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

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Table of Contents Page Introduction 4 Design Summary 6 Calculations 11 Slab Thickness 11 Loads 12

Direct Design Method 15 Mo 15 Plan View of Panels 19 Lateral Distribution of Moments 20 Plan View of Moment Regions 26 Flexural Design 27 One Way Shear 42 Corner Panels 42 E-W Edge Panels 43 N-S Edge Panels 44 Interior Panels 45 Punching Shear 46 Corner Columns 46 E-W Edge Columns 47 N-S Edge Columns 48 Interior Columns 49 Unbalanced Moment Transfer 50 Corner Columns 50 E-W Edge Columns 53 N-S Edge Columns 56 Interior Columns 59 Negative Moment Reinforcement Checks 62 Equivalent Frame Calculations 63 Exterior Frame 63 Interior Frame 66

EFM Analysis 69 Node Diagram with Coordinates 69 Exterior Frame 70 Output 70 Loaded Structure 79 Moment Diagrams 79 Shear Diagrams 79 Member 10 80 Member 21 81 Member 32 82 Interior Frame 83 Output 83 Loaded Structure 91 Moment Diagrams 91 Shear Diagrams 91


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Member 10 92 Member 21 93 Member 32 94

Moment Location Diagram 95 DDM to EFM Moment Comparison Chart 96 Drawings and Diagrams 97 Elevation Reinforcement Detailing 97 Column Strip Detailing 98 Middle Strip Detailing 99 Cost Breakdown 100 Rebar Quantities 100 Concrete Quantities 101 Total Costs 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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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 one-

way, 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


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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 ACI-

13.7.4.2 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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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 Unit Cost Amount Total Cost Cost Per Floor Concrete $100/yd3 161.774 yd3 $ 16,177.40 $ 16,177.40 Steel $1200/ton 11.378 tons $ 13,653.60 $ 13,653.60 Design $200/hr 140 hours $ 28,000.00 $ 4,666.67 Formwork $9/ft2 6612 ft2 $ 59,535.00 $ 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 thicknessln 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

hs_minln30

:= hs_min 8.4668in=

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

hs_minln33

:= hs_min 7.6971in=

So exterior panel controls. Round h s_min up to 8.5 inches

hs 8.5 in⋅:=


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Exterior Frame Load Calculations


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Summary of Direct Design Method Moments (kip-in)

Mo End span interior span

M- exterior

M-

interior M+ M- M+ Exterior Frame 2478 644 1734 1288 1610 867Interior Frame 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

CSwidth_extl22

9 in⋅+:= CSwidth_ext 11.813ft=

Middle strip width for interior equivalent frames and exterior equivalent framesMSwidth 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 1 2 3 4 5 6 7 8 9 10 11 12SE Col Strip M- 554 1,039 1,019 554 1,039 1,119 554 1,039 1,119 554 1,039 1,019E Col Strip M+ 1,330 448 1,330 1,330 448 1,330 1,330 448 1,330 1,330 448 1,330NE Col Strip M- 1,019 1,039 554 1,119 1,039 554 1,119 1,039 554 1,019 1,039 554SE Mid Strip M- 0 346 373 0 346 373 0 346 373 0 346 373E Mid Strip M+ 887 298 887 887 298 887 887 298 887 887 298 887NE Mid Strip M- 373 346 0 373 346 0 373 346 0 373 346 0SW Mid Strip M- 0 403 434 0 346 373 0 346 373 0 403 434W Mid Strip M+ 515 347 515 887 298 887 887 298 887 515 347 515NW Mid Strip M- 434 403 0 373 346 0 373 346 0 434 403 0SW Col Strip M- 644 1,208 1,301 554 1,039 1,119 554 1,039 1,119 644 1,208 1,301W Col Strip M+ 773 520 773 1,330 448 1,330 1,330 448 1,330 773 520 773NW Col Strip M- 1,301 1,208 644 1,119 1,039 554 1,119 1,039 554 1,301 1,208 644


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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)

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)

Member 10 Analysis (Interior Frame)


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Moment Location Diagram for EFM Analysis

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


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Reinforcement Detailing Diagram


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Column Strip Detailing


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Middle Strip Detailing


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Rebar Quantities

Distance between Steel

column faces Density ln (ft)

in L1 Direction (ft) (pcf)

20.5 21.92 490 Specified Rebar Spacing Requirements 50% 50% 100% 50% 50% Top Top Bottom Bottom Bottom Quantity Rebar Rebar Rebar Rebar Rebar Rebar Strip of Length Rebar Spacing Length Length Length Length Length Width Rebar No. in. ft. ft ft. ft. ft. ft ft T1 4 4 6.15 4.1 6.3 18.9 96.7T2 4 12 22.92 6.3 6.3 144.2T3 4 8 6.15 4.1 6.3 9.4 48.4T4 4 12 22.92 6.3 6.3 144.2T5 4 12 5.01 11.1 11.1 55.4T6 5 10 22.92 17.925 11.1 13.3 271.1T7 4 12 4.51 11.1 11.1 49.9T8 4 12 22.92 17.925 11.1 11.1 225.9T9 5 8 6.15 4.1 11.1 16.6 85.0T10 6 8 22.92 11.1 16.6 380.3T11 6 9 6.15 4.1 11.1 14.8 75.6T12 5 16 22.92 11.1 8.3 190.2T13 6 12 5.01 11.1 11.1 55.4T14 4 12 4.51 11.1 11.1 49.9T15 4 12 22.92 17.925 11.1 11.1 225.9 Quantities By Bar Size Bar # Length Volume Weight ft in^2 lbs

4 814.7 1955.2 6653.1 5 546.3 2032.4 6915.8 6 511.4 2700.0 9187.6

Total Steel (lbs) 22756.4


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Concrete Quantities

Slab Thickness Slab

Width Slab Depth Column Quantity

Column Area

in ft ft in2 8.5 74 90 20 324

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

Vol. Capital Vol. Capital

Capital Dimensions

Edge Capitals

Corner Capitals

in3 ft3 a b h ft3 ft3 3888 2.25 18 36 9 1.546875 0.984375

Slab Area Slab Volume

ft2 ft3 6660 4335.0

Item Item Vol. Quanitity Volume ft3 ft3 Corner Capitals 0.984375 4 3.9 Edge Capitals 1.546875 10 15.5 Interior Capitals 2.25 6 13.5 Slab 4335.0 1 4335.0 Total 4367.9

Formwork Area (ft2) 6615


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Total Costs Item Unit Cost Amount Total Cost Cost Per Floor Concrete $100/yd3 161.774 yd3 $ 16,177.40 $ 16,177.40 Steel $1200/ton 11.378 tons $ 13,653.60 $ 13,653.60 Design $200/hr 140 hours $ 28,000.00 $ 4,666.67 Formwork $9/ft2 6612 ft2 $ 59,535.00 $ 9,922.50 Total per floor $ 44,420.17 Total for building $ 266,521.00

two way slab design with column capital

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