Mathcad - Column C-1

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

A.

INPUT DATA

COLUMN:="C1"

A.1

MATERIAL PROPERTIES

Concrete:

Compressive Strength fc':=21MPa

Modulus of Elasticity Ec:=4700⋅ fc'⋅MPa Ec =21538MPa

Concrete strain εc:=0.003

Reinforcing Steel:

Yield Strength of Steel fy:=275MPa

Modulus of Elasticity Es:=2 10× 5MPa

Capacity Reduction Factor

Flexure and Compression ϕc:=0.70

Flexure and Tension ϕt:=0.90

A.2

COLUMN DIMENSIONS

Dimension parallel to x-axis b:=200mm Dimension parallel to y-axis t:=400mm

A.3

BAR DESIGNATIONS, SIZES AND AREAS

Table

No 0 1 2 3 4 5 6 7 8 9 10

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 No:=NoT dia:=d_bT mm As:=AsTmm2

Example for bar at bar:=4 No_bar=4 Bar diameter is: dia_bar=12mm Area of bar is: Asbar= 127mm2

A.4

COLUMN REINFORCEMENTS

Diameter of main reinforcements

5

Bar designation no.

Bar diameter ∅bar:=diabar ∅bar=16mm

Area of one (1) bar Ab:=Asbar Ab =200mm2

Diameter of ties

3

Tie bar designation no.

Tie diameter ∅ties:=diaties ∅ties= 10mm

Longitudinal bars arrangement No. of bars along b side

(one side only) Nbs:=2

No. of bars along t side

(one side only) Nts:=3

Total No. of bars Nr:=2 N⋅

(

bs+Nts−2

)

Nr=6

Concrete cover cc:=40mm

A.5

CROSS SECTION

SKETCH PLAN

Y-axis

X-axis

A.6

LIMITS OF REINFORCEMENTS, ACI 318 SEC. 5.10.9

Maximum area of reinforcement As Ag ≤ 0.08

Minimum area of reinforcement As Ag ≥ 0.01

Total steel areas of reinforcements A_st:=N_r⋅A_b A_st=1200mm2 Gross Area of Column A_g:=b t⋅ A_g =80000mm2

Reinforcement ratio Ast

ACI_318_5.10.9 "OK, reinforcement ratio is within the limits" 0.01 Ast Ag ≤ ≤0.08 if "N.G., overreinforced" Ast A_g >0.08 if "N.G., underreinforced" Ast A_g <0.01 if :=

ACI_318_5.10.9= "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_Case_col=206

Axial Force Pu_col= 86.44kN Moment about X-axis Muz_col= 33.13kN m Moment about Y-axis Muy_col= 0.53kN m Shear along X-axis Vuz_col=−0.23kN Shear along Y-axis Vuycol=12.84kN

C.

ANALYSIS RESULTS

C.1

INTERACTION DIAGRAMS

0 20 40 60 80 500 − 500 1 103× 1.5 103× Interaction Pt. 1 Interaction Pts 2 to 10 Load Case

X-Axis Interaction Diagram

øMnx (kN-m) øPnx (kN) 0 20 40 60 80 500 − 500 1 103× 1.5 103×

Y-Axis Interaction Diagram

øMny (kN-m)

øPny (kN)

X-axis Flexure and Axial Load Interaction Diagram Points

Location 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 Pt. #8 168.00 58.65 349.13 øPn = 0.1*fc'*Ag

Pt. #9 0.00 47.56 (infinity) Pure moment capacity Pt. #10 -297.00 0.00 0.00 Pure axial tension capacity

Y-axis Flexure and Axial Load Interaction Diagram Points

Location øPny (kN) øMny (kN-m) ex (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 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 Pt. #8 168.00 25.10 149.43 øPn = 0.1*fc'*Ag

Pt. #9 0.00 22.00 (infinity) Pure moment capacity Pt. #10 -297.00 0.00 0.00 Pure axial tension capacity

C.2

COLUMN CAPACITY

Member Uniaxial Capacity at Design Eccentricity, e _y :

Axial ϕPnx= 152.643kN

Moment ϕMnx=58.504kN m⋅

Eccentricity ey= 383.272mm

Member Uniaxial Capacity at Design Eccentricity, e _x :

Axial ϕPny= 972.485kN

Moment ϕMny=22.002kN m⋅

Eccentricity ex= 6.131mm

Biaxial Capacity and Stress Ratio for Pu≥0.1fc'⋅Ag

1 P_rxy 1 P_rx 1 P_ry + 1 ϕP_o − =

Factored Axial Resistance on the basis that only

eccentricity e_y is present Prx:=ϕPnx Prx=152.643kN

Factored Axial Resistance on the basis that only

eccentricity e_x is present Pry:=ϕPny Pry=972.485kN

Po=0.85fc'⋅

(

Ag−Ast

)

+fy⋅Ast Po=1736.58kN

Factored Maximum Axial

Resistance _{ϕPo:=ϕc⋅Po ϕPo=1215.606 kN}

Factored Axial Resistance in

Biaxial Flexure Prxy 1

1 P_rx 1 P_ry + 1 ϕPo −

⎛⎜

⎝

⎞⎟

⎠

:= Prxy =147.997kN

Stress Ratio SR if Pucol≥0.1fc'⋅Ag Pucol

P_rxy , , "Not Applicable"

⎛

⎜

⎝

⎞

⎟

⎠

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

Mutcol M_rx

⎛

⎜

⎝

⎞

⎟

⎠

1.15 _Mul col M_ry

⎛

⎜

⎝

⎞

⎟

⎠

1.15 + ≤ 1.0

Factored Moment of Resistance Mrx:=ϕMnx Mrx= 58.504kN m⋅

Stress Ratio

SR if Pu_col<0.1f_c'⋅A_g Muzcol M_rx

⎛

⎜

⎝

⎞

⎟

⎠

1.15 _Muy col M_ry

⎛

⎜

⎝

⎞

⎟

⎠

1.15 + , , "Not Applicable"

⎡

⎢

⎣

⎤

⎥

⎦

:= SR=0.534

D.

SUMMARY

COLUMN="C1" COLUMN DIMENSIONS

Dimension parallel to x-axis b=200mm Dimension parallel to y-axis t=400mm VERTICAL REINFORCEMENTS

VERTICAL_BARS="6 - 16mm Ø bars" TRANSVERSE REINFORCEMENT

Hoop/Stirrups Size ∅_ties= 10mm

Spacing of hoops Spc_hoops:=min 16 ∅

(

_bar, 48 ∅_ties, tb,

)

Spc_hoops=200mm