SKILLS Project October 2013

PORTAL FRAMES IN SINGLE STOREY BUILDINGS

LEARNING OUTCOMES  Structural

elastic analysis including second order effects and imperfections

 Design procedure of portal frames 

Design procedure of roof bracing and vertical bracing

3

LIST OF CONTENTS 





  

Introduction  Presentation of industrial steel buildings  Examples Global Analysis  General  Second order effects  Frame imperfection  Rigidity of joints Design Procedure of portal frames  Structural stability of frames  Stability of columns and rafters Vertical Bracing Roof Bracing Conclusion 4

INTRODUCTION

INTRODUCTION Typical design of single storey steel buildings

6

INTRODUCTION Purlins

7

INTRODUCTION Haunched portal frames

8

INTRODUCTION Roof bracing

Photo APK

9

INTRODUCTION Vertical bracing

Photo APK

10

INTRODUCTION

Photo APK – JP Muzeau

11

INTRODUCTION

12

GLOBAL ANALYSIS

GLOBAL ANALYSIS  Methods of structural analysis

EN 1993-1-1 § 5.4



Elastic analysis  Material is supposed to behave perfectly linear elastic



Plastic analysis  Material non linearity is taken into account  Redistribution of internal forces and moments

14

GLOBAL ANALYSIS  Effects to be taken into account when significant    

EN 1993-1-1 § 5.1

Effects of deformed geometry (2nd order effects) Imperfections Stiffness of joints Ground-structure interaction

15

GLOBAL ANALYSIS  First order and second order analysis



First order analysis: performed on the non deformed structure



Second order analysis: performed including effects of deformed geometry

16

GLOBAL ANALYSIS  Effects of deformed geometry/Second order effects

V



I

First order analysis of the structure gives:

H

MI  H  h

h

3 H  h I  3EI

I

M

17

GLOBAL ANALYSIS  Effects of deformed geometry/Second order effects

V

H

Second order analysis of the structure gives: M II  H  h  V   II

 iterative calculation of II necessary 

II

 n1II



 H  h  V  n

MII

18

II



h2  3EI

GLOBAL ANALYSIS  Effects of deformed geometry/Second order effects

V

H

 n1II



 H  h  V  n

MII



h2  3EI

h2  n Supposing: And: H  h   I 3EI 2 h 1 1 I  II  H  h     V 3EI Vh2 1  1 Vcr 3EI

 n1II

II

II

With: Vcr 

19

II

3EI h2

GLOBAL ANALYSIS  Effects of deformed geometry/Second order effects

 II   I 

V

H

1 V 1 Vcr Vcr   cr V

Substituting: II M

1

 II   I  II

1

1

 cr 1

M II  M I  1 20

1

 cr

GLOBAL ANALYSIS  Global and local second order effects 

Global 2nd order effects – P-D-effects P

D

Concerns the deformation of the whole structure 

Local 2nd order effects – P--effects P

Concerns the deformation  between member ends Generally covered by member checks EN 1993-1-1 § 6.3 21

GLOBAL ANALYSIS  Summarizing the effects of deformed geometry 

Taking the deformation of the structure into account generally leads to higher internal forces (shear force) and moments for portal frames.



The lesser the rigidity of the structure is, the higher are the deformation and therefore the 2nd order effects.



cr is representative for the influence of 2nd order effects (high values of cr stand for little influence of 2nd order effects )

22

GLOBAL ANALYSIS  Second order effects in EN-1993-1-1 

First order analysis is permitted if:  cr  10 for elastic analysis

EN 1993-1-1 § 5.2.1

 cr  15 for plastic analysis



If criterion is not respected  2nd order effects have to be accounted for

23

GLOBAL ANALYSIS  Accounting for second order effects in EN-1993-1-1

3   cr  10 

2nd order analysis (buckling length = member length) or



1st order analysis followed by amplification of sway effects (buckling length = member length) or



1st order analysis (buckling length according to sway buckling mode)

 cr  3 

2nd order analysis (buckling length = member length)

24

GLOBAL ANALYSIS  Amplification of sway effects 

Amplification factor: 1 1



Sway effects:

1

 cr

Horizontal loads (e.g. wind) Effects due to imperfection Effects due to geometry of the structure

25

GLOBAL ANALYSIS  Calculation of cr 

Simplified formula:  cr

 HEd  h   EN 1993-1-1 § 5.2.1 (4)        VEd   H,Ed  VEd H,Ed

h

HEd

if roof slop is swallow: < 26° Afy if axial force in the rafter is small:   0,3 or NEd  0,09Ncr NEd

26

GLOBAL ANALYSIS  Practical calulation of cr for portal frames Hunit

unit

VEd

 cr

 Hunit  h          VEd   unit 

h

VEd 0,5 Hunit

VEd 0,5 Hunit

0,25 Hunit

 unit   mean. column

0,5 Hunit

 unit   mean. column 27

0,25 Hunit

IMPERFECTIONS

GLOBAL ANALYSIS  Structural imperfections 

Due to: lack of verticality lack of straightness eccentricities in joints residual stresses inhomogeneity of material



Physical imperfection are replaced by equivalent geometric imperfection

29

GLOBAL ANALYSIS  Equivalent geometric imperfection 

Global initial sway imperfection f



f

Local bow imperfection

e0

e0

30

GLOBAL ANALYSIS  Global sway imperfection

f  f0hm

EN 1993-1-1 § 5.3.2

f0: Basic value f0  1/ 200  h: Reduction factor for the height of the columns 

h  

2 h

but

2  h  1 3

m: Reduction factor for the number of columns per row 

 m  0,51  

1  m

m is the number of columns carrying at least 50% of the average vertical load of the column row considered 31

GLOBAL ANALYSIS  Direction of sway imperfection 

Every possible direction has to be considered, but only one direction in a time f

f

f

f

f

f

32

f

f

GLOBAL ANALYSIS  System of equivalent forces replacing out-of-plumb

NEd

NEd

fNEd

f

fNEd NEd

NEd

33

GLOBAL ANALYSIS  System of equivalent forces replacing out-of-plumb

f

f

fNEd

fNEd

fNEd

fNEd 34

GLOBAL ANALYSIS  Possibility of disregarding global frame imperfection 

Relatively high horizontal loads HEd  0,15VEd



EN 1993-1-1 § 5.3.2

Frame stability check with equivalent column method (buckling length of columns are based on overall sway buckling mode) EN 1993-1-1 § 5.2.2

35

GLOBAL ANALYSIS  Local bow imperfection 

Local 2nd order effects are generally included in the member verification formulas of EN 1993-1-1 §6.3



Local bow imperfection has to be considered for slender members under high compression axial force

36

GLOBAL ANALYSIS frame is sensitive to 2nd order effects, local bow imperfection has to be applied on:

 If

 

compressed members that have at least one moment resistant joint and Afy EN 1993-1-1 § 5.3.2 whose reduced slenderness   0,5 NEd

 is calculated supposing a pin ended column: 

A fy Ncr

2

And Ncr

37

     EI L

GLOBAL ANALYSIS  Value of local bow imperfection

Elastic analysis

Plastic analysis

e0/L

e0/L

a0

1/350

1/300

a

1/300

1/250

b

1/250

1/200

c

1/200

1/150

d

1/150

1/100

Buckling curve

e0

EN 1993-1-1 § 5.3.2

38

GLOBAL ANALYSIS  System of equivalent forces replacing local bow imperfection

NEd

NEd

e0

4NEde0,d/L

8NEde0,d/L2

L

4NEde0,d/L NEd

NEd 39

GLOBAL ANALYSIS  System of equivalent forces replacing local bow imperfection

e0

e0

4NEd e0,d/L

8NEde0,d/L2

4NEd e0,d/L

L

8NEde0,d/L2 4NEd e0,d/L 40

4NEd e0,d/L

STIFFNESS OF JOINTS

GLOBAL ANALYSIS  Examples of Joints

Rigid joint

Nominally pinned joint

42

GLOBAL ANALYSIS  Classification of joints by stiffness

M

EN 1993-1-8 § 5.2.2

Joint A Joint B

Joint C f 43

GLOBAL ANALYSIS  Classification boundaries

M

Joint A

kb

EN 1993-1-8 § 5.2.2.5

EIbeam Lbeam

Semi-rigid joints Joint B

Rigid joints

0,5

EIbeam Lbeam

Joint C f 44

Nominally pinned joints

GLOBAL ANALYSIS  Value of kb for the classification of joints 

kb = 8 :



kb = 25 :

frames where the bracing system reduces the horizontal displacement by at least 80% other frames, provided that in every storey Kb/Kc ≥ 0,1 Kb: mean value of Ib/Lb for all beams at the top of the storey Kc: mean value of Ic/Lc for all columns of the storey Ic/b: second moment of area of a column/beam Lc/b: height/length of a column/beam 45

GLOBAL ANALYSIS  Practical comments 

The designer will probably choose the assumption of rigid rafter-to-column joints.



The designer will probably choose the assumption of either pinned or rigid column bases.



The assumptions will have to be checked afterwards.

46

DESIGN PROCEDURE OF PORTAL FRAMES

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

cr ≥ 10 :  1st Method:  1st order analysis without imperfections  Column in-plane stability check using buckling length according to sway buckling mode  2nd Method:  1st order analysis with global imperfection  Column in-plane stability check using member length

48

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

cr < 3 :  Check if introduction of local imperfection is necessary 

if necessary:  2nd order analysis with global imperfection if necessary  Column in-plane stability check = check of resistance of section

49

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

cr < 3 :  Check if introduction of local imperfection is necessary 

if not necessary:  2nd order analysis with global imperfection if necessary  Column in-plane stability check using member length

50

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

3 ≤ cr < 10 :  Check if introduction of local imperfection is necessary 

if necessary:  2nd order analysis with global imperfection if necessary  Column in-plane stability check = check of section resistance

51

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

3 ≤ cr < 10 :  Check if introduction of local imperfection is necessary  

if not necessary: 1st Method:  1st order analysis without imperfections  Column in-plane stability check using buckling length according to sway buckling mode  Verification of joints and rafters including second order effects (amplification of sway effects) 52

DESIGN PROCEDURE OF PORTAL FRAMES EN 1993-1-1 § 5.2.2

 Structural stability of frames 

3 ≤ cr < 10 :  Check if introduction of local imperfection is necessary  

if not necessary: 2nd Method:  1st order analysis with global imperfection if necessary  Amplification of sway effects  Column in-plane stability check using member length

53

DESIGN PROCEDURE OF PORTAL FRAMES Buckling length = Member length :

Lcr

Buckling length according to sway buckling mode :

Lcr

54

DESIGN PROCEDURE OF PORTAL FRAMES

Geometry + Boundary conditions + Loads

Calculation of cr

cr < 3

3 ≤ cr < 10

cr ≥ 10

Slide 58

Slide 59

Slide 57

55

DESIGN PROCEDURE OF PORTAL FRAMES

Geometry + Boundary conditions + Loads Calculation of cr cr ≥ 10

Global imperfection

1st order analysis

In plane stability check of columns using member length

In plane stability check of columns using buckling length according to global buckling mode

56

DESIGN PROCEDURE OF PORTAL FRAMES Geometry + Boundary conditions + Loads Calculation of cr cr < 3 Global imperfection if necessary: EN 1993-1-1 § 5.3.2 (4)

Local imperfection if necessary: EN 1993-1-1 § 5.3.2 (6)

Not necessary

Necessary

2nd order analysis In plane stability check of columns = resistance check of section 57

In plane stability check of columns using member length

DESIGN PROCEDURE OF PORTAL FRAMES Geometry + Boundary conditions + Loads Calculation of cr 3 ≤ cr < 10 Local imperfection if necessary: EN 1993-1-1 § 5.3.2 (6) Not necessary

Necessary

Global imperfection if necessary: EN 1993-1-1 § 5.3.2 (4) Necessary 2nd order analysis

In plane stability check of columns = resistance check of section

Amplification of sway effects In plane stability check of columns using member length 58

Not necessary 1st order analysis

In plane stability check of columns using buckling length according to sway buckling mode

STABILITY OF COLUMNS AND RAFTERS

DESIGN PROCEDURE OF PORTAL FRAMES  Stability of columns and rafters 

Columns and rafters are subjected to axial forces and moments EN 1993-1-1 § 6.3.3  Use of interaction formula M y ,Ed  DM y ,Ed M z,Ed  DM z,Ed NEd  k yy  k yz 1  y NRk M y ,Rk M z,Rk

 M1

 LT

 M1

 M1

M y ,Ed  DM y ,Ed M z,Ed  DM z,Ed NEd  k zy  k zz 1  z NRk M y ,Rk M z,Rk

 M1

 LT

 M1

 M1 60

DESIGN PROCEDURE OF PORTAL FRAMES  Simplification for common frames

Columns and rafters are not subjected to out-of-plane moments  Columns and rafters are usually double symmetric sections 

NEd  k yy  y NRk

 M1 NEd  k zy  z NRk

 M1

M y ,Ed 1 M y ,Rk

 LT

 M1

M y ,Ed 1 M y ,Rk

 LT 61

 M1

ROOF BRACING

ROOF BRACING

Photo APK

63

ROOF BRACING Rafters

Purlins transmitting horizontal loads to roof bracing

Roof bracing

64

ROOF BRACING  Ground view of roof bracing

Roof bracing

Purlins transmitting horizontal loads to roof bracing

6 Rafters

65

ROOF BRACING  Idealisation of roof bracing NEd

Fexterior

NEd

NEd

NEd

m rafters whose flanges are subjected to the axial force NEd

NEd

NEd

(including rafters acting as upper and lower flange of roof bracing)

NEd

NEd

Horizontal loads transmitted by purlins

Roof bracing 66

ROOF BRACING  Imperfection for roof bracing NEd

Fexterior

EN 1993-1-1 § 5.3.3 NEd

e0 NEd

NEd e0

NEd

NEd e0

NEd

m rafters whose flanges are subjected to the axial force NEd and that are subjected to imperfection e0

NEd e0

Horizontal loads due to imperfection e0 and axial forces NEd and to Fexterior Roof bracing 67

ROOF BRACING  Imperfection for roof bracing NEd

Fexterior

NEd

e0 NEd

NEd 

NEd e0

NEd

NEd

e0 

e0 NEd

MRafter,Ed hSection

 mL 500



AupFlange ASection

NEd e0

Horizontal loads due to imperfection e0 and axial forces NEd and to Fexterior Roof bracing

68



 m  0,51  

NRafter,Ed

1  m

ROOF BRACING  Calculation of roof bracing NEd

Fexterior

NEd

e0 NEd



Use of geometric imperfection and 2nd order analysis



Use of equivalent forces and 1st order analysis

NEd e0

NEd

NEd e0

NEd

NEd e0

69

ROOF BRACING  Equivalent load concept Fexterior

qd qdL/8 qdL/4 qdL/4 qdL/4 qdL/8

qdL/2

qdL/2

L

qd 

N

Ed 8

e0   g L2

g: deflection of the roof bracing due to exterior load Fexterior and equivalent load qd  iterative calculation of qd  1 or 2 iterations sufficient

70

VERTICAL BRACING

VERTICAL BRACING

Photo APK

72

VERTICAL BRACING  Design procedure

Calculation of cr  1st order or 2nd order theory  Determination of horizontal loads  Wind  Loads due to global imperfection if necessary  Calculation of internal forces and moments  Verification of stability in bracing plane  Verification of out of bracing plane stability as before 

73

VERTICAL BRACING  Calculation of cr for vertical bracings Vtotal

V

V

V

V

Hunit

h

H



h



  cr   unit   V   total  mean 

74

VERTICAL BRACING  In-plane loads on vertical bracing

V Ntotf + H

Ntotf

V

Ntot: Sum of axial forces of all columns stabilized by bracing H: External horizontal loads V: Vertical loads on columns f: Sway imperfection

75

CONCLUSION

CONCLUSION 2nd order effects and imperfections have to be accounted for in the design of portal frames.

 Generally

on the value of cr different calculation methods can be adopted.

 Depending

 For

portal frames it is convenient to account for global imperfection and global 2nd order effects in the global analysis.

77

CONCLUSION 2nd order effects are generally included in the member verification formulas of EN 1993-1-1 §6.3.

 Local



Physical imperfections are replaced by either equivalent geometric imperfections or equivalent loads.



Bracing systems are subjected to external horizontal loads and loads due to their function as stabilizing elements.

78

SKILLS training modules have been developed by a consortium of organisations whose logos appear at the bottom of this slide. The material is under a creative commons license The project was funded with support from the European Commission. This module reflects only the views of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.