Mathcad - Column C-1

ACI 318 RECTANGULAR COLUMN DESIGN OF REINFORCED CONCRETE RECTANGULAR COLUMN TABLE OF CONTENTS PAGE

CONTENTS

2

A.

INPUT DATA

4

B.

DESIGN LOADS

5

C.

ANALYSIS RESULTS

7

D.

SUMMARY

Page 1 of 7

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN A.

INPUT DATA A.1

COLUMN := "C1"

MATERIAL PROPERTIES Concrete: Compressive Strength

fc' := 21MPa

Modulus of Elasticity

Ec := 4700 ⋅ fc' ⋅ MPa

Concrete strain

εc := 0.003

Ec = 21538 MPa

Reinforcing Steel: Yield Strength of Steel

fy := 275MPa

Modulus of Elasticity

Es := 2 × 10 MPa

5

Capacity Reduction Factor

A.2

A.3

Flexure and Compression

ϕc := 0.70

Flexure and Tension

ϕt := 0.90

COLUMN DIMENSIONS Dimension parallel to x-axis

b := 200mm

Dimension parallel to y-axis

t := 400mm

BAR DESIGNATIONS, SIZES AND AREAS Table No

0

1

2

3

4

5

6

7

8

9

db (mm)

0

0

8

10

12

16

20

22

25

28

30

As (mm²)

0

0

50

100

127

200

300

387

500

616

700

T

T

No := No

Example for bar at

A.4

T

dia := db mm

10

2

As := As mm bar := 4

Nobar = 4

Bar diameter is:

diabar = 12mm

Area of bar is:

As bar = 127 mm

2

COLUMN REINFORCEMENTS Diameter of main reinforcements Bar designation no.

5

Bar diameter

∅ bar := diabar

∅ bar = 16mm

Area of one (1) bar

Ab := As bar

Ab = 200 mm

2

Diameter of ties Tie bar designation no. Tie diameter

∅ ties := diaties

∅ ties = 10mm

Area of one (1) tie bar

Aties := As ties

Aties = 100 mm

3

Page 2 of 7

2

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN Longitudinal bars arrangement No. of bars along b side (one side only) No. of bars along t side (one side only)

Nbs := 2 Nts := 3 Nr := 2 ⋅ ( Nbs + Nts − 2 )

Total No. of bars Concrete cover

A.5

Nr = 6

c c := 40mm

CROSS SECTION

X-axis

SKETCH PLAN

Y-axis

A.6

LIMITS OF REINFORCEMENTS, ACI 318 SEC. 5.10.9 Maximum area of reinforcement Minimum area of reinforcement

As Ag As Ag

≤ 0.08 ≥ 0.01 2

Total steel areas of reinforcements

Ast := Nr ⋅ Ab

Ast = 1200mm

Gross Area of Column

Ag := b ⋅ t

Ag = 80000 mm

Reinforcement ratio

Ast Ag

2

= 0.015

Page 3 of 7

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN ACI_3185.10.9 :=

"OK, reinforcement ratio is within the limits" if 0.01 ≤ Ast

"N.G., overreinforced" if

Ag

"N.G., underreinforced" if

Ast Ag

≤ 0.08

> 0.08

Ast Ag

< 0.01

ACI_3185.10.9 = "OK, reinforcement ratio is within the limits"

B.

DESIGN LOADS From STAAD Analysis and Design Output

STAAD_File := "2-Storey Residential.std"

Member := 53

Case Considered

Maximum Moment Z-axis

Load_Casecol = 206 Axial Force

Pucol = 86.44 kN

Moment about X-axis

Muzcol = 33.13 kN m

Moment about Y-axis

Muycol = 0.53kN m

Shear along X-axis

Vuzcol = − 0.23 kN

Shear along Y-axis

Vuycol = 12.84kN

Page 4 of 7

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN C.

ANALYSIS RESULTS C.1 1.5× 10

INTERACTION DIAGRAMS X-Axis Interaction Diagram

3

X-axis Flexure and Axial Load Interaction Diagram Points

øPnx (kN)

1× 10

Location

3

500

0

20

40

60

øPnx (kN)

øMnx (kN-m) ey (mm)

Comments

Pt. #1

1215.61

0.00

0.00

Nom. max. compression = øPo

Pt. #2

972.48

0.00

0.00

Allowable øPn (max) = 0.8*øPo

Pt. #3

972.48

38.34

39.43

Min. eccentricity

Pt. #4

863.22

49.93

57.84

0%

rebar tension = 0.0 MPa

Pt. #5

758.03

58.32

76.93

25%

rebar tension = 68.8 MPa

Pt. #6

666.76

64.01

96.00

50%

rebar tension = 137.5 MPa

Pt. #7

517.87

71.14

137.37

100%

rebar tension = 275.0 MPa

80 Pt. #8

168.00

58.65

349.13

Pt. #9

0.00

Pt. #10

-297.00

øPn = 0.1*f c'*A g

47.56 (infinity) 0.00

0.00

Pure moment capacity Pure axial tension capacity

− 500 øMnx (kN-m) Interaction Pt. 1 Interaction Pts 2 to 10 Load Case

1.5× 10

Y-Axis Interaction Diagram

3

Y-axis Flexure and Axial Load Interaction Diagram Points

øPny (kN)

1× 10

Location

3

500

0

20

40

60

øPny (kN)

øMny (kN-m) ex (mm) 0.00

Comments

Pt. #1

1215.61

0.00

Nom. max. compression = øPo

Pt. #2

972.48

Pt. #3

972.48

0.00

0.00

Allowable øPn (max) = 0.8*øPo

17.65

18.15

Min. eccentricity

Pt. #4

711.26

28.46

40.01

0%

rebar tension = 0.0 MPa

Pt. #5

620.37

30.57

49.27

25%

rebar tension = 68.8 MPa

Pt. #6

541.04

31.94

59.04

50%

rebar tension = 137.5 MPa

Pt. #7

392.56

33.06

84.23

100%

rebar tension = 275.0 MPa

80Pt. #8

168.00

25.10

149.43

Pt. #9

0.00

Pt. #10

-297.00

22.00 (infinity) 0.00

0.00

øPn = 0.1*f c'*A g Pure moment capacity Pure axial tension capacity

− 500 øMny (kN-m)

Page 5 of 7

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN C.2

COLUMN CAPACITY Member Uniaxial Capacity at Design Eccentricity, ey: Axial

ϕPnx = 152.643kN

Moment

ϕMnx = 58.504kN ⋅ m

Eccentricity

ey = 383.272mm

Member Uniaxial Capacity at Design Eccentricity, ex: Axial

ϕPny = 972.485kN

Moment

ϕMny = 22.002kN ⋅ m

Eccentricity

ex = 6.131 mm

Biaxial Capacity and Stress Ratio for P u ≥ 0.1fc' ⋅ Ag 1 1 1 1 = + − P rxy P rx Pry ϕPo Factored Axial Resistance on the basis that only eccentricity ey is present

P rx := ϕPnx

P rx = 152.643kN

Factored Axial Resistance on the basis that only eccentricity ex is present

P ry := ϕPny

P ry = 972.485kN

P o = 0.85fc' ⋅ ( Ag − Ast) + fy ⋅ Ast

P o = 1736.58kN

Factored Maximum Axial Resistance

ϕPo := ϕc ⋅ P o

ϕPo = 1215.606 kN

Factored Axial Resistance in Biaxial Flexure

P rxy :=

⎛ 1 ⎜P ⎝ rx

⎛ ⎝

1 1 ⎞ 1 + − ⎟ Pry ϕPo ⎠

SR := if ⎜Pucol ≥ 0.1fc' ⋅ Ag ,

Stress Ratio

Pucol P rxy

P rxy = 147.997kN

⎞ ⎠

, "Not Applicable" ⎟

SR = "Not Applicable" Biaxial Stress Ratio for Pu < 0.1fc' ⋅ Ag

⎛ Mutcol ⎞ ⎜ ⎟ ⎝ Mrx ⎠

1.15

⎛ Mulcol ⎞ ⎟ ⎝ Mry ⎠

+⎜

1.15

≤ 1.0

Factored Moment of Resistance

Mrx := ϕMnx

Mrx = 58.504kN ⋅ m

Factored Moment of Resistance

Mry := ϕMny

Mry = 22.002kN ⋅ m

Page 6 of 7

LNT4 - Dec.2011

ACI 318 RECTANGULAR COLUMN Stress Ratio

⎡

⎛ Muzcol ⎞ ⎟ ⎝ Mrx ⎠

SR := if ⎢Pucol < 0.1fc' ⋅ Ag , ⎜

⎣

1.15

⎛ Muycol ⎞ ⎟ ⎝ Mry ⎠

+⎜

1.15

⎤

, "Not Applicable"⎥

⎦

SR = 0.534

D.

SUMMARY

COLUMN = "C1"

COLUMN DIMENSIONS Dimension parallel to x-axis

b = 200 mm

Dimension parallel to y-axis

t = 400 mm

VERTICAL REINFORCEMENTS

VERTICAL_BARS = "6 - 16mm Ø bars" TRANSVERSE REINFORCEMENT Hoop/Stirrups Size

∅ ties = 10mm

Spacing of hoops

Spchoops := min ( 16 ∅ bar , 48 ∅ ties , b , t) Spchoops = 200 mm

TIES = "10mm Ø bars, 1 at 50, rests at 200mm O.C. to midheight"

Page 7 of 7

LNT4 - Dec.2011