##### Project 2 Final Report

Project 2
— *David Wagner 2007/07/04 18:28*

# Structural Configuration

The plan and elevation with overall dimensions provided by the architect has been simplified to what is presented here. For example, the 23”x12” rectangular structural columns used here are inscribed within the 26” circular columns planned, the diameter having been determined by dividing the 15' bay spacing by the golden ratio four times.^{1)}

Member Dimensions

- Roof Slab: 121½' x 36' x 8”
- Beams: 36' x 18” wide x 10” web height
- Structural Columns: 23” x 12” inscribed within 26” diameter circular architectural columns.
- Footing: 6' x 6' x 2'

Although the beam, columns, and footings of the two end frames may require less reinforcement than interior members, the full amount is specified to allow for future expansion of this building.

## Construction Sequence

The construction sequence is standard for reinforced concrete structures.

- After grading the site, frame and pour the column and wall footings.
- Frame and pour the columns.
- Install insulation and vapor barrier, and fill to new grade.
- Frame and pour the roof beams and roof slab.
- Frame and pour the floor slab.
- Install exterior walkways after erecting exterior walls.

## Materials

The structural materials needed are widely available standard construction materials.

- Soil fill is 120 lb/cf.
- The structural steel is grade 60.
- All structural concrete has f’c=4,000 psi. Reinforced concrete density is 150 lb/cf.
- The sod roof assembly includes vegetation and soil cover, lightweight concrete to taper the roof to slope laterally 2% from the center, waterproofing, and provisions for drainage. The entire assembly weighs 110 psf.
- Exterior walkways are permeable concrete lattice, 120 lb/cf.

## Loading

All loads act vertically downward except wind loads, which act transversely on the longitudinal face of the building.

## Dead Loads

- Slab Weight: 150 lb/cf x 8/12 = 100 psf
- Sod Roof: 110 psf
- Plaster Ceiling Finish: 5 psf
- Mechanical and Electrical: 4 psf
- Beam Web Weight: 150 lb/cf x 18/12 x 10/12 = 187.5 lb/ft
- Column Weight: 150 lb/cf x 242/12 x ¼π * (26/12)² = 11,150 lb

## Live Loads

In addition to the standard live and wind loads, when it rains the soil of the sod roof can become saturated equivalent to a uniform distribution of 6” of water.

- Roof: 10 psf
- Rain: 62 pcf x 0.5 = 31 psf
- Wind: 26.9⇒ psf

### Transverse Wind Load

Longitudinal wind loads are borne by shear walls not considered as part of this design. For simplicity, roof loads are not considered and the transverse wind load is approximated by a single uniform load acting on the 18' of exposed wall surface.

V=90 mph, I=1.00, K_z(20') = 0.90, K_{zt}=1, K_d = 1

z (ft) | K_z | q_z (psf) | Windward Wall p (psf) |
---|---|---|---|

0-15 | 0.85 | 17.6 | 11.968 ± 3.4 |

20 | 0.90 | 18.7 | 12.7 ± 3.4 |

h=21 ft | 0.91 | q_h=18.9 | 12.9 ± 3.4 |

The weighted average over the 18' exposed height of the windward wall is 12.1 ± 3.4

**Leeward Wall** (L/B=36/120=0.3→C_p=-0.5) p = -8.0 ± 3.4

Total w_{W,max} = 12.1+3.4 + 8.0+3.4 = 26.9⇒ psf

# Slab Shear and Bending Envelopes

The one-way roof slabs span 15'=180” center-to-center and have a clear span of 13.5'=162”. Longitudinal wind loads are borne by shear walls and not considered to effect the slab.

Total w_{D}: 219 psf,
w_{L} = 41 psf,
l = 15 ft = 180 in,
l_{n} = 13.5 ft = 162 in

- (9-1) U = 1.4D → 306.6 psf
- (9-3) U = 1.2D + 1.6L →
**328.4 psf**Controls

The maximum positive moment is near the middle of each end span. The maximum negative moment is at the exterior faces of the interior columns supporting the end spans. The maximum shear is at the face of the supporting beam.

- 4.275 kip-ft = 51.3 kip-in
- -5.985 kip-ft = -71.8 kip-in
- 2.217 kip

# Frame Shear and Bending Envelopes

Each beam and pair of supporting columns form a simple symmetrical frames with small cantilevered overhangs on either side. Assuming the base of the frame is pinned results in the most conservative maximum moment calculations.

The total dead loads on the beam are 219 psf + 187.5 lb/ft. the total live loads on the beam are 41 psf down and 26.9 psf lateral wind load.

w_{W,max} = 26.9⇒ psf

3.473 kip/ft

0.615 kip/ft

0.484⇒ kip/ft

- (9-1) U = 1.4D → 4.862 kip/ft
- (9-3) U = 1.2D + 1.6L + 0.8W →
**5.152 kip/ft down, and 0.387⇒ kip/ft laterally.**Controls - (9-6) U= 0.9D + 1.6W = 3.126 kip/ft down, and 0.774⇒ kip/ft laterally.

Since this building is much wider laterally (36') than it is tall (18'), by inspection (9-3) will produce the greatest shear and bending moment in the beam.

Ignoring more complicated wind effects, the negative moment on the beam at the exterior face of the column is simply the reaction to the cantilevered load, a much smaller value than the moment on the interior face.

46.53 kip-ft = 558.3 kip-in

h=20.25 ft, L=26 ft

0.5600

10.44 kip

-211.4 kip-ft

2.092

5.745

36.98 kip-ft

-42.36 kip-ft

-253.8 kip-ft = -3045 kip-in

12.5 kip

The maximum positive bending moment occurs near the center of the span. Consider a section from the face in which the positive moment is decreased by wind loads to a distance x from this face.

221.9 kip-ft = 2663 kip-in

Bending Envelope (kip-ft)

Shear Envelope (kip)

Dead and live loads are transfered into the columns, and the wind load causes a relatively small axial reaction.

- 74 kip
- 11 kip
- 1.2 kip
- 107 kip

The axial load may be insufficient to ensure column slenderness so a pinnacle tops each column. These pinnacles may be of any durable material so long as each weighs at least

The maximum bending moment in the column occurs at the top and is the difference between the interior and exterior face negative bending moments.

2487 kip-in

# Slab Design

, say t=8 in

d≈8-0.75-0.5=6.75

0.1728 in²/ft

Find grade 60 steel for the top to handle the larger negative moment.

0.2189 in²/ft

Try #4 bars (the smallest permissible on top), designing a 1' strip of slab as a 1' wide beam, b=b_{w}=12”.

→ #4@10”

0.24 in²/ft

7 in

0.3529 in

The β_{1} of 4000 psi concrete is 0.85.

0.04758 > 0.005 → φ=0.9

88.43 kip-in > 71.8✔

0.1564 in²/ft < 0.1728 ∴ Use A_{s,min}

13.89 → #4@12”

0.20 in²/ft

7 in

0.2941 in

0.05769 > 0.005 → φ=0.9

Longitudinal (121½') | Lateral (36') | |
---|---|---|

Slab Top Mat | #4 @10” | #4 @12” |

Slab Bottom Mat | #4 @12” | #4 @12” |

Though all the mid spans and the edge ends require less reinforcement in the top mat, it doesn't make sense to add unnecessary complexity to change the bar spacing at most 2” just for this. To simplify construction further, two kinds of welded wire mesh providing equivalent steel areas can be used for top and bottom mats in place of the rebar specified here.

If bars are used, the 121½' lengths can be made from a pair of 64' lengths, with one bar of each pair cut more-or-less randomly within its middle third and the resulting three pieces lap-spliced.

# Beam Design

The beams are T-beams, their flanges integral with the slab they support.

## T-beam Web Reinforcement

Start with positive bending moment (web steel).

h=18 in,
b_{w}=18 in

15.5

78 in

3.848 in

Try A_{s}= 4#9 = 4.00 in²,
b_{w} >= 12.2 in✔

15.4375 in

=0.9263 < 4.00✔

0.9050

0.0405 > 0.005 → φ=0.9

21.75 > 4.00✔

3237 kip-in > 2662✔

20' bars are sufficient to develop across the positive moments shown in the bending diagram.

T-Beam Web Reinforcement | 3#9, 20' |
---|

Shear Reinforcement

#3 U stirrups: A_{v}=0.22 in²

35.150 kip

70.3 kip

140.6 kip

26.36 kip; 13.18 kip

21.9 kip

14.67

15.46

**7.719 in** Controls Minimum spacing

24 in

Cantilever #3 U-stirrups: 1@2”, 7@7” |
---|

2.245 ft

52.3 kip

**58.5 kip** Controls

78.00 kip

42.85 kip

→ 35.15 < 42.85 < 70.3

**4.756 in** Controls initial spacing

Spacing may increase to 7” at V_{u} ≥ φV_{c}

11.044 ft = 132.5”

Distance | 0 to 27” | 27” to 133” | 133” to 156” | |
---|---|---|---|---|

V_{u} | 58.5 | 58.5-26.4 | 26.4-13.2 | 13.2-0 |

Minimum Spacing | 4” | 7” | 7” | none |

T-beam Stirrups: #3U 1@3”, 6@4”, 16 @7” |
---|

## T-beam Flange Reinforcement

15.5

4.042 in

With an effective flange width of 78”, these can be spread out quite a bit. Try 6#8 = 4.74 in²

15.5 in

4.647

3373 > 3045✔

Top Bar Length

37 in

Length Needed: 5' cantilever - 3” cover + 4' inside span + 37” development length ≈ 12'

T-beam Flange Reinforcement | 6#8, 12' (2 sets each beam) |
---|

# Column Design

23 in > 18 in → Use a different inscribed rectangle or the full circular section.

Try using an inscribed 23” x 12” rectangle. The diagonal is 25.9 in < 26 in✔

33.91

33.99

→ Short Column

1.0

0.1077

0.1077

0.6957

0.01

2.76 in²

Use 4#8 : 3.16 in²

0.01145

> 0.005 → φ=0.9

Column Reinforcement | 4#8, 20'9” long, 15” 90º Hook on top |
---|

(Development length calculations are the same as the footing dowel calculations below.)

Placing the centers of the bars on the corners of a 16” x 10” rectangle provides sufficient cover: 3.066 in > 3 in✔

Column Ties

V_{u,col} = 11.561 kip

34.91 > 11.561 → No additional shear reinforcement necessary. Use standard #3 column ties.

12”

Column Ties | 12”x23” 5#4@4” at the top, then #3@12” |
---|

# Footing Design

The circular column is treated as a square of same area (ACI 318-05 15.3). First, determine net pressure on the bottom of the footing.

23 in

Try h = 24 in

0.300 ksf

0.360 ksf

0.125 ksf

3.215 ksf

Estimate dimensions.

26.439 sf

6 ft = 72 in 36 sf

=2.956 ksf

20 in

=43 in = 3.583 ft

74.47 kip

870.3 kip

652.7 kip > V_u✔

0.375 ft

6.651 kip

136.6 kip > V_u✔

Footing Dimensions | 6' x 6' x 2' |
---|

Determine reinforcement.

24.5 in = 2.042 ft

36.97 ft-kip = 443.6 in-kip

0.4323 in²

3.11 in² Controls → φ=0.9

Try 6#7 = 3.60 in²

20.13

0.8824 in

3828 in-kip > 443.6✔

Footing Reinforcement | 6#7, 5½' (x2) |
---|

Place reinforcement uniformly across the bottom in both directions.

Check development length.

Length Available=21.5 in

3.437

7.857 > 2.5 → Use 2.5

12.45 in < 21.5✔

Check bearing strength.

9.8 > 2.0 → use 2.0

2338 kip

1169 kip

106.4 → Minimum dowels

2.645 in²

Use 4#8 = 3.16 in²

18.97 in

Available length in footing: in < 18.97 → use #4 ties @4” around the dowels in the footing.

14.23 < 18.25✔

Lap splice length: 30

45”

Footing Dowels | 4#8, 45” long, 15” 90º Hook |
---|

Dowels are tied by 4#4 ties and the dowels extend at least 15” into footing and least 30” into the column.

# Quantities and Costs

Number | 18 | 18 | 9 | 9 | 1 | |||
---|---|---|---|---|---|---|---|---|

Footings | Columns | Beam Web | Beam Flange | Slab | ||||

Length ft | 6 | 1.92 | 36 | 121.5 | ||||

Width ft | 6 | 1.92 | 1.5 | 36 | Quantity | Cost | ||

Height ft | 2 | 19.5 | 0.83 | 0.67 | Total | Unit | Total | |

Volume cy | 48 | 47.76 | 15 | 108 | 218.76 | $68.50 | $14,984.85 | |

#7, 5.5' | 12 | 216 | $11.22 | $2,423.52 | ||||

#8, 45”, 15” 90 Hook | 4 | 72 | $18.57 | $1,337.22 | ||||

#4 12”x23” Ties | 4 | 5 | 162 | $14.12 | $2,287.33 | |||

#3 12”x23” Ties | 18 | 324 | $12.32 | $3,991.25 | ||||

#8, 20'9”, 15” 90 Hook | 4 | 72 | $63.96 | $4,605.30 | ||||

#3 18”x18” U Stirrups | 31 | 279 | $11.75 | $3,279.55 | ||||

#9, 20' | 3 | 27 | $68.00 | $1,836.00 | ||||

#8, 12' | 12 | 108 | $32.04 | $3,460.32 | ||||

#4, 63' | 158 | 158 | $42.75 | $6,754.82 | ||||

#4, 35.5' | 242 | 242 | $23.71 | $5,738.79 | ||||

$50,698.95 |

# Final Drawings

^{1)}Φ = ½√5 + ½ ≈ 1.618; 15'Φ

^{-4}≈ 2.189' ≈ 26”

## Discussion