In dit artikel, we will develop a Slab Design Example using the last version of ACI-318-19: “Bouwvereisten voor constructief beton,” consisting of the modeling in SkyCiv of a Reinforced Concrete Low-Rise Building focusing on the comparison of software results and hand calculations by an accepted method by ACI: “De directe ontwerpmethode voor platen.” This procedure consists in assigning into different strips along the main directions and frames of the building the total moment by convenient factors to determine the quantity of reinforcement and the location in the slab.

We hope you have read the previous article, Plate Design in S3D, to introduce yourself to modeling and designing plates using SkyCiv. Another helpful piece of information we suggest you consider is found in How to model plates? Once you complete reading both docs, feel free to dive into the following full-worked slab comparison example!

General Building Layout

De volgende afbeeldingen tonen een isometrisch aanzicht en vlakafmetingen van het te berekenen voorbeeld. Het gebouw heeft twee verhoogde vlakke platen zonder balken tussen kolomsteunen.

Figuur 1. Isometrische weergave van het voorbeeld van een gebouw

Figuur 2. Afmetingen plaat

Directe ontwerpmethode voor tweerichtingsplaten (DDM)

Beperkingen

ACI 318 maakt het gebruik van de DDM mogelijk om platen van gewapend beton te ontwerpen voor zwaartekrachtbelastingen, die een aantal benodigdheden verzamelen volgens de geometrie, belastingsrelaties, symmetrie, enzovoort. We can summarize these limitations in the following list (PCA Notes):

• “Er moeten drie of meer ononderbroken overspanningen in elke richting zijn.”: Figuur 2 shows three spans in each main direction, longitudinal and transversal. OK!
• “Plaatpanelen moeten rechthoekig zijn met een verhouding van langere tot kortere overspanning (hartlijn-naar-hartlijn van steunen) not greater than 2.”: According to figure 2, the ratio is equal to \({\frac{l_1}{4}= frac{6m}{4m}=1.5 < 2}\). OK!
• “Opeenvolgende overspanningen (hartlijn-naar-hartlijn van steunen) in elke richting mag niet meer dan . verschillen 1/3 van de langere spanwijdte”. Span lengths are the same in each direction, 6m to longitudinal and 4m to transversal. OK!
• “Kolommen mogen niet meer dan . worden verschoven 10% van de overspanning (in de richting van offset) van beide assen tussen hartlijnen van opeenvolgende kolommen”. The building example doesn’t have offsets in columns. OK!
• “Belastingen moeten gelijkmatig worden verdeeld, met de niet-factored of service live load niet meer dan twee keer de niet-factored of service dead load (L/D 2)”. Taking the values of each gravity load, the ratio is defined as \({\frac{L}{D}= frac{2}{7.8}=0.256 < 2}\). OK!.
• “Voor in twee richtingen ondersteunde platen, relatieve stijfheid van liggers in twee loodrechte richtingen moet voldoen aan de minimum- en maximumvereisten die in de code worden vermeld.” Already satisfied; there are no beams in the slabs. OK!
• “Herverdeling van negatieve momenten per code is niet toegestaan.” Due to the simplicity of the example, it won’t be necessary to redistribute negative moments in the slabs. OK!.

Longitudinal and transverse strips definition

De plaat in DDM moet worden verdeeld in twee hoofdstroken voor de analyse en het ontwerp van een bepaald lijnraster: kolom en middelste stroken. The width for column strips is the lesser of \({\frac {l_1}{4}}\) en \({\frac{l_2}{4}}\), waar \({l_1}\) is the length of the span along the line grid and \({l_2}\) is the transverse length perpendicular.

Figuur 3. Longitudinal column and middle strips.

Figuur 4. Transverse column and middle strips.

Minimum thickness

ACI-318 suggests using the equation: \({t_{min}}= {\frac{l_n}{30}}={\frac{6m-0.50m}{30}}=0.1833m = 0.20m\)

Preliminary shear strength check

Before calculating the steel rebar reinforcement, it is recommended to check the shear capacity of the slab, one for direct shear in the connection and the other for the punching shear capacity on the connection slab column.

To calculate the shear demand, we use the following gravity loads:

• Self-weight slab: \({SW={\gamma_c}\keer {t_{plaat}}={24 {\frac{kN}{m^3}}}\keer {0.20m}=4.8{\frac{kN}{m^2}} }\)
• Bovenliggende dode belasting: \({SD={3 {\frac{kN}{m^2}}}}\)
• Total dead load (SW+SD): \({D={7.8 {\frac{kN}{m^2}}}}\)
• Live laden (Residential occupancy) : \({L={2 {\frac{kN}{m^2}}}}\)
• Factored strength load (1.2D+1.6L): \({q_{u}={12.56 {\frac{kN}{m^2}}}}\)

The first shear check is the “beam-shear” type, where the following image indicates the area to be considered to obtain the total shear. We inspect each direction, taking the more extensive area.

Figuur 5. Beam shear at interior column (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

Waar:

• Length span in longitudinal direction, \({l_1 = 6.0m }\)
• Length span in transverse direction, \({l_2 = 4.0m}\)
• Total tributary area, shear in longitudinal direction \({A_t = l_2 \times (\frac{l_1}{2}-\frac{c_1}{2}-d) = 4.0m \times (\frac{6.0m}{2}-\frac{0.50m}{2}-0.17m) = 10.32 m^2}\) (geselecteerd)
• Total tributary area, shear in transverse direction, \({A_t = l_1 \times (\frac{l_2}{2}-\frac{c_2}{2}-d) = 6.0m \times (\frac{4.0m}{2}-\frac{0.50m}{2}-0.17m) = 9.48 m^2}\)
• Square columns dimension, \({c_1 = c_2 = 0.50m}\)
• Distance d, \({d = h_{plaat} – cover = 0.20m – 0.03m = 0.17m }\)

Daarom, the maximum beam shear in the interior column is

\({V_u =q_u\times A_t =12.56 {\frac{kN}{m^2}}\keer 10.32 m^2 = 129.62 kN }\)

This is going to be compared against the shear resistance, \({\phi_sV_c}\)

• Betonsterkte, \({f’_c = 25 MPa}\)
• Yield rebar steel strength, \({f_y = 420 MPa}\)
• \({\phi_s = 0.75}\)
• \({\phi_sV_c = 0.17\phi_s \lambda \sqrt(f'_c) b_w d; b_w=l_2}\)

\({\phi_sV_c = 0.17\times 0.75\times 1\times \sqrt(25 MPa) \keer 4000 mm\times 170 mm = 433.50 kN }\)

We can see that the shear resistance is greater than the shear demand: \({\phi_sV_c = 433.50 kN > V_u = 129.62 kN }\) OK!.

According to the following images, we have to calculate the punching shear capacity and the force to be resisted by concrete in the interior slab-column connection. The code’s intention in checking punching shear is to maintain low shear stress values.

Figuur 6. Two-way shear at the interior column (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

• Total tributary area, ponsschaar, \({A_t = l_1 \times l_2 – (c_1+d)^2 = 6.0m \times 4.0m – (0.50m+0.17m)➔⡔ Koop generieke tadalafil 23.55 m^2}\) (same area for both main directions)

The total shear force to be resisted is

\({V_u =q_u\times A_t =12.56 {\frac{kN}{m^2}}\keer 23.55 m^2 = 295.79 kN }\)

To obtain the punching shear capacity in a two-way slab, we will use the empiric method established by code ACI-318, which considers the maximum shear stress available in the effective perimeter at the critical section. The more conservative expression for the interior column is

• Punching shear capacity, \({\phi_sV_c = 0.33\phi_s \lambda \sqrt(f'_c) b_0 d; b_0=4\times (c_1+d)}\)

Daarom, we have the shear resistance of

\({\phi_sV_c = 0.33\times 0.75 \keer 1 \sqrt(25 MPa) \keer (4\keer (500 mm+170 mm)\times 170mm) = 563.81 kN }\)

We can see that the shear resistance is greater than the shear demand: \({\phi_sV_c = 563.81 kN > V_u = 295.75 kN }\) OK!.

We have verified the one and the two-way shear demands at the interior column connection. Due to both demands being less than their respective capacities or resistances, we will now move to calculate the main rebar reinforcement for the slab bending.

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Total factored static moment per span.

The maximum moment that can be developed into a double fixed-end beam is an isostatic moment equal to \({M = frac{wtijden {l_1}^ 2}{8}}\) (Zie figuur 6).

Figuur 7. Bending moment in a fixed-end beam. (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

ACI-18 neemt dit principe over en, voor de directe ontwerpmethode (DDM), establishes the maximum static moment to be considered per span \({M_0}\)

Lengterichting:

• \({M_0 = frac {q_u\times l_2\times {l_{n,1}}^ 2}{8}}\)
• \({M_0 = frac {12.56 {\frac{kN}{m^2}}\times 4.0m\times (6m-0.50m)^ 2}{8}=189.97 kN-m}\)

Transverse direction:

• \({M_0 = frac {q_u\times l_1\times {l_{n,2}}^ 2}{8}}\)
• \({M_0 = frac {12.56 {\frac{kN}{m^2}}\times 6.0m\times (4m-0.50m)^ 2}{8}=115.40 kN-m}\)

The next step is to assign this total moment considering the panel type, interior or exterior. (Zie figuur 7). Daarna, due to the spans being continuous, it is necessary to divide also the moment into positive and negative. This last is shown in images 8 en 9.

Figuur 8. Definition of panels according to their relative position in a slab plan. (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

 

Figuur 9. Distribution of moments in an interior panel. (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

 

It is crucial to know the correct distribution of moments depending on the slab we are designing. In dit voorbeeld, we have the last case in the following image (figuur 9), “No beams,” applied to a flat slab or solid slab without any beam, neither on the edge nor between supports.

The main difference in the five cases shown in figure 9 is the moment fractions to be assigned on exterior panels, in which the relative restraint at the end changes the values to be calculated.

Figuur 10. Distribution of total static moment into negative and positive span moments. (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

Distribution of the total factored moment \({M_0}\) per span into negative and positive moments.

Eenmaal \({M_0}\) has been calculated, it is time to assign the fraction of moments into positive and negative in each design strip, dat is, kolom en middelste stroken. For more clarity, figuur 10 helps specify the appropriate factor to consider in the distribution of the total moment.

Figuur 11. Width of the equivalent rigid frame and distribution of moments in flat slabs. (Nadim Hassoun en Akthem AI-manager, “Structurele betontheorie en ontwerp”)

Using the previous factors indicated in figure 10, we obtain in the following table the ultimate moment.

Lengterichting: \({M_0 = 189.97 kN-m}\)

Span (ES:Exterior, IS:Interior)	Total moment (kN-m)	Column strip moment (kN-m)	Middle strip moment (kN-m)
Exterior Negative ES	0.26M0=49.39	0.26M0=49.39	0
Positive ES	0.52M0=98.78	0.31M0=58.89	0.21M0=39.89
Interior Negative ES	0.70M0=132.98	0.53M0=100.68	0.17M0=32.29
Positive IS	0.35M0=66.49	0.21M0=39.89	0.14M0=26.60
Negative IS	0.65M0=123.48	0.49M0=93.09	0.16M0=30.40

With the moment once distributed, it is time to determine the steel rebar reinforcement to be placed in the slab. We will only develop one calculation and then all the results into a table.

Moment in the exterior negative span in the column strip, \({M_u = 49.39 kN-m}\)

• Veronderstelde spanning gecontroleerde sectie. \({\phi_f = 0.9}\)
• Column strip width, \({b=2.0m}\)
• Stalen versterkingsgebied, \({A_s = frac{M_u}{\phi_ftimes 0.9dtimes fy}= frac{49.39kN-m}{0.9\keer 0.9(0.17m)\keer 420 MPa}=853.996 {mm}^ 2}\)
• \({\de opwaartse bodemdruk veroorzaakt bidirectionele buiging met trekspanningen aan het bodemoppervlak{min} = 0.0018}\). Stalen minimale wapeningsoppervlakte, \({EEN_{s,min}=rho_{min}\maal bmaal d = 0.0018 \times 2.0m \times 0.17m =612 {mm}^ 2}\). Nu, controleer of de sectie zich gedraagt ​​als spanningsgestuurd.
• \({a = frac{A_stimes f_y}{0.85\maal f'cmaal b} = frac{853.996 {mm}^2keer 420 MPa}{0.85\keer 25 MPa\times 2.0m }= 8.439 mm}\)
• \({c = frac{een}{\bèta_1}= frac{8.439 mm}{0.85} = 9.929mm }\)
• \({\varepsilon_t = (\frac{0.003}{c})\keer d – 0.003 = (\frac{0.003}{9.929mm})\times 170mm – 0.003 = 0.048 > 0.005 }\) OK!, het is een door spanning gecontroleerd gedeelte!.

Span(ES:Exterior, IS:Interior)	Column Strip Moment (kN-m)	\({EEN_{s,calc} ({mm}^ 2)}\)	\({EEN_{s,min} ({mm}^ 2)}\)	\({een (mm)}\)	\({c (mm)}\)	\({\varepsilon_t > 0.005}\)
Exterior Negative ES	49.39	853.996	612.0	8.439	9.929	0.048 > 0.005!
Positive ES	58.89	1018.259	612.0	10.063	11.839	0.040 > 0.005!
Interior Negative ES	100.68	1740.844	612.0	17.204	20.24	0.022 > 0.005!
Positive IS	39.89	689.733	612.0	6.816	8.019	0.06 > 0.005!
Negative IS	93.09	1609.607	612.0	15.907	18.714	0.024 > 0.005!

Moment in the exterior positive span in the middle strip, \({M_u = 39.89 kN-m}\)

• Veronderstelde spanning gecontroleerde sectie. \({\phi_f = 0.9}\)
• Middle strip width, \({b=2.0m}\)
• Stalen versterkingsgebied, \({A_s = frac{M_u}{\phi_ftimes 0.9dtimes fy}= frac{39.89kN-m}{0.9\keer 0.9(0.17m)\keer 420 MPa}=689.733 {mm}^ 2}\)
• \({\de opwaartse bodemdruk veroorzaakt bidirectionele buiging met trekspanningen aan het bodemoppervlak{min} = 0.0018}\). Stalen minimale wapeningsoppervlakte, \({EEN_{s,min}=rho_{min}\maal bmaal d = 0.0018 \times 2.0m \times 0.17m =612 {mm}^ 2}\). Nu, controleer of de sectie zich gedraagt ​​als spanningsgestuurd.
• \({a = frac{A_stimes f_y}{0.85\maal f'cmaal b} = frac{689.766 {mm}^2keer 420 MPa}{0.85\keer 25 MPa\times 2.0m }= 6.816 mm}\)
• \({c = frac{een}{\bèta_1}= frac{6.816 mm}{0.85} = 8.019 mm }\)
• \({\varepsilon_t = (\frac{0.003}{c})\keer d – 0.003 = (\frac{0.003}{8.019mm})\times 170mm – 0.003 = 0.0605 > 0.005 }\) OK!, het is een door spanning gecontroleerd gedeelte!.

Span(ES:Exterior, IS:Interior)	Middle Strip Moment (kN-m)	\({EEN_{s,calc} ({mm}^ 2)}\)	\({EEN_{s,min} ({mm}^ 2)}\)	\({een (mm)}\)	\({c (mm)}\)	\({\varepsilon_t > 0.005}\)
Exterior Negative ES	0	0.00	612.0	6.048	7.115	0.069 > 0.005!
Positive ES	39.89	689.733	612.0	6.816	8.019	0.061 > 0.005!
Interior Negative ES	32.29	558.322	612.0	6.048	7.115	0.069 > 0.005!
Positive IS	26.60	459.937	612.0	6.048	7.115	0.069 > 0.005!
Negative IS	30.40	525.642	612.0	6.048	7.115	0.069 > 0.005!

Transverse direction: \({M_0 = 115.40 kN-m}\)

Span (ES:Exterior, IS:Interior)	Total moment (kN-m)	Column strip moment (kN-m)	Middle strip moment (kN-m)
Exterior Negative ES	0.26M0=30.00	0.26M0=30.00	0
Positive ES	0.52M0=60.00	0.31M0=35.77	0.21M0=24.23
Interior Negative ES	0.70M0=80.78	0.53M0=61.16	0.17M0=19.62
Positive IS	0.35M0=40.39	0.21M0=24.23	0.14M0=16.16
Negative IS	0.65M0=75.01	0.49M0=56.55	0.16M0=18.46

With the moment once distributed, it is time to determine the steel rebar reinforcement to place in the slab. We will only develop one calculation and then all the results into a table.

Moment in the exterior negative span in the column strip, \({M_u = 30.00 kN-m}\)

• Veronderstelde spanning gecontroleerde sectie. \({\phi_f = 0.9}\)
• Column strip width, \({b=2.0m}\)
• Stalen versterkingsgebied, \({A_s = frac{M_u}{\phi_ftimes 0.9dtimes fy}= frac{30.00kN-m}{0.9\keer 0.9(0.17m)\keer 420 MPa}=518.726 {mm}^ 2}\)
• \({\de opwaartse bodemdruk veroorzaakt bidirectionele buiging met trekspanningen aan het bodemoppervlak{min} = 0.0018}\). Stalen minimale wapeningsoppervlakte, \({EEN_{s,min}=rho_{min}\maal bmaal d = 0.0018 \times 2.0m \times 0.17m =612 {mm}^ 2}\). Nu, controleer of de sectie zich gedraagt ​​als spanningsgestuurd.
• \({a = frac{A_stimes f_y}{0.85\maal f'cmaal b} = frac{518.726 {mm}^2keer 420 MPa}{0.85\keer 25 MPa\times 2.0m }= 6.048 mm}\)
• \({c = frac{een}{\bèta_1}= frac{6.048 mm}{0.85} = 7.115mm }\)
• \({\varepsilon_t = (\frac{0.003}{c})\keer d – 0.003 = (\frac{0.003}{7.115mm})\times 170mm – 0.003 = 0.069 > 0.005 }\) OK!, het is een door spanning gecontroleerd gedeelte!.

Span(ES:Exterior, IS:Interior)	Column Strip Moment (kN-m)	\({EEN_{s,calc} ({mm}^ 2)}\)	\({EEN_{s,min} ({mm}^ 2)}\)	\({een (mm)}\)	\({c (mm)}\)	\({\varepsilon_t > 0.005}\)
Exterior Negative ES	30.00	518.726	612.0	6.048	7.115	0.069 > 0.005!
Positive ES	35.77	618.494	612.0	6.112	7.191	0.068 > 0.005!
Interior Negative ES	61.16	1057.509	612.0	10.451	12.295	0.038 > 0.005!
Positive IS	24.23	418.958	612.0	6.048	7.115	0.069 > 0.005!
Negative IS	56.55	977.799	612.0	9.663	11.368	0.042 > 0.005!

Moment in the exterior positive span in the middle strip, \({M_u = 24.23 kN-m}\)

• Veronderstelde spanning gecontroleerde sectie. \({\phi_f = 0.9}\)
• Column strip width, \({b=4.0m}\)
• Stalen versterkingsgebied, \({A_s = frac{M_u}{\phi_ftimes 0.9dtimes fy}= frac{24.23 kN-m}{0.9\keer 0.9(0.17m)\keer 420 MPa}=418.958 {mm}^ 2}\)
• \({\de opwaartse bodemdruk veroorzaakt bidirectionele buiging met trekspanningen aan het bodemoppervlak{min} = 0.0018}\). Stalen minimale wapeningsoppervlakte, \({EEN_{s,min}=rho_{min}\maal bmaal d = 0.0018 \times 4.0m \times 0.17m =1224 {mm}^ 2}\). Nu, controleer of de sectie zich gedraagt ​​als spanningsgestuurd.
• \({a = frac{A_stimes f_y}{0.85\maal f'cmaal b} = frac{1224 {mm}^2keer 420 MPa}{0.85\keer 25 MPa\times 4.0m }= 6.048 mm}\)
• \({c = frac{een}{\bèta_1}= frac{6.048 mm}{0.85} = 7.115 mm }\)
• \({\varepsilon_t = (\frac{0.003}{c})\keer d – 0.003 = (\frac{0.003}{7.115mm})\times 170mm – 0.003 = 0.069 > 0.005 }\) OK!, het is een door spanning gecontroleerd gedeelte!.

Span(ES:Exterior, IS:Interior)	Middle Strip Moment (kN-m)	\({EEN_{s,calc} ({mm}^ 2)}\)	\({EEN_{s,min} ({mm}^ 2)}\)	\({een (mm)}\)	\({c (mm)}\)	\({\varepsilon_t > 0.005}\)
Exterior Negative ES	0.00	0.00	1224.00	6.048	7.115	0.069 > 0.005!
Positive ES	24.23	418.958	1224.00	6.048	7.115	0.069 > 0.005!
Interior Negative ES	19.62	339.247	1224.00	6.048	7.115	0.069 > 0.005!
Positive IS	16.16	279.420	1224.00	6.048	7.115	0.069 > 0.005!
Negative IS	18.46	319.189	1224.00	6.048	7.115	0.069 > 0.005!

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SkyCiv S3D Design Module

In deze sectie, we describe the design result using the module for plate design included in SkyCiv. We don’t explain how to model and analyze the structure (for these, see related articles on this topic in our documentation: How to model a structure in SkyCiv?, How to apply loads in my building model? en How to run a linear elastic analysis?)

It is convenient to apply a fine mesh to the slabs to obtain an accurate design result. Please take a look at the following image for more clarity.

Figuur 12. Finer mesh applied to slabs

The next step is to run the design module and select the options which calculate an optimized steel rebar area.

Figuur 13. Slab concrete properties definition before design optimization.

Figuur 14 represents the plate’s local axes orientation. Because local axis 3 is downward, de “top” is the bottom, en de “bodem” will be the top, thus correctly taking the data from the design.

Another important fact is the slab mesh size; it is a plate square element with plan dimensions of 500mm x 500mm. SkyCiv S3D gives the reinforcement area as an integrated value per finite element. Dus, if we want to obtain the total rebar area of a column or middle strip, we need to calculate the mean value from the number of elements that sum the strip width being analyzed. Bijvoorbeeld, for the column strip, four elements will be considered (4×0.5m = 2m).

Figuur 14. Local axes orientation in slab example.

Eerste, we analyze the reinforcement area required along the longitudinal direction in axis 1.

Column Strip

• Exterior negative moment (top reinforcement): \({EEN_{s,top} =(119.09\keer 2 + 952.72 + 833.64 )\frac{{mm}^ 2}{m} \times 0.50m = 1012.27 {mm}^ 2}\)
• Exterior positive moment (bottom reinforcement): \({EEN_{s,bot} = 4*463.90 \frac{{mm}^ 2}{m}\times 0.50m = 927.80 {mm}^ 2}\)
• Exterior interior negative moment (top reinforcement): \({EEN_{s,top} =(1071.82\keer 2 +714.54 \keer 2 )\frac{{mm}^ 2}{m} \times 0.50m = 1786.36 {mm}^ 2}\)
• Interior positive moment(bottom reinforcement): \({EEN_{s,bot} = 4*309.27 \frac{{mm}^ 2}{m}\times 0.50m = 618.54 {mm}^ 2}\)
• Interior negative moment (top reinforcement): \({EEN_{s,top} =(714.54\keer 2 +952.73 \keer 2 )\frac{{mm}^ 2}{m} \times 0.50m = 1667.27 {mm}^ 2}\)

Middle Strip

• Exterior negative moment (top reinforcement): \({EEN_{s,top} =(119.09\keer 4)\frac{{mm}^ 2}{m} \times 0.50m = 238.18 {mm}^ 2}\)
• Exterior positive moment (bottom reinforcement): \({EEN_{s,bot} = (463.90\keer 2 +412.36 \keer 2 ) \frac{{mm}^ 2}{m}\times 0.50m = 876.26 {mm}^ 2}\)
• Exterior interior negative moment (top reinforcement): \({EEN_{s,top} =(357.27\keer 2 +476.36 \keer 2 )\frac{{mm}^ 2}{m} \times 0.50m = 833.63 {mm}^ 2}\)
• Interior positive moment(bottom reinforcement): \({EEN_{s,bot} = 4*257.72 \frac{{mm}^ 2}{m}\times 0.50m = 515.44 {mm}^ 2}\)
• Interior negative moment (top reinforcement): \({EEN_{s,top} =(357.27\keer 2 +476.36 \keer 2 )\frac{{mm}^ 2}{m} \times 0.50m = 833.63 {mm}^ 2}\)

Figuur 15. Optimization results in direction “1” and the top side (Bottom side, actually).

Figuur 16. Optimization results in direction “1” and the bottom side (Top side, actually).

Tenslotte, we analyze the reinforcement area required along the transverse direction in axis 2.

Column Strip

• Exterior negative moment (top reinforcement): \({EEN_{s,top} =(91.55\keer 2 + 457.73 + 549.28 )\frac{{mm}^ 2}{m} \times 0.50m = 595.055 {mm}^ 2}\)
• Exterior positive moment (bottom reinforcement): \({EEN_{s,bot} = (269.68\keer 3+239.72) \frac{{mm}^ 2}{m}\times 0.50m = 524.38 {mm}^ 2}\)
• Exterior interior negative moment (top reinforcement): \({EEN_{s,top} =(823.92\keer 2 +549.28 +457.73)\frac{{mm}^ 2}{m} \times 0.50m = 1327.43 {mm}^ 2}\)
• Interior positive moment(bottom reinforcement): \({EEN_{s,bot} = (179.79\keer 3+149.82) \frac{{mm}^ 2}{m}\times 0.50m = 344.60 {mm}^ 2}\)
• Interior negative moment (top reinforcement): \({EEN_{s,top} =(823.92\keer 2 +549.28 +457.73)\frac{{mm}^ 2}{m} \times 0.50m = 1327.43 {mm}^ 2}\)

Middle Strip

• Exterior negative moment (top reinforcement): \({EEN_{s,top} =(183.09\times 2+91.55\times 6)\frac{{mm}^ 2}{m} \times 0.50m = 457.74 {mm}^ 2}\)
• Exterior positive moment (bottom reinforcement): \({EEN_{s,bot} = (209.75\keer 2 +179.79 \keer 2 +149.82 \keer 4) \frac{{mm}^ 2}{m}\times 0.50m = 689.18{mm}^ 2}\)
• Exterior interior negative moment (top reinforcement): \({EEN_{s,top} =(274.64\times 2+91.55\times 6)\frac{{mm}^ 2}{m} \times 0.50m = 549.29 {mm}^ 2}\)
• Interior positive moment(bottom reinforcement): \({EEN_{s,bot} = (119.86\keer 4 + 89.89\keer 4) \frac{{mm}^ 2}{m}\times 0.50m = 419.50 {mm}^ 2}\)
• Interior negative moment (top reinforcement): \({EEN_{s,top} =(274.64\times 2+91.55\times 6 )\frac{{mm}^ 2}{m} \times 0.50m = 549.29 {mm}^ 2}\)

Figuur 17. Optimization results in direction “2” and the top side (Bottom side, actually).

Figuur 18. Optimization results in direction “2” and the bottom side (Top side, actually).

 

Results comparison

The following table shows the results for the DDM (“Direct Design Method”) and the S3D steel rebar optimization.

Span (ES:Exterior, IS:Interior)	Column Strip (S3D Design) \({Als ({mm}^ 2)}\)	Column Strip (ACI-318 DDM) \({Als ({mm}^ 2)}\)	% Dif	Middle Strip (S3D Design) \({Als ({mm}^ 2)}\)	Middle Strip (ACI-318 DDM) \({Als ({mm}^ 2)}\)	% Dif
Exterior Negative ES	1012.27	853.996	15.636	238.18	0 (612.0)	100.00
Positive ES	927.80	1018.259	9.75	876.26	689.733	21.287
Interior Negative ES	1786.36	1740.844	2.48	833.63	558.322 (612.0)	26.586
Positive IS	618.54	689.733	11.51	515.44	459.937 (612.0)	18.734
Negative IS	1667.27	1609.607	3.459	833.63	525.642 (612.0)	26.586

Transverse direction

Span (ES:Exterior, IS:Interior)	Column Strip (S3D Design) \({Als ({mm}^ 2)}\)	Column Strip (ACI-318 DDM) \({Als ({mm}^ 2)}\)	% Dif	Middle Strip (S3D Design) \({Als ({mm}^ 2)}\)	Middle Strip (ACI-318 DDM) \({Als ({mm}^ 2)}\)	% Dif
Exterior Negative ES	595.055	518.726	12.827	457.74	0 (1224)	100.00
Positive ES	524.38	618.494	17.948	689.18	418.958	39.209
Interior Negative ES	1327.43	1057.509	20.334	549.29	339.247	38.239
Positive IS	344.60	418.958	21.578	419.50	279.42	33.392
Negative IS	1327.43	977.799	26.339	549.29	319.189	41.891

Gevolgtrekking

We have demonstrated in this article that SkyCiv module for plate design calculates the steel reinforcement for bending slab accordingly to the code ACI-318-19. Comparing the results from the analysis in the column strips, where because of their relative stiffness, the moments are highly concentrated, the differences between hand calculations and optimization by S3D round a value of 10 – 15%. This practicality indicates an excellent match between analysis and design procedures.

For middle strips, the results differ a bit more because the code only assigns the rest of the moment after taking the corresponding column strips. This will impact the match when we compare it with the analysis from the software, which is more accurate.

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