for one-way shear is loated at the fae of support (see Figure 7) Tributary area for one-way shear is A Tributary V u qu ATributary 0.70 10.00.70 kips 30 10 1 10.00 ft 1 V f ' b d ACI 318-14 (Eq..5.5.1) w Where 1 for normal weight onrete 8.51 V 0.75 1.0 5000 1 10.8 kips V 1000 Slab thikness of 10 in. is adequate for one-way shear for the seond ritial setion (from the edge of the drop panel). u Figure 7 Critial Setions for One-Way Shear. Slab shear strength two-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 8): 0 1.75 Tributary area for two-way shear is A Tributary (30 30) 89.6 ft 1 V u qu ATributary 0.334 89.6 97.9 kips V 4 f ' b d (For square interior olumn) ACI 318-14 (Table.6.5.(a)) o 8

1.75 V 4 5000 40 1.75 47 kips 1000 V 0.75 47 354 kips V u Slab thikness of 14.5 in. 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 8): 10 8.51 Tributary area for two-way shear is A Tributary (30 30) 785 ft 1 V u qu ATributary 0.70 785 1 kips V 4 f ' b d (For square interior olumn) ACI 318-14 (Table.6.5.(a)) o 8.51 V 4 5000 410 8.51 137 kips 1000 V 0.75137 98 kips V u Slab thikness of 10 in. is adequate for two-way shear for the seond ritial setion (from the edge of the drop panel). Figure 8 Critial Setions for Two-Way Shear 9

d. Column dimensions - axial load 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 3030 900ft Tributary Tributary area for interior olumn for self-weight of additional slab thikness due to the presene of the drop panel is A 1010 100 ft Tributary Assuming five story building Pu nqu ATributary 5 0.70 900 0.0638 100 147 kips Assume 0 in. square olumn with 4 No. 14 vertial bars with design axial strength, φp n,max of Pn,max 0.80 (0.85 f ' ( Ag Ast ) f y Ast ) ACI 318-14 (.4.) P n,max 0.80 0.65 0.85 6000 0 0 4.5 60000 4.5 1,317,730 lbs Pn,max 1,318 kips P 1, 47 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 318-14 (R8.10..3 & 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. 10

3) Elements of the struture (Torsional members) that provide moment transfer between the horizontal and vertial members..1.1. Limitations for use of equivalent frame method In EFM, live load shall be arranged in aordane with 6.4.3 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 318-14 (8.11.1. & 6.4.3) Complete analysis must inlude representative interior and exterior equivalent frames in both the longitudinal and transverse diretions of the floor. ACI 318-14 (8.11..1) Panels shall be retangular, with a ratio of longer to shorter panel dimensions, measured enter-to-enter of supports, not to exeed. ACI 318-14 (8.10..3).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 318-11. These alulations are shown below. a. Flexural stiffness of slab-beams at both ends, K sb. 0 0 0.056, 0.056 (301) (301) N1 N 1 For, stiffness fators, k k 5.59 PCA Notes on ACI 318-11 (Table A1) F1 N1 NF FN Es Is Es Is Thus, K k 5.59 PCA Notes on ACI 318-11 (Table A1) sb NF 1 1 30, 000 Ksb 360 Where, I 3 6 5.59 487 10 1, 997 10 in.-lb s 3 3 sh 360 (10) 30, 000 in. 1 1 4 E w 33 f 150 33 5000 487 10 psi ACI 318-14 (19...1.a) 1.5 1.5 3 s Carry-over fator COF = 0.578 PCA Notes on ACI 318-11 (Table A1) Fixed-end moment, FEM = m w l PCA Notes on ACI 318-11 (Table A1) n i1 NFi i 1 Uniform load fixed end moment oeffiient, m NF1 = 0.0915 Fixed end moment oeffiient for (b-a) = 0. when a = 0, m NF = 0.0163 Fixed end moment oeffiient for (b-a) = 0. when a = 0.8, m NF3 = 0.0163 11

b. Flexural stiffness of olumn members at both ends, K. Referring to Table A7, Appendix 0A, For the Bottom Column (Below): t a 10 / 4.5 9.5 in., t 10 / 5 in. b t t a b 9.5 1.85 5 H 13 ft 156 in., H 156 in. 9.5 in. 5 in. 11.81ft H 13 1.101 H 11.81 Thus, k 5.3 and C 0.54 by interpolation., bottom AB AB 5.3E I K PCA Notes on ACI 318-11 (Table A7) 13, 333 K, bottom 156 3 6 5.3 4696 10 135. 10 in.-lb 4 4 (0) Where I 13, 333 in. 1 1 1.5 1.5 3 4 E w 33 f 150 33 6000 4696 10 psi ACI 318-14 (19...1.a) l = 13 ft = 156 in. For the Top Column (Above): t t b a 5 0.54 9.5 H 13 1.101 H 11.81 Thus, k 4.88 and C 0.59 by interpolation. BA BA 4.88E I K PCA Notes on ACI 318-11 (Table A7) 13, 333 K, top 156 3 6 4.88 4696 10 1958.6 10 in.-lb. Torsional stiffness of torsional members, K t. K t 9EsC 3 [ (1 ) ] ACI 318-14 (R.8.11.5) 1

K t 3 9 487 10 1063 6 135310 in.-lb 3 301 (1 0 / (301)) 3 x x y Where C (1 0.63 )( ) ACI 318-14 (Eq. 8.10.5.b) y 3 14.5 0 C (1 0.63 )(14.5 ) 1063 in. 0 3 0 in., 30 ft = 360 in. Equivalent olumn stiffness K e. K e K e K K Kt K t 3 4 (135. 1958.6)( 1353) 10 [(135. 1958.) (1353)] K 1353 10 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 9 Torsional Member Figure 10 Column and Edge of Slab d. Slab-beam joint distribution fators, DF. At exterior joint, 1996 DF 0.551 (1996 169) At interior joint, 1996 DF 0.355 (1996 1996 169) COF for slab-beam =0.578 13

Figure 11 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 318-14 (6.4.3.) L 60 3 0.41 D (15 0) 4 a. Fatored load and Fixed-End Moments (FEM s). For slab: Fatored dead load qdu 1.(15 0) 174 psf Fatored live load qlu 1.6(60) 96 psf Fatored load q q q 70 psf For drop panels: u Du Lu Fatored dead load qdu 1.(150 4.5/ 1) 63.75 psf Fatored live load qlu 1.6(0) 0 psf Fatored load q q q 63.75 psf u Du Lu Fixed-end moment, FEM = m w l PCA Notes on ACI 318-11 (Table A1) n i1 NFi i 1 FEM 0.0915 0.70 30 30 0.0163 0.0.064 30 / 330 0.0163 0.0.064 30 / 3 30 FEM 677.6 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: 14

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 30 (0.064 30 / 6) 30 / 6 (331.7 807.6) (0.70 30) 30 / 6 30 30 / 8 30 M 349.6 ft-kips u Table 1 - Moment Distribution for Equivalent Frame Joint 1 3 4 Member 1- -1-3 3-3-4 4-3 DF 0.551 0.355 0.355 0.355 0.355 0.551 COF 0.578 0.578 0.578 0.578 0.578 0.578 FEM 677.6-677.6 677.6-677.6 677.6-677.6 Dist -373.1 0.0 0.0 0.0 0.0 373.1 CO 0.0-15.7 0.0 0.0 15.7 0.0 Dist 0.0 76.6 76.6-76.6-76.6 0.0 CO 44.3 0.0-44.3 44.3 0.0-44.3 Dist -4.4 15.7 15.7-15.7-15.7 4.4 CO 9.1-14.1-9.1 9.1 14.1-9.1 Dist -5.0 8. 8. -8. -8. 5.0 CO 4.8 -.9-4.8 4.8.9-4.8 Dist -.6.7.7 -.7 -.7.6 CO 1.6-1.5-1.6 1.6 1.5-1.6 Dist -0.9 1.1 1.1-1.1-1.1 0.9 CO 0.6-0.5-0.6 0.6 0.5-0.6 Dist -0.4 0.4 0.4-0.4-0.4 0.4 CO 0. -0. -0. 0. 0. -0. Dist -0.1 0. 0. -0. -0. 0.1 CO 0.1-0.1-0.1 0.1 0.1-0.1 Dist -0.1 0.1 0.1-0.1-0.1 0.1 CO 0.0 0.0 0.0 0.0 0.0 0.0 Dist 0.0 0.0 0.0 0.0 0.0 0.0 M, k-ft 331.7-807.6 71.9-71.9 807.6-331.7 Midspan M, 349.6 197.4 349.6 ft-kips 15

.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 1. 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 0.175 l 1 from the enters of supports. ACI 318-14 (8.11.6.1) 0 in. 0.83 ft 0.175 30 5.5 ft (use fae of supporting loation) 1 16

Figure 1 - Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Fatored Live Load) 17

.1.5. Fatored moments in slab-beam strip a. Chek whether the moments alulated above an take advantage of the redution permitted by ACI 318-14 (8.11.6.5): If the slab system analyzed using EFM within the limitations of ACI 318-14 (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 8.10.3. in the ACI 318-14. Chek Appliability of Diret Design Method: 1. There is a minimum of three ontinuous spans in eah diretion. ACI 318-14 (8.10..1). Suessive span lengths are equal. ACI 318-14 (8.10..) 3. Long-to-Short ratio is 30/30 = 1.0 <.0. ACI 318-14 (8.10..3) 4. Column are not offset. ACI 318-14 (8.10..4) 5. Loads are gravity and uniformly distributed with servie live-to-dead ratio of 0.41 <.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 318-14 (8.10..5 and 6) 6. Chek relative stiffness for slab panel. ACI 318-14 (8.10..7) Slab system is without beams and this requirement is not appliable. All limitation of ACI 318-14 (8.10.) are satisfied and the provisions of ACI 318-14 (8.11.6.5) may be applied: M o n qu (30 0 /1) 0.70 30 81.8 ft-kips ACI 318-14 (Eq. 8.10.3.) 8 8 331.7 807.6 End spans: 349.6 919.3 ft-kips 71.9 71.9 Interior span: 197.3 919.ft-kips To illustrate proper proedure, the interior span fatored moments may be redued as follows: Permissible redution = 81.8/919.3 = 0.884 Adjusted negative design moment = 71.9 0.884 = 638. ft-kips Adjusted positive design moment = 197.3 0.884 = 174.4 ft-kips M o 638. 638. 174.4 81.8 ft-kips 18

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 1), the moment values obtained from EFM will be used for omparison reasons. 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 318-14 (8.11.6.6) 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 46.5 100 46.5 0 0.0 Positive 349.6 60 09.8 40 139.8 Interior Negative 695.9 75 51.9 5 174.0 Negative 63.4 75 467.6 5 155.9 Positive 197.3 60 118.4 40 78.9.1.6. Flexural reinforement requirements a. Determine flexural reinforement required for strip moments The flexural reinforement alulation for the olumn strip of end span exterior negative loation is provided below. M 46.5 ft-kips u Use d avg = 1.75 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 1.11 in. Column strip width, b 301 / 180 in. Middle strip width, b 301 180 180 in. Mu 46.51000 As 4.5 in. f jd 0.960000 1.11 y Af s y 5.09 60000 Realulate ' a' for the atual As 4.5 in. a 0.355 in. 0.85 f ' b 0.855000 180 19

a 0.355 0.417 in. 0.85 1 0.003 0.003 t ( ) dt 0.003 ( ) 1.75 0.003 0.089 0.005 0.417 Therefore, the assumption that setion is tension-ontrolled is valid. Mu 46.51000 As 4.357 in. f ( d a / ) 0.9 60000 (1.75 0.355 / ) y The slab have two thiknesses in the olumn strip (14.5 in. for the slab with the drop panel and 10 in. for the slab without the drop panel). The weighted slab thikness: h w 30 / 3 30 / 30 / 3 14.5 30 / 3 10 30 / 30 / 3 1.83 in. As,min 0.0018 1801.83 4.157 in. < 4.357 in. ACI 318-14 (4.4.3.) smax h w 1.83 5.66 in. > 18 in. ACI 318-14 (8.7..) s 18 in. max Provide 10 - #6 bars with A s = 4.40 in. and s = 180/10 = 18 in. s max Based on the proedure 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 46.5 180 1.75 4.357 4.157 10-#6 4.40 Positive 09.8 180 8.50 5.631 3.40 13-#6 5.7 Interior Negative 51.9 180 1.75 9.366 4.157 -#6 9.68 Exterior Negative 0.0 180 8.50 0.0 3.40 10-#6 * ** 4.40 Positive 139.8 180 8.50 3.719 3.40 10-#6 ** 4.40 Interior Negative 174.0 180 8.50 4.649 3.40 11-#6 4.84 Interior Span Column Strip Positive 118.4 180 8.50 3.141 3.40 10-#6 * ** 4.40 Middle Strip Positive 78.9 180 8.50.083 3.40 10-#6 * ** 4.40 * Design governed by minimum reinforement. ** Number of bars governed by maximum allowable spaing. 0

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 x 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 318-14 (8.4..3) Portion of the unbalaned moment transferred by flexure is γ f x M s ACI 318-14 (8.4..3.1) Where 1 ACI 318-14 (8.4..3.) f 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 13). b = Dimension of the ritial setion b o measured in the diretion perpendiular to b 1 in ACI 318, Chapter 8 (see Figure 13). b b = Effetive slab width = 3h ACI 318-14 (8.4..3.3) Figure 13 Critial Shear Perimeters for Columns 1

For exterior support: d = h over d/ = 14.5 0.75 0.75/ = 13.13 in. b1 1 + d/ = 0 + 13.13/ = 6.56 in. b + d = 0 + 13.13 = 33.13 in. b 0 314.5 6.75 in. b 1 0.66 f 1 ( / 3) 6.56 / 33.13 0.66 331.7 07.7 ft-kips f M s Using the same proedure in.1.7.a, the required area of steel: A 3.63 in. s However, the area of steel provided to resist the flexural moment within the effetive slab width b b: 6.75 As, provided 4.40 1.534 in. 180 Then, the required additional reinforement at exterior olumn for moment transfer between slab and olumn: A, 3.63-1.534.096 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 331.7 0.66 07.7 6.75 13.13 3.63 1.534 5-#6 Interior Negative 85.7 0.60 51.4 6.75 13.13 0.877 3.375 - *M s is taken at the enterline of the support in Equivalent Frame Method solution. Add l Reinf..1.7. 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 1, the unbalaned moment at the exterior and interior joints are: Exterior Joint = +331.7 ft-kips Joint = -807.6 + 71.9 = -85.7 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.

Figure 14 - Column Moments (Unbalaned Moments from Slab-Beam) In summary: For Top olumn (Above): For Bottom olumn (Below): M ol,exterior= 150.61 ft-kips M ol,exterior= 157.1 ft-kips M ol,interior = 38.91 ft-kips M ol,interior = 40.6 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. 3

Mu kips-ft Table 5 Fatored Moments in Columns Column Loation Interior Exterior Corner Mux 40.6 157.1 157.1 Muy 40.6 40.6 157.1 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. 3.1. Determination of fatored loads Interior Column: Assume 5 story building Tributary area for interior olumn for live load, superimposed dead load, and self-weight of the slab is A 3030 900 ft Tributary Tributary area for interior olumn for self-weight of additional slab thikness due to the presene of the drop panel is A 1010 100 ft Tributary Assuming five story building Pu nqu ATributary 5 0.70 900 0.0638 100 147 kips M u,x = 40.6 ft-kips (see the previous Table) M u,y = 40.6 ft-kips (see the previous Table) Edge (Exterior) Column: Tributary area for edge olumn for live load, superimposed dead load, and self-weight of the slab is A Tributary 30 0 / 30 475 ft 1 Tributary area for edge olumn for self-weight of additional slab thikness due to the presene of the drop panel is A Tributary 10 0 / 10 58.33 ft 1 Pu nqu ATributary 5 0.70 475 0.0638 58.33 660 kips M u,x = 157.1 ft-kips (see the previous Table) 4

M u,y = 40.6 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 A Tributary 30 0 / 30 0 / 51ft 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 10 0 / 10 0 / 34.03 ft 1 1 Pu nqu ATributary 5 0.70 51 0.0638 34.03 350 kips M u,x = 157.1 ft-kips (see the previous Table) M u,y = 157.1 ft-kips (see the previous Table) 5

3.. Moment Interation Diagram Interior Column: 6

Edge Column: 7

Corner Column: 8

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. 4.1. One-Way (Beam ation) Shear Strength ACI 318-14 (.5) One-way shear is ritial at a distane d from the fae of the olumn as shown in Figure 3. Figures 15 and 16 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 318-14 (Eq..5.1.1) n s s Where: φv φλ f ' b d ACI 318-14 (Eq..5.5.1) w Note: The alulations below follow one of two possible approahes for heking one-way shear. Refer to the onlusions setion for a omparison with the other approah. 4.1.1. At distane d from the supporting olumn h weighted 14.5 30 / 6 10 30 30 / 6 11.4 in. 30 d 11.4 0.75 0.75/ 10.9 in. w 1 for normal weight onrete 5000 V 0.75 1.0 (301) 10.9 39.91 kips 1000 Beause V V at all the ritial setions, the slab has adequate one-way shear strength. u Figure 15 One-way shear at ritial setions (at distane d from the fae of the supporting olumn) 9

4.1.. At the fae of the drop panel h 10 in. d 10 0.75 0.75/ 8.88 in. 1 for normal weight onrete 5000 V 0.75 1.0 (301) 8.88 339.1 kips 1000 Beause V V at all the ritial setions, the slab has adequate one-way shear strength. u Figure 16 One-way shear at ritial setions (at the fae of the drop panel) 30

4.. Two-Way (Punhing) Shear Strength ACI 318-14 (.6) 4..1. 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 13. 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). 6.56 33.13 Vu V q b b 105.64 0.334 103.60 kips u 1 144 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. 6.56 8.18 0 / Munb M V b / 331.7 103.60 59 ft-kips u 1 AB 1 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 (6.56 13.136.56 / ) 8.18 in. area of the sides 6.56 13.13 33.1313.13 The polar moment J of the shear perimeter is: 3 3 b d db b 1 1 1 J b d b 1 d 1 1 AB J AB 3 3 6.56 13.13 13.136.56 6.56 6.56 13.13 8.18 33.1313.13 8.18 1 1 J 98,315 in. γ v 4 1 γ 1 0.66 0.374 ACI 318-14 (Eq. 8.4.4..) f The length of the ritial perimeter for the exterior olumn: b 6.56 33.13 86.6 in. o The two-way shear stress (v u) an then be alulated as: 31

v u V γ M b d J o u v unb AB ACI 318-14 (R.8.4.4..3) v u 103.60 1000 0.374 (5911000) 8.18 91.47 96.71 188.3 psi 86.6 13.13 98, 315 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v 4 3013.13 min 41 5000, 1 5000, 1 5000 1 86.6 v min 8.8, 44.3, 464.3 psi 8.8 psi φ v 0.75 8.8 1.1 psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength at this joint. b. Interior olumn: 33.13 33.13 Vu V q b b 137.36 11.5 0.334 56.35 kips u 1 144 M M V b / 807.6 71.9 56.31 0 85.70 ft-kips unb u 1 AB 1 For the interior olumn in Figure 13, the loation of the entroidal axis z-z is: b 1 33.13 AB 16.57in. The polar moment J of the shear perimeter is: 3 3 b d db b 1 1 1 J b d b 1 d 1 1 AB J AB 3 3 33.1313.13 13.1333.13 33.13 33.1313.13 16.57 33.1313.13 16.57 1 1 J 330,800 in. γ v 4 1 γ 1 0.600 0.400 ACI 318-14 (Eq. 8.4.4..) f The length of the ritial perimeter for the interior olumn: 3

b 33.13 33.13 13.5 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 318-14 (R.8.4.4..3) v u 56.311000 0.400 (85.70 11000) 16.57 147.3 0.6 167.9 psi 13.5 13.13 330,800 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v v 4 4013.13 min 41 5000, 1 5000, 1 5000 1 13.5 min 8.8, 44.3, 41.7 psi 8.8 psi φ v 0.75 8.8 1.1psi 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 eduational purposes. Same proedure is used to find the reation and fatored unbalaned moment used for shear transfer at the entroid of the ritial setion for the orner support for the exterior equivalent frame strip. 6.56 6.56 Vu V qu b1 b 61.93 0.334 60.9 kips 144 6.56 6.64 0 / Munb M Vu b1 AB 1 / 187.51 60.9 137.65 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 (6.56 13.136.56 / ) 6.64 in. in. area of the sides 6.56 13.13 6.5613.13 The polar moment J of the shear perimeter is: J b d b d 1 1 3 3 b1 d db1 b1 1 AB AB 33

J 3 3 6.56 13.13 13.136.56 6.56 6.56 13.13 6.64 6.56 13.13 6.64 1 1 J 56,9 in. 4 γ v 1 γ 1 0.600 0.400 ACI 318-14 (Eq. 8.4.4..) f The length of the ritial perimeter for the orner olumn: b 6.56 6.56 53.13 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 318-14 (R.8.4.4..3) v u 60.9 1000 0.400 (137.65 11000) 6.64 86.4 77.9 164.40 psi 53.1313.13 56, 9 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v v 4 013.13 min 41 5000, 1 5000, 1 5000 1 53.13 min 8.8, 44.3, 490.9 psi 8.8 psi φ v 0.75 8.8 1.1 psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength at this joint. 4... Around drop panels Two-way shear is ritial on a retangular setion loated at d/ away from the fae of the drop panel. Note: The two-way shear stress alulations around drop panels do not have the term for unbalaned moment sine drop panels are a thikened portion of the slab and are not onsidered as a support. a. Exterior drop panel: 74.44 18.88 Vu V q A 105.64 0.70 87.65 kips u 144 The length of the ritial perimeter for the exterior drop panel: b 74.44 18.88 77.76 in. o 34

The two-way shear stress (v u) an then be alulated as: v u V u b d o ACI 318-14 (R.8.4.4..3) v u 87.651000 35.5 psi 77.76 8.88 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v v 4 308.88 min 41 5000, 1 5000, 1 5000 1 77.76 min 8.8, 44.3, 09. psi 09. psi φ v 0.75 09. 156.9 psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength around this drop panel. b. Interior drop panel: V V q A u V u u 18.88 18.88 137.36 11.5 0.70 7.7 kips 144 The length of the ritial perimeter for the interior drop panel: b 18.88 18.88 515.5 in. o The two-way shear stress (v u) an then be alulated as: v u V u b d o ACI 318-14 (R.8.4.4..3) v u 7.7 1000 49.7 psi 515.5 8.88 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v v 4 408.88 min 41 5000, 1 5000, 1 5000 1 515.5 min 8.8, 44.3, 190.1 psi 190.1 psi 35

φ v 0.75190.1 14.6 psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength around this drop panel.. Corner drop panel: V V q A u V u u 74.44 74.44 61.93 0.70 51.54 kips 144 The length of the ritial perimeter for the orner drop panel: b 74.44 74.44 148.88 in. o The two-way shear stress (v u) an then be alulated as: v u V u b d o ACI 318-14 (R.8.4.4..3) v u 51.541000 38.98 psi 148.88 8.88 ' 4 ' αd s ' v min 4 λ f, λ f, λ f β bo ACI 318-14 (Table.6.5.) v v 4 08.88 min 41 5000, 1 5000, 1 5000 1 148.88 min 8.8, 44.3, 5.8 psi 5.8 psi φ v 0.75 5.8 169.3 psi Sine φ v v u at the ritial setion, the slab has adequate two-way shear strength around this drop panel. 36

5. Servieability Requirements (Defletion Chek) Sine the slab thikness was seleted below the minimum slab thikness tables in ACI 318-14, the defletion alulations of immediate and time-dependent defletions are required and shown below inluding a omparison with spslab model results. 5.1. 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 318-14 (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 318-14 (Eq. 4..3.5a) 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 17. 37

Figure 17 Maximum Moments for the Three Servie Load Levels (No live load is sustained in this example) For positive moment (midspan) setion: M Craking moment. r M f r r fi r g 530.33 30000 1 65.17 ft-kips y 5 11000 t = Modulus of rapture of onrete. ACI 318-14 (Eq. 4..3.5b) f r I g 7.5 f ' 7.51.0 5000 530.33 psi ACI 318-14 (Eq. 19..3.1) = Moment of inertia of the gross unraked onrete setion. 38

I g 3 3 lh 301 10 30000 in. 1 1 y t = Distane from entroidal axis of gross setion, negleting reinforement, to tension fae, in. h 10 y 5 in. t I = Moment of inertia of the raked setion transformed to onrete. PCA Notes on ACI 318-11 (9.5..) r As alulated previously, the positive reinforement for the end span frame strip is 3 #6 bars loated at 1.15 in. along the setion from the bottom of the slab. Two of these bars are not ontinuous and will be onservatively exluded from the alulation of I r sine they might not be adequately developed or tied (1 bars are used). 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. 1.5 1.5 3 s E w 33 f 150 33 5000 487 10 psi ACI 318-14 (19...1.a) E s 9000000 n 6.76 PCA Notes on ACI 318-11 (Table 10-) E 487000 s b 301 B 5.76 in. na 6.76 10.44 s 1 PCA Notes on ACI 318-11 (Table 10-) db 11 8.88 5.76 11 kd 1.59 in. PCA Notes on ACI 318-11 (Table 10-) B 5.76 b( kd) I na d kd 3 3 r s ( ) PCA Notes on ACI 318-11 (Table 10-) I r 3 301 (1.59) 6.76 1 0.44 8.88 1.59 380 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 3 #6 bars loated at 1.15 in. along the setion from the top of the slab. 39

M r fi r g 530.33 53445 1 401.4 ft-kips y 5.88 11000 t ACI 318-14 (Eq. 4..3.5b) f r 7.5 f ' 7.51.0 5000 530.33 psi ACI 318-14 (Eq. 19..3.1) I 53445 in. g y 5.88 in. t Figure 19 I g alulations for slab setion near support 1.5 1.5 3 s E w 33 f 150 33 5000 487 10 psi ACI 318-14 (19...1.a) E s 9000000 n 6.76 PCA Notes on ACI 318-11 (Table 10-) E 487000 s b b 101 B 1.6 in. na 6.76 30.44 s 1 PCA Notes on ACI 318-11 (Table 10-) db 11 13.131.6 11 kd 3.84 in. PCA Notes on ACI 318-11 (Table 10-) B 1.6 3 b ( kd) b I na ( d kd) PCA Notes on ACI 318-11 (Table 10-) r 3 s I r 3 101 (3.84) 6.76 3 0.44 13.13 3.84 10471 in. 3 4 40

Figure 0 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 318-14 (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 318-14 (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 401.4 ft-kips < M a =546.5 ft-kips e M g M r a a ACI 318-14 (4..3.5a) Where I e - is the effetive moment of inertia for the ritial negative moment setion (near the support). 3 3 401.4 401.4 I e 53445 1 10471 7503 in. 546.5 546.5 I I M M e g 4 30000 in., sine r 65.17 ft-kips > a =15.03 ft-kips 4 Where I e + is the effetive moment of inertia for the ritial positive moment setion (midspan). 41

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. Both the midspan stiffness (I + e ) and averaged span stiffness (I e,avg) an be used in the alulation of immediate (instantaneous) defletion. 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 PCA Notes on ACI 318-11 (9.5..4()) I, 0.85 I 0.15 I for end span PCA Notes on ACI 318-11 (9.5..4(1)) e avg e e However, these expressions lead to improved results only for ontinuous prismati members. The drop panels in this example result in non-prismati members and the following expressions should be used aording to ACI 318-89: e, avg e e, l e, r I 0.50 I 0.5 I I for interior span ACI 435R-95 (.14) For the middle span (span with two ends ontinuous) with servie load level (D+LL full): I, 0.50 30000 0.5 7503 7503 4173 in. e avg 4 I, 0.50 I 0.50 I for end span ACI 435R-95 (.14) e avg e e For the end span (span with one end ontinuous) with servie load level (D+LL full): Ie, avg 0.50 6503 0.50 649 37190 in. 4 Where: I el, I el, = The effetive moment of inertia for the ritial negative moment setion near the left support. = The effetive moment of inertia for the ritial negative moment setion near the right support. I e = The effetive moment of inertia for the ritial positive moment setion (midspan). 4

Table 6 provides a summary of the required parameters and alulated values needed for defletions for exterior and interior spans. Table 6 Averaged Effetive Moment of Inertia Calulations For Frame Strip Span zone I g, in. 4 I r, in. 4 D M a, kips-ft D + LL Sus D + L full M r, k-ft D I e, in. 4 I e,avg, in. 4 D + LL Sus D + L full D D + LL Sus D + L full Ext Int Left 53445 7170-179.7-179.7-5.8 401.4 53445 53445 53445 Midspan 30000 3797 197.3 197.3 78.14 65.17 30000 30000 6503 Right 53445 10471-434.41-434.41-611.14 401.4 44379 44379 649 Left 53445 10471-388.46-388.46-546.48 401.4 53445 53445 7503 Mid 30000 3317 107.56 107.56 15.03 65.17 30000 30000 30000 Right 53445 10471-388.46-388.46-546.48 401.4 53445 53445 7503 37190 37190 4576 4173 4173 875 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 in 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 318-11 (9.5.3.4 Eq. 8) 43

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. Figure Δ x alulation proedure 44

For end span - servie dead load ase: 4 wl frame, fixed PCA Notes on ACI 318-11 (9.5.3.4 Eq. 10) 384EI frame, averaged Where: frame, fixed = Defletion of olumn strip assuming fixed end ondition. w (0 15010 / 1)(30) 4350 lb/ft E w 33 f 487 10 psi ACI 318-14 (19...1.a) 1.5 3 I frame,averaged = The averaged effetive moment of inertia (I e,avg) for the frame strip for servie dead load ase from Table 6 = 37190 in. 4 4 3 (4350)(30) (1) frame, fixed 0.0995 in. 3 384(487 10 )(37190) I frame, averaged, fixed LDF frame, fixed PCA Notes on ACI 318-11 (9.5.3.4 Eq. 11) Ig, For this example and like in the spslab program, the effetive moment of inertia at midspan will be used. 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, 0.75, and 0.60, respetively (From Table of this doument). Thus, the load distribution fator for the olumn strip for the end span is given by: LDF 1.0 0.75 0.6 0.738 I,g = The gross moment of inertia (I g) for the olumn strip for servie dead load = 15000 in. 4 45

30000, fixed 0.738 0.0995 0.1468 in. 15000 ( M ) net, L frame L, PCA Notes on ACI 318-11 (9.5.3.4 Eq. 1) K Where: e L, = Rotation of the span left support. ( M ) 179.7 ft-kips = Net frame strip negative moment of the left support. net, L frame K e = effetive olumn stiffness = 6 135310 in.-lb (alulated previously). L, 179.7 11000 0.00159 rad 6 135310 I l PCA Notes on ACI 318-11 (9.5.3.4 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. 301 30000 L, 0.00159 0.05786 in. 8 37190 R, Where ( M ) net, R frame (611.14 546.48) 11000 0.00041 rad K 135310 6 e R, = Rotation of the end span right support. ( M ) Net frame strip negative moment of the right support. net, R frame I l g 301 30000, R, R 0.00041 0.01479 in. 8 I 8 37190 e frame Where: R, = Midspan delfetion due to rotation of right support. 46

x x, fixed x, R x, L PCA Notes on ACI 318-11 (9.5.3.4 Eq. 9) x 0.1468 0.01479 0.05786 0.19 in. 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). Sine in this example the panel is squared, Δ x = Δ y= 0.19 in. and Δ mx = Δ my= 0.15 in. The average Δ for the orner panel is alulated as follows: ( ) ( ) x my y mx x my y mx 0.19 0.15 0.344 in. 47

Table 7 Immediate (Instantaneous) Defletions in the x-diretion Column Strip Middle Strip Span LDF Δ frame-fixed, in. Δ -fixed, in. θ 1, rad D θ, rad Δθ 1, in. Δθ, in. Δ x, in. LDF Δ frame-fixed, in. Ext 0.738 0.0995 0.1468 0.00159 0.00041 0.05786 0.01479 0.19 0.6 0.0995 0.051 0.00159 0.00041 0.05786 0.01479 0.15 Int 0.675 0.0886 0.1197 0.00041 0.00041-0.01319-0.01319 0.093 0.35 0.0886 0.0576 0.00041 0.00041-0.01319-0.01319 0.031 Δ m-fixed, in. θ m1, rad D θ m, rad Δθ m1, in. Δθ m, in. Δ mx, in. Span LDF Δ frame-fixed, in. Δ -fixed, in. θ 1, rad D+LL sus θ, rad Δθ 1, in. Δθ, in. Δ x, in. LDF Δ frame-fixed, in. Ext 0.738 0.0995 0.1468 0.00159 0.00041 0.05786 0.01479 0.19 0.6 0.0995 0.051 0.00159 0.00041 0.05786 0.01479 0.15 Int 0.675 0.0886 0.1197 0.00041 0.00041-0.01319-0.01319 0.093 0.35 0.0886 0.0576 0.00041 0.00041-0.01319-0.01319 0.031 Δ m-fixed, in. θ m1, rad D+LL sus θ m, rad Δθ m1, in. Δθ m, in. Δ mx, in. Span LDF Δ frame-fixed, in. Δ -fixed, in. θ 1, rad D+LL full θ, rad Δθ 1, in. Δθ, in. Δ x, in. LDF Δ frame-fixed, in. Ext 0.738 0.18 0.775 0.004 0.00057 0.1316 0.0315 0.43 0.6 0.18 0.0985 0.004 0.00057 0.1316 0.0315 0.53 Int 0.675 0.1819 0.455 0.00057 0.00057-0.0693-0.0693 0.19 0.35 0.1819 0.118 0.00057 0.00057-0.0693-0.0693 0.064 Δ m-fixed, in. θ m1, rad D+LL full θ m, rad Δθ m1, in. Δθ m, in. Δ mx, in. Span LDF LL Δ x, in. Ext 0.738 0.13 0.6 0.18 Int 0.675 0.098 0.35 0.033 LDF LL Δ mx, in. 48

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 318-11 (9.5..5 Eq. 4) s sust Inst The total time-dependent (long-term) defletion is alulated as: ( ) ( ) (1 ) [( ) ( ) ] CSA A3.3-04 (N9.8..5) total lt Where: sust Inst total Inst sust Inst ( ) Immediate (instantaneous) defletion due to sustained load, in. sust Inst 1 50 ' ACI 318-14 (4..4.1.1) ( ) 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 318-14 (Table 4..4.1.3) ' = 0, onservatively. 1500 0.19 0.439 in. s 0.19 1 0.43 0.19 0.871 in. 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. Table 8 - Long-Term Defletions Column Strip Span (Δsust)Inst, in. λδ Δs, in. (Δtotal)Inst, in. (Δtotal)lt, in. Exterior 0.19.000 0.439 0.43 0.871 Interior 0.093.000 0.187 0.19 0.378 Middle Strip Exterior 0.15.000 0.50 0.53 0.503 Interior 0.031.000 0.06 0.064 0.17 49

6. spslab Software Program Model Solution spslab program utilizes the Equivalent Frame Method desribed and illustrated in details here for modeling, analysis and design of two-way onrete floor slab systems with drop panels. 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 (ACI 318-14 (R8.11.4)). spslab Program models the equivalent 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 are provided below for both input and output of the spslab model. 50

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7. Summary and Comparison of Design Results Table 9 - Comparison of Moments obtained from Hand (EFM) and spslab Solution (ft-kips) Hand (EFM) spslab Exterior Span Exterior Negative * 46.5 44.8 Column Strip Positive 09.8 19.7 Interior Negative * 51.9 517.6 Exterior Negative * 0.0 0.0 Middle Strip Positive 139.8 146.5 Interior Negative * 174.0 17.5 Interior Span Column Strip Interior Negative * 467.6 463.6 Positive 118.4 10.1 Middle Strip Interior Negative * 155.9 154.5 Positive 78.9 80.1 * negative moments are taken at the faes of supports Table 10 - Comparison of Reinforement Results Span Loation Reinforement Provided for Flexure Additional Reinforement Provided for Unbalaned Moment Transfer* Total Reinforement Provided Column Strip Middle Strip Column Strip Middle Strip Exterior Negative Hand spslab Hand spslab Hand spslab Exterior Span 10-#6 10-#6 5-#6 5-#6 15-#6 15-#6 Positive 13-#6 13-#6 n/a n/a 13-#6 13-#6 Interior Negative Exterior Negative -#6 1-#6 --- --- -#6 1-#6 10-#6 10-#6 n/a n/a 10-#6 10-#6 Positive 10-#6 10-#6 n/a n/a 10-#6 10-#6 Interior Negative 11-#6 11-#6 n/a n/a 11-#6 11-#6 Interior Span Positive 10-#6 10-#6 n/a n/a 10-#6 10-#6 Positive 10-#6 10-#6 n/a n/a 10-#6 10-#6 71

Table 11 - Comparison of One-Way (Beam Ation) Shear Chek Results Span Vu @ d, kips Vu @ drop panel, kips φv @ d, kips φv @ drop panel, kips Hand spslab Hand spslab Hand spslab Hand spslab Exterior 13.7 16.7 96.9 96.7 39.91 39.97 339.10 338.88 Interior 107.8 110.9 81.0 81.0 39.91 39.97 339.10 338.88 * x u alulated from the enterline of the left olumn for eah span Table 1 - Comparison of Two-Way (Punhing) Shear Chek Results (around Columns Faes) Support b1, in. b, in. bo, in. Vu, kips AB, in. Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior 6.56 6.56 33.13 33.13 86.6 86.5 103.6 114.6 8.18 8.18 Interior 33.13 33.13 33.13 33.13 13.5 13.50 56.4 63.0 16.57 16.56 Corner 6.56 6.56 6.56 6.56 53.13 53.1 60.3 60.6 6.64 6.64 Support J, in. 4 γv Munb, ft-kips vu, psi φv, psi Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior 98,315 98,39 0.374 0.374 59 49.5 188.3 194.4 1.1 1.1 Interior 330,800 330,50 0.400 0.400 85.70 85.07 167.9 171.7 1.1 1.1 Corner 56,9 56,49 0.400 0.400 137.65 137.40 164.4 164.8 1.1 1.1 Table 13 - Comparison of Two-Way (Punhing) Shear Chek Results (around Drop Panels) Support b1, in. b, in. bo, in. Vu, kips AB, in. Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior 74.44 74.44 18.88 18.88 77.76 77.75 87.65 98.4 19.95 19.95 Interior 18.88 18.88 18.88 18.88 515.5 515.5 5.5 33.91 64.44 64.44 Corner 74.44 74.44 74.44 74.44 148.88 148.87 51.54 51.53 18.61 18.61 Support J, in. 4 γv Munb, ft-kips vu, psi φv, psi Hand spslab Hand spslab Hand spslab Hand spslab Hand spslab Exterior 1,468,983 1,468,000 N.A. N.A. N.A. N.A. 35.5 39.9 156.9 156.9 Interior 1,688,007 1,679,000 N.A. N.A. N.A. N.A. 49.7 51.1 14.6 14.6 Corner 767,460 766,930 N.A. N.A. N.A. N.A. 38.98 39.00 169.3 169.3 Note: Shear stresses from spslab are higher than hand alulations sine it onsiders the load effets beyond the olumn enterline known in the model as right/left antilevers. This small inrease is often negleted in simplified hand alulations like the one used here. 7

Span Table 14 - Comparison of Immediate Defletion Results (in.) Column Strip D D+LLsus D+LLfull LL Hand spslab Hand spslab Hand spslab Hand spslab Exterior 0.19 0.07 0.19 0.07 0.43 0.395 0.13 0.188 Interior 0.093 0.089 0.093 0.089 0.19 0.185 0.098 0.096 Span Middle Strip D D+LLsus D+LLfull LL Hand spslab Hand spslab Hand spslab Hand spslab Exterior 0.15 0.10 0.15 0.10 0.53 0.18 0.18 0.098 Interior 0.031 0.030 0.031 0.030 0.064 0.071 0.033 0.040 Table 15 - Comparison of Time-Dependent Defletion Results Column Strip Span λδ Δs, in. Δtotal, in. Hand spslab Hand spslab Hand spslab Exterior.0.0 0.439 0.414 0.871 0.808 Interior.0.0 0.187 0.178 0.378 0.363 Middle Strip Span λδ Δs, in. Δtotal, in. Hand spslab Hand spslab Hand spslab Exterior.0.0 0.50 0.41 0.503 0.459 Interior.0.0 0.06 0.060 0.17 0.131 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. 73

8. Conlusions & Observations 8.1. One-Way Shear Distribution to Slab Strips In one-way shear heks above, shear is distributed uniformly along the width of the design strip (30 ft.). StruturePoint finds it neessary sometimes to alloate the one-way shears with the same proportion moments are distributed to olumn and middle strips. spslab allows the one-way shear hek using two approahes: 1) alulating the one-way shear apaity using the average slab thikness and omparing it with the total fatored one-shear load as shown in the hand alulations above; ) distributing the fatored one-way shear fores to the olumn and middle strips and omparing it with the shear apaity of eah strip as illustrated in the following figures. An engineering judgment is needed to deide whih approah to be used. Figure 3a Distributing Shear to Column and Middle Strips (spslab Input) 74