Two Way Slab Design With Column Capital

Two-way Slab Design with Column Capitals Wayne Hoklas Brendan Nee Justin Zimmerman Dec 16, 2003

Two Way Slab Design with Column Capitals

Table of Contents Introduction Design Summary Calculations Slab Thickness Loads Direct Design Method Mo Plan View of Panels Lateral Distribution of Moments Plan View of Moment Regions Flexural Design One Way Shear Corner Panels E-W Edge Panels N-S Edge Panels Interior Panels Punching Shear Corner Columns E-W Edge Columns N-S Edge Columns Interior Columns Unbalanced Moment Transfer Corner Columns E-W Edge Columns N-S Edge Columns Interior Columns Negative Moment Reinforcement Checks Equivalent Frame Calculations Exterior Frame Interior Frame EFM Analysis Node Diagram with Coordinates Exterior Frame Output Loaded Structure Moment Diagrams Shear Diagrams Member 10 Member 21 Member 32 Interior Frame Output Loaded Structure Moment Diagrams Shear Diagrams

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Page 4 6 11 11 12 15 15 19 20 26 27 42 42 43 44 45 46 46 47 48 49 50 50 53 56 59 62 63 63 66 69 69 70 70 79 79 79 80 81 82 83 83 91 91 91

Two Way Slab Design with Column Capitals Member 10 Member 21 Member 32 Moment Location Diagram DDM to EFM Moment Comparison Chart Drawings and Diagrams Elevation Reinforcement Detailing Column Strip Detailing Middle Strip Detailing Cost Breakdown Rebar Quantities Concrete Quantities Total Costs

92 93 94 95 96 97 97 98 99 100 100 101 102

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Introduction The goal of this project was to design an intermediate floor of a six story concrete building in one direction using the Direct Design Method (DDM) outlined in ACI-318-02. In addition, the Equivalent Frame Method (EFM) for obtaining bending moments in the slab, also outlined in ACI-318-02, was performed. The bending moments obtained from the EFM were then compared to those found using the equations of the DDM. The following information was given to our design team: c/c story height = 12 ft min. c/c column spacing = 22 ft cladding weight = 250 plf partition weight = 20 psf electrical/mechanical system weight = 6 psf service live load = 80 psf fc’ = 5 ksi fy = 60 ksi preliminary dimensions: columns – 18x18 in In addition to this information, our design team was instructed to follow a flat slab design that had no beams between columns and included column capitals. Our preliminary estimates of the shear capacity of the slab showed that column capitals were probably not needed. However, since their use was required, we arbitrarily chose to use 9” column capitals. Upon making this decision, the minimum slab thickness allowed by ACI-318-02 was used and the general procedures of the DDM were followed for the North-South direction of the floor. Following this, checks for one and two way shear were made, as well as a check for unbalanced moment transfer. For the EFM analysis, two equivalent frames were analyzed. One frame consisted of a column line on an exterior edge of the building, and the other frame consisted of an interior column line. Five computer programs were used to assist in our design. Mathcad was used to assist performing the general calculations. Excel was used for designing the flexural reinforcement and performing cost and quantity

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Two Way Slab Design with Column Capitals calculations. Fast Frame 2D frame analysis software was used for the EFM analysis. Adobe Photoshop 7.0 and Autocad 2002 were used to prepare figures and diagrams for this document.

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Design Summary Initial Design Work The first step in the design process was to take the given information and determine the geometry of the floor system. In order to make calculations simpler, all center to center column spaces in the same direction were made equal for all panels. This was accomplished by subtracting two one half column widths from the out to out dimensions in the North-South and East-West directions. The remaining dimension was then divided into thirds in the North-South direction and into four panels in the East-West direction to obtain center to center column spacing. Next, some preliminary estimates of the required column capital size were made to ensure adequate capacity for punching shear, because this often controls the acceptable slab thickness and the need for drop panels and column capitals. It was determined that column capitals would likely not be needed. Because of this, relatively small, nine inch column capitals were chosen. After defining the columns and capitals, the minimum allowable slab thickness was determined using the clear span distance. From Table 9.5 (C) in ACI-318-02, the controlling minimum thickness was for exterior panels without drop panels and without edge beams. This thickness was rounded up to 8.5 inches and used for the rest of the design. Once all dimensions of the floor system were known, the widths of column and middle strips and the factored dead and live loads were calculated. To handle the effects of the cladding load on the exterior equivalent frame, all area loads were multiplied by the width of the frame to create line loads. The line load of the cladding was then added to the dead weight line load and the resulting dead and live line loads were subsequently factored. The effects from cladding located on East-West building edges were neglected in the DDM calculations since they will not create significant bending moments in the North-South direction. The effect of this cladding must be taken into account when the building is designed in the East-West direction.

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Direct Design Method The next step in the design process was to use the DDM to determine the bending moments for which the slab system in the North-South direction must be reinforced for. Mo was calculated and distributed to positive and negative moment regions and between column and middle strips. Using these distributed moments, a map of where moments occurred was developed and the design moments were determined. According to ACI-13.6.3.4, negative moment regions must be designed for the larger of the two moments that they are subjected to, thus both moments were compared and the largest was selected for design. Fifteen different moment regions were identified and labeled Type1-15. The moments in each region were divided by width of their region to obtain moments per foot. By inputting these moments per width into an Excel spreadsheet, a design for reinforcement for all fifteen regions was developed. The spreadsheet required the input of Mu, hs, fy, fc’, β1, clear cover depth, and an initial assumption of a bar size. Using a series of If() statements and equations, the spread sheet retrieved the correct bar diameter and area from a table, calculated d, and then solved a quadratic equation for the required reinforcement ratio to resist the specified moment. This reinforcement ratio was multiplied by b*d to obtain As_req. From this As_req per foot, the spreadsheet displayed the required spacing for bar sizes from 3 to 18 to provide the necessary area of steel per foot. Using this information, a spacing and bar size could be specified causing the spreadsheet to calculate φMn, the depth of the Whitney stress block, As_min, As_max, the strain in the tension steel, and the maximum allowable spacing for shrinkage and temperature as well as flexural requirements. Lastly, a series of If() statements checked this output against code specifications and displayed a corresponding text box stating if the results were acceptable. Thus, with half a dozen key strokes per region, our team rapidly designed the slab reinforcement for the 15 different sections.

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Shear Checks The next step was to check the slab system to see if it possessed adequate shear capacity. First, the oneway, or beam shear, method of failure was checked. After some consideration, it was decided to assume that the cladding weight was distributed over the entire panel. While this is probably not an ideal assumption, it should be satisfactory because the one-way shear capacity was three to four times greater than the applied shear loading. ACI-318-02 provided no guidance on this issue, thus it is up to the designers discretion. Two-way, or punching shear, was the next check performed. Four separate regions were identified: corner panel columns, E-W edge panel columns, N-S edge panel columns, and interior panel columns. The edge columns have the same shear capacity but not the same loading. All regions were found to have excess shear capacity.

Unbalanced Moment Transfer In accordance with the DDM, the slab system’s capacity for transferring unbalanced moment was checked. Four separate regions were identified for this check: corner columns, E-W edge columns, N-S edge columns, and interior columns. All regions were found to have sufficient shear capacity to transfer the shear portion of the unbalanced moment. However, several columns were found to have insufficient flexural capacity to transfer the flexural portion of unbalanced moment. The total amount of steel required per foot in these regions was calculated. The amount of steel in the column strips was increased where needed to provide adequate flexural capacity. Specifically, all of the edge column strips’ areas of steel per foot were increased to handle the moment due to unbalanced moment transfer.

Equivalent Frame Method Having the slab system completely designed, the bending moments for slab system were determined using the EFM from ACI-318-02 for comparison purposes. First, the slab/column system was idealized as a two dimensional frame. This frame was constructed of a series of individual members with varying moments of -8-

Two Way Slab Design with Column Capitals inertia connected rigidly together. Once the members’ lengths were calculated, a sketch of the frame was drawn and the moment of inertia for each member was calculated. Some of these, such as the moment of inertia of the slab away from the supports, could be calculated directly. However, most of the moments of inertia were more complicated. Given that the idealized two dimensional frame was really a complex and non-homogenous three dimensional frame, special considerations were necessary for many members. The equations for the EFM from ACI-318-02 were followed where applicable for these calculations. One area where the code provided no guidance was the column capital region. The code specifies in ACI13.7.4.2 that “Variation in the moment of inertia of along the axis of columns shall be taken into account”, but provides the designer with no recommended means of doing so. The technique used was to average the moments of inertia for the columns in the column capital region. First, the moment of inertia for an 18 inch square column and slab system was calculated. Next, a column having dimensions of the actual column plus the column capital width was considered. The moment of inertia of this fictitious composite column was calculated. These two values were then averaged and used as the moment of inertia for the entire 9 inch region of the column where the capital is located. Once all needed properties and dimensions were determined, two frame models were constructed using FastFrame, the powerful and user friendly two dimensional frame analysis software available at no cost from Enercalc®. The analysis was run using the loads calculated for the DDM design. However, the area at the end of each frame between the center of the column and the edge of the floor had been neglected in the DDM design. The contribution from dead and live load was factored and added into the EFM model. The most significant load in this area was the cladding weight. A point load of 6.31 kips and a moment of 3.70 ft-kips was applied to the corner columns. To the south and north edge columns, a point load of 11.28 kips and moment of 6.72 ft-kips was applied. These loads can be clearly seen in the loading diagram for each EFM analysis. After running frame analysis, the moments at the i and j ends of members 10, 21, and 32 were compared to those found using the DDM. The moments on the East-West oriented edge columns were approximately twice

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Two Way Slab Design with Column Capitals the magnitude of the same moments calculated using the DDM. At first it seemed that the slab might have insufficient flexural capacity in these regions. However, the amount of steel in these regions was significantly increased when the effects of unbalanced moment transfer were taken into account. The net amount of reinforcement in these regions would likely be equivalent whether the reinforcement was designed for EFM or DDM moments.

Costs and Detailing Detailing of the slab steel was done using figure 13.3.8 in ACI 318-02. Once the detailing was completed, the quantities of steel and concrete used were calculated. From these quantities, material and design costs were determined. The total costs: Item Concrete Steel Design Formwork

Unit Cost $100/yd3 $1200/ton $200/hr $9/ft2

Amount 161.774 yd3 11.378 tons 140 hours 6612 ft2

Total Cost $ 16,177.40 $ 13,653.60 $ 28,000.00 $ 59,535.00

Cost Per Floor $ 16,177.40 $ 13,653.60 $ 4,666.67 $ 9,922.50

Total per floor

$

44,420.17

Total for building

$ 266,521.00

It can be seen that the formwork and design add significant costs to the floor system. However, the formwork is reusable for each of the six floors and the design need only be performed once, so long as the column size does not vary. This allows these costs to be divided among all six floors. The material cost applies to each floor. Thus, the material cost of steel and concrete makes up only 68% of the total flooring cost. This cost estimate does not consider columns, cladding, roof material, foundations, or partitions. Finally, drawings and diagrams to make our design clear and understandable were constructed using Autocad, and our results were reviewed for errors. These drawings can be seen throughout the calculations section of this report.

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Selection of Slab thickness ln := 24.167ft ⋅ − 18⋅ in − 2⋅ 9⋅ in ln = 21.167ft From table 9.5 (C) for exterior panels with no drop panels and without edge beams

h s_min :=

ln 30

h s_min = 8.4668in

From table 9.5 (C) for interior panels with no drop panels and without edge beams

h s_min :=

ln 33

h s_min = 7.6971in

So exterior panel controls. Round h

s_min

up to 8.5 inches

h s := 8.5⋅ in

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Exterior Frame Load Calculations

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Summary of Direct Design Method Moments (kip-in)

Mo Mexterior Exterior Frame Interior Frame

End span Minterior

interior span M+

M-

M+

2478

644

1734

1288

1610

867

4262.4

1108

2984

2217

2771

1492

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Middle Strip and Column Strip Calculations Column strip width for interior equivalent frames l1 := 24.167ft ⋅

l2 := 22.125ft ⋅

(

)

CSwidth_int := 2⋅ .25⋅ min l1 , l2

CSwidth_int = 11.063ft

Column Strip width for exterior equivalent frames l2 CSwidth_ext := + 9⋅ in 2

CSwidth_ext = 11.813ft

Middle strip width for interior equivalent frames and exterior equivalent frames MSwidth := l2 − CSwidth_int

MSwidth = 11.063ft

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Plan View of Panels

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Summary of Lateral Moment Distributions in kip-in Pannel SE Col Strip ME Col Strip M+ NE Col Strip MSE Mid Strip ME Mid Strip M+ NE Mid Strip MSW Mid Strip MW Mid Strip M+ NW Mid Strip MSW Col Strip MW Col Strip M+ NW Col Strip M-

1 554 1,330 1,019 0 887 373 0 515 434 644 773 1,301

2 1,039 448 1,039 346 298 346 403 347 403 1,208 520 1,208

3 1,019 1,330 554 373 887 0 434 515 0 1,301 773 644

4 554 1,330 1,119 0 887 373 0 887 373 554 1,330 1,119

5 1,039 448 1,039 346 298 346 346 298 346 1,039 448 1,039

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6 1,119 1,330 554 373 887 0 373 887 0 1,119 1,330 554

7 554 1,330 1,119 0 887 373 0 887 373 554 1,330 1,119

8 1,039 448 1,039 346 298 346 346 298 346 1,039 448 1,039

9 1,119 1,330 554 373 887 0 373 887 0 1,119 1,330 554

10 554 1,330 1,019 0 887 373 0 515 434 644 773 1,301

11 1,039 448 1,039 346 298 346 403 347 403 1,208 520 1,208

12 1,019 1,330 554 373 887 0 434 515 0 1,301 773 644


Plan View of Moment Regions

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EFM Node Diagram

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Exterior Frame EFM Output

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Loaded Structure (Exterior Frame)

Moment Diagram (Exterior Frame)

Shear Diagram (Exterior Frame)

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Member 10 Analysis (Exterior Frame)



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Exterior Frame EFM Output


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Loaded Structure (Interior Frame)

Moment Diagram (Interior Frame)

Shear Diagram (Interior Frame)

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Member 10 Analysis (Interior Frame)



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Moment Location Diagram for EFM Analysis

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Comparison of DDM to EFM moments Moments (see preceding diagram for locations)

DDM moments (k*ft) for exterior frame

EFM moments (k*ft) for exterior frame

DDM moments (k*ft) for interior frame

EFM moments (k*ft) for interior frame

M1

53.7

113.6

92.3

231.9

M2

107.3

109.1

184.8

190

M3

144.5

81.7

248.7

88.2

M4

134.2

111.6

230.9

173.8

M5

72.3

94.9

124.3

172.6

M6

134.2

111.6

230.9

173.8

M7

144.5

81.7

248.7

88.2

M8

107.3

109.1

184.8

190

M9

53.7

113.6

92.3

231.9


Reinforcement Detailing Diagram

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Column Strip Detailing

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Middle Strip Detailing

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Rebar Quantities

ln (ft) 20.5

Distance between column faces in L1 Direction (ft) 21.92

Steel Density (pcf) 490

Specified Rebar Spacing Requirements 50% 50% Top Top Rebar Rebar Rebar Spacing Length Length No. in. ft. ft T1 4 4 6.15 4.1 T2 4 12 T3 4 8 6.15 4.1 T4 4 12 T5 4 12 T6 5 10 T7 4 12 T8 4 12 T9 5 8 6.15 4.1 T10 6 8 T11 6 9 6.15 4.1 T12 5 16 T13 6 12 T14 4 12 T15 4 12 Quantities By Bar Size Bar # Length Volume ft in^2 4 814.7 1955.2 5 546.3 2032.4 6 511.4 2700.0 Total Steel (lbs)

100% Bottom Rebar Length ft.



22.92 22.92 5.01 22.92

17.925

22.92

17.925

22.92

17.925

4.51

22.92 22.92 5.01 4.51

Weight lbs 6653.1 6915.8 9187.6 22756.4

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Strip Width ft 6.3 6.3 6.3 6.3 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1 11.1

Quantity of Rebar

Rebar Length ft

18.9 6.3 9.4 6.3 11.1 13.3 11.1 11.1 16.6 16.6 14.8 8.3 11.1 11.1 11.1

96.7 144.2 48.4 144.2 55.4 271.1 49.9 225.9 85.0 380.3 75.6 190.2 55.4 49.9 225.9


Concrete Quantities Slab Thickness in

Slab Width ft

Slab Depth ft

8.5

74

90

Column Quantity

Column Area in2

20

324

V=1/3(a2+a*b+b2)*h-182*h Vol. Capital ft3

Vol. Capital in3 3888

2.25

Capital Dimensions a

b

18

36

Quanitity

Volume ft3

Slab Volume ft3

Slab Area ft2 6660 Item Corner Capitals Edge Capitals Interior Capitals Slab

4335.0 Item Vol. ft3 0.984375 1.546875 2.25 4335.0

4 10 6 1

Total

4367.9

Formwork Area (ft2)

3.9 15.5 13.5 4335.0

6615

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h

Edge Capitals ft3

Corner Capitals ft3

9

1.546875

0.984375


Total Costs Item Concrete Steel Design Formwork

Unit Cost $100/yd3 $1200/ton $200/hr $9/ft2

Amount 161.774 yd3 11.378 tons 140 hours 6612 ft2

Total Cost $ 16,177.40 $ 13,653.60 $ 28,000.00 $ 59,535.00

Cost Per Floor $ 16,177.40 $ 13,653.60 $ 4,666.67 $ 9,922.50

Total per floor

$

Total for building

$ 266,521.00

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44,420.17

Two Way Slab Design With Column Capital

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