Design the multi-storey building with the flat slab
construction
DATA:
- site location: Lębork
- structural class: S6
- exposure class: X0
kN
qk := 4.0
- category of use: D1 - general retail shops
m2
- concrete: C20/25
- steel grade: BSt500, class: C
lcol := 3.7m
n := 3
lx := 5.2m
nx := 5
ly := 5.8m
ny := 5
qf := 180kPa
Ly := ly Å" ny = 29 m
Lx := lx Å" nx = 26 m
H := lcol Å" n = 11.1 m
1. PRELIMINARY ANALYSIS AND DESIGN
1.1. Preliminary analysis and design - first floor plan of building
hslab := 0.24m
- Columns
ëÅ‚ öÅ‚
ly lx lcol
ëÅ‚ öÅ‚
lcmx := 0.5hslab = 0.12 m
bcol := max , , , 35cm = 0.35 m
ìÅ‚ ÷Å‚
20 20 15
íÅ‚ Å‚Å‚
hcol := bcol = 0.35 m
- Cantiliver slab
lcx := 0.15 Å" lx = 0.78 m lcx := 0.25 Å" lx = 1.3 m
lcx := 1m
lcy := 0.15 Å" ly = 0.87 m lcy := 0.25 Å" ly = 1.45 m
lcy := 1.15m
- Total depth of solid RC flat slab:
lmax := max lx, ly = 5.8 m
( )
lmax lmax
ëÅ‚ öÅ‚
hslab := max , , 0.16m = 0.24 m
ìÅ‚ ÷Å‚
24 30
íÅ‚ Å‚Å‚
1.1.1 Non bearing walls - curtain walls
Clinker brick 0,120m
1 2,28
19kN/m3x0,12m
Foamed polystyrene 0,100m
2 0,045
0,45kN/m3x0,1m
3 brick wall (porotherm) 0,250m 1,994
Cement lime plaster 0,015m
4 0,285
19kN/m3x0,015m
gk[kN/m2] 4,604
gksup[kN/m2] Å‚=1,1
5,0644
gkinf[kN/m2] Å‚=0,9
4,1436
kN
gd.sup.curt.1wall := 5.064
kN
gd.supcurt := gd.sup.curt.1wall Å" lcol - hslab
( )
m2 gd.supcurt := 19.4
m
kN
kN
gd.inf.curt.1wall := 4.144
gd.infcurt := gd.inf.curt.1wall Å" lcol - hslab = 14.331 Å"
( )
m2
m
1.2 Preliminary design cross section of building
1.2.1 Snow loads PN EN 1991-1-3
Localization : Lębork III --> 32m n.p.m.
S := źi Å" Ce Å" Ct Å" sk
źi
A := 32
> 1.2
sk := 0.006 Å" A - 0.6 = -0.408
kN kN
>
sk := 1.2 1.2
m2 m2
Ce := 1
Ct := 1
ź1 := 0.4
ź2 := 0.8
Å‚f := 1.5
kN kN
sd1 := Å‚f Å" Ce Å" Ct Å" sk Å" ź1 = 0.72 Å" sd2 := Å‚f Å" Ce Å" Ct Å" sk Å" ź2 = 1.44 Å"
m2 m2
1.2.2 Roof structure
1 Water proof insulation x2 0,12
Thermal insulation PAROC ROS 30 0,02m
2 0,036
1,8kN/m3x0,02m
3 Vapour insulation 0,06
Thermal insulation PAROC ROB 60 0,16m
4 2,88
1,8kN/m3x0,16m
Weak concrete 0,04m
5 0,84
21kN/m3x0,04m
Fall layer - mineral wool
6 0,324
1,8kN/m3x0,18m
RC slab 0,23m
7 5,75
25kN/m3x0,23m
Cement lime plaster 0,015m
8 0,285
19kN/m3x0,015m
gk[kN/m2] 10,295
gksup[kN/m2] Å‚=1,1
11,3245
gkinf[kN/m2] Å‚=0,9
9,2655
kN
gdinf.roof := 6.67
m2
kN
gdsup.roof := 1.35 Å" gdinf.roof = 9.005 Å"
m2
1.2.3. Floor layers
Flooring panels 0,015m
1 0,105
7,0kN/m3x0,015m
Cement mortar 0,04m
2 0,84
21kN/m3x0,04m
Polystyrene foam 0,03m
3 0,014
0,45kN/m3x0,03m
RC slab 0,23m
4 5,75
25kN/m3x0,23m
Cement lime plaster 0,015m
5 0,285
19kN/m3x0,015m
gk[kN/m2] 6,994
gksup[kN/m2] Å‚=1,1 7,6934
gkinf[kN/m2] Å‚=0,9 6,2946
kN
gdinf.floor := 7.59
m2
kN
gdsup.floor := 1.5 Å" gdinf.floor = 11.385 Å"
m2
kN
gdinf.gf := 6.83
m2
kN
gdsup.gf := 1.5 Å" gdinf.gf = 10.245 Å"
m2
1.2.5 Imposed loads
Cathegory of load area D1
kN
qk := 4 Å‚f = 1.5
m2
kN
qd := qk Å" Å‚f = 6 Å"
m2
1.2.4. Stairs
b := 320mm
h := 160mm
2 Å" h + b = 0.64 m
1.2.5. Column self weight
kN
Gdinf := 25 Å" bcol Å" hcol Å" lcol = 11.331 Å" kN
m3
Gdsup := Gdinf Å" 1.5 = 16.997 Å" kN
1.2.7 Spot footing
Nd := + gdsup.roof + gdsup.gf + qd Å" n + gdsup.floor Å" (n - 1) Å" lx Å" ly + n Å" Gdsup = 1.905 × 103 Å" kN
îÅ‚sd2 Å‚Å‚
ðÅ‚ ûÅ‚
Nd
>
= 15.548 Å" MPa fcd := 14.3MPa
bcol Å" hcol
Nd
hcol := = 0.381 m
bcol Å" fcd
hcol := 0.4m bcol := 0.4m
kN
Gdsup := 1.5 Å" 25 Å" bcol Å" hcol Å" lcol = 22.2 Å" kN
m3
Nd := + gdsup.roof + gdsup.gf + qd Å" n + gdsup.floor Å" (n - 1) Å" lx Å" ly + n Å" Gdsup = 1.92 × 103 Å" kN
îÅ‚sd2 Å‚Å‚
ðÅ‚ ûÅ‚
kN kN
Å‚soil := 20 qf = 180 Å" m := 0.81 hdf := 1.3m
m3 m2
Nd
Af := = 16.029 m2
m Å" qf - Å‚soil Å" hdf
B := Af = 4.004 m
B := 4m
hssf1 := 0.3 Å" B - bcol = 1.08 m
( )
->
hssf := 1.2m
hssf2 := 0.4 Å" B - bcol = 1.44 m
( )
2 STRUCTURAL ANALYSIS - "EQUIVALENT FRAME METHOD"
2.1. Specification of actions on frame 1 (internal frame along x axis) - middle strip
2.1.1. Permanent action.
gdsup.floor Å" ly gdsup.roof Å" ly
kN kN
gd.supcurt Å" ly
= 44.022 Å" = 38.686 Å"
= 83.348 Å" kN
1.5 m 1.35 m
1.35
Gdsup kN gdsup.gf Å" ly Å" lx
= 4 Å" = 205.993 Å" kN
gd.supcurt Å" ly Å" lcx
lcol Å" 1.5 m 1.5
= 83.348 Å" kN Å" m
1.35
2.1.2. Imposed load on building and movable partitions
kN
kN kN
qd = 6 Å"
gd.partition := 0.375 gd.partition Å" ly = 2.175 Å"
m2
m
m2
kN kN
Assumed self weight of partition < 3 kN/m - qk' := 1.2 qk Å" ly + qk' Å" ly = 30.16 Å"
( )
m
m2
2.1.3. Snow load.
2.1.4. Snow load-on face.
sd1 Å" ly sd2 Å" ly
kN kN
= 2.784 Å" = 5.568 Å"
1.5 m 1.5 m
2.1.5. Wind action.
2.1.5.1. Wind action - wind from the left.
b := 29m
cs := 1 cd := 1 cs Å" cd = 1
h := 11.1m
kN
cpi := 0.2
qp := 0.64
e := min(b , 2 Å" h) = 22.2 m
m2
roof
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" ly = 5.197 Å" a := = 2.22 m
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" ly = 3.341 Å" a := - = 8.88 m
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" ly = 1.485 Å"
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" ly Å" lcol = 8.241 Å" kN
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" ly Å" lcol = 9.614 Å" kN
b := 29m
cs := 1 cd := 1 cs Å" cd = 1
h := 11.1m
kN
cpi := -0.3
qp := 0.64
e := min(b , 2 Å" h) = 22.2 m
m2
roof
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" ly = 3.341 Å" a := = 2.22 m
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" ly = 1.485 Å" a := - = 8.88 m
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" ly = 0.371 Å"
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" ly Å" lcol = 15.108 Å" kN
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" ly Å" lcol = 2.747 Å" kN
2.1.5.2. Wind action - wind from the right - mirror of previous scheme
2.1.5.3. Wind action - front wind.
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
cpi := 0.2
roof
kN
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" ly = 3.341 Å"
m
wall
cpeB := -0.8 wBk := cs Å" cd Å" qp Å" cpeB - cpi Å" ly Å" lcol = 13.734 Å" kN
cpi := -0.3
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" ly = 1.485 Å"
m
wall
cpeB := -0.8 wBk := cs Å" cd Å" qp Å" cpeB - cpi Å" ly Å" lcol = 6.867 Å" kN
2.1.6. Thermal action.
"t1 := -15 Å" °C "t2 := 15 Å" °C
°C °C
2.2. Specification of actions on frame 2 (internal frame along y axis) - middle strip
2.2.1. Permanent action.
gd.supcurt Å" lx gdsup.floor Å" lx gdsup.roof Å" lx
kN kN
= 74.726 Å" kN = 39.468 Å" = 34.684 Å"
1.35 1.5 m 1.35 m
Gdsup kN
gdsup.gf Å" lx Å" ly
kN
= 4 Å"
= 205.993 m Å"
lcol Å" 1.5 m
1.5 m
2.2.2. Imposed load on building and movable partitions
kN kN kN
gd.partition := 0.375 gd.partition Å" lx = 1.95 Å" qd = 6 Å"
m
m2 m2
kN kN
Assumed self weight of partition < 2 kN/m - qk' := 0.8 qk Å" lx + qk' Å" lx = 24.96 Å"
( )
m
m2
2.2.3. Snow load.
2.2.4. Snow load-on face.
sd1 Å" lx sd2 Å" lx
kN kN
= 2.496 Å" = 4.992 Å"
1.5 m 1.5 m
2.2.5. Wind action.
2.2.5.1. Wind action - wind from the left.
:= := := , Å" =
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
cpi := 0.2
roof
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" lx = 4.659 Å" a := = 2.22 m
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" lx = 2.995 Å" a := - = 8.88 m
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" lx = 1.331 Å"
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" lx Å" lcol = 7.388 Å" kN
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" lx Å" lcol = 8.62 Å" kN
cpi := -0.3
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" lx = 2.995 Å" a := = 2.22 m
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" lx = 1.331 Å" a := - = 8.88 m
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" lx = 0.333 Å"
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" lx Å" lcol = 13.545 Å" kN
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" lx Å" lcol = 2.463 Å" kN
2.2.5.2. Wind action - wind from the right - mirror of previous scheme
2.2.5.3. Wind action - front wind.
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
cpi := 0.2
roof
kN
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" lx = 2.995 Å"
m
wall
cpeB := -0.8 wBk := cs Å" cd Å" qp Å" cpeB - cpi Å" lx Å" lcol = 12.314 Å" kN
cpi := -0.3
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" lx = 1.331 Å"
m
wall
cpeB := -0.8 wBk := cs Å" cd Å" qp Å" cpeB - cpi Å" lx Å" lcol = 6.157 Å" kN
2.2.6. Thermal action.
"t1 := -15 Å" °C
°C
"t2 := 15 Å" °C
°C
2.3. Specification of actions on frame 3 (edge frame along y axis)
2.3.1. Permanent action.
gdsup.floor Å" 0.5lx + lcx
( )
kN
gd.supcurt Å" 0.5lx + lcx
( )
= 27.324 Å"
= 51.733 Å" kN
1.5 m
1.35
Gdsup kN
gdsup.roof Å" 0.5lx + lcx gdsup.gf Å" 0.5lx + lcx
( ) ( )
kN kN
= 4 Å"
= 24.012 Å" = 24.588 Å"
lcol Å" 1.5 m
1.35 m 1.5 m
2.3.2. Imposed load on building and movable partitions
kN
kN kN
qd = 6 Å"
gd.partition := 0.375 gd.partition Å" 0.5lx + lcx = 1.35 Å"
( )
m2
m
m2
kN kN
Assumed self weight of partition < 2 kN/m - qk' := 0.8 Å" 0.5lx + lcx + qk' Å" 0.5lx + lcx = 17.28 Å"
( ) ( )ûÅ‚
îÅ‚qk Å‚Å‚
ðÅ‚
m
m2
2.3.3. Snow load.
2.3.4. Snow load-on face.
sd2 Å" 0.5lx + lcx
( )
kN
sd1 Å" 0.5lx + lcx
( )
kN
= 3.456 Å"
= 1.728 Å"
1.5 m
1.5 m
2.3.5. Wind action.
2.3.5.1. Wind action - wind from the left.
cpi := 0.2
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5lx + lcx = 4.608 Å" a := = 2.22 m
( )
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" 0.5lx + lcx = 2.074 Å" a := - = 8.88 m
( )
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" 0.5lx + lcx = 0.922 Å"
( )
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" 0.5lx + lcx Å" lcol = 5.115 Å" kN
( )
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" 0.5lx + lcx Å" lcol = 5.967 Å" kN
( )
cpi := -0.3
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5lx + lcx = 3.456 Å" a := = 2.22 m
( )
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" 0.5lx + lcx = 0.922 Å" a := - = 8.88 m
( )
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" 0.5lx + lcx = 0.23 Å"
( )
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" 0.5lx + lcx Å" lcol = 9.377 Å" kN
( )
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" 0.5lx + lcx Å" lcol = 1.705 Å" kN
( )
2.3.5.2. Wind action - wind from the right - mirror of previous scheme
2.3.5.3. Wind action - front wind.
cpi := 0.2
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5lx + lcx = 4.608 Å" = 5.55 m
( )
m 4
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" 0.5lx + lcx = 3.226 Å" = 11.1 m
( )
m 2
wall
cpeA := -1.2 wAk := cs Å" cd Å" qp Å" cpeA - cpi Å" 0.5lx + lcx Å" lcol = 11.935 Å" kN
( )
cpi := -0.3
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5lx + lcx = 3.456 Å" = 5.55 m
( )
m 4
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" 0.5lx + lcx = 2.074 Å" = 11.1 m
( )
m 2
wall
cpeA := -1.2 wAk := cs Å" cd Å" qp Å" cpeA - cpi Å" lx Å" lcol = 11.082 Å" kN
2.3.6. Thermal action.
"t1 := -15 Å" °C "t2 := 15 Å" °C
°C °C
2.4. Specification of actions on frame 4 (edge frame along x axis)
2.4.1. Permanent action.
Gdsup kN
gd.supcurt Å" 0.5ly + lcy gdsup.floor Å" 0.5ly + lcy
( ) ( )
kN
= 4 Å"
= 58.2 Å" kN = 30.739 Å"
lcol Å" 1.5 m
1.35 1.5 m
gdsup.roof Å" 0.5ly + lcy gdsup.gf Å" 0.5ly + lcy Å" 0.5lx + lcx
( ) ( ) ( )
kN kN
= 27.013 Å" = 99.581 m Å"
1.35 m 1.5 m
2.4.2. Imposed load on building and movable partitions
kN kN kN
gd.partition := 0.375 gd.partition Å" 0.5ly + lcy = 1.519 Å" qd = 6 Å"
( )
m
m2 m2
kN kN
Assumed self weight of partition < 2 kN/m - qk' := 0.8 Å" 0.5ly + lcy + qk' Å" 0.5ly + lcy = 19.44 Å"
( ) ( )ûÅ‚
îÅ‚qk Å‚Å‚
ðÅ‚
m
m2
2.4.3. Snow load.
2.4.4. Snow load-on face.
sd1 Å" 0.5ly + lcy sd2 Å" 0.5ly + lcy
( ) ( )
kN kN
= 1.944 Å" = 3.888 Å"
1.5 m 1.5 m
2.4.5. Wind action.
2.4.5.1. Wind action - wind from the left.
cpi := 0.2
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5ly + lcy = 5.184 Å" a := = 2.22 m
( )
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" 0.5ly + lcy = 2.333 Å" a := - = 8.88 m
( )
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" 0.5ly + lcy = 1.037 Å"
( )
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" 0.5ly + lcy Å" lcol = 5.754 Å" kN
( )
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" 0.5ly + lcy Å" lcol = 6.713 Å" kN
( )
cpi := -0.3
b := 29m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof
kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5ly + lcy = 3.888 Å" a := = 2.22 m
( )
m 10
kN e e
cpeH := -0.7 wHk := cs Å" cd Å" qp Å" cpeH - cpi Å" 0.5ly + lcy = 1.037 Å" a := - = 8.88 m
( )
m 2 10
kN
cpeI := -0.2 wIk := cs Å" cd Å" qp Å" cpeI - cpi Å" 0.5ly + lcy = 0.259 Å"
( )
m
left wall
cpeD := 0.8 wDk := cs Å" cd Å" qp Å" cpeD - cpi Å" 0.5ly + lcy Å" lcol = 10.549 Å" kN
( )
right wall
cpeE := -0.5 wEk := cs Å" cd Å" qp Å" cpeE - cpi Å" 0.5ly + lcy Å" lcol = 1.918 Å" kN
( )
2.4.5.2. Wind action - wind from the right - mirror of previous scheme
2.4.5.3. Wind action - front wind.
cpi := 0.2
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5ly + lcy = 5.184 Å" = 5.55 m
( )
m 4
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" 0.5ly + lcy = 3.629 Å" = 11.1 m
( )
m 2
wall
cpeA := -1.2 wAk := cs Å" cd Å" qp Å" cpeA - cpi Å" 0.5ly + lcy Å" lcol = 13.427 Å" kN
( )
cpi := -0.3
b := 26m h := 11.1m e := min(b , 2 Å" h) = 22.2 m
roof kN e
cpeF := -1.8 wFk := cs Å" cd Å" qp Å" cpeF - cpi Å" 0.5ly + lcy = 3.888 Å" = 5.55 m
( )
m 4
kN e
cpeG := -1.2 wGk := cs Å" cd Å" qp Å" cpeG - cpi Å" 0.5ly + lcy = 2.333 Å" = 11.1 m
( )
m 2
wall
cpeA := -1.2 wAk := cs Å" cd Å" qp Å" cpeA - cpi Å" 0.5ly + lcy Å" lcol = 8.631 Å" kN
( )
2.4.6. Thermal action.
"t1 := -15 Å" °C "t2 := 15 Å" °C
°C °C
3. CALCULATION OF SLAB REINFORCEMENT.
Concrete C20/25
fcd := 13.3MPa fctm := 2.2MPa fctd := 1.47MPa fck := 20MPa
Steel BSt500
fyd := 420MPa fyk := 500MPa ¾efflim := 0.50
Diameter of main reinforcement:
Õ := 12mm
Concrete cover for
cmin := 10mm
enviroment X0
allowable deviation of
"h := 5mm
concrete cover:
sitrrups diameter:
Õs := 6mm
c := cmin + Õs = 0.016 m
3.1.1. INTERNAL STRIP OF THE SLAB-equivalent frame in direction X
COLUMN STRIP
Distance of center of gravity from tensile edge:
ly
b := = 2.9 m Õ := 12mm
2
hslab := 0.24m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 170.46kN Å" m
MSd60 := MSd Å" 60% = 102.276 Å" kN Å" m
MSd60
scceff := scceff = 0.058 Õ = 0.012 m
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.06 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.179 × 10- 3 m2
fyd
As1req
n := n = 10.423
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
accepted 12
n := 11 Õ
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.244 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 7.066 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 8.03 × 10- 4 m2
b
= 263.636 Å" mm
n
Assumed 11 fi 12 every 260mm
3.1.2. INTERNAL STRIP OF THE SLAB-equivalent frame in direction X
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
ly
Õ := 8mm
b := = 2.9 m hslab = 0.24 m
2
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective depth
d := hslab - a1 d = 0.215 Å" m
According to results:
MSd := 170.46kN Å" m
MSd40 := MSd Å" 40% = 68.184 Å" kN Å" m
MSd40
scceff := scceff = 0.038
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.039 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 7.701 × 10- 4 m2
fyd
As1req
n := n = 15.321
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 8
n := 16 Õ
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 8.042 × 10- 4 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 7.133 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 8.105 × 10- 4 m2
b
= 181.25 Å" mm
n
Assumed 16 fi 8 every 180mm
3.1.3. INTERNAL STRIP OF THE SLAB-equivalent frame in direction X
INTERNAL COLUMN STRIP
Distance of center of gravity from tensile edge:
ly
b := = 1.45 m hslab = 0.24 m
Õ := 20mm
4
a1 := cmin + "h + Õs + 0.5Õ
a1 = 0.031 Å" m
Effective depth
d := hslab - a1 = 0.209 m d = 0.209 Å" m
According to results:
MSd := 426.52kN Å" m
MSd50 := MSd Å" 50% = 213.26 Å" kN Å" m
MSd50
scceff := scceff = 0.253
fcd Å" b Å" d2
¾eff := 1 - 1 - 2 Å" scceff ¾ = 0.297 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 2.854 × 10- 3 m2
fyd
As1req
n := n = 9.084
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 20
Õ:=
n 10 Õ
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 3.142 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 3.467 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 3.94 × 10- 4 m2
b
= 145 Å" mm
n
Assumed 10 fi 20 every 145mm
3.1.4. INTERNAL STRIP OF THE SLAB-equivalent frame in direction X
EXTERNAL COLUMN STRIP
Distance of center of gravity from tensile edge:
ly
b := = 1.45 m hslab = 0.24 m Õ := 12mm
4
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 426.52kN Å" m
MSd25 := MSd Å" 25% = 106.63 Å" kN Å" m
MSd25
scceff := scceff = 0.122
fcd Å" b Å" d2
¾eff := 1 - 1 - 2 Å" scceff ¾ = 0.13 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.275 × 10- 3 m2
fyd
As1req
n :=
n = 11.274
ëÅ‚ öÅ‚
accepted 12
Ä„ Å" Õ2 n := 12 Õ
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.357 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 3.533 × 10- 4 m2
fyk
b
As1.min2 := 0.0013 Å" b Å" d = 4.015 × 10- 4 m2
= 120.833 Å" mm
n
Assumed 12 fi 12 every 120mm
3.1.5. INTERNAL STRIP OF THE SLAB-equivalent frame in direction X
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
ly
Õ := 12mm
b := = 2.9 m hslab = 0.24 m
2
a1 := cmin + "h + Õs + 0.5Õ
a1 = 0.027 Å" m
Effective depth
d := hslab - a1
d = 0.213 Å" m
According to results:
MSd := 426.52kN Å" m
MSd12.5 := MSd Å" 25% = 106.63 Å" kN Å" m
MSd12.5
scceff :=
scceff = 0.061
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.063 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d
As1req = 1.231 × 10- 3 m2
fyd
As1req
accepted 12
n := n = 10.881 n := 11 Õ
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.244 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 7.066 × 10- 4 m2
fyk
b
As1.min2 := 0.0013 Å" b Å" d = 8.03 × 10- 4 m2
= 263.636 Å" mm
n
Assumed 11 fi 12 every 260mm
3.2.1. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction X
COLUMN STRIP
Distance of center of gravity from tensile edge:
lcy = 1.15 m
ly
ly = 5.8 m
b := lcy + = 2.6 m
Õ := 12mm
4
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 118.2kN Å" m
MSd80 := MSd Å" 80% = 94.56 Å" kN Å" m
MSd80
scceff := scceff = 0.06
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.062 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.091 × 10- 3 m2
fyd
As1req
n := n = 9.646
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 12
n := 10 Õ
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.131 × 10- 3 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 6.335 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 7.199 × 10- 4 m2
b
= 260 Å" mm
n
Assumed 10 fi 12 every 260mm
3.2.2. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction X
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
ly
b := = 1.45 m
Õ := 8mm
4
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective depth
d := hslab - a1 d = 0.215 Å" m
According to results:
MSd := 118.2kN Å" m
MSd20 := MSd Å" 20% = 23.64 Å" kN Å" m
MSd20
scceff := scceff = 0.027
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.027 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 2.654 × 10- 4 m2
fyd
As1req
n := n = 5.279
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
accepted
n := 6 Õ 8
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 3.016 × 10- 4 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 3.566 × 10- 4 m2
fyk
b
As1.min2 := 0.0013 Å" b Å" d = 4.053 × 10- 4 m2
= 241.667 Å" mm
n
Assumed 6 fi 8 every 240mm
3.2.3. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction X
COLUMNS STRIP
Distance of center of gravity from tensile edge:
ly
b := lcy + = 2.6 m
4
Õ := 20mm
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.031 Å" m
Effective depth
d := hslab - a1 d = 0.209 Å" m
According to results:
MSd := 299.84kN Å" m
MSd87.5 := MSd Å" 87.5% = 262.36 Å" kN Å" m
MSd87.5
scceff := scceff = 0.174
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.192 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d
fyd
As1req = 3.307 × 10- 3 m2
As1req
accepted 20
n := n = 10.525 n := 11 Õ
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 3.456 × 10- 3 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 6.216 × 10- 4 m2
fyk
b
= 236.364 Å" mm As1.min2 := 0.0013 Å" b Å" d = 7.064 × 10- 4 m2
n
Assumed 11 fi 20 every 235mm
3.2.4. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction X
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
ly
b := = 1.45 m hslab = 0.24 m Õ := 8mm
4
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective depth
d := hslab - a1 d = 0.215 Å" m
:= Å"
According to results:
MSd := 299.84kN Å" m
MSd12.5 := MSd Å" 12.5% = 37.48 Å" kN Å" m
MSd12.5
scceff := scceff = 0.042
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.043 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 4.242 × 10- 4 m2
fyd
As1req
n := n = 8.439
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted
n := 9 Õ 8
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 4.524 × 10- 4 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 3.566 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 4.053 × 10- 4 m2
b
= 161.111 Å" mm
n
Assumed 9 fi 8 every 160mm
3.3.1. INTERNAL STRIP OF THE SLAB-equivalent frame in direction Y
COLUMN STRIP
Distance of center of gravity from tensile edge:
lx
b := = 2.6 m Õ := 12mm
2
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective
depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 162.38kN Å" m
MSd60 := MSd Å" 60% = 97.428 Å" kN Å" m
MSd60
scceff := scceff = 0.062 Õ = 0.012 m
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.064 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.125 × 10- 3 m2
fyd
As1req
n := n = 9.949
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
accepted 12
n := 10 Õ
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.131 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 6.335 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 7.199 × 10- 4 m2
b
= 260 Å" mm
n
Assumed 10 fi 12 every 260mm
3.3.2. INTERNAL STRIP OF THE SLAB-equivalent frame in direction Y
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
lx
Õ := 8mm
b := = 2.6 m hslab = 0.24 m
2
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective
depth
d := hslab - a1 d = 0.215 Å" m
According to results:
MSd := 162.38kN Å" m
MSd40 := MSd Å" 40% = 64.952 Å" kN Å" m
MSd40
scceff := scceff = 0.041
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.041 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 7.345 × 10- 4 m2
fyd
As1req
n := n = 14.613
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 8
n := 15 Õ
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 7.54 × 10- 4 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 6.395 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 7.267 × 10- 4 m2
b
= 173.333 Å" mm
n
Assumed 15 fi 8 every 170mm
3.3.3. INTERNAL STRIP OF THE SLAB-equivalent frame in direction Y
COLUMN STRIP
Distance of center of gravity from tensile edge:
lx
b := = 1.3 m hslab = 0.24 m
Õ := 20mm
4
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.031 Å" m
Effective
depth
d := hslab - a1 d = 0.209 Å" m
According to results:
MSd := 468.46kN Å" m
MSd50 := MSd Å" 50% = 234.23 Å" kN Å" m
MSd50
scceff := scceff = 0.31
fcd Å" b Å" d2
¾eff := 1 - 1 - 2 Å" scceff ¾ = 0.384 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 3.302 × 10- 3 m2
fyd
As1req
n := n = 10.511
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 20
Õ:=
n 11 Õ
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 3.456 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 3.108 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 3.532 × 10- 4 m2
b
= 118.182 Å" mm
n
Assumed 11 fi 20 every 115mm
3.3.4. INTERNAL STRIP OF THE SLAB-equivalent frame in direction Y
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
lx
b := = 1.3 m hslab = 0.24 m Õ := 12mm
4
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective
depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 468.46kN Å" m
MSd25 := MSd Å" 25% = 117.115 Å" kN Å" m
MSd25
scceff := scceff = 0.149
fcd Å" b Å" d2
¾eff := 1 - 1 - 2 Å" scceff ¾ = 0.163 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.425 × 10- 3 m2
fyd
As1req
n :=
accepted 12
n = 12.599 n := 13 Õ
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 1.47 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 3.168 × 10- 4 m2
fyk
b
= 100 Å" mm As1.min2 := 0.0013 Å" b Å" d = 3.6 × 10- 4 m2
n
Assumed 13 fi 12 every 100mm
3.3.5. INTERNAL STRIP OF THE SLAB-equivalent frame in direction Y
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
lx
b := = 2.6 m hslab = 0.24 m Õ := 12mm
2
a1 := cmin + "h + Õs + 0.5Õ
a1 = 0.027 Å" m
Effective depth
d := hslab - a1
d = 0.213 Å" m
According to results:
MSd := 468.46kN Å" m
MSd12.5 := MSd Å" 25% = 117.115 Å" kN Å" m
MSd12.5
scceff :=
scceff = 0.075
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.078 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d
As1req = 1.362 × 10- 3 m2
fyd
As1req
accepted 12
n := n = 12.043 n := 13 Õ
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å" As1.prov = 1.47 × 10- 3 m2
4
fctm
As1.prov > As1.min As1.min1 := 0.26 Å" Å" b Å" d = 6.335 × 10- 4 m2
fyk
b
As1.min2 := 0.0013 Å" b Å" d = 7.199 × 10- 4 m2
= 200 Å" mm
n
Assumed 13 fi 12 every 200mm
3.4.1. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction Y
COLUMN STRIP
Distance of center of gravity from tensile edge:
lx
Õ := 12mm
b := lcx + = 2.3 m
4
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.027 Å" m
Effective depth
d := hslab - a1 d = 0.213 Å" m
According to results:
MSd := 113.36kN Å" m
MSd80 := MSd Å" 80% = 90.688 Å" kN Å" m
MSd80
scceff := scceff = 0.065
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.068 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 1.049 × 10- 3 m2
fyd
As1req
n := n = 9.277
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
accepted 12
n := 10 Õ
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 1.131 × 10- 3 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 5.604 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 6.369 × 10- 4 m2
b
= 230 Å" mm
n
Assumed 10 fi 12 every 230mm
3.4.2. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction Y
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
lx
b := = 1.3 m
Õ := 8mm
4
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective depth
d := hslab - a1 d = 0.215 Å" m
According to results:
MSd := 113.36kN Å" m
MSd20 := MSd Å" 20% = 22.672 Å" kN Å" m
MSd20
scceff := scceff = 0.028
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.029 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 2.547 × 10- 4 m2
fyd
As1req
n := n = 5.068
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
8
accepted
n := 6 Õ
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 3.016 × 10- 4 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 3.197 × 10- 4 m2
fyk
As1.min2 := 0.0013 Å" b Å" d = 3.633 × 10- 4 m2
b
= 216.667 Å" mm
n
Assumed 6 fi 8 every 215mm
3.4.3. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction Y
COLUMNS STRIP
Distance of center of gravity from tensile edge:
lx = 5.2 m
lx
b := lcx + = 2.3 m
Õ := 20mm
4
hslab = 0.24 m
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.031 Å" m
Effective depth
d := hslab - a1 d = 0.209 Å" m
According to results:
MSd := 351.59kN Å" m
MSd87.5 := MSd Å" 87.5% = 307.641 Å" kN Å" m
MSd87.5
scceff := scceff = 0.23
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.265 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d
fyd
As1req = 4.041 × 10- 3 m2
As1req
accepted
n := n = 12.863 n := 13 Õ 20
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 4.084 × 10- 3 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 5.499 × 10- 4 m2
fyk
b
As1.min2 := 0.0013 Å" b Å" d = 6.249 × 10- 4 m2
= 176.923 Å" mm
n
Assumed 13 fi 20 every 175mm
3.4.4. EXTERNAL STRIP OF THE SLAB-equivalent edge frame in direction Y
STRIP BETWEEN COLUMNS
Distance of center of gravity from tensile edge:
lx
b := = 1.3 m hslab = 0.24 m Õ := 8mm
4
a1 := cmin + "h + Õs + 0.5Õ a1 = 0.025 Å" m
Effective depth
d := hslab - a1 d = 0.215 Å" m
According to results:
MSd := 351.59kN Å" m
MSd12.5 := MSd Å" 12.5% = 43.949 Å" kN Å" m
MSd12.5
scceff := scceff = 0.055
fcd Å" b Å" d2
<
¾eff := 1 - 1 - 2 Å" scceff ¾eff = 0.057 ¾eff.lim := 0.35
fcd
As1req := ¾eff Å" Å" b Å" d As1req = 5.009 × 10- 4 m2
fyd
As1req
accepted
n := n = 9.964 n := 10 Õ 8
ëÅ‚ öÅ‚
Ä„ Å" Õ2
ìÅ‚ ÷Å‚
4
íÅ‚ Å‚Å‚
Ä„ Å" Õ2
As1.prov := n Å"
As1.prov = 5.027 × 10- 4 m2
4
fctm
As1.prov > As1.min
As1.min1 := 0.26 Å" Å" b Å" d = 3.197 × 10- 4 m2
fyk
b
= 130 Å" mm
n
As1.min2 := 0.0013 Å" b Å" d = 3.633 × 10- 4 m2
Assumed 10 fi 8 every 130mm
4. The design procedure for punching shear.
4.1.1. Internal column
d := 0.209m bcol = 0.4 m u := 4 Å" (b + 4d) = 8.544 m V := 1917.49kN - 1228.432kN = 689.058
M := 133.26kN Å" m
c1
c1 := bcol = 0.4 m c2 := hcol = 0.4 m = 1 k := 0.6
c2
c12
w1 := + c1 Å" c2 + 4 Å" c2 Å" d + 16d2 + 2 Å" Ä„ Å" d Å" c1 = 1.799 m2
2
M u
² := 1 + k Å" Å" = 1.551
V w1
V
VEd := ² = 0.599 Å" MPa
u Å" d
0.18
CRdc := = 0.12
1.5
200mm
ëÅ‚2 öÅ‚
k := min , 1 + = 1.978
ìÅ‚ ÷Å‚
As1y := 3.456 Å" 10- 3m2
d
íÅ‚ Å‚Å‚
As1z := 3.142 Å" 10- 3m2
As1y As1z
Ály := = 0.0056 Álz := = 0.0051
(b + 6d) Å" hslab (b + 6d) Å" hslab
Ál := min 0.02 , Álz Å" Ály = 0.0054
( )
1228.43kN
1278.36kN
Ãcz := = 7.678 Å" MPa
Ãcy := = 7.99 Å" MPa
bcol2
bcol2
Ãcy + Ãcz
Ãcp := = 7.834 Å" MPa
Ãcp
2
Ãcp :=
MPa
fck 0.5
ëÅ‚ öÅ‚
vmin := 0.035 Å" k1.5 Å" = 0.436 vmin + 0.1 Å" Ãcp = 1.219
ìÅ‚ ÷Å‚ ( )
MPa
íÅ‚ Å‚Å‚
0.333
îÅ‚ Å‚Å‚
ïÅ‚CRdc ëÅ‚100 fck öÅ‚ + 0.1 Å" Ãcp , vmin + 0.1 Å" Ãcpśł
vRd.c := max Å" k Å" Å" Ál Å" Å" MPa = 1.307 Å" MPa
ìÅ‚ ÷Å‚
MPa
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
<
VEd = 0.599 Å" MPa vRd.c = 1.307 Å" MPa
Punching shear reinforcement is not necessary
4.1.1. Edge column
d := 0.209m bcol = 0.4 m u := 4 Å" (b + 4d) = 8.544 m V := 1328.41kN - 856.49kN = 471.92 Å"
M := 115.85kN Å" m
M u
² := 1 + k Å" Å" = 3.307
V w1
V
VEd := ² = 0.874 Å" MPa
u Å" d
0.18
CRdc := = 0.12
1.5
200mm
ëÅ‚2 öÅ‚
k := min , 1 + = 1.978
ìÅ‚ ÷Å‚
As1z := 3.456 Å" 10- 3m2
d
íÅ‚ Å‚Å‚
As1y := 4.084 Å" 10- 3m2
As1y As1z
Ály := = 0.0067 Álz := = 0.0056
(b + 6d) Å" hslab (b + 6d) Å" hslab
Ál := min 0.02 , Álz Å" Ály = 0.0061
( )
856.49kN
777.59kN
Ãcz := = 5.353 Å" MPa
Ãcy := = 4.86 Å" MPa
bcol2
bcol2
Ãcy + Ãcz
Ãcp := = 5.106 Å" MPa
Ãcp
2
Ãcp :=
MPa
fck 0.5
ëÅ‚ öÅ‚
vmin := 0.035 Å" k1.5 Å" = 0.436
ìÅ‚ ÷Å‚
vmin + 0.1 Å" Ãcp = 0.946
MPa
íÅ‚ Å‚Å‚
0.333
îÅ‚ Å‚Å‚
ïÅ‚CRdc ëÅ‚100 fck öÅ‚ + 0.1 Å" Ãcp, vmin + 0.1 Å" Ãcpśł
vRd.c := max Å" k Å" Å" Ál Å" Å" MPa = 1.058 Å" MPa
ìÅ‚ ÷Å‚
MPa
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
<
VEd = 0.874 Å" MPa vRd.c = 1.058 Å" MPa
Punching shear reinforcement is not necessary
5. Column sizing
5.1 Ground floor internal column
column height:
lcol := 3.7m
column cross-section dimensions:
b := 0.4m
h := 0.4m
internal forces:
MSd := 206.12 Å" kNm
NSd := 1994.36 Å" kN
concrete class C20/25 steel class A-III N
concrete properties: steel properties:
fcd := 13.3MPa fyd := 420MPa
fck := 20MPa
fyk := 500MPa
fcm := fck + 8MPa
fcm = 28 Å" MPa Es := 210GPa
Ecm := 30GPa ¾eff.lim := 0.5
buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h Ac = 0.16 m2
u := 2 Å" (b + h) u = 1.6 m
2 Å" Ac
h0 := h0 = 200 Å" mm
u
RH := 50%
2. STRENGTH CALCULATIONS - BUCKLING
2.1 Effective column height
l0 := ² Å" lcol l0 = 4.317 m
2.2 Assumption of cross-section dimensions
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õ := 20mm Õs := 8mm
c' := cmin + "c = 0.02 m
a1 := cmin + "c + Õs + 0.5 Å" Õ a1 = 0.038 m
a2 := a1 = 0.038 m
d := h - a1 d = 0.362 m
Limit realive depth of the compression zone:
¾efflim := 0.5
creep coefficient (28-day concrete
Õto := 2.458
and RH=50%)
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d xeff.lim = 0.181 m
2.5 Column slenderness
l0
> 7,0 slender member
= 10.792
h
2.6 Eccentrics
2.6.a static eccentric
MSd
ee := ee = 0.103 m
NSd
2.6.b accidental eccentric
lcol 1 h
îÅ‚ Å‚Å‚
ea := max Å" + , , 0.01m ea = 0.013 m
ïÅ‚600 ëÅ‚1 5öÅ‚ śł
ìÅ‚ ÷Å‚
30
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
2.6.c initial eccentric
e0 := ee + ea e0 = 0.117 m
2.6.d total eccentric
assumption of reinforcement ratio:
Ácrt := 1.2%
NSdt
NSdt
influence of long term loading on the value of critical force:
kt := 1 + 0.5 Å" Å" Õto
NSd
kt
kt =
influence of bending moments and cracking on the value of critical force:
e0 l0 fcd
ëÅ‚ öÅ‚
² := max , 0.05 , 0.5 - 0.01 Å" - 0.01 Å" ² = 0.292
ìÅ‚ ÷Å‚
h h MPa
íÅ‚ Å‚Å‚
second moment of area for concrete section I.c about axis coming thorough centroid of
the concrete section:
b Å" h3
Ic := Ic = 2.133 × 10- 3 Å" m4
12
second moment of area for reinforcement I.s about axis coming through centroid of the
concrete section:
Is := Ácrt Å" b Å" d Å" 0.5 Å" h - a1 Is = 4.56 × 10- 5 m4
( )2
Å" Ic 0.11
9 îÅ‚Ecm Å‚Å‚
ëÅ‚ öÅ‚
conventional critical force:
Ncrt := Å" Å" + 0.1 + Es Å" Is
ïÅ‚ śł
ìÅ‚ ÷Å‚
kt
l02 2 Å" kt 0.1 + e0
ïÅ‚ śł
ìÅ‚ ÷Å‚
h
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
Ncrt
Ncrt = Å" kN
1
factor increasing eccentric:
· := · =
·
NSd
1 -
Ncrt
Ncrt
total eccentric
etot := · Å" e0 etot =
· etot
2.7 Internal forces for section sizing
MSdcorr := NSd Å" etot MSdcorr = Å" kNm
etot MSdcorr
3. STRENGTH CALCULATIONS - SECTION SIZING
3.1 Eccentric about reinforcement As1 and As2 centroids
e1 := etot + 0.5 Å" h - a1
etot
e1
e1 =
e2 := etot - 0.5 Å" h + a2
etot
e2
e2 =
3.2 Minimal reinforcement
ëÅ‚0.15 NSd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h Asmin = 7.123 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area A.s2 - moment condition (situation C2 assumed)
NSd Å" e1 - fcd Å" b Å" xeff.lim Å" d - 0.5 Å" xeff.lim
e1
( )
As2 :=
fyd Å" d - a2
( )
As2
As2 =
3.4 Reinforcement area A.s2 acceptance
Ä„ Å" Õ2
a2prov :=
a2prov = 3.142 × 10- 4 m2
4
As2
As2
assumed
= n := 9
a2prov
As2prov := a2prov Å" n = 2.827 × 10- 3 m2
As2prov = 2.827 × 10- 3 m2 > As2 =
As2
bar clearance:
c1 := max(20mm , Õ) c1 = 20 Å" mm
20mm
a2 := 20mm + 6mm + = 0.036 m
2
3.5 Correction of compression zone depth x.eff
2 Å" Å" e1 - fyd Å" As2 Å" d - a2
e1
( )ûÅ‚ xeff =
îÅ‚NSd Å‚Å‚
ðÅ‚
xeff := d - d2 - xeff
fcd Å" b
2 Å" a2 = 0.072 m
2a2 < xeff < xeff.lim
xeff.lim = 0.181 m
3.6 Reinforcement area A.s1 acceptance
-NSd + fcd Å" b Å" xeff + fyd Å" As2
xeff
As1 := As1 =
As1
fyd
3.7 Reinforcement area A.s1 acceptance
Ä„ Å" Õ2
a1prov :=
a1prov = 3.142 × 10- 4 m2
4
As1
As1
assumed
= n := 2
a1prov
As1prov := a1prov Å" n = 6.283 × 10- 4 m2
As1prov = 6.283 × 10- 4 m2 > As1 =
As1
bar clearance:
c1 := max(20mm , Õ) c1 = 20 Å" mm
20mm
a1 := 20mm + 6mm + = 0.036 m
2
3.8 Total reinforcement ratio
As1
As1 + As2
Áprov :=
b Å" d
Áprov
Áprov = Å" %
4. CHECKING CORRECTNESS OF BUCKLING PREDICTIONS
4.1 Second moment of area for reinforcement l.s correction
Iscorr := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 Iscorr =
As1 Iscorr
( )2 ( )2
4.2 Conventional buckling force correction
Å" Ic 0.11
9 îÅ‚Ecm Å‚Å‚
ëÅ‚ öÅ‚
Ncrit.corr := Å" Å" + 0.1 + Es Å" Iscorr Ncrit.corr = Å" kN
Ncrit.corr
ïÅ‚ śł
ìÅ‚ ÷Å‚
kt
l02 2 Å" kt 0.1 + e0
ïÅ‚ śł
ìÅ‚ ÷Å‚
h
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
factor increasing total eccentric 1
·corr := ·corr =
·corr
NSd
1 -
Ncrit.corr
Ncrit.corr
etot.corr := ·corr Å" e0 etot.corr =
·corr etot.corr
4.3 Internal forces for section sizing
NSd = 1.994 × 103 Å" kN
MSd := NSd Å" etot.corr MSd = Å" kNm
etot.corr MSd
4.4 Total eccentrics comparison
etot.corr
etot.corr
0.9 < < 1.03
=
etot
5. ULTIMATE CAPACITY OF THE SECTION
Column cross-section dimensions
b = 0.4 m
h = 0.4 m
Tension reinforcement
As1
As1 =
a1 = 0.036 m
Compression reinforcement
As2
As2 =
a2 = 0.036 m
Internal forces
NSd = 1.994 × 103 Å" kN
NSdt
NSdt = Å" kN
MSd := 265kNm
5.1 Effective depth of the section
d := h - a1 d = 0.364 m
5.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d xeff.lim = 0.182 m
5.3 Total eccentric
etot := etot.corr etot =
etot.corr etot
5.4 Eccentrics about centroids of tension and compression reinforcement
e1 := etot + 0.5 Å" h - a1 e1 =
etot e1
e2 := etot - 0.5 Å" h + a2 e2 =
etot e2
5.5 Depth of the compression zone of concrete (for situation C2)
- Å" + Å"
NSd - fyd Å" As2 + fyd Å" As1
As2
xeff := xeff =
xeff
fcd Å" b
2a2 = 0.072 m
2a2 < xeff < xeff.lim
xeff.lim = 0.182 m
5.6 Ultimate capacity
NSd Å" e1 = Å" kNm
e1
MRd := fcd Å" b Å" xeff Å" d - 0.5 Å" xeff + fyd Å" As2 Å" d - a2 MRd = Å" kNm
xeff MRd
( ) ( )
<
NSd Å" e1 = Å" kNm MRd = Å" kNm
e1 MRd
5.2 Ground floor edge column
5.2 Ground floor edge column
column height:
lcol := 3.7m
column cross-section dimensions:
b := 0.4m
h := 0.4m
internal forces:
MSd := 188.54 Å" kNm
NSd := 1386.71 Å" kN
NSdt := 0.6 Å" NSd
2. STRENGTH CALCULATIONS - BUCKLING
2.1 Effective column height
l0 := ² Å" lcol l0 = 4.317 m
2.2 Assumption of cross-section dimensions
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õ := 20mm Õs := 8mm
c' := cmin + "c = 0.02 m
a1 := cmin + "c + Õs + 0.5 Å" Õ a1 = 0.038 m
a2 := a1 = 0.038 m
d := h - a1 d = 0.362 m
Limit realive depth of the compression zone:
¾efflim := 0.5
creep coefficient (28-day concrete
Õto := 2.458
and RH=50%)
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d xeff.lim = 0.181 m
2.5 Column slenderness
l0
> 7,0 slender member
= 10.792
h
2.6 Eccentrics
2.6.a static eccentric
MSd
ee := ee = 0.136 m
NSd
2.6.b accidental eccentric
lcol 1 h
îÅ‚ Å‚Å‚
ea := max Å" + , , 0.01m ea = 0.013 m
ïÅ‚600 ëÅ‚1 5öÅ‚ śł
ìÅ‚ ÷Å‚
30
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
2.6.c initial eccentric
e0 := ee + ea e0 = 0.149 m
2.6.d total eccentric
assumption of reinforcement ratio:
Ácrt := 1%
NSdt
influence of long term loading on the value of critical force:
kt := 1 + 0.5 Å" Å" Õto
NSd
kt = 1.737
influence of bending moments and cracking on the value of critical force:
e0 l0 fcd
ëÅ‚ öÅ‚
² := max , 0.05 , 0.5 - 0.01 Å" - 0.01 Å" ² = 0.373
ìÅ‚ ÷Å‚
h h MPa
íÅ‚ Å‚Å‚
second moment of area for concrete section I.c about axis coming thorough centroid of
the concrete section:
b Å" h3
Ic := Ic = 2.133 × 10- 3 Å" m4
12
second moment of area for reinforcement I.s about axis coming through centroid of the
concrete section:
Is := Ácrt Å" b Å" d Å" 0.5 Å" h - a1 Is = 3.8 × 10- 5 m4
( )2
Å" Ic 0.11
9 îÅ‚Ecm Å‚Å‚
ëÅ‚ öÅ‚
conventional critical force:
Ncrt := Å" Å" + 0.1 + Es Å" Is
ïÅ‚ śł
ìÅ‚ ÷Å‚
l02 2 Å" kt 0.1 + e0
ïÅ‚ śł
ìÅ‚ ÷Å‚
h
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
Ncrt = 6.812 × 103 Å" kN
1
factor increasing eccentric:
· := · = 1.256
NSd
1 -
Ncrt
total eccentric
etot := · Å" e0 etot = 0.187 m
2.7 Internal forces for section sizing
MSdcorr := NSd Å" etot MSdcorr = 259.948 Å" kNm
3. STRENGTH CALCULATIONS - SECTION SIZING
3.1 Eccentric about reinforcement As1 and As2 centroids
e1 := etot + 0.5 Å" h - a1
e1 = 0.349 m
e2 := etot - 0.5 Å" h + a2
e2 = 0.025 m
3.2 Minimal reinforcement
ëÅ‚0.15 NSd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h Asmin = 4.953 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area A.s2 - moment condition (situation C2 assumed)
NSd Å" e1 - fcd Å" b Å" xeff.lim Å" d - 0.5 Å" xeff.lim
( )
As2 :=
fyd Å" d - a2
( )
As2 = 1.64 × 10- 3 m2
3.4 Reinforcement area A.s2 acceptance
Ä„ Å" Õ2
a2prov :=
a2prov = 3.142 × 10- 4 m2
4
As2
I assumed
= 5.22 n := 6
a2prov
As2prov := a2prov Å" n = 1.885 × 10- 3 m2
>
As2prov = 1.885 × 10- 3 m2 As2 = 1.64 × 10- 3 m2
bar clearance:
c1 := max(20mm , Õ) c1 = 20 Å" mm
20mm
a2 := 20mm + 6mm + = 0.036 m
2
3.5 Correction of compression zone depth x.eff
2 Å" Å" e1 - fyd Å" As2 Å" d - a2
( )ûÅ‚ xeff = 0.18 m
îÅ‚NSd Å‚Å‚
ðÅ‚
xeff := d - d2 -
fcd Å" b
2 Å" a2 = 0.072 m
2a2 < xeff < xeff.lim
xeff.lim = 0.181 m
3.6 Reinforcement area A.s1 acceptance
-NSd + fcd Å" b Å" xeff + fyd Å" As2
As1 := As1 = 6.129 × 10- 4 m2
fyd
3.7 Reinforcement area A.s1 acceptance
Ä„ Å" Õ2
a1prov :=
a1prov = 3.142 × 10- 4 m2
4
As1
I assumed
= 1.951 n := 2
a1prov
As1prov := a1prov Å" n = 6.283 × 10- 4 m2
>
As1prov = 6.283 × 10- 4 m2 As1 = 6.129 × 10- 4 m2
bar clearance:
c1 := max(20mm , Õ) c1 = 20 Å" mm
20mm
a1 := 20mm + 6mm + = 0.036 m
2
3.8 Total reinforcement ratio
As1 + As2
Áprov :=
b Å" d
Áprov = 1.556 Å" %
4. CHECKING CORRECTNESS OF BUCKLING PREDICTIONS
4.1 Second moment of area for reinforcement l.s correction
Iscorr := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 Iscorr = 6.059 × 10- 5 m4
( )2 ( )2
4.2 Conventional buckling force correction
Å" Ic 0.11
9 îÅ‚Ecm Å‚Å‚
ëÅ‚ öÅ‚
Ncrit.corr := Å" Å" + 0.1 + Es Å" Iscorr Ncrit.corr = 9.103 × 103 Å" kN
ïÅ‚ śł
ìÅ‚ ÷Å‚
l02 2 Å" kt 0.1 + e0
ïÅ‚ śł
ìÅ‚ ÷Å‚
h
ðÅ‚ íÅ‚ Å‚Å‚ ûÅ‚
factor increasing total eccentric 1
·corr := ·corr = 1.18
NSd
1 -
Ncrit.corr
etot.corr := ·corr Å" e0 etot.corr = 0.176 m
4.3 Internal forces for section sizing
4.3 Internal forces for section sizing
NSd = 1.387 × 103 Å" kN
MSd := NSd Å" etot.corr MSd = 244.235 Å" kNm
4.4 Total eccentrics comparison
etot.corr
0.9 < < 1.03
= 0.94
etot
5. ULTIMATE CAPACITY OF THE SECTION
Column cross-section dimensions
b = 0.4 m
h = 0.4 m
Tension reinforcement
As1 = 6.129 × 10- 4 m2
a1 = 0.036 m
Compression reinforcement
As2 = 1.64 × 10- 3 m2
a2 = 0.036 m
Internal
NSd = 1.387 × 103 Å" kN
forces
NSdt = 832.026 Å" kN
MSd := 265kNm
5.1 Effective depth of the section
d := h - a1 d = 0.364 m
5.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d xeff.lim = 0.182 m
5.3 Total eccentric
etot := etot.corr etot = 0.176 m
5.4 Eccentrics about centroids of tension and compression reinforcement
e1 := etot + 0.5 Å" h - a1 e1 = 0.34 m
e2 := etot - 0.5 Å" h + a2 e2 = 0.012 m
5.5 Depth of the compression zone of concrete (for situation C2)
5.5 Depth of the compression zone of concrete (for situation C2)
NSd - fyd Å" As2 + fyd Å" As1
xeff := xeff = 0.18 m
fcd Å" b
2a2 = 0.072 m
2a2 < xeff < xeff.lim
xeff.lim = 0.182 m
5.6 Ultimate capacity
NSd Å" e1 = 471.655 Å" kNm
MRd := fcd Å" b Å" xeff Å" d - 0.5 Å" xeff + fyd Å" As2 Å" d - a2 MRd = 487.883 Å" kNm
( ) ( )
<
NSd Å" e1 = 471.65 Å" kNm MRd = 487.88 Å" kNm
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1942.57kN
forces
M0Ed1 := 133.26kNm M0Ed2 := 120.81kNm
M0Edqp := 10.07kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.116
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed2
1
A := = 0.977 rm := = 0.907
1 + 0.2 Å" Õeff M0Ed1
C := 1.7 - rm = 0.793
B := 1.1
NEd
n := = 0.913
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 17.855
n
> slender column
 = 37.383 lim = 17.855
2.6.a Initial eccentric
M0Ed1
eo := = 0.069 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.201 k2 := 0.2
170
k1 Å" k2
Kc := = 0.179
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 20.645 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 10935.21 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.266
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.112 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 217.842 Å" kNm
NEd = 1.943 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.269 m
e2 := etot - 0.5 Å" h + a2 = -0.045 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 4.625 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.365 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.088 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = -7.807 × 10- 5 Ãs1 := min µs1 Å" Es , fyd = -16.395 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 8.903 × 10- 4 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 2.115 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.325 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.037 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 3.446 × 10- 4 Ãs1 := min µs1 Å" Es , fyd = 72.369 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.609 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.566 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsN = 1.609 × 10- 3 m2 As2 := As1 = 1.609 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 5.122
Ä„ Õg2
n1 := 6
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.885 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.026 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 9.412 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.932 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.553 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.176
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.104 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.943 × 103 Å" kN
MEd := NEd Å" e'tot = 202.375 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.929 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.885 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.885 × 10- 3 m2
internal forces:
NEd = 1.943 × 103 Å" kN
M0Ed1 = 133.26 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.104 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.263 m
e2 := etot - 0.5 Å" h + a1 = -0.055 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.456 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.456 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.312 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.312 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 511.244 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 587.038 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1942.57kN
forces
M0Ed1 := 120.81kNm M0Ed2 := 133.26kNm
M0Edqp := 5.93kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.075
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed1
1
A := = 0.985 rm := = 0.907
1 + 0.2 Å" Õeff M0Ed2
C := 1.7 - rm = 0.793
B := 1.1
NEd
n := = 0.913
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 17.998
n
> slender column
 = 37.383 lim = 17.998
2.6.a Initial eccentric
M0Ed1
eo := = 0.062 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.201 k2 := 0.2
170
k1 Å" k2
Kc := = 0.186
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 21.007 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 11126.6 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.26
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.104 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 201.2 Å" kNm
NEd = 1.943 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.261 m
e2 := etot - 0.5 Å" h + a2 = -0.053 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 4.625 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.365 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.088 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = -7.807 × 10- 5 Ãs1 := min µs1 Å" Es , fyd = -16.395 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 8.903 × 10- 4 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.852 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.333 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.048 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 2.523 × 10- 4 Ãs1 := min µs1 Å" Es , fyd = 52.973 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.431 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.435 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsM = 1.435 × 10- 3 m2 As2 := As1 = 1.435 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 4.566
Ä„ Õg2
n1 := 5
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.571 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.022 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 7.843 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.639 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.398 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.199
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.099 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.943 × 103 Å" kN
MEd := NEd Å" e'tot = 191.36 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.951 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.571 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.571 × 10- 3 m2
internal forces:
NEd = 1.943 × 103 Å" kN
M0Ed1 = 120.81 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.099 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.258 m
e2 := etot - 0.5 Å" h + a1 = -0.06 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.456 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.456 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.322 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.322 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 500.228 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 547.62 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1942.57kN
forces
M0Ed1 := 140.15kNm M0Ed2 := 130.45kNm
M0Edqp := 11.45kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.125
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed2
1
A := = 0.976 rm := = 0.931
1 + 0.2 Å" Õeff M0Ed1
C := 1.7 - rm = 0.769
B := 1.1
NEd
n := = 0.913
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 17.279
n
> slender column
 = 37.383 lim = 17.279
2.6.a Initial eccentric
M0Ed1
eo := = 0.072 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.201 k2 := 0.2
170
k1 Å" k2
Kc := = 0.178
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 20.566 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 10892.87 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.267
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.117 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 226.787 Å" kNm
NEd = 1.943 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.274 m
e2 := etot - 0.5 Å" h + a2 = -0.04 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 4.625 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.365 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.088 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = -7.807 × 10- 5 Ãs1 := min µs1 Å" Es , fyd = -16.395 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 8.903 × 10- 4 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 2.256 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.324 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.035 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 3.565 × 10- 4 Ãs1 := min µs1 Å" Es , fyd = 74.861 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.633 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.67 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsM = 1.67 × 10- 3 m2 As2 := As1 = 1.67 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 5.314
Ä„ Õg2
n1 := 6
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.885 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.026 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 9.412 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.924 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.549 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.176
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.108 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.943 × 103 Å" kN
MEd := NEd Å" e'tot = 210.575 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.929 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.885 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.885 × 10- 3 m2
internal forces:
NEd = 1.943 × 103 Å" kN
M0Ed1 = 140.15 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.108 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.267 m
e2 := etot - 0.5 Å" h + a1 = -0.051 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.456 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.456 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.312 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.312 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 519.444 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 587.038 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1942.57kN
forces
M0Ed1 := 130.45kNm M0Ed2 := 140.15kNm
M0Edqp := 6.58kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.077
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed1
1
A := = 0.985 rm := = 0.931
1 + 0.2 Å" Õeff M0Ed2
C := 1.7 - rm = 0.769
B := 1.1
NEd
n := = 0.913
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 17.442
n
> slender column
 = 37.383 lim = 17.442
2.6.a Initial eccentric
M0Ed1
eo := = 0.067 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.201 k2 := 0.2
170
k1 Å" k2
Kc := = 0.186
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 20.988 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 11116.46 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.26
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.11 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 213.396 Å" kNm
NEd = 1.943 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.267 m
e2 := etot - 0.5 Å" h + a2 = -0.047 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 4.625 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.365 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.088 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = -7.807 × 10- 5 Ãs1 := min µs1 Å" Es , fyd = -16.395 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 8.903 × 10- 4 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 2.045 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.328 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 3.041 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 3.095 × 10- 4 Ãs1 := min µs1 Å" Es , fyd = 64.985 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.54 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.541 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsM = 1.541 × 10- 3 m2 As2 := As1 = 1.541 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 4.906
Ä„ Õg2
n1 := 5
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.571 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.022 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 7.843 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.637 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.397 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.199
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.104 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.943 × 103 Å" kN
MEd := NEd Å" e'tot = 202.942 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.951 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.571 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.571 × 10- 3 m2
internal forces:
NEd = 1.943 × 103 Å" kN
M0Ed1 = 130.45 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.104 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.263 m
e2 := etot - 0.5 Å" h + a1 = -0.055 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.456 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.456 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.322 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.322 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 511.81 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 547.62 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1386.71kN
forces
M0Ed1 := 178.95kNm M0Ed2 := 152.23kNm
M0Edqp := 10.49kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.09
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed2
1
A := = 0.982 rm := = 0.851
1 + 0.2 Å" Õeff M0Ed1
C := 1.7 - rm = 0.849
B := 1.1
NEd
n := = 0.652
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 22.737
n
> slender column
 = 37.383 lim = 22.737
2.6.a Initial eccentric
M0Ed1
eo := = 0.129 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.143 k2 := 0.2
170
k1 Å" k2
Kc := = 0.183
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 20.874 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 11056.05 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.176
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.175 m
( )
2.8 internal forces for section sizing
NEd = 1.387 × 103 Å" kN MEd := NEd Å" etot = 243.143 Å" kNm
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.332 m
e2 := etot - 0.5 Å" h + a2 = 0.018 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 3.302 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.261 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.923 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.294 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 271.657 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.87 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.261 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN > AsM
> small eccentricity
xeff := 0.27 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.943 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.128 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 236.833 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.297 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.333 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsM = 1.333 × 10- 3 m2 As2 := As1 = 1.333 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 4.242
Ä„ Õg2
n1 := 5
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.571 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.022 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 7.843 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.626 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.391 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.136
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.169 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.387 × 103 Å" kN
MEd := NEd Å" e'tot = 234.841 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.966 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.571 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.571 × 10- 3 m2
internal forces:
NEd = 1.387 × 103 Å" kN
M0Ed1 = 178.95 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.169 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.328 m
e2 := etot - 0.5 Å" h + a1 = 0.01 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.326 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.326 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.267 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.267 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 455.328 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 528.771 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1386.71kN
forces
M0Ed1 := 152.23kNm M0Ed2 := 178.95kNm
M0Edqp := 5.14kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.052
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed1
1
A := = 0.99 rm := = 0.851
1 + 0.2 Å" Õeff M0Ed2
C := 1.7 - rm = 0.849
B := 1.1
NEd
n := = 0.652
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 22.909
n
> slender column
 = 37.383 lim = 22.909
2.6.a Initial eccentric
M0Ed1
eo := = 0.11 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.143 k2 := 0.2
170
k1 Å" k2
Kc := = 0.19
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 21.229 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 11244.09 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.173
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.152 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 211.104 Å" kNm
NEd = 1.387 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.309 m
e2 := etot - 0.5 Å" h + a2 = -0.005 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 3.302 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.261 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.923 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.294 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 271.657 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.87 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 9.66 × 10- 4 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.275 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.953 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.044 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 219.164 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.077 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.054 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsN = 1.077 × 10- 3 m2 As2 := As1 = 1.077 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 3.428
Ä„ Õg2
n1 := 4
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.257 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.018 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 6.274 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.332 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.235 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.156
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.15 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.387 × 103 Å" kN
MEd := NEd Å" e'tot = 207.962 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.985 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.257 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.257 × 10- 3 m2
internal forces:
NEd = 1.387 × 103 Å" kN
M0Ed1 = 152.23 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.15 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.309 m
e2 := etot - 0.5 Å" h + a1 = -9.032 × 10- 3 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.326 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.326 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.273 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.273 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 428.449 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 489.73 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1386.71kN
forces
M0Ed1 := 188.54kNm M0Ed2 := 163.04kNm
M0Edqp := 13.98kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.114
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed2
1
A := = 0.978 rm := = 0.865
1 + 0.2 Å" Õeff M0Ed1
C := 1.7 - rm = 0.835
B := 1.1
NEd
n := = 0.652
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 22.256
n
> slender column
 = 37.383 lim = 22.256
2.6.a Initial eccentric
M0Ed1
eo := = 0.136 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.143 k2 := 0.2
170
k1 Å" k2
Kc := = 0.18
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 20.664 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 10945.11 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.178
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.184 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 254.867 Å" kNm
NEd = 1.387 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.341 m
e2 := etot - 0.5 Å" h + a2 = 0.027 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 3.302 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.261 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.923 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.294 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 271.657 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.87 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.369 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.267 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.936 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.18 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 247.753 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.453 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.421 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsN = 1.453 × 10- 3 m2 As2 := As1 = 1.453 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 4.627
Ä„ Õg2
n1 := 5
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.571 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.022 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 7.843 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.605 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.38 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.137
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.177 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.387 × 103 Å" kN
MEd := NEd Å" e'tot = 246.001 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.965 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.571 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.571 × 10- 3 m2
internal forces:
NEd = 1.387 × 103 Å" kN
M0Ed1 = 188.54 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.177 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.336 m
e2 := etot - 0.5 Å" h + a1 = 0.018 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.326 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.326 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.267 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.267 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 466.488 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 528.771 Å" kN Å" m
( ) ( )
condition satisfied
DATA
Column hight
lcol := 3.7m
Column dimensions
b := 0.40m h := 0.4m
Internal
NEd := 1386.71kN
forces
M0Ed1 := 163.04kNm M0Ed2 := 188.54kNm
M0Edqp := 7.96kNm
Exposure class
X0
Concrete class C20 / 25 Steel class
AIIIN
µcu2 := 0.0035
fyk := 500MPa
fck := 20MPa
fyd := 420MPa
fcd := 13.3MPa
Es = 210 Å" GPa
fcm := fck + 8MPa = 28 Å" MPa
Ecm = 30 Å" GPa
Buckling factor:
b Å" h3
Ic := = 2.133 × 10- 3 m4
12
EcmÅ"Ic
lcol
1
kA := = 1 ² := 1 + = 1.167 l0 := ² Å" lcol = 4.317 m
Ecm Å" Ic 5kA + 1
ëÅ‚ öÅ‚
ìÅ‚ ÷Å‚
lcol
íÅ‚ Å‚Å‚
Creep coefficient:
Õ := 2.5
Ac := b Å" h = 0.16 m2
u := 2 Å" (b + h) = 1.6 m
2 Å" Ac
h0 := = 0.2 m
u
RH := 50%
age of concrete at loading (days)
t0 := 60
RH
100
ÕRH := 1 + 1 - = 1.145
3
0.1 Å" h0
16.8
²fcm := = 3.175
fcm
1
²t0 := = 0.422
0.1 + t00.2
Õ0 := ÕRH Å" ²fcm Å" ²t0 = 1.535
M0Edqp
Õeff := Õ0 Å" = 0.075
M0Ed1
2. Strength calculations - buckling
2.1 Effective column height
lo := ² Å" lcol = 4.317 m
2.2 Assumption of cross-section dimensions (declared in data set)
b = 0.4 m
h = 0.4 m
2.3 Assumption of reinforcement centroid location
concrete cover
cmin := 15mm "c := 5mm Õg := 20mm Õs := 8mm c1 := max 20mm , Õg = 20 Å" mm
( )
cmin := max Õg, 10mm , 10mm = 20 Å" mm
( )
"cdev := 5mm cnom := cmin + "cdev = 25 Å" mm
Õw1 := 6mm c := cnom + Õs = 33 Å" mm
a1 := cmin + "c + Õs + 0.5 Å" Õg a1 = 0.043 m
a2 := a1 = 0.043 m
d := h - a1 = 0.357 m
¾efflim := 0.5
Limit realive depth of the compression zone:
2.4 Limit depth of the compression zone of concrete
xeff.lim := ¾efflim Å" d = 0.179 m
2.5 a Column slenderness
Ic
b Å" h3
Ac := b Å" h = 0.16 m2 Ic := = 2.133 × 10- 3 m4 i := = 0.115 m
12 Ac
lo
 := = 37.383
i
2.5b Boundary slenderness
M0Ed1
1
A := = 0.985 rm := = 0.865
1 + 0.2 Å" Õeff M0Ed2
C := 1.7 - rm = 0.835
B := 1.1
NEd
n := = 0.652
fcd Å" Ac
20 Å" A Å" B Å" C
lim := = 22.427
n
> slender column
 = 37.383 lim = 22.427
2.6.a Initial eccentric
M0Ed1
eo := = 0.118 m
NEd
2.6.b Accidental eccentric
lo
= 0.011 m
400
h
= 0.013 m
30
lo
ëÅ‚ h öÅ‚
ei := max , , 0.02m = 0.02 m
ìÅ‚ ÷Å‚
400 30
íÅ‚ Å‚Å‚
2.7 II order influences
Áb := 0.015 reinforcement ratio asumption
fck
k1 := = 1
20MPa

k2 := n Å" = 0.143 k2 := 0.2
170
k1 Å" k2
Kc := = 0.186
1 + Õeff
Ks := 1
Second moment of area Is
Is := Áb Å" b Å" d Å" 0.5 Å" h - a1 = 5.28 × 10- 5 m4
( )2
Nominal sifness
EI := Kc Å" Ecd Å" Ic + Ks Å" Es Å" Is = 21.011 Å" MN Å" m2
2.7 b Buckling force NB
Ä„2 Å" EI
NB := = 11128.58 Å" kN
lo2
2.7 c Factor increasing eccentric ·
Co := 8
Ä„2
² := = 1.003 Å" 1.23
Co
² := 1.23
²
· := 1 + = 1.175
NB
- 1
NEd
2.7 d total eccentric
etot := · Å" eo + ei = 0.162 m
( )
2.8 internal forces for section sizing
MEd := NEd Å" etot = 224.176 Å" kNm
NEd = 1.387 × 103 Å" kN
3. Strength calculations section sizing
3.1 Eccentic about reinforcement A.si i A.s2
e1 := etot + 0.5 Å" h - a1 = 0.319 m
e2 := etot - 0.5 Å" h + a2 = 0.005 m
3.2 Minimal total reinforcement
ëÅ‚0.1 NEd öÅ‚
Asmin := max Å" , 0.002 Å" b Å" h = 3.302 × 10- 4 m2
ìÅ‚ ÷Å‚
fyd
íÅ‚ Å‚Å‚
3.3 Reinforcement area
Assumed symmetrical reinforcement
As1 := As2
NEd
> small eccentricity
xeff := = 0.261 m xeff.lim = 0.179 m
fcd Å" b
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.923 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.294 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 271.657 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.87 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.086 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
II iteration:
AsN < AsM
> small eccentricity
xeff := 0.275 Å" m xeff.lim = 0.179 m
µcu := µcu2 = 3.5 × 10- 3  := 0.8 · := 1
µcu Å" xeff - a2
( )
µs2 := = 2.953 × 10- 3 Ãs2 := min µs2 Å" Es , fyd = 420 Å" MPa
( )
xeff
µcu Å" h - xeff - a2
( )
µs1 := = 1.044 × 10- 3 Ãs1 := min µs1 Å" Es , fyd = 219.164 Å" MPa
( )
xeff
NEd - · Å" fcd Å" b Å"  Å" xeff
AsN := = 1.077 × 10- 3 m2
Ãs2 - Ãs1
MEd - 0.5 Å" · Å" fcd Å" b Å"  Å" xeff Å" h -  Å" xeff
( )
AsM := = 1.184 × 10- 3 m2
h
ëÅ‚ öÅ‚
- a2 Å" Ãs2 + Ãs1
( )
ìÅ‚ ÷Å‚
2
íÅ‚ Å‚Å‚
As1 := AsM = 1.184 × 10- 3 m2 As2 := As1 = 1.184 × 10- 3 m2
Number of bars
Õg := 20mm
4As1
n1 := = 3.77
Ä„ Õg2
n1 := 4
Õg
a1 := cnom + Õw1 + = 0.041 m
2
d := h - a1 = 0.359 m
Ä„ Å" Õg2
As1 := n1 Å" = 1.257 × 10- 3 m2 As2 := As1
4
3.7 Total reinforcement ratio
As1 + As2
Áprov := = 0.018 Ácrit := 0.015
>
b Å" d
4. Checking correctness of buckling prediction
4.1 Second moment of area for reinforcement correction
l's
I's := As1 Å" 0.5 Å" h - a1 + As2 Å" 0.5 Å" h - a2 = 6.274 × 10- 5 m4
( )2 ( )2
EI' := Kc Å" Ecd Å" Ic + Ks Å" Es Å" I's = 2.31 × 107 m3 Å" kg Å" s- 2
Ä„2 Å" EI'
N'B := = 1.223 × 107 m Å" kg Å" s- 2
lo2
²
·' := 1 + = 1.157
N'B
- 1
NEd
e'tot := ·' Å" eo + ei = 0.159 m
( )
4.3 Internal forces for section sizing (after correction of buckling influence)
NEd = 1.387 × 103 Å" kN
MEd := NEd Å" e'tot = 220.77 m Å" kN
4.4 Total eccentrices (for assumed and provided reinforcement) comparision
e'tot
= 0.985 - eccentric ratio including, that assumption was on safe side
etot
6. Ultimate capacity of section
Column cross section:
b = 0.4 m h = 0.4 m
Tensioned reinforcement:
As1 = 1.257 × 10- 3 m2 a1 = 0.041 m
a2 = 0.043 m
Compression reinforcement:
As2 = 1.257 × 10- 3 m2
internal forces:
NEd = 1.387 × 103 Å" kN
M0Ed1 = 163.04 m Å" kN
6.1 Effective depth of the section:
d := h - a1 = 0.359 m
6.2 Limit depth of the compression zone of concrete
xeff.lim := ¾eff.lim Å" d = 0.18 m
6.3 total eccentric
etot := e'tot = 0.159 m
6.4 Eccentrics about centroids of tensioned As1 and compression As2 reinforcement
e1 := etot + 0.5 Å" h - a1 = 0.318 m
e2 := etot - 0.5 Å" h + a1 = 2.039 × 10- 4 m
6.5 Depth of the compression zone of the concrete
NEd - fyd Å" As2 - As1
( )
xeff := = 0.326 m
 Å" · Å" fcd Å" b
<
xeff.lim = 0.18 m xeff = 0.326 m
1
( - ¾efflim Å" NEd - fyd Å" As2 + 1 + ¾efflim Å" fyd Å" As1
) ( ) ( )
xeff := d = 0.273 m
 Å" Å" fcd Å" b Å" d Å" 1 - ¾efflim + 2 Å" fyd Å" As1
( )
îÅ‚· Å‚Å‚
ðÅ‚ ûÅ‚
< <
xeff.lim = 0.18 m xeff = 0.273 m d = 0.359 m
6.6 Ultimate capacity
<
NEd Å" e1 = 441.257 Å" kN Å" m MRd := fcd Å" b Å" xeff Å" d - 0.5xeff + fyd Å" As2 Å" d - a2 = 489.73 Å" kN Å" m
( ) ( )
condition satisfied
689.058 Å" kN
kN
kNm := kN Å" m
1
² := 1 + = 1.167
5kA + 1
kNm := kN Å" m
INTERNAL MX +
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
INTERNAL MX -
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
INTERNAL MY +
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
INTERNAL MY -
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
EDGE MX +
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
EDGE MX -
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
EDGE MY +
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28
EDGE MY -
Ecm
Ecd :=
1.2
h0 := 200
RH := 50
fcm := 28