verificare stalp cu factorii de interactiune simplificat

NCCI: Verification of columns in simple construction- a simplified interaction criterion SN048b-EN-GB

Localized resource for UK

NCCI: Verification of columns in simple construction a simplified interaction criterion

This NCCI proposes a new expression for the verification of columns that will avoid the calculation of k factors in Annexes A and B of EN 1993-1-1.

Contents

1. Simplified interaction criterion 2

2. Background: derivation of simplified expression from EN 1993-1-1, Annex B 2

Page 1

NCCI: Verification of columns in simple construction - a simplified interaction criterion (GB)Cr

eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


1. Simplified interaction criterion For the design of columns in the following situations:

The column is a hot rolled I or H section, or rectangular hollow section, The cross section is class 1, 2 or 3 under compression, The bending moment diagrams about each axis are linear, The column is restrained laterally in both the y and z directions at each floor but is

unrestrained between floors;

The following expression may be used to verify the member:

15.1Rdcb,z,

Edz,

Rdb,y,

Edy,

Rdb,min,

Ed ++MM

MM

NN

This expression may be used within the limitations on value given in section 2.2.5. This expression is a simplified version of expression (6.62) in section 6.3.3 of EN1993-1-1. In deriving this expression kzy has been taken as 1.0 and kzz has been taken as 1.5. The parameters in this expression are as follows:

NEd, My,Ed, Mz,Ed are as defined in EN1993-1-1 section 6.3.3.

Nmin,b,Rd is the lesser of M1

yy

Af

and M1

yz

Af

.

My,b,Rd is equal toM1

plyLT

Wf.

Mz,cb,Rd is given by M1

plyRdcb,z,

WfM = , which is the same as the design value of the bending

resistance of the cross section M (= c,RdM0

ply

Wf ) when = . M1 M0

2. Background: derivation of simplified expression from EN 1993-1-1, Annex B

2.1 Design expressions in EN 1993-1-1 Designing columns to EN 1993-1-1 requires verification according to expressions 6.61 and 6.62, given in clause 6.3.3(4) as shown below:

1

M1

Rkz,

Edz,Edz,yz

M1

Rky,LT

Edy,Edy,yy

M1

Rky

Ed ++++

M

MMk

MMM

kN

N (6.61)

Page 2


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


1

M1

Rkz,

Edz,Edz,zz

M1

Rky,LT

Edy,Edy,zy

M1

Rkz

Ed ++++

M

MMk

MMM

kN

N (6.62)

For Class 1, Class 2 and Class 3 cross sections, 0Edz,Edy, == MM .

WritingM1

Rky,y

N as , Rdb,y,N

M1

Rkz,z

Nas , Rdb,z,N

M1

Rky,LT

M as and Rdb,y,M

M1

Rkz,

M

as

, equations 6.61 and 6.62 become: Rdcb,z,M

1Rdcb,z,

Edz,yz

Rdb,y,

Edy,yy

Rdb,y,

Ed ++M

Mk

M

Mk

N

N (ex 6.61)

1Rdcb,z,

Edz,

Rdb,y,

Edy,zy

Rdb,z,

Ed ++M

Mk

M

Mk

N

Nzz (ex 6.62)

2.2 Simplification The determination of interaction factors k is made using Annex A and Annex B of EN1993-1-1. The values of these factors are often difficult to determine, especially when using Annex A. In order to avoid the time-consuming calculation of these factors, a simplified expression derived from (ex 6.61) and (ex 6.62) is needed.

The simplified expression should use specific values for the interaction parameters that will be conservative over the practical range of bending of columns. For reasons discussed below, the chosen values are:

kyy = kzy = 1.0 and

kyz = kzz = 1.5.

The range of bending moment diagram for which these values are valid is examined below.

Because the same values are chosen for kyy and kzy, and for kyz and kzz, the use of a single expression is justified because with restraints at the same level in both directions, Nz,b,Rd will always be less than Ny,b,Rd. The expression in Section 1 therefore uses only Nz,b,Rd , which it refers to simply as Nb,Rd.

2.2.1 Use of kyy = 1.0 For Class 1 and 2 sections:

The value of kyy is given by Table B.2 and Table B.1.

( )

+=

Rdb,y,

Edymyyy 2.01

N

NCk

Page 3


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


but for plastic properties,

+

Rdb,y,

Edmyyy 8.01 N

NCk

For valid design situations, 1Rdb,y,

Ed N

N, hence ( ) mymyyy 8.118.01 CCk =+

Therefore, when 0.1yy k 556.08.10.1

my =C

From Table B.3 for straight-line bending moment diagrams, ymy 4.06.0 +=C (but 0.4)

Therefore 11.04.0044.0

4.06.0556.0556.04.06.0 yy ==+

Therefore, using kyy = 1.0 is safe for 11.0y For the case where the member is pin-ended (y = 0 and thus Cmy = 0.6), the use of kyy = 1 will still be safe if a lower limit to the value of NEd/Ny,b,Rd applies:

0.18.016.0Rdb,y,

Ed

+

NN and thus 83.0

48.06.00.1

Rdb,y,

Ed =N

N

For Class 3 sections:

Table B.2 says that kyy should be found from Table B.1

Table B.1 gives:

+=

Rdb,y,

Edymyyy 6.01

N

NCk

but for elastic properties,

+

Rdb,y,

Edmyyy 6.01

N

NCk

Because 1Rdb,y,

Ed N

N, ( ) mymyyy 6.116.01 CCk =+

Therefore, for , 0.1yy k 625.06.10.1

6.1

yymy ==

kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0 ymy += C

Therefore 0625.04.0

025.04.0

6.0625.04.06.0625.0 yy ==+

Page 4


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


Therefore, kyy = 1.0 is valid for 0625.0y

2.2.2 Use of kyz = 1.5 I and H sections

For Class 1 and 2 sections:

Table B.2 says that kyz should be found from Table B.1

Table B.1 gives kyz = 0.6kzz,

Therefore:

( )

+=

Rdb,z,

Edzmzyz 6.0216.0

N

NCk

but for plastic properties of I and H sections,

+

Rdb,z,

Edmzyz 4.116.0

N

NCk

For 1Rdb,z,

Ed N

N, ( ) mzmzyz 44.114.116.0 CCk =+

For , 5.1yz k 04.144.15.1

44.1

yzmz ==

kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0 zmz += C And 11 Therefore kyz = 1.5 is valid for all cases

RHS sections

Table B.2 says that kyz should be found from Table B.1

Table B.1 gives kyz = 0.6kzz,

Therefore:

( )

+=

Rdb,z,

Edzmzyz 2.016.0

N

NCk

but for plastic properties of RHS sections,

+

Rdb,z,

Edmzyz 8.016.0

N

NCk

Page 5


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


For 1Rdb,z,

Ed N

N, ( ) mzmzyz 08.118.016.0 CCk =+

For , 5.1yz k 389.108.15.1

08.1

yzmz ==

kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0 zmz += C And 11 Therefore kyz = 1.5 is valid for all cases


Table B.1 gives kyz = kzz,

Therefore for elastic properties of RHS sections, :

+=

Rdb,z,

Edmzyz 6.01

N

NCk z

+

Rdb,z,

Edmzyz 6.01

N

NCk

1Rdb,z,

Ed N

N, ( ) mzmzyz 6.116.01 CCk =+

For , 5.1yz k 938.06.15.1

6.1

yzmz ==

kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0mLT += C

Therefore 845.04.0

338.04.0

6.0938.04.06.0938.0 z ==+

Therefore kyz = 1.5 for 845.0z 2.2.3 Use of kzy = 1.0 For Class 1 and 2 sections:

From Table B.2, for plastic sections:

( )

= Rdb,z,Ed

LTm,

zzy

25.01.01

N

N

Ck but ( )

Rdb,z,Ed

LTm, 25.01.01

N

N

C, so that kzy is less than 1.0

Page 6


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


Therefore kyz = 1.0 is valid for all cases.


From Table B.2, for elastic sections:

( )

Rdb,z,Ed

LTm,

z

25.005.01

N

N

C but ( )

Rdb,z,Ed

LTm, 25.005.01

N

N

C, so that kzy is less than 1.0

Therefore kzy = 1.0 is valid for all cases.

2.2.4 Use of kzz = 1.5 I and H sections

For Class 1 and 2 sections:

Table B.2 says that kzz should be found from Table B.1

Table B.1 gives kzz for plastic I and H sections:

( )

+

Rdb,z,

Edzmz 6.021

N

NC

+

Rdb,z,

Edmz 4.11

N

NC

For 1Rdb,z,

Ed N

N, ( ) mzmzzz 4.214.11 CCk =+

For , 5.1zz k 625.04.25.1

4.2zz

mz == kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0mz += C

Therefore 0625.04.0

025.04.0

6.0625.04.06.0625.0 z ==+

Therefore kzz = 1.5 for 0625.0z RHS sections

Table B.1 gives kzz for plastic RHS sections:

( )

+=

Rdb,z,

Edzmzzz 2.01

N

NCk

Page 7


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


but

+

Rdb,z,

Edmzzz 8.01

N

NCk

For 1Rdb,z,

Ed N

N, ( ) mzmzzz 8.118.01 CCk =+

For , 5.1yz k 833.08.15.1

8.1

yzmz ==

kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0 zmz += C

Therefore 583.04.0

233.04.0

6.0833.04.06.0833.0 yy ==+

Therefore, kyz = 1.5 is valid for 583.0z For Class 3 sections:

Table B.1 gives for elastic sections kzz:

+=

Rdb,z,

Edzmzzz 6.01

N

NCk

+

Rdb,z,

Edmzzz 6.01

N

NCk

1Rdb,z,

Ed N

N, ( ) mzmzzz 6.116.01 CCk =+

For , 5.1zz k 938.06.15.1

6.1zz

mz == kC

From Table B.3 for straight-line bending moment diagrams, 4.04.06.0mLT += C

Therefore 845.04.0

338.04.0

6.0938.04.06.0938.0 z ==+

Therefore kzz = 1.5 for 845.0z 2.2.5 Summary of limitations for proposed factors The following tables summarise the maximum i values for the validation of the simplified interaction formulae using specific values for the interaction parameters (i.e. kyy = kzy = 1.0 and kyz = kzz = 1.5).

Page 8


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


Table 2.1 i limits for validity of kyy = kzy = 1.0 and kyz = kzz = 1.5 for class 1 and 2 sections I and H sections RHS

kyy = 1.0 -0.11 (see note) -0.11(see note)

kyz = 1.5 Valid for all cases Valid for all cases

kzy = 1.0 Valid for all cases Valid for all cases

kzz = 1.5 0.0625 0.583

Note: If the column is pin-ended ( = 0), the simplified expression is still valid with k = 1,0 yyif NEd/Ny,b,Rd 0,83. This limit will usually be easily achieved.

Table 2.2 i limits for validity of kyy = kyz = 1.0 and kyz = kzz = 1.5 for class 3 sections I and H sections RHS

kyy = 1.0 0.0625 0.0625

kyz = 1.5 0.845 0.845

kzy = 1.0 Valid for all cases Valid for all cases

kzz = 1.5 0.845 0.845

2.2.6 Summary of parametric study A parametric study has been conducted of columns using the following sections:

HD 400 237 HD 400 677 HD 260 93 HD 260 172 UC 356 368 202 UC 254 254 107 UC 102 102 30 HE 200B HE 340A IPE 330 IPE 500

All the sections were analysed in lengths of 4 and 8 m, and using both S275 and S355 steel grades.

The applied forces were increased for each section until all of them were in the critical load, this is, until the sum of the three terms of the governing equation (whichever it was for each particular case) was 1. This lead to the following range of loads:

Page 9


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent


998.0725.0zRd,b,

Ed N

N

138.00Rdb,y,

Edy,zy

M

Mk

2737.00Rdcb,z,

Edz,zz

M

Mk

This showed that for H-section columns with their height, h, approximately equal to their breadth, b, the proposed equation is safe for (y,z) of (0,0); (0,-0.5); (-0.5,0); (-0.5,-0.5).

Page 10


eate

d on

Frid

ay, D

ecem

ber 0

2, 2

011

This

mat

eria

l is

copy

right

- al

l rig

hts

rese

rved

. Use

of t

his

docu

men

t is

subje

ct to

the te

rms a

nd co

nditio

ns of

the A

cces

s Stee

l Lice

nce A

greem

ent

1. Simplified interaction criterion2. Background: derivation of simplified expression from EN1993-1-1, Annex B2.1 Design expressions in EN1993-1-12.2 Simplification 2.2.1 Use of kyy = 1.02.2.2 Use of kyz = 1.52.2.3 Use of kzy = 1.02.2.4 Use of kzz = 1.52.2.5 Summary of limitations for proposed factors Table2.1 (i limits for validity of kyy = kzy = 1.0 and kyz = kzz = 1.5 for class 1 and 2 sectionsTable2.2 (i limits for validity of kyy = kyz = 1.0 and kyz = kzz = 1.5 for class 3 sections

2.2.6 Summary of parametric study

verificare stalp cu factorii de interactiune simplificat

Documents