Mathcad - Beam RB-1

ACI 318RECTANGULAR BEAM DESIGN

DESIGN OF REINFORCED CONCRETE BEAMTABLE OF CONTENTS

PAGE CONTENTS2 A. INPUT DATA

4 B. DESIGN LOADS

5 C. ANALYSIS RESULTS

7 D. SUMMARY

Design of Reinforced Concrete Beam Page 1 of 7 LNT4 - Dec.2011


A. INPUT DATA

A.1 MATERIAL PROPERTIESConcrete:

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 105MPa×:=

Capacity Reduction Factor

Shear ϕv 0.85:=

Flexure ϕf 0.90:=

A.2 BAR DESIGNATIONS, SIZES AND AREASTableNo 0 1 2 3 4 5 6 7 8 9 10db (mm) 0 0 8 10 12 16 20 22 25 28 30As (mm²) 0 0 50 100 127 200 300 387 500 616 700

No NoT

:= dia dbT

mm:= As AsT

mm2:=

Example for bar at bar 4:= Nobar 4=

Bar diameter is: diabar 12mm=

Area of bar is: Asbar 127mm2=

A.3 BEAM DIMENSIONSBeam width b 200mm:=

Beam height h 300mm:=

A.4 BEAM REINFORCEMENTSCONCRETE COVER TO STIRRUPS cc 40mm:=

STIRRUPS

3Tie bar designation no.

Tie diameter diaTies 10mm=

Area of one (1) tie bar AsTies 100mm2=



TENSILE REINFORCEMENTS

3No. of bars

5Bar designation no.

Bar diameter diabarT16mm=

Area of one (1) bar AsbarT200mm2=

Total area of tension bars As NT AsbarT⋅:= As 600mm2=

Depth to tensile reinforcements d h cc diaTies+diabarT

2+

⎛⎜⎝

⎞⎟⎠

−:= d 242mm=

COMPRESSION REINFORCEMENTS

2No. of bars

5Bar designation no.

Bar diameter diabarC16mm=

Area of one (1) bar AsbarC200mm2=

Total area of compression bars As' NC AsbarC⋅:= As' 400mm2=

Depth to compression reinf. d' cc diaTies+diabarT

2+:= d' 58mm=

A.5 CROSS SECTION

BASIC SKETCH

Beam Width, b

Beam

Hei

ght,

h



B. DESIGN LOADS Beam "RB1":=

From STAAD Analysis and Design Output STAAD_File "2-Storey Residential.std":= Member 50:=LC 200,201,202,203,204...

End Axial Shear-Y Shear-Z Torsion Mom-Y Mom-Z Msagpos

Joint kN kN kN kN·m kN·m kN·m Msagneg

1.4D 200 27 5.19 31.51 -0.02 1.14 -0.14 21.38 -21.32

1.4D 200 158 -5.19 -18.72 0.02 -1.14 0.17 21.32 21.38

1.2D + 1.6L + 0.5Lr 201 27 5.78 34.47 -0.22 1.39 -0.03 24.06 -25.21

1.2D + 1.6L + 0.5Lr 201 158 -5.78 -23.50 0.22 -1.39 0.41 25.21 24.06

1.2D + 1.6Lr + 0.5L 202 27 4.86 29.34 -0.08 1.10 -0.09 20.12 -20.44

1.2D + 1.6Lr + 0.5L 202 158 -4.86 -18.38 0.08 -1.10 0.23 20.44 20.12

1.2D + 1.6Lr + 0.8Wx 203 27 6.40 27.61 0.06 0.99 -0.22 19.33 -18.29

1.2D + 1.6Lr + 0.8Wx 203 158 -6.40 -16.64 -0.06 -0.99 0.11 18.29 19.33

1.2D + 1.6Lr + 0.8Wz 204 27 4.02 29.22 -3.91 0.23 2.44 20.06 -20.30

1.2D + 1.6Lr + 0.8Wz 204 158 -4.02 -18.26 3.91 -0.23 4.20 20.30 20.06

1.2D + 1.3Wx + 0.5L + 0.5Lr 205 27 8.04 30.31 0.04 1.12 -0.25 21.75 -20.46

1.2D + 1.3Wx + 0.5L + 0.5Lr 205 158 -8.04 -19.35 -0.04 -1.12 0.18 20.46 21.75

1.2D + 1.3Wz + 0.5L + 0.5Lr 206 27 4.18 32.93 -6.41 -0.10 4.07 22.93 -23.74

1.2D + 1.3Wz + 0.5L + 0.5Lr 206 158 -4.18 -21.97 6.41 0.10 6.82 23.74 22.93

0.9D + 1.3Wx 207 27 6.51 21.23 0.11 0.75 -0.25 15.37 -13.72

0.9D + 1.3Wx 207 158 -6.51 -13.00 -0.11 -0.75 0.06 13.72 15.37

0.9D + 1.3Wz 208 27 2.65 23.85 -6.34 -0.47 4.07 16.56 -17.00

0.9D + 1.3Wz 208 158 -2.65 -15.63 6.34 0.47 6.71 17.00 16.56

1.2D + 1.0Ex + 0.5L 209 27 6.93 33.05 -3.56 1.22 -0.12 22.81 -24.07

1.2D + 1.0Ex + 0.5L 209 158 -6.93 -22.09 3.56 -1.22 6.17 24.07 22.81

1.2D + 1.0Ez + 0.5L 210 27 4.34 32.11 -4.95 0.18 3.11 22.28 -22.98

1.2D + 1.0Ez + 0.5L 210 158 -4.34 -21.14 4.95 -0.18 5.30 22.98 22.28

0.9D + 1.0Ex 211 27 5.40 23.97 -3.49 0.84 -0.12 16.43 -17.33

0.9D + 1.0Ex 211 158 -5.40 -15.75 3.49 -0.84 6.05 17.33 16.43

0.9D + 1.0Ez 212 27 3.76 19.36 2.95 1.16 -2.24 12.96 -12.97

0.9D + 1.0Ez 212 158 -3.76 -11.14 -2.95 -1.16 -2.76 12.97 12.96

Combination for Strength Design

Load Comb



C. ANALYSIS RESULTSC.1 LIMITS OF REINFORCEMENTS

Steel reinforcement ratio ρAs

b d⋅:= ρ 0.01240=

Minimum steel reinforcement ratioNSCP Section 410.6.1

ρmin maxfc' MPa⋅

4 fy⋅1.4fy

MPa, ⎛⎜⎝

⎞⎟⎠

:= ρmin 0.00509=

Whitney stress block factorNSCP Section 410.3.7.3

β1 if fc' 30MPa≤ 0.85, max 0.850.05

7

fc'MPa

30−⎛⎜⎝

⎞⎟⎠

⋅− 0.65, ⎡⎢⎣

⎤⎥⎦

, ⎡⎢⎣

⎤⎥⎦

:=

β1 0.85=

Balanced steel reinforcement ratiofor singly reinforced beam

ρb 0.85fc'fy

β1⋅600MPa

600 MPa fy+⎛⎜⎝

⎞⎟⎠

⋅:= ρb 0.03783=

Maximum steel ratio for singly reinforcedbeam ρmax 0.75ρb:= ρmax 0.02837=

Analyse_as "SINGLY REINFORCED BEAM" ρ 0.75ρb≤if

"DOUBLY REINFORCED BEAM" otherwise

:=

Analyse_as "SINGLY REINFORCED BEAM"=

C.2 SINGLY REINFORCED BEAM ANALYSISGuess values

c

a

⎛⎜⎝

⎞⎟⎠

0.3 h

β1 c

⎛⎜⎝

⎞⎟⎠

:= fs fy:=

T C=

Given As fs⋅ 0.85 fc'⋅ a⋅ b⋅=

where fs min fy 600 MPad c−

c⋅, ⎛⎜⎝

⎞⎟⎠=

ca

β1=

Solve for a and c

fs

c

a

⎛⎜⎜⎝

⎞⎟⎟⎠

Find fs c, a, ( ):=

Hence c 54.4mm=

a 46.2mm=

fs 275.0MPa=

Solve for moment capacity of beam:

C 0.85fc' a⋅ b⋅:= C 165.0kN=

Mu.SRB ϕf C⋅ da2

−⎛⎜⎝⎞⎟⎠

⋅:= Mu.SRB 32.5kN m=



C.3 MOMENT CAPACITY CHECK

The ultimate moment capacity of beam is equal to Mu.cap 32.5kN m=

Check positive moment

Mu.pos min Musag( ):= Mu.pos 25.21− kN m=

if Mu.cap Mu.pos≥ "Beam is SAFE for Positive Bending", "REDESIGN", ( ) "Beam is SAFE for Positive Bending"=

Check negative moment

Mu.neg max Musag( ):= Mu.neg 24.06kN m=

if Mu.cap Mu.neg≥ "Beam is SAFE for Negative Bending", "REDESIGN", ( ) "Beam is SAFE for Negative Bending"=

C.4 SHEAR REINFORCEMENTArea of shear reinforcement provided:

Av 2 AsTies⋅:= Av 200mm2=

Factored shear force:Vu max Vuy( ):= Vu 34.47kN=

Shear strength provided by concrete:

Vc16

fc' MPa⋅⋅ b⋅ d⋅:= Vc 36.97kN=



Calculate required spacing of shear reinforcement:

S

sAv fy⋅ d⋅

Vu

ϕvVc−

←

s min sd2

, 600mm, ⎛⎜⎝⎞⎟⎠

←Vu

ϕvVc−

13

fc' MPa⋅⋅ b⋅ d⋅≤if

s min sd4

, 300mm, ⎛⎜⎝⎞⎟⎠

←Vu

ϕvVc−

13

fc' MPa⋅⋅ b⋅ d⋅>if

Vu

ϕvVc−

23

fc' MPa⋅ b⋅ d⋅( )⋅≤if

"Adjust the size of beam"Vu

ϕvVc−

23

fc' MPa⋅ b⋅ d⋅( )⋅>if

Vu ϕv Vc⋅>if

s3

fyMPa

⋅ Av⋅

b←

s min sd2

, 600mm, ⎛⎜⎝⎞⎟⎠

←

12

ϕv Vc⋅ Vu< ϕv Vc⋅<if

"Stirrups are not required"

mind2

500mm, ⎛⎜⎝⎞⎟⎠

Vu12

ϕv Vc⋅<if

:= S 121mm=

D. SUMMARY BEAM DIMENSIONS

Width b 200mm=

Height h 300mm=

REINFORCEMENTS

TENSION_BARS "3 - 16mm Ø bars"=

COMPRESSION_BARS "2 - 16mm Ø bars"=

STIRRUPS "10mm Ø bars, 1 at 50, rests at 100mm O.C. to midspan"=

ULTIMATE MOMENT CAPACITY

Mu.cap 32.5kN m=


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Mathcad - Beam RB-1