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First order bending moments with imperfection

The calculation of first order moment runs according to the following procedure:

first order eccentricity without effect of imperfection is calculated,
eccentricity caused by imperfection is calculated ,
first order eccentricity including effect of imperfection is calculated.

Calculation first order eccentricity without effect of imperfection

The first order eccentricity without effect of imperfection is calculated according to formulas below

where

e_1,y(z) the 1st order eccentricity in direction of y (z) axis of LCS

M_0Ed,y(z) the 1st order moment around of y (z) axis of LCS (in SCIA Engineer value My(z))

N_Ed the design axial force in calculated section of the column (in SCIA Engineer value N)

e_0e,y(z) the 1st order equivalent eccentricity in direction of y (z) axis of LCS

M_0e,y(z) the 1st order equivalent moment around of y (z) axis of LCS

N_0,e the equivalent axial force

The 1st order equivalent moment is calculated according to clause 5.8.8.2 (2) in EN 1992-1-1

M_0,ey = max (0,6*M_02,y +0,4*M01,y; 0,4* M_02,y)

M_0,ez = max (0,6*M_02,z +0,4*M01,z; 0,4*M_02,z)

where

M_01y(z) is first end bending moments around y(z) axis of LCS with lesser absolute value as second end bending moment. |M01y(z)|< |M02y(z)| The same values are used for calculation limit slenderness

M_02y(z) is second end bending moments around y(z) axis of LCS with greater absolute value as first end bending moment. |M02y(z)|≥ |M01y(z)| The same values are used for calculation limit slenderness

The equivalent axial force is calculated according to same rules as equivalent bending moments

N_0,e = max (0,6*N₀₂ +0,4*N₀₁; 0,4*N₀₂)

where

N₀₁ It is first end axial force with lesser absolute value as second end axial force. |N01|< |N02|.

N₀₂ It is second end axial force with greater absolute value as first end axial force. |N01|< |N02|.

The equivalent values of bending moment and axial force are the same for all sections in column and the values of equivalent eccentricity and equivalent bending moments can be presented in numerical output after adding the selected values to table (via icon Table composer in Preview)

Calculation eccentricity caused by imperfection

The imperfection in SCIA Engineer is represented by an inclination according to clause 5.2(5) in EN 1992-1-1. The inclination is calculated around both axis ( axis y and z) of LCS according to formula:

where

F₀ is the basic value of inclination. The value is National parameter, it means that value can be different for each country. The value can be set in Concrete setup ( Manager for national annex > EN 1992-1-1 > General > ULS > General > Theta_0 )

α_h is the reduction factor for length of column or height of structure. The value is calculated according to formula

α_m,y(z) is the reduction factor for numbers of members calculated according to formula

l is length of column or height of structure defined via parameter Tot.height in tab-sheet Buckling data in dialogue Buckling and relative lengths (member properties > parameter Buckling and relative length > button Edit ), see chapter 4.3

m_y(m_z) is the number of vertical members contributing to the total effect of the imperfection perpendicular to y(z) of LCS. It means, that value is used for recalculation of bending moment around y(z) axis of LCS. These value can be defined in tab-sheet Buckling data in dialogue Buckling and relative lengths (member properties > parameter Buckling and relative length > button Edit )

The effect of imperfection for isolated column and for structure too is taken into account always as an eccentricity according to clause 5.2(7a) in EN 1992-1-1. At first step basic eccentricity caused by imperfection in both direction of local axis are calculated according to formula

where

F_i,y(z) is the inclination around y(z) axis of LCS (perpendicular to y (z) axis of LCS)

l_0,y(z) is the effective length of the member (column) around y(z) axis of LCS (perpendicular to y (z) axis of LCS), which can be defined via Member buckling data

In second step basic value of eccentricity caused by imperfection around of both local axis are recalculated depending on slenderness and first order eccentricities:

Recalculation eccentricity caused by imperfection depending on slenderness
- in direction of y-axis
  e_i,y,1 = ei,y .... if ly ≥ lz
  
  ei,y,1 = 0m ..... if ly < lz
- in direction of z-axis
  ei,z,1 = ei,z .... if ly ≤ lz
  
  ei,z,1 0m .... if ly > lz
Recalculation eccentricity caused by imperfection in depending on first order eccentricities
- in direction of y-axis
  e_i,y,2 = e_i,y .... if e_1,z=0
  
  otherwise

in direction of z-axis
e_i,y,2 = e_i,z .... if e_1,y=0

otherwise

where

e_1,y(z) is the 1st order eccentricity in direction of y (z) axis of LCS (see chapter Calculation first order eccentricity without effect of imperfection)

e_i,1 is eccentricity caused by imperfection recalculated to direction of resultant first order eccentricity e_i,1 = e_i,y*cos(α)+ e_i,z*sin(α). For simplification in SCIA Engineer the value e_i,1 is calculated according to formula: e_i,1 = max(e_i,y;e_i,z)

α is angle between direction of resultant first order eccentricity and y-axis of LCS

The final value of eccentricity caused by imperfection is calculated according to formulas below and these values are presented in numerical output for value
M_y,recal and M_z,recal

e_i,y = max(e_i,y,1; e_i,y,2)

e_i,z = max(e_i,z,1; e_i,z,2)

The table of recalculated and final eccentricity caused by imperfection for different values of slenderness and first order eccentricities according to inputted values and previous formulas is below (example Tutorial_column_02.esa)

Type	Column	B1	B2	B3	B4	B5	B6	B7
Basic	ei,y [mm]	9	9	9	9	9	9	9
	ei,z [mm]	9	9	9	9	9	9	9
	e1,y [mm]	50	0	50	50	0	50	50
	e1,z [mm]	50	50	0	50	50	0	50
	slenderness	ly = lz	ly < lz	ly < lz	ly < lz	ly > lz	ly > lz	ly > lz
Recalculated	ei,y,1 [mm]	9	9	9	9	0	0	0
	ei,z,1 [mm]	9	0	0	0	9	9	9
	ei,y,2 [mm]	9	0	9	6	0	9	6
	ei,z,2 [mm]	9	9	0	6	9	0	6
Final	ei,y [mm]	9	9	9	9	0	9	6
Final	ei,z [mm]	9	9	0	6	9	9	9

The results in SCIA Engineer are the following:

The direction (sign) of final value of eccentricity caused by imperfection has to be same as direction (sign) of first order eccentricity.

Calculation first order eccentricity including effect of imperfection

First order eccentricity including effect of imperfection is calculated according to formula below

e_o,y(z) = e_1,y(z) + e_i,y(z) > e_0,min,y(z)

The minimum first order eccentricity similar as eccentricity caused by imperfection is calculated in direction of both local axis according to formulas below:

where

e_1,y(z) Is the 1st order eccentricity in direction of y (z) axis of LCS (see chapter Calculation first order eccentricity without effect of imperfection)

e_i,y(z) is eccentricity caused by imperfection in direction of y (z) axis of LCS (see chapter Calculation eccentricity caused by imperfection)

e_0,min,y(z) is minimum first order eccentricity in direction of y (z) axis of LCS according to clause 6.1(4) in EN 1992-1-1. This minimum value will be taken into account only in case, that check box Take into account eccentricity according to chapter 6.1.4 is ON (Concrete solver >General > Calculation > Columns >Advanced mode), see chapter Take into account eccentricity according to chapter 6.1.4

b is maximum width of cross-section (maximum dimension of cross-section in direction of y axis of LCS)

h is height of the cross-section ( dimension of cross-section in direction of z axis of LCS))

After calculation of first order eccentricity including effect of imperfection, the 1st order moment, including the effect of imperfections around y (z) axis of LCS is calculated:

M_0Ed,y(z) = N_Ed* e_o,z(y)

The table of recalculated and final second order eccentricities for different value of first order eccentricities according to inputted values and previous formulas is below (example Tutorial_column_03.esa)

Type	Column b/h =400/700 mm	B1 (x=0)	B1 (x=1.8)	B2 (x=0)	B2 (x=1.8)	B3 (x=0)	B3 (x=1.8)
Basic	M0Ed,z/NEd [mm]	-50	0	0	0	-50	0
	M0Ed,y/NEd [mm]	-50	0	-50	0	0	0
	e0e,y[mm]	20		0		20
	e0e,z[mm]	20		20		0
	e1,y [mm]	-50	20	0	0	-50	20
	e1,z [mm]	-50	20	-50	20	0	0
Recalculated	ei,y [mm]	-9	9	9	9	-9	9
	ei,z [mm]	-6	6	-9	9	0	0
	e0,min,y [mm]	-20	20	20	20	-20	20
	e0,min,z [mm]	-23	23	-23	23	23	23
Final	e0,y [mm]	-59	29	20	20	-59	29
Final	e0,z [mm]	-56	26	-59	29	23	23

The results in SCIA Engineer are the following:

• First order moment and eccentricity including effect of imperfection are presented in numerical output for value My recal and Mz recal
• The direction (sign) of minimum first order eccentricity has to be same as direction (sign) of first order eccentricity

Calculation nominal second order moment

Nominal second order moment is calculated according to clause 5.8.8.2(3) in EN 1992-1-1

M_2,y(z) = N_Ed* e_2,z(y)

where

N_Ed is the design axial force in calculated section of the column (in SCIA Engineer value N)

e_2,z(y) is the second order eccentricity in direction of z (y) axis of LCS

The basic value of second order eccentricity is taken into account only if:

check box Use buckling data is ON in concrete setup (see chapter Buckling data) or in concrete member data (see chapter Group Column calculation), if concrete member data is inputted on checked column
calculated slenderness is greater than limit slenderness

The basic values of second order eccentricity are calculated according to formulas below in first step:

l_z >l_z,lim	check box Use buckling data	Second order eccentricity
YES	ON	e2,y = (1/r)z.l0,z2 / cz
YES	OFF	e2,y = 0
NO	ON
NO	OFF
l_y >l_y,lim	check box Use buckling data	Second order eccentricity
YES	ON	e2,z = (1/r)y.l0,y2 / cy
YES	OFF	e2,z = 0
NO	ON
NO	OFF

where

(1/r)_y(z) is the curvature around y(z) axis of LCS (perpendicular to y (z) axis of LCS) calculated according to clause 5.8.8.3 in EN 1992-1-1

l_0,y(z) is the effective length of the member (column) around y(z) axis of LCS (perpendicular to y (z) axis of LCS), which can be defined via Member buckling data

c_y(z) is a factor depending on the curvature distribution around y(z) axis of LCS according to clause 5.8.8.2(4) in EN 1992-1-1.

c_y(z) = 8 (for constant first order bending moment at whole length of the column around y(z) axis of LCS)

c_y(z) = 10 (for the other cases)

l_y(z) Slenderness ratio around y(z) axis of LCS

l_y(z),lim Limit slenderness ratio around y(z) axis of LCS

In second step basic value of second order eccentricities in direction of both local axis are recalculated depending on slenderness and first order eccentricities

Recalculation second order eccentricities in depending on slenderness
- in direction of y-axis
  e_2,y,1 = e_2,y .... if ly ≥ lz
  e_2,y,1 = 0m .... if ly < lz

in direction of z-axis
e_2,z,1 = e_2,z .... if ly ≤ lz
e_2,z,1 = 0m .... if ly > lz

Recalculation second order eccentricities in depending on first order eccentricities
- in direction of y-axis
  e_2,y,2 = e_2,y .... if e_1,z=0
  otherwise

in direction of z-axis
e_2,z,2 = e_2,z .... if e_1,y=0
otherwise

where

e_1,y(z) is the 1st order eccentricity in direction of y (z) axis of LCS (see chapter Calculation first order eccentricity without effect of imperfection)

e_i,1 is second order eccentricity recalculated to direction of resultant first order eccentricity ei,1 = e2,y•cos(α)+ e2,z•sin(α). For simplification in SCIA Engineer the value ei,1 is calculated according to formula: ei,1 = max(e2,y;e2,z)

α is angle between direction of resultant first order eccentricity and y-axis of LCS

The final value of eccentricity caused by imperfection is calculated according to formulas below and these values are presented in numerical output for value My,recal and Mz,recal

e_2,y = max(e_2,y,1; e_2,y,2)

e_2,z = max(e_2,z,1; e_2,z,2)

The table of recalculated and final second order eccentricities for different value of slenderness and first order eccentricities according to inputted values and previous formulas is below (example Tutorial_column_02.esa)

Type	Column	B1	B2	B3	B4	B5	B6	B7
Basic	e2,y [mm]	22	22	22	22	22	22	22
	e2,z [mm]	22	22	22	22	22	22	22
	e1,y [mm]	50	0	50	50	0	50	50
	e1,z [mm]	50	50	0	50	50	0	50
	slenderness	ly = lz	ly < lz	ly < lz	ly < lz	ly > lz	ly > lz	ly > lz
Recalculated	e2,y,1 [mm]	22	22	22	22	0	0	0
	e2,z,1 [mm]	22	0	0	0	22	22	22
	e2,y,2 [mm]	22	0	22	15	0	22	15
	e2,z,2 [mm]	22	22	0	15	22	0	15
Final	e2,y [mm]	22	22	22	22	0	22	15
Final	e2,z [mm]	22	22	0	15	22	22	22

The results in SCIA Engineer are the following:

The direction (sign) of final value of second order eccentricity has to be same as direction (sign) of first order eccentricity

Calculation of curvature for nominal second order moment

The curvature for calculation of second order eccentricity is calculated according to clause 5.8.8.3 in EN 1992-1-1.

(1/r)_y(z) = K_r*Kf_,y(z)*(1/r0)_y(z)

where

K_r is a correction factor depending on axial load, see clause 5.8.8.3 (3) in EN 1992-1-1. This factor depends on relative normal force (n) and mechanical ratio of reinforcement (ω).

Kf_,y(z) is factor for taking into account of creep around y(z) axis of LCS , see clause 5.8.8.3 (4) in EN 1992-1-1. This factor depends on effective creep ratio (φ_ef) and factor (β_y(z)) depending on slenderness

(1/r0)y(z) is the basic value of curvature around y(z) axis of LCS. This value depends on effective depth of cross-sections and strain in reinforcement at reaching design yield strength of reinforcement

It follows that the calculation of curvature depends on many parameters and factors, but the most important are the following:

relative normal force
mechanical ratio of reinforcement
effective creep ratio
slenderness of the column
effective depth of cross-section
strain in reinforcement at reaching design yield strength of reinforcement

Relative normal force

Relative normal force is used for calculation correction factor K_r and it is calculated according to formula

n = N_Ed / A_c•f_cd

where

N_Ed is design value of the applied axial force in compression in calculated sections

A_c is cross sectional area of concrete

f_cd is design value of concrete compressive strength

Mechanical reinforcement ratio

Mechanical reinforcement ratio, which is used for calculation correction factor Kr, depends on total area of longitudinal reinforcement. It follows that for design of reinforcement to column and for check of column different mechanical ratio can be taken into account.

for design of reinforcement (Type of check = Design ULS in service Internal forces) total area of reinforcement is calculated from ratio loaded from concrete setup (Concrete solver >General > Calculation > column > Advanced setting > User estimate of reinf. for design of reinforcement), see chapter User estimate of reinf. for design of reinforcement. The mechanical reinforcement ratio is the same at whole length of the column and it is calculated according to formula:
for ULS checks of columns (Type of check = Non-prestressed check ULS in service Internal forces) total area of reinforcement is calculated from inputted reinforcement via REDES or Free bars. The mechanical reinforcement ratio is calculated according to formula below. The mechanical reinforcement can be different at whole length of the column and in each section of the member is calculated according to formula:

where

Ratio_lon is reinforcement ratio loaded from concrete setup (Concrete solver >General > Calculation > column > Advanced setting > User estimate of reinf. for design of reinforcement), see chapter User estimate of reinf. for design of reinforcement

f_yd is design yield strength of reinforcement. The quality of reinforcement can be input in Project data or in concrete member data (see chapter 4.2.34.2.4), if concrete member data is inputted on checked column

f_cd is design value of concrete compressive strength

A_si is cross-sectional area of i-th reinforcement in the cross-section inputted via REDES or Free bars

f_ydi is design yield strength of i-th reinforcement in the cross-section inputted via REDES or Free bars inputted via REDES or Free bars

Effective creep ratio

In SCIA Engineer for calculation factor Kf_,y(z) is used creep ratio loaded from concrete setup (if member data is not defined on column) or concrete member data. It means that if user wants to take into account effective creep ratio according to clause 5.8.4 in EN 1992-1-1, the value of this creep ratio has to be directly input to concrete setup or to concrete member data. Otherwise, the final creep ratio will be taken into account.

Concrete solver > SLS > Creep	Concrete member data

Slenderness

Slenderness of the column for calculation of factor Kf_,y(z) is taken into account by parameter (b_y(z)) , which is calculated according to formula:

where

f_ck is characteristic value of concrete compressive strength

ly(z) is slenderness ratio around y(z) axis of LCS

Effective depth of cross-section

Effective depth of cross-section is used for calculation basic value of curvature and it is calculated according to clause 5.8.8.3(2) in EN 1992-1-1.

where

i_sy(z) is the radius of gyration of the total reinforcement area in direction of y(z) axis of LCS

b is maximum width of cross-section (maximum dimension of cross-section in direction of y axis of LCS)

h is height of the cross-section ( dimension of cross-section in direction of z axis of LCS))

The radius of gyration of the total reinforcement depends on selected item in combo box Type of check (tree Concrete Advanced > 1D member > Internal forces). It means, the radius of gyration of the total reinforcement can be different for design of reinforcement to column and for check of column. The calculation for design of reinforcement to column depends on type of method and shape of concrete cross-section too. There are the following methods for calculation the radius of gyration of the total reinforcement:

for design of reinforcement according to method Only corner design (Type of check = Design ULS in service Internal forces) the radius of gyration of the total reinforcement area is calculated according to following steps:
- total area of reinforcement
  A_s = Ratio_lon *A_c
- area of reinforcement of one bar
  A_1si = A_s/n_s

second moment of reinforcement area

radius of gyration of the total reinforcement area

for design of reinforcement according to other methods and for rectangular section (Type of check = Design ULS in service Internal forces) the radius of gyration of the total reinforcement area is calculated according to following steps:
- total area of reinforcement
  A_s = Ratio_lon *A_c
- calculation of ratio of reinforcement in y and z direction
  
  if then
- area of reinforcement in direction of y(z) axis of LCS
  A_s,y(z) = ratio_y(z)*A_s
- second moment of reinforcement area
- radius of gyration of the total reinforcement area

for design of reinforcement for circular section (Type of check = Design ULS in service Internal forces) the radius of gyration of the total reinforcement area is calculated according to following steps:
- total area of reinforcement
  A_s = Ratio_lon *A_c
- second moment of reinforcement area
- radius of gyration of the total reinforcement area

for ULS checks of columns (Type of check = Non-prestressed check ULS in service Internal forces the radius of gyration of the total reinforcement area is calculated according to following steps:
- total area of reinforcement
- second moment of reinforcement area
- radius of gyration of the total reinforcement area

where

Ratio_lon is reinforcement ratio loaded from concrete setup (Concrete solver >General > Calculation > column > Advanced setting > User estimate of reinf. for design of reinforcement), see chapter User estimate of reinf. for design of reinforcement

A_c is cross sectional area of concrete

n_s is number of bars in cross-section (the number of bars for Only corner design depends on shape of cross-section, see chapter Corner design only)

A_1si is the cross-sectional area of i-th bar of reinforcement

z_s,y(z)i is the position of i-th bar of reinforcement from centroid of concrete cross-section in direction of y(z) axis of LCS (the positions of bars for Only corner design depends on shape of cross-section, see chapter Corner design only)

sy(z) is the bending stress in concrete calculated for uncracked concrete cross-section according to formulas:

M_0Ed,y(z) is the 1st order moment around of y (z) axis of LCS (in SEN value My(z)) without imperfection

W_c,y(z) is the section modulus of concrete cross-section around y (z) axis of LCS

z_s,y(z) is the position of reinforcement from centroid of concrete cross-section in direction of y (z) axis for rectangular section (the positions of bars for Only corner design depends on shape of cross-section, see chapter Corner design only)

z_s is th position of reinforcement from centroid of concrete cross- for circular section

b is width of rectangular cross-section

h is height of the rectangular cross-section

D is diameter of circular cross-section

c_nom is nominal concrete cover loaded from concrete setup (Design default) or from concrete member data, if member data are defined on calculated column

d_s is diameter of longitudinal reinforcement loaded from concrete setup (Design default) or from concrete member data, if member data are defined on calculated column

d_ss is diameter of transverse reinforcement (stirrup) loaded from concrete setup (Design default) or from concrete member data, if member data are defined on calculated column

In the table below is calculation effective depth of cross-section for different type of check, different cross-section and different method

c_nom = 35 mm, d_s = 20 mm, d_ss = 8 mm

Type of check = Design ULS

Only corner design = ON

Cross-section = standard

n_s = 4; b = h = 0,4 m; A_c = 0,4·0,4 = 0,16 m²

Ratio_lon = 2% = 0,02

A_s = 0,02·0,16 = 0,0032 m²; A_1si=0,0032/4 = 0,0008 m²

z_s,y = z_s,z = 0,5· 0,4-(0,0350,008+0,5·0,02) = 0,147 m

z_s,y1 = z_s,y3 = z_s,y = 0,147 m; z_s,y2 = z_s,y4 = -z_s,y = -0,147 m

z_s,z1 = z_s,z2 = z_s,z = 0,147 m; z_s,z3 = z_s,z4 = -z_s,z = -0,147 m

I_sy = ΣA_si•z_s,zi² = 69,15·10^-6 m⁴; I_sz = ΣA_si•z_s,yi² = 69,15·10^-6 m⁴

i_sy = i_sz = 0,147 m

d_y = d_z = 0,5·0,4 + 0,147 = 0,347 m

Type of check = Design ULS

Only corner design = OFF

Cross-section = rectangle

b = h = 0,4 m; Ac = 0,4·0,4 = 0,16 m²

Ratio_lon = 2% = 0,02; A_s = 0,02·0,16 = 0,0032 m²

M_0Ed,y = -100 kNm, M_0Ed,z = -100 kNm

W_c,y = W_c,z = 0,4·0,42/6 = 0,01067 m³

σ_y = 100/0,01067 = 9,372 MPa; σ_z = 0/0,01067 = 0 MPa

Ratio_y = 1, Ratio_z = 0

A_sy = 1,0·0,0032 = 0,0032 m², Asz = 0 m²

z_s,y = z_s,y = 0,5· 0,4-(0,035+0,008+0,5·0,02) = 0,147 m

I_sy = 69,15·10^-6 m⁴; I_sz = 23,05·10^-6 m⁴

i_sy = 0, 085 m; i_sz = 0,147 m

d_y = 0,5·0,4+0,085 = 0,285 m; d_z = 0,5·0,4+0,147 = 0,347 m

Type of check = Design ULS

Only corner design = OFF

Cross-section = rectangle

b = h = 0,4 m; A_c = 0,4·0,4 = 0,16 m²

Ratio_lon = 2% = 0,02; A_s = 0,02·0,16 = 0,0032 m²

M_0Ed,y = -100 kNm, M_0Ed,z = -60 kNm

W_c,y = W_c,z = 0,4·0,42/6 = 0,01067 m³

σ_y = -100/0,01067 = -9,37 MPa; σ_z = -60/0,01067 = -5,62 MPa

Ratio_y = 0,625, Ratio_z = 0,375

A_sy = 0,625·0,0032 = 0,002 m²; A_sz = 0,375·0,0012 = 0,0012 m²

z^s,y = z_s,z = 0,5· 0,4-(0,035+0,008+0,5·0,02) = 0,147 m

I_sy = 51,86·10^-6 m⁴; I_sz = 40,33·10^-6 m⁴

i_sy = 0,112 m, i_sz = 0,127 m

d_y = 0,5·0,4+0,112 = 0,312 m; d_z = 0,5·0,4+0,127= 0,327 m

Type of check = Design ULS

Only corner design = OFF

Cross-section = circular

D =0,4 m; A_c = 0,25•π•D²= 0,126 m²

Ratio_lon = 2% = 0,02; A_s = 0,02·0,126 = 0,0025 m²

z_s = 0,5· 0,4-(0,035+0,008+0,5·0,02) = 0,147 m

I_sy = I_sz=27,15·10^-6 m⁴

i_sy = i_sz = 0,104 m

d_y = d_z = 0,5·0,4 + 0,104 = 0,304 m

Type of check = Non-prestressed ULS checks

Only corner design = OFF

Cross-section = all

b = h = 0,4 m; A_c = 0,4·0,4 = 0,16 m²

Reinforcement = 6ϕ20 (B400B); ns = 6

A_1si = 0,000314 m² ; As = 6• 0,000314 = 0,00188 m²

z_s,y = z_s,z = 0,5· 0,4-(0,035+0,008+0,5·0,02) = 0,147 m

z_s,y1 = z_s,y4 = z_s,y = 0,147 m; z_s,y3 = z_s,y6 = -z_s,y = -0,147 m

z_s,y2 = z_s,y4 = 0 m;

z_s,z1 = z_s,z2 = z_s,z3 = z_s,z = 0,147 m; z_s,z4 = z_s,z5 = z_s,z6 = -z_s,z = -0,147 m

I_sy = ΣA_si•z_s,zi² = 40,71·10^-6 m⁴; I_sz= ΣA_si•z_s,yi²= 27,14·10^-6 m⁴

i_sy = 0,120 m; i_sz = 0,147 m

d_y = 0,5·0,4 + 0,120 = 0,320 m; d_y = 0,5·0,4 + 0,14 = 0,347 m

The user (real) reinforcement defined via REDES and free bars are not taken into account for calculation effective depth of cross-section for design reinforcement to column (Type of check = Design ULS in service Internal forces)

Calculation of strain in reinforcement at reaching design yield strength of reinforcement

Strain in reinforcement at reaching design yield strength of reinforcement is used for calculation basic value of curvature. This value depends on material properties of reinforcement. It follows that for design of reinforcement to column and for check of column different value can be taken into account.

for design of reinforcement (Type of check = Design ULS in service Internal forces) strain in reinforcement at reaching design yield strength of reinforcement is calculated from default material properties defined in dialogue Project data or in concrete member data (if member data are defined on calculated column) according to formula:
for ULS checks of columns (Type of check = Non-prestressed check ULS in service Internal forces) strain in reinforcement at reaching design yield strength of reinforcement is calculated from material properties of inputted reinforcement via REDES or Free bars according to formula:

where

f_yd is design yield strength of reinforcement. The quality of reinforcement can be input in Project data or in concrete member data (see chapter Group Design and Group Column calculation), if concrete member data is inputted on checked column

E_s is design value of modulus of elasticity of reinforcement. The quality of reinforcement can be input in Project data or in concrete member data (see chapter Group Design and Group Column calculation), if concrete member data is inputted on checked column

f_ydi is design yield strength of i-th reinforcement in the cross-section inputted via REDES or Free bars inputted via REDES or Free bars

E_si is design value of modulus of elasticity of i-th reinforcement in the cross-section inputted via REDES or Free bars inputted via REDES or Free bars

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