calculation of the moment resistance of z- and c-shaped cold-formed sections according to eurocode 3

8/13/2019 Calculation of the Moment Resistance of Z- And C-Shaped Cold-Formed Sections According to Eurocode 3

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Calculation of the moment resistance

of Z- and C-shaped cold-formed sections

according to Eurocode 3

Eurocode 3 version:EN1993-1-3: 20XX, Final Draft, 27 September 2002

prepared by Sandor ADANY

December 2003.



Sandor ADANY Moment resistance for Z-C sections EN1993-1-3, Final Draft

code for production mode (1:rolled, 2:pressed) prod_mod 1:=

reduction for design thickness∆t 0mm:=

partial safety factor γM0 1:=

Other input data

Poisson ratioν 0.3:=

modulus of elasticityE 210000MPa:=

ultimate strengthf u 642.2MPa:=

basic yield strengthf yb 507.4MPa:=

Input material data

Section name: D11.5Z082-4

Input geometrical data

tg 2.06mm:=

hg 290mm:=

bg1 86mm:=

bg2 87mm:=

cg1 22mm:=

cg2 22mm:=

α1 130.1:=α2 131.6:=

r 11 5.94mm:=

r 12 5.94mm:=

r 21 5.94mm:=

r 22 5.94mm:=

ZC 1−:= ZC = 1 for C, -1 for Z section

2




c2 21.54mm=c1 21.52mm=

b2 85.51mm= b1 84.49mm=

h 287.94 mm=

t tg ∆t−:=

c2 cg2 ∆2−:=

c1 cg1 ∆1−:=

b2 bg2

tg

2− ∆2−:=

b1 bg1

tg

2− ∆1−:=

h hg tg−:=

∆2

tg

2tan φ2( )⋅:=

∆1

tg

2tan φ1( )⋅:=

φ2

π α2−( )2

:=

φ1

π α1−( )2

:=

α2 α2π

180⋅:=

α1 α1π

180⋅:=

Calculation for cross-section mid-line data

3




c p2 21.26mm=c p2 c2 gr22−:=

c p1 21.22mm=c p1 c1 gr12−:=

h p 283.86 mm=h p h gr11− gr21−:=

b p2 83.19mm= b p2 b2 gr21− gr22−:=

b p1 82.15mm= b p1 b1 gr11− gr12−:=

Nominal plate widths:

gr22 0.28 mm=

gr21 2.04 mm=

gr12 0.30 mm=

gr11 2.04 mm=

gr22 r 22

tg

2+

tan φ2( ) sin φ2( )−( )⋅:=

gr21 r 21

tg

2+

tan

π

4

sinπ

4

−

⋅:=

gr12 r 12

tg

2+

tan φ1( ) sin φ1( )−( )⋅:=

g

r11

r

11

tg

2

+

tan

π

4

sin

π

4

−

⋅:=

Gaps at the corners

Calculation of nominal plate dimensions

4




Co-ordinate system and numbering for the section properties calculation

Co-ordinates of the nominal gross cross-section

elem x_start x_end y_start y_end x_CG y_CG

# mm mm mm mm mm mm

1 -84.69 -98.35 0.23 16.46 -91.52 8.35

2 -2.04 -84.19 0.00 0.00 -43.11 0.00

3 0.00 0.00 2.04 285.90 0.00 143.97

4 2.04 85.23 287.94 287.94 43.64 287.94

5 85.69 99.81 287.73 271.83 92.75 279.78

5




Wy.g 15740mm3

=Elastic section modulus about y:

Wx.g 86700mm3

=

Elastic section modulus about x:ymax 288.97 mm=Max extreme fibre distance in y-dir:

ymin 1.03− mm=Min extreme fibre distance in y-dir:

xmin 99.14− mm=Min extreme fibre distance in x-dir:

xmax 100.58 mm=Max extreme fibre distance in x-dir:

Iy 1579323mm4

=Iy Iy Ag xCG.g2

⋅−:=Moment of inertia about y:

Ix 12599726 mm4=Ix Ix Ag yCG.g2⋅−:=Moment of inertia about x:

yCG.g 144.30 mm=yCG.g

Sx

Ag

:=Mass center in y-dir

xCG.g 0.24 mm=xCG.g

Sy

Ag

:=Mass center in x-dir:

Ag 1013 mm2

=Gross area:

s t A yCG Sx Ix xCG Sy Iy

mm mm mm2 mm mm3 mm4 mm mm3 mm4

1 21.22 2.1 43.7 8.35 365 4011 -91.52 -4000 366790

2 82.15 2.1 169.2 0.00 0 60 -43.11 -7296 409725

3 283.86 2.1 584.7 143.97 84186 16046547 0.00 0 207

4 83.19 2.1 171.4 287.94 49345 14208429 43.64 7478 425153

5 21.26 2.1 43.8 279.78 12254 3429512 92.75 4062 377506

1012.8 146150 33688558 244 1579382

Calculation of the section properties of the gross section

6




Notation for the effective cross-section calculation

Effective and ineffective portions of the (nominal) cross-section:

Stresses:

7




λrel.fl1 1.551=λrel.fl1

λ p

λel

:=

λel 0 .5 0.25 0.055 3 ψ+( )⋅−+:=

Slenderness ratio

b2.e1 31.59mm= b2.e1 0.5ρ b p2⋅:=

Effective portion:

ρ 0.759=ρ min ρ 1,( ):=

ρ 1 λ p.red 0.673≤if

1 0.055 3 ψ+( )

⋅λ p.red−

λ p.red

0.18λ p λ p.red−( )

λ p 0.6−⋅+ λ p.red 0.673>if

:=

Reduction factor:

λ p.red 1.040=λ p.red λ p

σfl γM0⋅

f yb

⋅:=

Reduced plate slenderness:

λ p 1.044=λ p

b p2

t

12 1 ν2

−( )⋅ f yb⋅

π2

E⋅ k σ⋅⋅:=

k σ 4:=ψ 1:=

Plate slenderness:

σfl 503.80 MPa=σfl σ t.max σc.max σt.max−( ) h 0.5t+

h t+⋅+:=

Stress at the mid-line of the compression flange:

σ t.max 507.4− MPa=

σc.max 507.4 MPa=σ t.max σc.max−:=σc.max

f yb

γM0

:=

Initial assumption for the extreme fibre stresses:

(the correct values will be determined later by iteration)

Calculation of the flange effective width adjacent to web

8




ht 141.93 mm=ht h p hc−:=

hc 141.93 mm=hc h p ψ 0≥if

h p

1 ψ− ψ 0<if

:=

Effective and ineffective portions:

ρ 0.644=ρ min ρ 1,( ):=


1 0.055 3 ψ+( )⋅

λ p.red

−

λ p.red

0.18λ p.w λ p.red−( )

λ p.w 0.6−⋅+ λ p.red 0.673>if

:=

Reduction factor:

λ p.red 1.442=λ p.red λ p.w

σw2 γM0⋅

f yb

⋅:=

Reduced plate slenderness:

λ p.w 1.457=λ p.w

h p

t

12 1 ν2

−( )⋅ f yb⋅

π2

E⋅ k σ⋅

⋅:=

k σ 23.90=

k σ 4 ψ 1=if

8.2

1.05 ψ+ 1 ψ> 0>if

7.81 ψ 0=if

7.81 6.29 ψ⋅− 9.78 ψ 2⋅+( ) 0 ψ> 1−>if

23.9 ψ 1−=if

5.98 1 ψ−( )2⋅ 1− ψ> 3−>if

95.68 3− ψ≥if

:=

ψ 1.00−=ψ

σw1

σw2

:=

Plate slenderness:

σw2 496.65 MPa=σw2 σ t.max σc.max σt.max−( )h 0.5 t⋅+ g

r21−

h t+⋅+:=

σw1 496.65− MPa=σw1 σ t.max σc.max σt.max−( )

0.5t gr11+

h t+⋅+:=

Stress at the web edges:

Calculation of the effective width of the web

9




λrel.fl2 1.551=λrel.fl2

λ p

λel

:=

λel 0 .5 0.25 0.055 3 ψ+( )⋅−+:=

Slenderness ratio

b2.i b2 b2.e2− b2.e1−:=

b2.e2 31.45mm= b2.e2 0.5ρ b p2⋅:=

Effective portion:

ρ 0.756=ρ min ρ 1,( ):=

ρ 1 λ p 0.673≤if

1 0.055 3 ψ+( )⋅

λ p

−

λ p

λ p 0.673>if

:=Reduction factor:

λ p 1.044=λ p

b p2

t

12 1 ν2

−( )⋅ f yb⋅

π2

E⋅ k σ⋅

⋅:=

k σ 4:=ψ 1:=

Plate slenderness:

Here only initial values are calculated which then may be modified due to distorsional buckling. Final

values will be determined later by iteration.

Calculation of the flange effective width adjacent to edge stiffener

λrel.w 1.667=λrel.w

λ p.w

λel

:=

λel 0 .5 0.25 0.055 3 ψ+( )⋅−+:=

Slenderness ratio

hi 50.54mm=hi hc he1− he2−:=

h

e2

54.83mm=he2 ρ hc⋅ he1−:=

he1 36.56mm=he1

2 ρ⋅ hc⋅

5 ψ−

ψ 0≥if

0.4 ρ⋅ hc⋅( ) ψ 0<if

:=

10




λrel.st 1.121=λrel.st

λ p

λel

:=

λel 0.673:=

Slenderness ratio

c2.i 0.11 mm=c2.i c p2 c2.ef −:=

c2.ef 21.15mm=c2.ef c p2 ρ⋅:=

Effective and ineffective portions:

ρ 0.995=ρ min ρ 1,( ):=


10.188

λ p.red−

λ p.red

0.18λ p λ p.red−( )

λ p 0.6−⋅+ λ p.red 0.673>if

:=

Reduction factor:

λ p.red 0.755=λ p.red λ p

σdist γM0⋅

f yb

⋅:=

σdist

f yb

γM0

:=

λ p 0.755=λ p

c p2

t

12 1 ν2

−( )⋅ f yb⋅

π2

E⋅ k σ⋅

⋅:=

k σ 0.500=

k σ 0.5c p2

b p2

sin α 2( )⋅

0.35≤if

0.5 0.83

3

c p2

b p2

sin α 2( )⋅ 0.35−

2

⋅+ otherwise

:=

Plate slenderness:

Here only initial values are calculated which then may be modified due to distorsional buckling. Final

values will be determined later by iteration.

Calculation of the effective width of the edge stiffener

11




χd 0.4195=

χd 1 λd 0.65≤if

1.47 0.723 λd⋅−( ) 0.65 λd< 1.38<if

0.66

λd

λd 1.38≥if

:=

Reduction factor

λd 1.573=

λd

f yb

σcr.st

:=

Slenderness

σcr.st 204.96 MPa=σcr.st

2 K 2 E⋅ Ix.st⋅⋅( )Ast

:=

Critical stress

K 2 0.221 N

mm2

=K 2

E t3

⋅( )4 1 ν

2−( )⋅

1

b2.12

h⋅ b2.13

+

⋅:=

b2.1 78.84mm= b2.1 b2 xst.CG−:=

Spring stiffness per unit length

Ix.st 2654.4 mm4

=Ix.st Ix.st c2.ef t⋅ 0.5c2.ef gr22+( ) sin α 2( )⋅

2⋅+ Ast y st.CG

2⋅−:=

Ix.st b2.e2t3

12⋅

c2.ef t3

12⋅ c2.ef

3 t

12⋅+

2

c2.ef t3

12⋅ c2.ef

3 t

12⋅−

2

cos 2 α2⋅( )⋅+

+:=

yst.CG 3.26 mm=yst.CG

c2.ef t⋅ 0.5c2.ef gr22+( )⋅ sin α 2( )⋅

Ast:=

xst.CG 6.67 mm=xst.CG

b2.e2 t⋅ 0.5b2.e2 gr22+( )⋅ c2.ef t⋅ 0.5c2.ef gr22+( )⋅ cos π α2−( )⋅−

Ast

:=

Ast 108.4mm2

=Ast b2.e2 c2.ef +( ) t⋅:=

Section properties of the stiffener:

Here only initial values are calculated which then may be

modified due to distorsional buckling. Final values will be

determined later by iteration.

Calculation of the reduction factor for the edge stiffener

12




λrel.d 2.502=λrel.d

λd

0.65:=Relative slenderness for distorsional buckling:

λd 1.626=λd

f yb

σcr.st

:=Distorsinal buckling slenderness:

σcr.st 191.83 MPa=Critical stress for distorsional buckling:

tred 0.858mm=tred t min

Ast0

Ast redd

f yb

γM0 σ st⋅⋅,

⋅:=

Reduced thickness

σst 494.14 MPa=σst σt.max σc.max σ t.max−( )

h 0.5 t⋅+ yst.CG−

h t+⋅+:=

Stress at the centroid of edge stiffener:

yst.CG 2.759mm=Stiffener's mass center distance:

Ast 129.5mm2

=Stiffener's area after iteration:

Ast0 108.4mm2

=Stiffener's area before iteration:

redd 0.4058=The reduction factor due to distorsional buckling:

iter b2.e2 c2.ef Ast Ist yCG.st cr reduction

# mm mm mm2 mm4 mm MPa factor

1 31.4462 21.1546 108.36 2654.4 3.2638 204.96 0.41947

2 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

3 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

4 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

5 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

6 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

7 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

8 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

9 41.5952 21.2618 129.49 2887.7 2.7587 191.83 0.40582

The following Table shows some important results of the iteration to find the final nominal dimensions of

the edge stiffener and the value of the reduction factor which counts for the distorsional buckling. It is

interesting to mention that all these parameters are NOT dependent on the real stresses, but only on

geometrical and material data. The effect of stresses will be considered later in the determination of the

reduced thickness of the plate elements of edge stiffener.

Iteration for the reduction factor of the edge stiffener

13




Elements for the section properties calculation of effective section

Co-ordinates of the nominal effective cross-section

elem x_start x_end y_start y_end x_CG y_CG

# mm mm mm mm mm mm

1 -84.69 -98.35 0.23 16.46 -91.52 8.35

2 -2.04 -84.19 0.00 0.00 -43.11 0.00

3 0.00 0.00 2.04 285.90 0.00 143.97

4 2.04 85.23 287.94 287.94 43.64 287.94

5 85.69 99.81 287.73 271.83 92.75 279.78

6

7

14




σ t.max 349.93− MPa=Max tensile stress:

σc.max 507.40 MPa=Max compressive stress:

Wy.ef 9868 mm3

=Effective section modulus about y:

Wx.ef 54758mm3=Effective section modulus about x:

yten 118.37− mm=Tensioned extreme fibre distance from the centroid:

ycom 171.63 mm=Compressed extreme fibre distance from the centroid:

Iy.ef 1067077mm4

=Iy.ef Iy.ef Aef xCG.ef 2

⋅−:=Moment of inertia about y:

Ix.ef 9398316mm4

=Ix.ef Ix.ef Aef yCG.ef 2

⋅−:=Moment of inertia about x:

Mass center in y-dir yCG.ef 117.34 mm=yCG.ef

Sx.ef

Aef :=

Mass center in x-dir: xCG.ef 7.56− mm=xCG.ef

Sy.ef

Aef

:=

Aef 812.6mm2

=Effective area:

s t A yCG Sx Ix xCG Sy Iy

mm mm mm2 mm mm3 mm4 mm mm3 mm4

1 21.22 2.06 43.7 8.35 365 4011 -91.52 -4000 366790

2 82.15 2.06 169.2 0.00 0 60 -43.11 -7296 409725

3 196.76 2.06 405.3 100.42 40704 5395335 0.00 0 143

4 36.56 2.06 75.3 267.62 20153 5401781 0.00 0 27

5 31.59 2.06 65.1 287.94 18738 5395558 17.84 1161 26117

641.60 0.86 35.7 287.94 10281 2960365 64.43 2301 153391

7 21.26 0.86 18.3 279.78 5106 1429099 92.75 1693 157307

812.6 95348 20586209 -6142 1113500

The following Table shows the details of the effective section properties calculation. Here only initial values

are calculated based on the assumed extreme fibre stresses. Later, all these parameters will be

re-calculated to find the final values of extreme fibre stresses and section properties.

Calculation of section properties of the effective section

15




Iteration for the effective section properties

In the following Table some results of the iteration for the effective section properties are shown. The

iteration is governed by the extreme fibre stresses. In each iteration step the effective portions of the

compressed flange and web are determined, then the section properties are calculated in accordance with

the principles and formulae presented above.

c.max t.max xCG yCG A Ix Iy Wx Wy

MPa MPa mm mm mm2 mm4 mm4 mm3 mm3

0 507.40 -507.40 -7.558 117.337 812.6 9398316 1067077 54758.2 9868.0

1 507.40 -349.93 -8.147 112.184 756.2 9101045 1061956 51480.6 9767.5

2 507.40 -324.94 -8.265 111.382 745.7 9066274 1060987 51052.2 9748.0

3 507.40 -321.18 -8.284 111.263 744.0 9061433 1060837 50990.8 9745.0

4 507.40 -320.62 -8.286 111.245 743.8 9060727 1060814 50981.8 9744.5

5 507.40 -320.54 -8.287 111.243 743.8 9060624 1060811 50980.4 9744.4

6

7

The elastic modulus of the effective cross-section: Wx.ef 50980.4mm3

=

Maximum compressive stress: σc.max 507.40 MPa=

Maximum tensile stress:σ t.max 320.54− MPa=

Relative slenderness of the web:

λrel.w

λ p.w

0.5 0.25 0.055 3σ t.max

σc.max

+

⋅−+

:=

λrel.w 1.723=

Maximal relative slenderness:

λrel.max max λ rel.fl1 λrel.fl2, λrel.w, λrel.st, λrel.d,( ):= λrel.max 2.502=

16




Wx.pl 102407.4 mm3

=

Wx.pl 2 c p1 t⋅ yCG.g 0.5 c p1⋅ gr12+( ) sin α 1( )⋅−⋅ b p1 t⋅ yCG.g⋅+ hw.pl t⋅ yCG.g 0.5 hw.pl⋅− gr11−( )⋅+⋅:=

The plastic section modulus:

hw.pl 142.47 mm=hw.pl

0.5Ag c p1 b p1+( )t−

t:=

Web portion for the plastic modulus:

Calculation of the plastic modulus of the gross cross-section about x-axis

f ya 519.62 MPa=f ya min f ya

f u f yb+2

,

:=

f ya f yb f u f yb−( ) k n⋅ t

2⋅

Ag

⋅+:=

n n11 n12+ n21+ n22+:=

n22π α2−

0.5π r 22 5t≤if

0 otherwise

:=

n12

π α1−

0.5π r 12 5t≤if

0 otherwise

:=

n21 1 r 21 5t≤if

0 otherwise

:=

n11 1 r 11 5t≤if

0 otherwise

:=

k 7 prod_mod 1=if

5 otherwise

:=

Calculation of the average yield strength

17




Calculation of the moment resistance of the cross-section

The effective, gross and plastic section muduli: Wx.ef 50980mm3

=

Wx.g 86700mm

3

=

Wx.pl 102407 mm3

=

Maximal relative slenderness: λrel.max 2.502=

The moment resistance:

Mc.Rd Wx.ef

f yb

γM0

⋅ Wx.ef Wx.g<if

min

f ya

γM0Wx.g Wx.pl Wx.g−( )4 1 λrel.max−( )+⋅ Wx.pl

f yb

γM0⋅,

Wx.ef Wx.g=( ) ZC =(∧if

Wx.g

f ya

γM0

⋅ Wx.ef Wx.g=( ) ZC 1−=( )∧if

:=

Mc.Rd 25.87 kNm=

Remarks:

It is assumed that a C-section is approx. symmetrical. Thus, utilization of plastic capacity•is allowed.

In the first part of equation (6.5) of EC3 "fyb" is written, which seems to be meaningless.•Thus, it is changed to "fya".

18

calculation of the moment resistance of z- and c-shaped cold-formed sections according to eurocode 3

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