EN1992: Design of Concrete Structures

Formula sheet for the design of rectangular concrete sections e.g. beams and slabs.
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School of Civil Engineering

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Concrete Design, Formulae and Design Charts to EN1992

Table 1: Flexural Design Formulae for Rectangular Sections (Beams & Slabs)

Lever Arm (Z)

When K ≤ K' (0.168), then

Z= d 2 [ 1+ 13.53K ] 0.95d

When K > K' (0.168), then

Z= d 2 [ 1+ 13.53 K ' ] 0.95d
Where      K= M Ed b d 2 f ck
and,      K ' =0.168

Area of Compression Reinforcement

A s2,req'd = ( K K ' ) f ck b d 2 f sc ( d d 2 ) Where       f sc =700 [ x d 2 x ] f yd
and,     x= dz 0.4 0.6d

Area of Tension Reinforcement

When K ≤ K' (0.168), then

A s,req'd = M Ed f yd Z

When K > K' (0.168), then

A s,req'd = K' f ck b d 2 f yd Z A s2,req'd f sc f yd

Maximum & Minimum Reinforcement Requirement

A s,max =0.04 A c
A s,min = 0.26 f ctm b t d f yk
    where    f ck 25N/m m 2
fck fctm Minimum
percentage
28 2.8 0.14%
30 2.9 0.15%
32 3.0 0.16%
35 3.2 0.17%
fck & fctm in N/mm2

Table 2: Shear Design Formulae for Rectangular Sections (Beams & Slabs)

Design Shear Stress

v Ed = V Ed b w Z

Shear Stress Capacity

v Rd,c =0.12k ( 100 ρ 1 f ck ) 1 3 0.035 k 1.5 f ck 0.5
where   k=1+ ( 200 /d ) 2
and    ρ 1 = A s,req'd bd 0.2

Shear Strut Angle

Where

v Rd,max cotθ=2.5 v Ed v Rd,max cotθ=1.0
θ=0.55 Sin 1 [ v Ed 0.20 f ck ( 1 f ck / 250 ) ]
fck vRd,max cotθ=2.5 vRd,max cotθ=1.0
28 3.43 1.97
30 3.64 5.28
32 3.84 5.58
35 4.15 6.02
All figures are in N/mm2

Maximum Spacing or Shear Reinforcement

s L,max =0.75d

Area of Shear Reinforcement

A sw = v Ed b w S l f ywd Cotθ

Table 3: Deflection Design Formulae for Rectangular Sections (Beams & Slabs)

Factors F1, F2 & F3

F1=1,  F2=1,  F3= A s,prov A s,req'd 1.5           NOTE: Where L (Span)>7m,  F2= 7 L eff

Reference Reinforcement Ratio

ρ 0 = f ck 0.5 1000

Tension Reinforcement Ratio

ρ= A s,req'd bd

Compression Reinforcement Ratio

ρ'= A s2,req'd bd

Basic Span to Effective Depth Ratios

ρ ρ 0 l d =K[ 11+ 1.5 f ck × ρ 0 ρ +3.2 f ck ( ρ 0 ρ 1 ) 1.5 ]

For simply supported slabs & beams, K = 1.0

K = 1.5 for the interior span condition

K = 1.3 for the end span condition

K = 0.4 for cantilevers

ρ> ρ 0 l d =K[ 11+ 1.5 f ck × ρ 0 ( ρρ' ) + f ck 12 ρ' ρ 0 ]

Table 4: Bending Moment and Shear Coefficients for Continuous One-way Spanning Slabs

End Support / Slab Connection

First Interior Support

Interior Span

Interior Supports

Pinned

Continuous

End Support

End Span

End Support

End Span

Moment

0

0.086Fl

-0.04Fl

0.075Fl

-0.086Fl

0.063

-0.063Fl

Shear

0.40F

0.46F

0.6F

0.5F

Notes

1

Applicable to one-way spanning slabs where the area of eachbay exceeds 30m2

2

Qk ≤ 1.25 Gk and qk ≤ 5.0kN/m2

3

F is the total design load, l id the span

4

Minimum span ≥ 0.85 longest span, minimum 3 No. spans

5

Based on 20% redistribution at supports and no decrease in the span moments

Table 5: Bending Moment Coefficients for Two-way Spanning Slabs Simply Supported on Four Sides

ly/lx

1.0

1.1

1.2

1.3

1.4

1.5

1.75

2.0

αsx

0.062

0.074

0.084

0.093

0.099

0.104

0.113

0.118

αsy

0.062

0.061

0.059

0.055

0.051

0.046

0.037

0.029

Table 6: Bending Moment Coefficients for Two-way Spanning Continuous Slabs Supported by Beams on Four Sides

Type of Panel and Moment Considered

Short Span Coefficient for Values of ly/lx

Long Span Coefficient for all Values of ly/lx

1.0

1.25

1.5

1.75

2.0

Interior Panel

     Negative Moment at Continuous Edge

0.031

0.044

0.053

0.059

0.063

0.032

     Positive Moment at Mid-span

0.024

0.034

0.040

0.044

0.048

0.024

One Short Edge Discontinuous

     Negative Moment at Continuous Edge

0.039

0.050

0.058

0.063

0.067

0.037

     Positive Moment at Mid-span

0.029

0.038

0.043

0.047

0.050

0.028

One Long Edge Discontinuous

     Negative Moment at Continuous Edge

0.039

0.059

0.073

0.083

0.089

0.037

     Positive Moment at Mid-span

0.030

0.045

0.055

0.062

0.067

0.028

Two Adjacent Edges Discontinuous

     Negative Moment at Continuous Edge

0.047

0.066

0.078

0.087

0.093

0.045

     Positive Moment at Mid-span

0.036

0.049

0.059

0.065

0.070

0.034

Table 7: Bending Moment and Shear Coefficients for Continuous Beams

Moment

Shear

Outer Support

25% of span moment

0.45(Gd + Qd)

Near Middle of End Support

0.090Gdl + 0.100Qdl

At First Interior Support

-0.094(Gd + Qd)l

0.63(Gd + Qd)a

At Middle Interior Support

0.066Gdl + 0.086Qdl

At Interior Support

-0.075(Gd + Qd)l

0.5(Gd + Qd)

Key

a

0.55(Gd + Qd) may be used adjacent to the interior span.

Notes

1

Redistribution of support moments by 15% has been included.

2

Applicable to three or more spans only, and where Qk ≤ Gk.

3

Minimum span ≥ 0.85 longest span.

4

l is the span, Gd is the design value of the permanent actions at ULS and Qd is the design value of the variable actions at ULS.