Elastic Analysis

Document Ref: SX029a-EN-EU Sheet 1 of 28 Title

Example: Elastic analysis of a single bay portal frame

Eurocode Ref Made by Valérie Lemaire Date April 2006

CALCULATION SHEET

Checked by Alain Bureau Date April 2006


A single bay portal frame made of rolled profiles is designed according to EN 1993-1-1. This worked example includes the elastic analysis of the frame using first order theory, and all the verifications of the members under ULS combinations.

30,00

5,98

8

α

[m]

7,20

7,30 72,00

1 Basic data

• Total length : b = 72,00 m • Spacing: s = 7,20 m • Bay width : d = 30,00 m • Height (max): h = 7,30 m • Roof slope: α = 5,0°

1

3,00 3,00 3,00 3,00 3,00

1 : Torsional restraints

Example: Elastic analysis of a single bay portal frameC

reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent




CALCULATION SHEET


2 Loads

EN 1991-1-12.1 Permanent loads

• self-weight of the beam

• roofing with purlins G = 0,30 kN/m2

for an internal frame: G = 0,30 × 7,20 = 2,16 kN/ml

2.2 Snow loads EN 1991-1-3Characteristic values for snow loading on the roof in [kN/m] S = 0,8 × 1,0 × 1,0 × 0,772 = 0,618 kN/m²

⇒ for an internal frame: S = 0,618 × 7,20 = 4,45 kN/m

α

7,30

30,00

s = 4,45 kN/m

[m]


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent




CALCULATION SHEET


2.3 Wind loads EN 1991-1-4Characteristic values for wind loading in kN/m for an internal frame

30,00e/10 = 1,46 1,46

Zone D:w = 4,59

Zone G:w = 9,18

Zone H:w = 5,25

Zone J:w = 5,25 Zone I:

w = 5,25

Zone E:w = 3,28

wind direction

3 Load combinations EN 1990

3.1 Partial safety factor

• γGmax = 1,35 (permanent loads) • γGmin = 1,0 (permanent loads) EN 1990

Table A1.1• γQ = 1,50 (variable loads)

• ψ0 = 0,50 (snow)

• ψ0 = 0,60 (wind)

= 1,0 • γM0

= 1,0 • γM1

3.2 ULS Combinations EN 1990 Combination 101 : γGmax G + γ QsQ

Combination 102 : γGmin G + γQ Qw

Combination 103 : γGmax G + γ QQ s + γQ ψ0

Combination 104 : γGmin G + γQ Qs + γQ ψ0 Qw

Combination 105 : γGmax G + γ ψQ 0 Qs + γQ Qw

Combination 106 : γGmin G + γQ ψ0 Qs + γQ Qw

3.3 SLS Combinations EN 1990 Combinations and limits should be specified for each project or by National Annex.


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

http://www.access-steel.com/discovery/linklookup.aspx?id=EC291





CALCULATION SHEET


4 Sections

hw

y y

z

z

tf

tw

b

h

4.1 Column Try IPE 600 – Steel grade S275

Depth h = 600 mm

Web Depth hw = 562 mm

Depth of straight portion of the web

dw = 514 mm

Width b = 220 mm

Web thickness tw = 12 mm

Flange thickness t = 19 mm f

Fillet r = 24 mm

Mass 122,4 kg/m

Section area A = 156 cm2

Second moment of area /yy Iy = 92080 cm4

Second moment of area /zz Iz = 3386cm4

Torsion constant I = 165,4 cm4t

Warping constant Iw = 2845500 cm6

Elastic modulus /yy Wel,y = 3069 cm3

Plastic modulus /yy Wpl,y = 3512 cm3

Elastic modulus /zz W = 307,80 cm3el,z

Plastic modulus /zz Wpl,z = 485,60 cm3


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent




CALCULATION SHEET


4.2 Rafter Try IPE 500 – Steel grade S275

Depth h = 500 mm

Web Depth hw = 468 mm

Depth of straight portion of the web

dw = 426 mm

Width b = 200 mm

Web thickness tw = 10,2 mm

Flange thickness tf = 16 mm

Fillet r = 21 mm

Mass 90,7 kg/m

Section area A = 115,50 cm2


Second moment of area /zz Iz = 2141 cm4

Torsion constant It = 89,29 cm4

Warping constant Iw = 1249400 cm6


Plastic modulus /yy Wpl,y = 2194 cm3

Elastic modulus /zz Wel,z = 214,1 cm3

Plastic modulus /zz Wpl,z = 335,90 cm3

5 Global analysis The joints are assumed to be:

• pinned for column bases

• rigid for beam to column.

EN 1993-1-1 § 5.2

The frame has been modelled using the EFFEL program.


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent






CALCULATION SHEET


5.1 Buckling amplification factor α EN 1993-1-1cr § 5.2.1In order to evaluate the sensitivity of the frame to 2nd order effects, a buckling

analysis is performed to calculate the buckling amplification factor αcr for the load combination giving the highest vertical load: γ G + γmax Q QS (combination 101).

For this combination, the amplification factor is: αcr = 14,57

The first buckling mode is shown hereafter.

EN 1993-1-1 So : αcr = 14,57 > 10 § 5.2.1First order elastic analysis may be used. (3)

5.2 Effects of imperfections EN 1993-1-1 § 5.3.2The global initial sway imperfection may be determined from (3)

310204,3866,0740,0

2001 −⋅=×× φ = φ0 αh α = m

where φ0 = 1/200

740,030,7

22==

h αh =

866,0)11(5,0 =+m

m = 2 (number of columns) =α m

Sway imperfections may be disregarded where H ≥ 0,15 V EN 1993-1-1 Ed Ed.

§ 5.3.2The effects of initial sway imperfection may be replaced by equivalent horizontal forces:

(4)

H = φ V in the combination where H < 0,15 ⎢Veq Ed Ed Ed ⎢


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent












CALCULATION SHEET


The following table gives the reactions at supports.

Left column 1 Right column 2 Total ULS

Comb. HEd,1

kN

VEd,1

kN

HEd,2

Kn

VEd,2

kN

HEd

kN

VEd

kN

0,15 ⎢VEd ⎢

101 -125,5 -172,4 125,5 -172,4 0 -344,70 51,70

102 95,16 80,74 -24,47 58,19 70,69 138,9 20,83

103 -47,06 -91,77 89,48 -105,3 42,42 -197,1 29,56

104 -34,59 -73,03 77,01 -86,56 42,42 -159,6 23,93

105 43,97 11,97 26,72 -10,57 70,69 1,40 0,21

106 56,44 30,71 14,25 8,17 70,69 38,88 5,83

The sway imperfection has only to be taken into for the combination 101:

VEd kN

Heq = φ.VEd kN

172,4 0,552

⇒ Modelling with Heq for the combination 101

EN 1993-1-1 § 5.3.2 (7)

5.3 Results of the elastic analysis 5.3.1 Serviceability limit states Combinations and limits should be specified for each project or in National Annex.

For this example, the deflections obtained by modeling are as follows:

EN 1993-1-1 § 7 and

EN 1990

Vertical deflections:

G + Snow: Dy = 124 mm = L/241

Snow only: Dy = 73 mm = L/408

Horizontal deflections: Deflection at the top of column by wind only

Dx = 28 mm = h/214


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent








CALCULATION SHEET


5.3.2 Ultimate limit states Moment diagram in kNm

Combination 101:

Combination 102:

Combination 103:

Combination 104:


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent




CALCULATION SHEET


Combination 105:

Combination 106:

6 Column verification

Profile IPE 600 - S275 (ε = 0,92) The verification of the member is carried out for the combination 101 : N = 161,5 kN (assumed to be constant along the column) Ed

= 122,4 kN (assumed to be constant along the column) VEd M = 755 kNm (at the top of the column) Ed

6.1 Classification of the cross section • Web: The web slenderness is c / tw = 42,83 EN 1993-1-1

§ 5.5mm94,48

27512161500

yw

EdN =

×==

ftN

d

548,05142

94,485142 w

Nw =×+

=+

=d

ddα > 0,50

49,591548,013

92,0396=

−×× The limit for Class 1 is : 396ε / (13α − 1) =

Then : c / tw = 42,83 < 59,49 The web is class 1.


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent






CALCULATION SHEET


• Flange: The flange slenderness is c / tf = 80 / 19 = 4,74 The limit for Class 1 is : 9 ε = 9 × 0,92 = 8,28

Then : c / t = 4,74 < 8,28 The flange is Class 1 f

So the section is Class 1. The verification of the member will be based on the plastic resistance of the cross-section.

6.2 Resistance of cross section Verification for shear force Shear area : A = A - 2btv f + (tw+2r)t > η.hf w.t EN 1993-1-1w

§ 6.2.6 η may be conservatively taken equal to 1 (3)

838019)24212(19220215600 =××++××−=vA mm2 > η.hw.tw = 6744 mm2

Vpl,Rd = Av (fy / 3 ) /γM0 = (8380×275/ 3 ).10-3

V = 1330 kN pl,Rd

V / VEd pl,Rd = 122,40/1330 = 0,092 < 0,50

The effect of the shear force on the moment resistance may be neglected.

Verification to axial force EN 1993-1-1 § 6.2.4-3 N = A fpl,Rd y / γ = (15600 × 275/1,0).10M0 N = 4290 kN pl,Rd N = 161,5 kN < 0,25 NEd pl,Rd = 4290 x 0,25 = 1073 kN EN 1993-1-1

and NEd = 161,5 kN < 3,92710001

275125625,05,0

M0

yww =×

×××=

γfth

§ 6.2.8kN (2)

The effect of the axial force on the moment resistance may be neglected.

Verification to bending moment EN 1993-1-1 § 6.2.5-3 M = Wpl,y,Rd pl,y fy / γ = (3512 × 275/1,0).10 M0

M = 965,8 kNm pl,y,Rd

My,Ed = 755 kNm < M = 965,8 kNm pl,y,Rd


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent












CALCULATION SHEET


6.3 Buckling resistance The buckling resistance of the column is sufficient if the following conditions are fulfilled (no bending about the weak axis, M

EN 1993-1-1z,Ed = 0):

§ 6.3.3

1

M1

Rky,LT

Edy,yy

M1

Rky

Ed ≤+

γχ

γχ M

MkN

N

1

M1

Rky,LT

Edy,zy

M1

Rky

Ed ≤+

γχ

γχ M

MkN

N

The k and kyy zy factors will be calculated using the Annex A of EN 1993-1-1.

The frame is not sensitive to second order effects (αcr = 14,57 > 10). Then the buckling length for in-plane buckling may be taken equal to the system length.

EN 1993-1-1 § 5.2.2

(7)

Lcr,y = 5,99 m Note: For a single bay symmetrical frame that is not sensitive to second order effects, the check for in-plane buckling is generally not relevant. The verification of the cross-sectional resistance at the top of the column will be determinant for the design.

Regarding the out-of-plane buckling, the member is laterally restrained at both ends only. Then :

L = 5,99 m for buckling about the weak axis cr,z L = 5,99 m for torsional buckling cr,T and L = 5,99 m for lateral torsional buckling cr,LT

• Buckling about yy Lcr,y = 5,99 m

EN 1993-1-1 Buckling curve : a (αy = 0,21) § 6.3.1.2

10005990

10000920802100002

22

ycr,

y2ycr, ×

××== ππ

LEI

N (2)

=53190kN Table 6.1

284,0

10.5319027515600

3ycr,

yy =

×==

NAf

λ EN 1993-1-1 § 6.3.1.3 (1)


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent













CALCULATION SHEET


( )[ ] ( )[ ]22yyy 284,02,0284,021,015,02,015,0 +−+×=+−+= λλαφ EN 1993-1-1 = 0,5491

§ 6.3.1.2 (1)

9813,0284,05491,05491,0

11222

y2

yy

y =−+

=−+

=λφφ

χ

• Buckling about zz L = 5,99 m EN 1993-1-1 cr,z § 6.3.1.2 Buckling curve : b (αz = 0,34)

(2)

Table 6.1

10005990100003386210000

22

2zcr,

z2zcr, ×

××== ππ

LEI

N = 1956 kN

EN 1993-1-1 481,1

10.195627515600

3zcr,

yz =

×==

NAf

λ § 6.3.1.3 (1)

( )[ ] ( )[ ]22zzz 481,12,0481,134,015,02,015,0 +−+×=+−+= λλαφ EN 1993-1-1 = 1,814

§ 6.3.1.2 (1)

3495,0481,1814,1814,1

11222

z2zz

z =−+

=−+

=λφφ

χ

• Lateral torsional buckling Lcr,LT = 5,99 m

EN 1993-1-1 Buckling curve : c (αLT = 0,49) § 6.3.2.3 Moment diagram with linear variation : ψ = 0 C1 = 1,77 Table 6.5

Z2

t2

LTcr,

Z

W2

LTcr,

Z2

1cr EIGIL

II

LEICM

ππ

+=

NCCI SN003

42

42

4

6

62

2

cr 10.338621000010.4,165807705990

10.338610.2845500

10599010000338621000077,1

××××

+×

××××=

ππM

Mcr = 1351 kNm

8455,010.1351

27510.35126

3

cr

yypl,LT =

×==

MfW

λ

( )[ ]2

LTLT,0LTLTLT 15,0 βλλλαφ +−+= EN 1993-1-1 § 6.3.2.3 (1)

and β =0,75 40,0LT,0 =λ with


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent











http://www.access-steel.com/discovery/linklookup.aspx?id=SN003






CALCULATION SHEET


( )[ ] 8772,08455,075,04,08455,049,015,0 2LT =×+−+×=φ

7352,08455,075,08772,08772,0

11222

LT2LTLT

LT =×−+

=−+

=βλφφ

χ

7519,033,033,1

1=

− ψEN 1993-1-1 k (ψ = 0) c = § 6.3.2.3

( )[ ]2LTc 8,021)1(5,01 −−−×−= λkf

(2) Table 6.6

( )[ ] 8765,08,08455,021)7519,01(5,01 2 =−−−×−=f < 1

8388,08765,07352,0

==fLTχ χLT,mod = < 1

and kCalculation of the factors kyy zy according to Annex A of EN 1993-1-1 EN 1993-1-1 Annex A

9999,0

531905,1619813,01

531905,1611

1

1

ycr,

Edy

ycr,

Ed

y =×−

−=

−

−

=

NN

NN

χμ

9447,0

19565,1613495,01

19565,1611

1

1

zcr,

Edz

zcr,

Ed

z =×−

−=

−

−=

NN

NN

χμ

EN 1993-1-1 144,1

30693512

yel,

ypl,y ===

WW

w < 1,5 Annex A

578,1

8,3076,485

zel,

zpl,z ===

WW

w > 1,5 ⇒ w z = 1,5

NCCI Critical axial force in the torsional buckling mode

SN003)( 2

Tcr,

w2

t0

Tcr,L

EIGIIAN π

+=

For a doubly symmetrical section,

cm4 95466338692080)( 20

20zy0 =+=+++= AzyIII


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent











CALCULATION SHEET


⎟⎟⎠

⎞⎜⎜⎝

⎛ ×+××

×= 2

624

4cr, 599010.284550021000010.4,16580770

100010.9546615600 πTN

Ncr,T = 4869 kN

NCCI

Z2

t2

LTcr,

Z

W2

LTcr,

Z2

1cr,0 EIGIL

II

LEICM

ππ

+= SN003

0λMcr,0 is the critical moment for the calculation of for uniform bending moment as specified in Annex A.

⇒ C1 = 1

42

42

4

6

62

42

cr,0 10.338621000010.4,165807705990

10.338610.2845500

10599010.33862100001

××××

+×××

×=π

πM

kNmM 3,763ocr, =

EN 1993-1-1 6

3

ocr,

yypl,0

10.3,76327510.3512 ×

==M

fWλ = 1,125 Annex A

4

TFcr,

Ed

zcr,

Ed1lim0 )1)(1(2,0

NN

NN

C −−=λ

with N = Ncr,TF cr,T (doubly symmetrical section)

4lim0 )4869

5,1611)(1956

5,1611(77,12,0 −−=λ = 0,2582

0λ > lim0λ

LTy

LTymy,0my,0my 1

)1(a

aCCC

ε

ε

+−+=

yel,Ed

Edy,y W

AN

M=ε

y

tLT I

Ia −=1= 23,76 (class 1) and = 0,9982 with


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent







CALCULATION SHEET


Calculation of the factor C my,0

EN 1993-1-1

ycr,

Edyymy,0 )33,0(36,021,079,0

NN

C −++= ψψ Annex A

Table A2

ycr,

Edmy,0 1188,079,0

NN

C −= = 0,7896 0y =ψ

Calculation of the factors C and Cmy m,LT :

LTy

LTymy,0my,0my 1

)1(a

aCCC

ε+

ε−+=

9641,09982,076,231

9982,076,23)7896,01(7896,0Cmy =

×+

×−+=

EN 1993-1-1 1

)1)(1(Tcr,

Ed

zcr,

Ed

LT2mymLT ≥

−−

=

NN

NN

aCC Annex A

9843,0)

48695,1611)(

19565,1611(

9982,09641,0 2mLT =

−−×=C < 1

⇒ C = 1 mLT

Calculation of the factors Cyy and C : EN 1993-1-1 zy Annex A

ypl,

yel,LTpl

2max

2my

y

max2my

yyyy ])6,16,12)[(1(1

WW

bnCw

Cw

wC ≥−λ−λ−−+=

03765,01/27515600

161500/ M1Rk

Edpl =

×==

γNNn

Mz,Ed = 0 ⇒ and 0LT =b 0LT =d 4810,1zmax =λ=λ

]03765,0)481,19641,0144,1

6,1481,19641,0144,1

6,12[()1144,1(1 222yy ×××−××−×−+=C

8739,035123069

ypl,

yel, ==>WW

Cyy = 0,9849


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent










CALCULATION SHEET


ypl,

yel,

z

yLTpl

2max

2my5

yyzy 6,0])142)[(1(1

WW

ww

dnCw

wC ≥−λ−−+=

9318,0]03765,0)481,19641,0144,1142)[(1144,1(1 22

5zy =×××−−+=C

4579,035123069

5,1144,16,06,0

ypl,

yel,

z

y ==WW

ww

>= 9318,0zyC

Calculation of the factors kyy and kzy : EN 1993-1-1 Annex A

yy

ycr,

Ed

ymLTmyyy

1

1 CNNCCk

−

μ=

9818,09849,01

531905,1611

9999,019641,0yy =×−

××=k

z

y

zy

ycr,

Ed

zmLTmyzy 6,01

1 ww

CNNCCk

−=

μ

5138,050,1

144,16,09318,01

531905,1611

9447,019641,0zy =×××−

××=k

Verification with interaction formulae

EN 1993-1-1 1

M1

Rky,LT

Edy,

M1

Rky

Ed ≤+

γχ

γχ M

MkN

Nyy

§ 6.3.3

9534,0

127510.35128388,0

10.7559818,0

1275156009813,0

1615003

6=

××

×+×

× <1 OK

1

M1

Rky,LT

Edy,zy

M1

Rkz

Ed ≤+

γχγ

χ MM

kNN


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent








CALCULATION SHEET


5867,0

127510.35128388,0

10.7555138,0

1275156003495,0

1615003

6=

××

×+×

×<1 OK

So the buckling resistance of the member is satisfactory.

7 Rafter verification

7.1 Classification Case with maximum compression in beam: (Combination 101)

• Web: h = 468 mm w

tw = 10,2 mm c / tw = 41,76

c = 426 mm

NEd= 136 kN mmft

Nd 5,482752,10

136000

yw

EdN =

×==

557,04262

5,484262

N =×+

=+

=dddα > 0,5

c / tw = 41,76 < 38,581557,013

92,0396113

396=

−××

=−αε ⇒ class 1

EN 1993-1-1 § 5.5

• Flanges b = 200 mm tf = 16 mm r = 21 mm c / tf = 4,44 c = 71 mm

part to compression

c / tf < 9ε = 8,28 (S275 ⇒ ε = 0,92 )

c / t = 4,44 class 1 f

So the section is Class 1. The verification of the member will be based on the plastic resistance of the cross-section.


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent






CALCULATION SHEET


7.2 Resistance of cross-section Verification with maximum moment along the member in cross-section of IPE 500: Combination 101 Maximum force in IPE 500 at the end of the haunch: N = 136,00 kN Ed V = 118,50 kN Ed M = 349,10 kNm y,Ed

305,23 kNm

349,10 kNm 754,98 kNm

Combination 101: Bending moment diagram along the rafter

Shear V = 118,50 kN EN 1993-1-1Ed § 6.2 A = A - 2btv f + (tw+2r)t η = 1 f

598516)2122,10(16200211550 =××++××−=vA mm2

EN 1993-1-1 Av > η.hw.tw = 468×10,2 = 4774mm2

§ 6.2.8 (2)

3 3Vpl,Rd = Av (f / ) /γ = 5985×275/ /1000 = 950,3 kN y M0

V / VEd pl,Rd = 118,5/950,3 = 0,125 < 0,50

⇒ its effect on the moment resistance may be neglected!

Compression EN 1993-1-1 § 6.2.4Npl, = 11550 x 275/1000 = 3176 kN Rd N = 136 kN < 0,25 NEd pl, = 3176 × 0,25 = 794,1 kN Rd and EN 1993-1-1

kN4,65610001

2752,104685,05,0

M0

yww =×

×××=

γfth § 6.2.8NEd = 136 kN < (2)

⇒ its effect on the moment resistance may be neglected!


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent












CALCULATION SHEET


Bending EN 1993-1-1 § 6.2.5 M = 2194 × 275/1000 = 603,4 kNm pl,y,Rd M = 349,10 kNm < My,Ed pl,y,Rd = 603,4 kNm

7.3 Buckling resistance Uniform members in bending and axial compression EN 1993-1-1

§ 6.3.3Verification with interaction formulae

1

M1

Rky,LT

Edy,yy

M1

Rky

Ed ≤+

γχ

γχ M

Mk

NN

and 1

M1

Rky,LT

Edy,zy

M1

Rkz

Ed ≤+

γχγ

χ MM

kN

N

• Buckling about yy:

For the determination of the buckling length about yy, a buckling analysis is performed to calculate the buckling amplification factor αcr for the load combination giving the highest vertical load, with a fictitious restraint at top of column:

EN 1993-1-1 § 6.3.1.2 (2)

Table 6.1 Combination 101 ⇒ αcr = 37,37

EN 1993-1-1 § 6.3.1.3 (1)

EN 1993-1-1 Buckling curve : a (h/b>2) ⇒ αy = 0,21 § 6.3.1.2

kNNN 508213637,37Edcrycr, =×=α= (2)

Table 6.1

7906,010.508227511550

3ycr,

yy =

×==λ

NAf


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent














CALCULATION SHEET


( ) ⎥⎦⎤

⎢⎣⎡ +−+=

2yyy 2,015,0 λλαφ

= 0,8745 ( )[ ]2y 7906,02,07906,021,015,0 +−×+×=φ

8011,07906,08745,08745,0

11222

y2

yy

y =−+

=−+

=λφφ

χ

• Buckling about zz: For buckling about zz and for lateral torsional buckling, the buckling length is taken as the distance between torsional restraints: Lcr = 6,00m Note: the intermediate purlin is a lateral restraint of the upper flange only. Its influence could be taken into account but it is conservatively neglected in the following. Flexural buckling EN 1993-1-1

L = 6,00 m § 6.3.1.3cr,z

10006000100002141210000

22

2zcr,

z2zcr, ×

××== ππ

LEIN

= 1233kN

NCCI Torsional buckling

Lcr,T = 6,00 m SN003

)( 2Tcr,

w2

t0

Tcr, LEIGI

IAN π

+=

with yo = 0 and zo = 0 (doubly symmetrical section)

cm450340214148199)( 20

20zy0 =+=+++= AzyIII

)

600010.124937021000010.29,8980770(

100010.5034011550

2

624

4Tcr,×

+×××

= πN

Ncr,T = 3305 kN

Ncr = min ( N ; Ncr,z cr,T ) = 1233 kN EN 1993-1-1 § 6.3.1.3

605,110.123327511550

3cr

yz =

×==

NAf

λ (1)


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent









CALCULATION SHEET


Buckling curve : b EN 1993-1-1 § 6.3.1.2αz = 0,34 (1)

( )[ ]2zzz 2,015,0 λλαφ +−+=

Table 6.1

( )[ ]2

z 605,12,0605,134,015,0 +−×+×=φ =2,027

3063,0605,1027,2027,2

11222

z2zz

z =−+

=−+

=λφφ

χ

• Lateral torsional buckling : EN 1993-1-1 § 6.3.1.3 L , = 6,00 m cr LT

Table 6.5 = 0,49 Buckling curve : c αLT

Moment diagram along the part of rafter between restraints:

Combination 101

NCCI Calculation of the critical moment:

SN003 ψ = - 0,487

MqL8

2

q = - 9,56 kN/m μ = = - 0,123

⇒ C1 = 2,75

NCCI

Z2

t2

LTcr,

Z

W2

LTcr,

Z2

1cr EIGIL

II

LEICM

ππ

+=

42

42

4

6

62

42

cr 10.214121000010.29,89807706000

10.214110.1249400

10600010214121000075,2

××××

+×

××××=

ππM

Mcr = 1159 kNm


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent









CALCULATION SHEET


7215,0

10.115927510.21956

3

cr

yypl,LT =

×==

MfW

λ

( )[ ]2

LTLT,0LTLTLT 15,0 βλλλαφ +−+=

EN 1993-1-1 § 6.3.2.3 with 40,0LT,0 =λ and β =0,75 (1)

( )[ ] 7740,07215,075,04,07215,049,015,0 2LT =×+−×+×=φ

8125,07215,075,07740,07740,0

11222

LT2LTLT

LT =×−+

=−+

=λβφφ

χ

kc = 0,91

( )[ ]28,021)1(5,01 −−−×−= LTckf λ

EN 1993-1-1 § 6.3.2.3

( )[ ] 9556,08,07215,021)91,01(5,01 2 =−×−×−×−=f (2)

< 1 Table 6.6

8503,09556,08125,0

==fLTχ χLT,mod = < 1

Combination 101 N = 136 kN compression Ed

M = 349,10 kNm y,Ed

M z,Ed = 0

Section class1 ⇒ ΔM = 0 et ΔMy,Ed z,Ed = 0 EN 1993-1-1 § 6.3.3

1

M1

Rky,LT

Edy,yy

M1

Rky

Ed ≤+

γχ

γχ M

MkN

N 1

M1

Rky,LT

Edy,zy

M1

Rkz

Ed ≤+

γχγ

χ MM

kNN


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent










CALCULATION SHEET


EN 1993-1-1

9946,0

50821368011,01

50821361

1

1

ycr,

Edy

ycr,

Ed

y =×−

−=

−

−=

NN

NN

χμ

Annex A

9208,0

12331363063,01

12331361

1

1

zcr,

Edz

zcr,

Ed

z =×−

−=

−

−=

NN

NN

χμ

EN 1993-1-1 138,1

19282194

yel,

ypl,y ===

WW

w < 1,50 Annex A

569,1

1,2149,335

zel,

zpl,z ===

WW

w > 1,50 ⇒ w = 1,5 z

NCCI

Z2

t2

LTcr,

Z

W2

LTcr,

Z2

1cr,0 EIGIL

II

LEICM

ππ

+= SN003

0λMcr,0 is the critical moment for the calculation of for uniform bending moment as specified in Annex A.

⇒ C1 = 1

42

42

4

6

62

42

cr,0 10.214121000010.29,89807706000

10.214110.1249400

10600010.21412100001

××××

+×××

×=π

πM

kNmM 5,421ocr, =

EN 1993-1-1 6

3

ocr,

yypl,0

10.5,42127510.2195 ×

==M

fWλ = 1,196

Annex A

4

TFcr,

Ed

zcr,

Ed1lim0 )1)(1(2,0

NN

NNC −−=λ with C1 = 2,75

with N = Ncr,TF cr,T (doubly symmetrical section)

4lim0 )33051361)(

12331361(75,22,0 −−=λ = 0,3187

0λ =1,196 > lim0λ =0,3187


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent











CALCULATION SHEET


EN 1993-1-1

LTy

LTymy,0my,0my 1

)1(a

aCCC

ε

ε

+−+= Annex A

with 3

6

yel,Ed

Edy,y 101928

11550136000

10.10,349×

×==W

AN

Mε =15,38 (class 1)

and 48200

29,8911y

tLT −=−=

IIa = 0,9981

Calculation of the factor C EN 1993-1-1 my,0 Annex AMoment diagram along the rafter: Table A2

My,Ed = maximum moment along the rafter = 755kNm

δ = maximum displacement along the rafter = 179mm 30m

ycr,

Ed

Edy,2

xy2

my,0 11NN

MLEI

C⎥⎥⎦

⎤

⎢⎢⎣

⎡−+=

δπ

50821361

10755300001791048200210000

1 62

42

my,0⎥⎥⎦

⎤

⎢⎢⎣

⎡−

××××××

+=π

C =0,9803

Calculation of the factors C and C my m,LT :

LTy

LTymy,0my,0my 1

)1(a

aCCC

ε

ε

+−+=

996,09982,038,151

9982,038,15)9803,01(9803,0my =×+

×−+=C


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent









CALCULATION SHEET


EN 1993-1-1 1

)1)(1(Tcr,

Ed

zcr,

Ed

LT2mymLT ≥

−−=

NN

NN

aCC Annex A

072,1)

33051361)(

12331361(

9981,0996,0 2mLT =

−−×=C > 1

Calculation of the factors Cyy and C EN 1993-1-1 zyAnnex A

ypl,

yel,LTpl

2max

2my

y

max2my

yyyy ])6,16,12)[(1(1

WW

bnCw

Cw

wC ≥−−−−+= λλ

0428,01/27511550

136000/ M1Rk

Edpl =

×==

γNNn

M 605,1max == zλλz,Ed = 0 ⇒ and 0LT =b 0LT =d

]0428,0)605,1996,0138,1

6,1605,1996,0138,1

6,12)[(1138,1(1 222yy ×××−××−−+=C

Cyy = 0,9774

ypl,

yel,

z

yLTpl

2max

2my5

yyzy 6,0])142)[(1(1

WW

ww

dnCw

wC ≥−−−+= λ

9011,0]0428,0)605,1996,0138,1142)[(1138,1(1 22

5zy =×××−−+=C

Calculation of the factors kyy and kzy : EN 1993-1-1 Annex A

yy

ycr,

Ed

ymLTmyyy

1

1 CNNCCk

−=

μ

116,19774,01

50821361

9946,0072,1996,0yy =×−

××=k


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent










CALCULATION SHEET


z

y

zy

ycr,

Ed

zmLTmyzy 6,01

1 ww

CNNCCk

−=

μ

5859,050,1

138,16,09011,01

50821361

9208,0072,1996,0zy =×××−

××=k

Verification with interaction formulae EN 1993-1-1 § 6.3.3

1

M1

Rky,LT

Edy,yy

M1

Rky

Ed ≤+

γχ

γχ M

MkN

N (6.61)

8131,0

127510.21948503,0

10.1,349116,1

1275115508011,0

1360003

6=

××

×+×

× < 1 OK

1

M1

Rky,LT

Edy,zy

M1

Rkz

Ed ≤+

γχγ

χ MM

kNN

(6.62)

5385,0

127510.21948503,0

10.1,3495859,0

1275115503063,0

1360003

6=

××

×+×

×< 1 OK

8 Haunch verification

For the verification of the haunch, the compression part of the cross-section is considered as alone with a length of buckling about the zz-axis equal to 3,00m (length between the top of column and the first restraint).

Maximum forces and moments in the haunch:

N = 139,2 kN EdV = 151,3 kN EdM = 755 kNm Ed


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent






CALCULATION SHEET


Properties of the whole section:

The calculation of elastic section properties for this case is approximate, ignoring the middle flange.

1000 mm

200 mm

Section area A = 160,80 cm2


Second moment of area /zz Iz = 2141 cm4


Elastic modulus /zz W = 214 cm3el,z

Properties of the compression part:

Section at the mid-length of the haunch including 1/6th of the web depth

Section area A = 44 cm2 120 mm


Second moment of area /zz Iz =1068 cm4

⇒ cmi 93,444

1068z == 200 mm

7044,0

39,8630,493000

1z

fz =

×==

λλ

iL z

Buckling of welded I section with h/b > 2 :

⇒ curve d ⇒ α = 0,76

( )[ ] ( )[ ] 9397,07044,02,07044,076,015,02,015,0 22zzz =+−×+×=+−+= λλαφ

640,07044,09397,09397,0

11222

z2zz

z =−+

=−+

=λφφ

χ


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent




CALCULATION SHEET


Compression in the bottom flange:

kNN 7604400100010.4610

100075500016080440024,139 3fEd, =×

××

+×=

Verification of buckling resistance of the bottom flange:

981,02754400640,0

760000

Rkz

fEd, =××

=N

Nχ

< 1 OK


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

Example: Elastic analysis of a single bay portal frame SX029a-EN-EU

Quality Record

Example: Elastic analysis of a single bay portal frame RESOURCE TITLE

Reference(s) T2703

ORIGINAL DOCUMENT

Name Company Date

Created by Valérie LEMAIRE CTICM 25/10/2005

Technical content checked by Alain BUREAU CTICM 26/10/2005

Editorial content checked by

Technical content endorsed by the following STEEL Partners:

1. UK G W Owens SCI 10/04/06

2. France A Bureau CTICM 10/04/06

3. Sweden B Uppfeldt SBI 10/04/06

4. Germany C Muller RWTH 10/04/06

5. Spain J Chica Labein 10/04/06

Resource approved by Technical Coordinator

G W Owens SCI 18/09/06

TRANSLATED DOCUMENT

This Translation made and checked by:

Translated resource approved by:


reat

ed o

n F

riday

, Sep

tem

ber

03, 2

010

Thi

s m

ater

ial i

s co

pyrig

ht -

all

right

s re

serv

ed. U

se o

f thi

s do

cum

ent i

s su

bjec

t to

the

term

s an

d co

nditi

ons

of th

e A

cces

s S

teel

Lic

ence

Agr

eem

ent

Elastic Analysis

Documents

Transcript of Elastic Analysis