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I n - p l a n e S t a b i l i t y o fP o r t a l F r a m e s t o B S

C h a r l e s K i n g B S c M S c D I C M l S t r u c t E

P u b l i s h e d b y :

T h e S t e e l C o n s t r u c t i o n I n s t i t u t eS i l w o o d P a r k

A s c o tB e r k s h i r e S L 5 7 0 N

T e l : 0 1 3 4 4 6 2 3 3 4 5F a x :0 1 3 4 4 6 2 2 9 4 4

SCI PUBLICATIONP292

In-plane Stabil ity o f

Portal Frames t o BS 5950-1 :2000

Char les King BSc MSc DIC MlStructE

Published by:

The Steel Construction Insti tuteSilwood Park

AscotBerkshire SL5 7QN

Tel: 01 344 623345Fax: 01344 22944

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2 0 0 1 T h e S t e e l C o n s t r u c t i o n I n s t i t u t e

A p a r t f r o m a n y f a i r d e a l i n g f o r t h e p u r p o s e s o f r e s e a r c hp e r m i t t e d u n d e r t h e C o p y r i g h t D e s i g n s a n d P a t e n t s Ar e p r o d u c e d , s t o r e d o r t r a n s m i t t e d , i n a n y f o r m o r b y a nw r i t i n g o f t h e p u b l i s h e r s , o r i n t h e c a s e o f r e p r o g r a p h i c t e r m s o f t h e l i c e n c e s i s s u e d b y t h e U K C o p y r i g h t L i c et e r m s o f l i c e n c e s i s s u e d b y t h e a p p r o p r i a t e R e p r o d u c t i o n

E n q u i r i e s c o n c e r n i n g r e p r o d u c t i o n o u t s i d e t h e t e r m s s t a tT h e S t e e l C o n s t r u c t i o n I n s t i t u t e , a t t h e a d d r e s s g i v e n o n

A l t h o u g h c a r e h a s b e e n t a k e n t o e n s u r e , t o t h e b e si n f o r m a t i o n c o n t a i n e d h e r e i n a r e a c c u r a t e t o t h e e x t e n to r a c c e p t e d p r a c t i c e o r m a t t e r s o f o p i n i o n a t t h e t i m eI n s t i t u t e , t h e a u t h o r s a n d t h e r e v i e w e r s a s s u m e nm i s i n t e r p r e t a t i o n s o f s u c h d a t a a n d / o r i n f o r m a t i o n o r a nt o t h e i r u s e .P u b l i c a t i o n s s u p p l i e d t o t h e M e m b e r s o f t h e I n s t i t u t e a t

P u b l i c a t i o n N u m b e r : S C I P 2 9 2

I S B N 1 8 5 9 4 2 1 2 3 7

B r i t i s h L i b r a r y C a t a l o g u i n g - i n - P u b l i c a t i o n D a t a .

A c a t a l o g u e r e c o r d f o r t h i s b o o k i s a v a i l a b l e f r o m t h e B r

0 0 0 1 T h e S t e e l C o n s t r u c t i o n I n s t i t u t e

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A catalogue record for this book is available from the Brit ish Library

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F O R E W O R D

T h ec h e c k i n g o f t h e i n - p l a n e s t a b i l i t y o f s i n g l e - s t oa p p r o a c h e s t o t h o s e c o m m o n l y u s e d f o r m u l tB S 5 9 5 0 - 1 : 2 0i n t r o d u c e s m o r e r i g o r o u s r e c o m m e n d a t i o n s f o r t ht h e 1 9 9 0 v e r s i o n . T h i s i s n e c e s s a r y b e c a u s e p os u c c e s s f u l s t r u c t u r a l f o r m t h a t m o r e f r a m e s a r e a r e b e y o n d t h e r a n g e f o r e s e e n w h e n t h e r e c o mp r e p a r e d .

T h i s d o c u m e n t i s i n t e n d e d f o r t h e d e s i g n o f p o r t a ll o a d e d p r e d o m i n a n t l y w i t h r o o f l o a d i n g t h a t c a ua n d t h e e x t e r n a l c o l u n m s . I t i s n o t i n t e n d e d f o r pa s u s e d w h e r e c r o s s - b r a c i n g i s n o t p o s s i b l e , b u t t ht h e d e s i g n o f s u c h f r a m e s .T h i s p u b l i c a t i o n w a s w r i t t e n b y M r C h a r l e s K i n g

T h e S C I w o u l d l i k e t o a c k n o w l e d g e w i t h s p e c i a l tC S C ( U K ) L t d , p a r t i c u l a r l y M r A J R a t h b o n e , im e t h o d s a n d c h e c k i n g t h e c o n t e n t s o f t h e d o c u m e n

T h e S C I w o u l d a l s o l i k e t o e x p r e s s i t s t h a n k s t o :

P r o f e s s o r J M D a v i e s ( U n i v e r s i t y o f M a n c h er e v i e w o f t h e d o c u m e n t a n d t h e m e t h o d s , a s t h e y

M r M B a r k u s a n d M r J K n o t t ( b o t h o f W e s c o l( I m p e r i a l C o l l e g e o f S c i e n c e , T e c h n o l o g y a n d MS o f t w a r e ) f o r t h e i r c o m m e n t s o n t h e d r a f t d o c u m

F u n d i n g f o r t h i s p r o j e c t w a s g r a t e f u l l y r e cE n v i r o n m e n t , T r a n s p o r t a n d t h e R e g i o n s ( D E T R )

I I I

FOREWORD

Thechecking of the in-planestability of single-storeyportal rames equiresdifferentapproacheso those commo nly used formulti-storeyuildings. BS 5950 -1:2000introduces more rigorous recomm endations for the stability checks for portal frames thanthe 1990version.This is necessarybecauseportal rames have proved o be suchasuccessfulstructuralform that moreframesare being constructed with geom etries hatare beyond the range oreseen when the recomm endations inBS 5950-1: 1990 wereprepared.

This document is intended for the design of portal frames used for single-storey buildingsloaded predom inantly with roof load ing that cause large bendin g mom ents in the raftersand the external colum ns. It is not intend ed for portals used to stabilise buildings, suchas used wh ere cross-bracing is not possible, but the principles described are applicable to

the design of such frames.

This publication was written by Mr Charles King of The Steel Construction Institute.

The SCI would like to acknowledge with special thanks, the extensive work conducted byCSC (UK) Ltd, particularly Mr A J Rathbo ne, in the de velopm ent and checking of hemethods and checking the contents of the document.

The SCI would also like to express its thanks to:

Professor J MDav ies Univ ersity of M anchester)and Mr Y Galea CTICM ) orreview of the docum ent and the methods, as they were develop ed.

Mr M Barkus and Mr J Kno tt (both of W escol G losford), ProfessorD A Nethercot(Imperial College of Science, Technology and Medicine) and Mr P Bennett (QuickportSoftware) for their comments on the draft documents.

Fundingorhisroject was gratefullyeceivedrom the Dep artment ofheEnviron men t, Transpo rt and the Regions (DET R) and Corus (formerly, British Steel)

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C O N T E N T S

P a g e

F O R E W O R D i i i

S U M M A R Y v i i

T H E I N - P L A N E S T A B I L I T Y C H E C K S I N B11 . 1 C h e c k s f o r p o r t a l f r a m e s 11 . 2 T h e m e t h o d s i n b r i e f 11 . 3 S e l e c t i n g m e t h o d s f o r d i f f e r e n t t y p e s o31 . 4 R e q u i r e d l o a d f a c t o r , X 1 41 . 5 B a s e s t i f f n e s s 41 . 6 N o t i o n a l h o r i z o n t a l f o r c e s 51 . 7 L o c a l c o n c e n t r a t e d l a t e r a l l o a d s i n b u i l d7

2 I N T R O D U C T I O N T O I N - P L A N E S T A B I L I T Y82 . 1 W h y a r e t h e r e i n - p l a n e s t a b i l i t y c h e c k s ?82 . 2 A x i a l c o m p r e s s i v e f o r c e s i n f r a m e s82 . 3 E l a s t i c c r i t i c a l b u c k l i n g o f f r a m e s i i2 . 4 S e c o n d o r d e r ( P - d e l t a ) e f f e c t s 1 2

3 S W A Y - C H E C K M E T H O D 2 13 . 1 I n t r o d u c t i o n 2 13 . 2 G e o m e t r i c a l l i m i t a t i o n s 2 13 . 3 T h e h / i 0 0 0 c h e c k 2 23 . 4 T h e f o r m u l a m e t h o d 2 83 . 5 S n a p - t h r o u g h c h e c k 3 1

4 A M P L I F I E D M O M E N T S M E T H O D 3 34 . 1 A p p l i c a t i o n d e s i g n s t e p s 3 34 . 2 B a c k g r o u n d t o m e t h o d 3 44 . 3 C a l c u l a t i o n o f X , f o r B S 5 9 5 0 - i 3 54 . 4 S i m p l i f i e d h a n d s o l u t i o n s f o r X c r 3 8

5 S E C O N D - O R D E R A N A L Y S I S 4 45 . 1 I n t r o d u c t i o n 4 45 . 2 D e s i g n s t e p s 4 45 . 3 S t r u c t u r e m o d e l 4 55 . 4 A n a l y s i s m e t h o d s 4 7

6 M E M B E R C H E C K S 5 36 . 1 G e n e r a l 5 36 . 2 A d d i t i o n a l b e n d i n g m o m e n t s f r o m s t r u t5 36 . 3 I n - p l a n e m e m b e r c h e c k s 5 46 . 4 B e n d i n g m o m e n t s f o r f r a m e s u s i n g p l a s5 76 . 5 B e n d i n g m o m e n t s f o r f r a m e s u s i n g e l a s t5 96 . 6 O t h e r m e m b e r c h e c k s 5 9

7 R E F E R E N C E S 6 1

V

CONTENTS

FOREWORD

SUMMARY

Page

...I l l

vii

THE IN-PLANE STABILITY CHECKS IN BS 5950-1 20001 .l Checks or portal rames1.2 Themethods nbrief1.3 Selecting methods or different ypes of rames1.4 Required oad actor, hr1.5 Base stiffness

1.6 Notional orizontal orces1.7 Local concentrated ateral oads nbuildings

INTRODUCTION TO IN-PLANE STABILITY2.1 Why are there n-plane stability checks?2.2 Axialcompressive orces n rames2.3 Elastic critical buckling of rames2.4 Second order P-delta) effects

SWAY-CHECK METHOD3.1 Introduction3.2 Geometrical imitations3.3 The h/1000 check3.4 The ormulamethod3.5 Snap-through heck

AMPLIFIED MOMENTS METHOD4.1 Application - design teps4.2 Background omethod4.3 Calculation of hcr or BS 5950-14.4 Simplified hand solutions or hcr

SECOND-ORDER ANALYSIS

5.1 Introduction5.2 Design teps5.3 Structure odel5.4 Analysis ethods

MEMBER CHECKS6.1 General6.2 Additional bending moments rom strut action6.3 In-plane member hecks6.4 Bending moment s or rames using plastic design6.5 Bending momen ts or rames using elastic design6.6 Othermember hecks

REFERENCES

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44444547

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A P P E N D I X A S e c o n d - o r d e r a n a l y s i s o f c o m m6 3A . 1 R a n g e o f a p p l i c a t i o n a n d d e s i g n s t e p s6 3A . 2 B a s i s o f m e t h o d 6 4A . 3 D e f l e c t i o n s o f t h e ' e l a s t i c ' f r a m e 6 9A . 4 D e f l e c t i o n s o f t h e ' p l a s t i c ' f r a m e 7 2A . 5 A x i a l f o r c e s 7 8A . 6 R e s e r v e f a c t o r a t U l t i m a t e L i m i t S t a t e7 8

A P P E N D I X B S e c o n d o r d e r a n a l y s i s o f t i e d p7 9B . 1 R a n g e o f a p p l i c a t i o n a n d d e s i g n s t e p s7 9B . 2 B a s i s o f m e t h o d 8 0B . 3 B e n d i n g d e f l e c t i o n s o f t h e ' e l a s t i c ' f r a m e8 1B . 4 B e n d i n g d e f l e c t i o n s o f t h e ' p l a s t i c ' f r a m e8 1B . 5 D e f l e c t i o n s o f t h e r a f t e r s / t i e ' t r u s s ' s y s t e8 6B . 6 A x i a l f o r c e s 9 0B . 7 R e s e r v e f a c t o r a t U l t i m a t e L i m i t S t a t e9 1

A P P E N D I X C E f f e c t i v e s t i f f n e s s o f m e m b e r s9 2A P P E N D I X D D e f l e c t i o n s f r o m h o r i z o n t a l l o a

c a l c u l a t i o n s 9 5D . 1 G e n e r a l 9 5D . 2 ' E l a s t i c ' f r a m e s w a y d e f l e c t i o n 9 5D . 3 ' P l a s t i c ' f r a m e s w a y d e f l e c t i o n 1 0 0D . 4 N o m i n a l l y p i n n e d b a s e s 1 0 3

A P P E N D I X E H i n g e d e f l e c t i o n s b y i n t e r p o l a t i1 0 4E . 1 V e r t i c a l d e f l e c t i o n s 1 0 4E 2 H o r i z o n t a l d e f l e c t i o n 1 0 5

W O R K E D E X A M P L E S 1 0 7S i n g l e s p a n s t e e p r o o f p o r t a l f r a m e 1 0 9T i e d p o r t a l f r a m e 1 2 7T w o - s p a n p o r t a l f r a m e 1 5 1T w o - s p a n p o r t a l f r a m e w i t h h i t / m i s s i n t e r n a1 7 3

v i

APPENDIX A econd-order nalysis fommon ortalsbyand' 63A . l Range ofpplicationndesignteps 63A. 2 Basis ofethod 64A.3 Deflectionsfheelastic'rame9A.4eflectionsfheplastic'e 72A.5A.6 Reserve facto r a t Ultimateimittate 78

APPENDIX B Second order analysis of ied portals by hand'B. l Range of application and design stepsB.2 Basis of methodB.3 Bending deflections of he elastic' rameB.4 Bending deflections of he plastic' rameB.5 Deflections of he afterdtie truss' systemB.6 AxialorcesB.7 Reserve facto r a t Ultimate LimitState

7 97 98 0818 18 69 091

APPENDIX C Effectivetiffnessfembers2

APPENDIX D Deflections rom horizontal loads for 'hand' second-ordercalculations 95

D. General 95D.2Elastic'ramewayD.3Plastic'ramewayonD.4ominallyd bases 10 3

APPENDIX E Hinge deflections by nterpolationE. 1Vertical eflectionsE.2 Horizontal eflection

WORKED EXAMPLESSingle span steep roof portal frameTied portal frameTwo-span portal frameTwo-span portal frame with hit/miss internal columns

1041 0 4105

107109127151173

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S U M M A R Y

T h i sd o c u m e n t i n t r o d u c e s d e s i g n e r s t o t h e i n - p l aB S 5 9 5 0 - 1 : 2 0 0 0 f o r s i n g l e - s t o r e y p o r t a l f r a m e s

a n a l y s i s . T h e s e c a l c u l a t i o n s a r e a n e s s e n t i a l pv e r i f i c a t i o n s o f p o r t a l f r a m e s . I n a d d i t i o n t o a r e vs e c o n d - o r d e r c a l c u l a t i o n s c a n b e p e r f o r m e d e va v a i l a b l e .

T h i s d o c u m e n t i n c l u d e s :

A n i n t r o d u c t i o n t o t h e i n - p l a n e s t a b i l i t y o f s i n g A c o m m e n t a r y o n t h e t h r e e m e t h o d s o f c h e

f r a m e s g i v e n i n B S 5 9 5 0 - 1 : 2 0 0 0 , t h a t i s :( a ) T h e S w a y - c h e c k m e t h o d( b ) T h e A m p l i f i e d M o m e n t m e t h o d( c ) S e c o n d - o r d e r a n a l y s i s

W o r k e d e x a m p l e s o f a s i m p l e m e t h o d f o r s e c o nw h e r e s e c o n d - o r d e r a n a l y s i s s o f t w a r e i s n o t a v a

T h e i n s t a n c e s i n w h i c h i n d i v i d u a l m e m b e r s n e e d a l s o e x p l a i n e d . S e c o n d - o r d e r a n a l y s i s b y a p p l i c ai n a f o r m t h a t c a n b e a p p l i e d i n h a n d c a l c u l a t i o n se x a m p l e s .

S t a b i l i t e n P l a n d e s P o r t i q u e s s e l o n l a N o r m e BR s u m C e d o c u m e n t p r s e n t e a u x c a l c u l a t e u r s l e s m t h ol a n o r m e B S 5 9 5 0 - 1 : 2 0 0 0 p o u r d e s p o r t i q u e s a ua n a l y s e e l a s t i q u e , s o i t u n e a n a l y s e p l a s t i q u e . C e sv e r i f i c a t i o n s a u x E t a t s L i m i t e s U / t i m e s ( E L Up r e s e n t a t i o n d e t o u t e s c e s m t h o d e s , i i e s t m o n t rp e u v e n t t r e e f f e c t u e s s a n s a v o i r r e c o u r s a u n l o g i

C e d o c u m e n t c o m p r e n d :

U n e i n t r o d u c t i o n a l a s t a b i l i t e n p l a n d e s p o r t i D e s c o m m e n t a i r e s s u r l e s t r o i s m t h o d e s d e v e

p o r t i q u e s d o n n e s d a n s l a n o r m e B S 5 9 5 0 - 1 : 2 0

( a ) L a m t h o d e d e v e r i f i c a t i o n a v e c l o n g u e u r s d( b ) L a m t h o d e p a r a m p l i f i c a t i o n d e s m o m e n t s .( c ) L ' a n a l y s e a u s e c o n d o r d r e .

v i i

This ocumentntroduces esignerso the in-plane stability calculation methods inBS 5950-1:2000 orsingle-storeyportal rames designed using either elastic or plasticanalysis.These alculation s re n ssentialpart ofhe Ultimate Limit State (ULS)verifications of portal frames. In addition to a review of all these methods, it show s howsecond -order alculations an be performed ven when second -order oftware is notavailable.

This document includes:

An introduction to he in-plane stability of single-storey portal frames.

A ommentaryon the three methods of checking the in-plane stability of portal

frames given in BS 5950-1:2000, that is:(a) TheSway-check method

(b) The Am plified Mom ent method

(c) Secon d-order nalysis

W orked exam ples of a simple method for second-o rder calculations that can be usedwhere second-order analysis software is not available.

The instances in which individual membe rs need to be ch ecked for in-plane buckling arealso explained . Second -order analysis by application of the energy method is explainedin a form thatcan be applied in hand calculations, and this is illustrated by four w orkedexamples.

Stabilit6 en Plan des Portiques selon la Norme BS 5950-1:2000

R6sum6

Ce document prksente aux calculateursles mkthodes de calcul de stabilitken plan selonla norme BS 5950-l:2000 pour des portiques aun niveau calculks en utilisant soit uneanalyse klastique, soit une analyse plastique. Ces calculs sont une par tie essentielle desvkrijications a m Etats Limites Ultimes(ELU) des portiques. En omplkment eaprksentation de toutes ces mkthodes,il est montrk comment des calculs au second ordrepeuvent &re efSectuks sans avoir recours a un logiciel d'analyse au second ordre.

Ce document comprend:

Une introduction a la stabilitk en plan des portiques aun niveau.

Descommentaires sur les trois mkthodes de vkrijication de lastabilitken plan desportiques donnkes dans la normeBS 5950-1:2000, c 'est-a-dire:

(a) La mkthode de vkrijication avec longueurs de Jambement a noeuds dip lq ab les .(b) La mkthode par ampl@cation des moments.

(c) L'ana lyse au second ordre.

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D e s e x e m p l e s d ' a p p l i c a t i o n d ' u n e m t h o d e s i m p t r e u t i l i s e e n l ' a b s e n c e d e l o g i c i e l d ' a n a l y s e a u

L e s c a s p o u r l e s q u e l s l e s b a r r e s d o i v e n t t r e v r fp l a n s o n t a u s s i e x p l i c i t s . L ' a n a l y s e a u s e c o n d o

1 ' n e r g i e e s t d c r i t e d e f a c o n a c e q u ' e l l e p u i s si l l u s t r e p a r q u a t r e e x e m p l e s .

E b e n e S t a b i l i t t v o n R a h m e n t r a g w e r k e n n a c h B SZ u s a m m e n f a s s u n gD i e s e s D o k u m e n t f u h r t T r a g w e r k s p l a n e r i n d i e Bd e r T r a g w e r k s e b e n e v o n e i n g e s c h o s s i g e n R a h m e n td i e e n t w e d e r e l a s t i s c h o d e r p l a s t i s c h b e r e c h n e t ww i c h t i g e rT e l l d e r U b e r p r u f u n g d e s G r e n z z u s t av o nR a h m e n t r a g w e r k e n . Z u s t z l i c h z u m U b e r b l i c kB e r e c h n u n g e n n a c h T h e o r i e I I . O r d n u n g d u r c he n t s p r e c h e n d e S o f t w a r e n i c h t v e r f g b a r i s t .

D i e s e s D o k u m e n t e n t h l t :

e i n e E i n f u h r u n g i n d i e S t a b i l i t t v o n e i n g e s c h oE b e n e ,

e i n e n K o m m e n t a r z u d e n d r e i M e t h o d e n d eT r a g w e r k s e b e n e v o n R a h m e n t r a g w e r k e n n a c h B S( a ) U b e r p r u f u n g d e r S e i t e n s t e i j h e i t / - w e i c h h e i t ,( b ) M e t h o d e d e r m i t e i n e m V e r g r o J i e r u n g s f a k t o r( c ) B e r e c h n u n g n a c h T h e o r i e I I . O r d n u n g .

B e r e c h n u n g s b e i s p i e l e e i n e r e i n f a c h e n M e t h o d e O r d n u n g , d i e b e n u t z t w e r d e n k a n n , w e n n e n t s p r e c

D i e F l l e , i n w e i c h e n f r e i n z e l n e B a u t e i l e e m K n ii s t , w e r d e n e r k l r t . D i e B e r e c h n u n g n a c h T h e o r i

E n e r g i e m e t h o d e w i r d i n e i n e r w e i s e e r k l r t , d a J 3 s id i e s w i r d a n h a n d v o n v i e r B e r e c h n u n g s b e i s p i e l e n i l

E s t a b i l i d a d d e p r t i c o s e n s u p i a n o s e g t i n B S 5 9 5R e s u m e nC o n e s t e d o c u m e n t o s e d e s c r i b e n a l o s p r o y e c t i s t a sd e I a B S 5 9 5 0 - 1 : 2 0 0 0 p a r a p o r t i c o s s e n c i l l o s d e e l s t i c o s o p l s t i c o s q u e s o n u n a p a r t e e s e n c i a l d e

( E L U ) . A d e m s d e r e v i s a r e s o s m t o d o s s e m u ec O l c u l o s d e s e g u n d o o r d e n i n c l u s o s i n s o f t w a r e d e s

v i i i

Des exemples d'application d'une d t h o d e simple de calcul au second ordre qui peutetre utiliske en l'absence de logiciel d'analyse au second ordre.

Les cas pour lesquels les barres doivent ktre vkrifikes vis-a-vis du jlambement dans leplan sont aussi explicites. L'analyse au second ordre par application de la dthode de

l'tnergie est decrite de fagona cequ'ellepuisse 2tre appl iqute manuellement et estillustree pa r quatre exemples.

Ebene Stabilitat von Rahmentragwerken nach BS 5950-1:2000

Zusammenfassung

Dieses Dokument fuhrt Tragwerksplaner in die Berechnungsmethoden der Stabilitat inderTragwerksebene von eingeschossigen Rahmentragwerken nach BS5950-1 2000 ein,die entweder elastisch oder plastisch berechnet wurden. Diese Berechnungen sind einwichtiger Teil der Uberpriifung desrenuu stands der TragfiihigkeitonRahmentragwerken. Zusatzlichzum Uberblick dieser Methoden wird gezeigt, wieBerechnungen nach TheorieII. Ordnung durchgefuhrt werden kiinnen, auch wennentsprechende Software nicht vequgbar ist.

Dieses Dokument enthalt:

eine Einfuhrung in die Stabilitat von eingeschossigen Rahmentragwerken in ihrerEbene,

einen Kommentar zu den drei Methoden der Uberpriifung der Stabilitat inTragwerksebene von Rahmentragwerken nach BS5950-1 2000, welche sind:

(a) Uberprufung derSeitensteifheit/-weichheit,

(b) Methode der mit einem VergrMerungsfaktor erhohten Momente,

(c) Berechnung nach TheorieII. Ordnung.

Berechnungsbeispiele einer einfachen Methode fur Berechnungen nach TheorieII.Ordnung, die benutzt werdenkann, wenn entsprechende Software nicht vequgbar ist.

Die Falle, in welchen fu r einzelne Bauteile ein Knick-/ Biegehichachweis eqorderlichist, werden erklart. Die Berechnung nach TheorieII. Ordnung durch Anwendung der

Energiemethode wird in einer weise erklart, d@ sie von Hand durchgefuhrt werdenkann;dies wird anhand von vier Berechnungsbeispielen illustriert.

Estabilidad de p6rticos en su plan0 s e g h BS 5950-1:2000

Resumen

Con este documento se describen a10s proyectistas 10s d t o d o s de calculo de estabilidadde laBS 5950-1:2000 para pbrticos sencillos de una planta calculados segun d t o d o selasticos o plasticos que son una parte esencial de las comprobaciones de estados limiteultimos (ELU ). Ad emis de revisaresos &todos se muestra comose pueden llevar a cabocalculos de segundo orden incluso sin software de segundo orden.

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L a p u b l i c a c i n i n c l u y e :

U n a i n t r o d u c c i n a I a e s t a b i l i d a d e n s u p i a n o d C o m e n t a r i o s s o b r e l o s t r e s m t o d o s d e c o m p r o b

e s t o e s :( a ) E l m t o d o d e l a c o m p r o b a c i n d e i a d e r i v a( b ) E l i n t o d o d e a m p 4 f i c a c i O n d e m o m e n t o s( c ) C a / c u b e n s e g u n d o o r d e n .

E j e m p l o s d e s a r r o l l a d o s d e u n m t o d o s e n c i lu t i l i z a b l e s i n s o f t w a r e d e s e g u n d o o r d e n .

T a m b i n s e e x p l i c a n l o s c a s o s e n q u e p i e z a s i n d iL o s m t o d o s b a s a d o s e n l a E n e r g I a s e e x p l i c a nm a n u a l m e n t e , l o q u e s e i l u s t r a c o n c u a t r o e j e m p l o

S t a b i i t n e l p i a n o d i p o r t a l i i n a c c o r d o a l l aB S5 9 2 0 - 1 : 2 0 0 0S o m i n a r i oQ u e s t a p u b b l i c a z i o n e a f f r o n t a i l p r o b l e m a d e l i a s tr i v o l t a p r e v a l e n t e m e n t e a i p r o g e t t i s t i , r i p o r t a i me l a s t i c a s i a p l a s t i c a i n a c c o r d o a / l a B S 5 9 5 0 - 1 : 2 0

m e t o d i c o s t i t u i s c o n o u n a p a r t e e s s e n z i a i e d e l / a v ed i p o r t a l i i n a c c i a i o . I n a g g i u n t a a d u n a p r e s e n t am o s t r a t o c o m e e f f e t t u a r e a n a l i s i d e l s e c o n d o o r d i ns p e c i f i c i s t r u m e n t i s o f t w a r e i n g r a d o d i e f f e t t u a rQ u e s t a p u b b l i c a z i o n e i n c l u d e :

u n ' i n t r o d u z i o n e a i i a s t a b i l i t n e l p i a n o d i p o r t a i u n c o m m e n t a r i o a i t r e m e t o d i d i v e r f i c a p e r 1

a c c o r d o a / l a B S 5 9 5 0 - 1 : 2 0 0 0 , c i o :( a ) i l m e t o d o d i c o n t r o l l o d e l l o s p o s t a m e n t o t r a

( b ) i i m e t o d o d i a m p i t f i c a z i o n e d e i m o m e n t i ;( c ) 1 ' a n a l i s i d e l s e c o n d o o r d i n e .

e s e m p i a p p i i c a t i v i d i u n m e t o d o s e m p l f i c a t o u s a r e q u a n d o n o n s o n o d i s p o n i b i i i m e t o dd i r e t t a m e n t e i n c o n t o g l i e f f e t t i d e l s e c o n d o o r d i

V i e n e a n c h e t r a t t a t o i t c a s o i n c u i i e v e r i f i c h e c o n d o t t e s u i s i n g o / i e l e m e n t i . I n a g g i u n t a , p r o pb a s e d e i m e t o d i e n e r g e t i c i i n u n a f o r m a a n c h e a pr i f e r i m e n t o a i q u a t t r o e s e m p i a p p l i c a t i v i r i p o r t a t i n

i x

La publicacion incluye:

Una introduccion a la estabilidad ensu plano de porticos de una planta.

9 Comentarios sobre10s tres mktodos de comprobacion incluidos enBS 5950-1:2000,

esto es:(a) El mktodo de la comprobacibn de la deriva

(b) El d t o d o de ampl@acidn de momentos

(c) Calculo en segundo orden.

Ejernplos desarrollados de un mktodo sencillo para calculos de segundo ordenutilizable sin sofmare de segundo orden.

TambiCn se explican10s casos en que piezas individuales deben comprobarse a pandeo.Los m ktodos basados en la Energa e explican de form a quepuedan ser aplicadosmanualmente,lo que se ilustra con cuatro ejemplos totalmente resueltos.

Stabilith ne1 piano di portali in accordo alla BS 5920-1:2000

Sommario

Questa pubblicazione affronta il pr o b le m della stabilita ne1 piano di portaliin acciaio e,rivolta prevalentemente ai progettisti, riporta i metodi di calcolo pe r la progettazione siaelastica sia plastica in accordo allaBS 5950-1:2000. I calcoli effettuati in accordo a tali

metodi costituiscono una parte essenziale della verifica agli Stati Limite Ultimi (S.L.U.)di portali in acciaio.In aggiunta ad una presentazione generale di questi metodi, vienemostrato come effettuare analisi del secondo ordine anche quandonon si siano disponibilispecifici strumenti software in grado di effettuare automaticamente tale tip0 di analisi.Questa pubblicazione include:

un introduzione alla stabilita ne1 piano di portali in acciaio.

un commentario ai tre metodi di verifica perl instabilita ne1 piano dei portali inaccordo alla BS 5950-1:2000; ciot?:

(a) il metodo di controllo dell0 spostamento trasversale;

(b) il metodo di amplijicazione dei momenti;(c) lanalisi del secondo ordine.

esempi applicativi di un metodo sernplifcato per i calcoli del secondo ordine dausare quando non sono disponibili metodi piuafinati in grado di teneredirettamente in conto gli efSetti del secondo ordine.

Viene anche trattato il cas0 in cui le verifiche di stabilita ne1 piano debbano esserecondotte sui singoli elementi.In aggiunta, t proposta lanalisi del secondo ordine sullabase dei metodi energeticiin una forma anche applicabile manualmente, con esplicitoriferimento ai quattro esempi applicativi riportati nella pubblicazione.

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1 T H E I N - P L A N E S T A B I LB S 5 9 5 0 - 1 : 2 0 0 0

1 . 1 C h e c k s f o r p o r t a l f r a m e sS i n g l e - s t o r e yp o r t a l f r a m e s o f e c o n o m i c p r o p o r t i oe n s u r e t h a t t h e y h a v e a d e q u a t e i n - p l a n e s t a b i lp l a s t i c m e t h o d s . T h i s t y p e o f f r a m e c a i m o t bf o r m u l t i - s t o r e y f r a m e s i n B S 5 9 5 0 - 1 C l a ua x i a l c o m p r e s s i o n i n t h e r a f t e r i s n o t c o n s i d ep h e n o m e n a i n v o l v e d i n i n - p l a n e s t a b i l i t y o f s iS e c t i o n 2 t o g e t h e r w i t h a c o m p a r i s o n w i t h m u

B S 5 9 5 0 - 1 : 2 0 0 0 g i v e s t h r e e m e t h o d s f o r cs i n g l e - s t o r e y f r a m e s : T h e S w a y - c h e c k m e t h o d T h e A m p l i f i e d M o m e n t m e t h o d S e c o n d - o r d e r a n a l y s i sT h e m e t h o d s a p p l y t o p o r t a l f r a m e s d e s i gC l a u s e 5 . 5 . 2 o f B S 5 9 5 0 - 1 ) o r b y p l a s tB S 5 9 5 0 - 1 ) .

I t w i l l a l m o s t a l w a y s b e p r e f e r a b l e t o p e r f o rp o s s i b l e t o p e r f o r m t h e c h e c k s b y ' h a n d ' , b u tb e l e s s e c o n o m i c a l . T h e o n l y b e n e f i t o f t ha n a l y s i s i s t o g a i n a g r e a t e r u n d e r s t a n d i n g os e c o n d - o r d e r ( P - d e l t a ) e f f e c t s a n d t h e l o sf o r m a t i o n o f p l a s t i c h i n g e s .

1 . 2 T h e m e t h o d s i n b r i e f1 . 2 . 1 T h e S w a y - c h e c k m e t h o d

R a n g e o f a p p l i c a t i o nT h eS w a y - c h e c k m e t h o d m a y b e u s e d f o r p o r tw h i c h s a t i s f y t h e f o l l o w i n g g e o m e t r i c a l l i m i t a t S p a n / h e i g h t t o e a v e s i s n o t m o r e t h a n 5 . R i s e o f a p e x a b o v e c o l u m n t o p s i s n o t m o

s p a n s o r a v a l u e g i v e n b y a f o r m u l a f o r a s y m E i t h e r t h e n o t i o n a l s w a y d e f l e c t i o n f r o m n o t

o r d e r a n a l y s i s ) i s n o t m o r e t h a n h / b O O ,r a f t e r s i s w i t h i n a l i m i t g i v e n b y a f o r m u l a .n o t t o b e c o n s i d e r e d i n c a l c u l a t i n g t h e p r e d o m i n a n t l y g r a v i t y l o a d c a s e s ( e . g . C o m b

A d v a n t a g e sa n d d i s a d v a n t a g e sT h eS w a y - c h e c k m e t h o d i s t h e s i m p l e s t m e t h o d t h e f r a m e i s s u f f i c i e n t l y s t i f f t o s a t i s f y e i t h e r

l THE N-PLANE STABILITY CHECKS INBS 5950-1 Z OO0

1.1 Checks o r port al ramesSingle-storey portal frames of economic proportions need to be checked toensure that they have adequate in.-plane stability, whether designed by elastic orplastic metho ds. This typeof frame cannot be checked by he simple methodsfor multi-storey frames inBS S9S0-1r1 Clauses 2.4 .2.6 and 2.4.2 .7 becauseaxial comp ression in the rafter is not considered in that metho d. The structuralpheno men a involved in in-plane stability of single-storey frame s are described inSection 2 together with a comparison with multi-storey frames.

BS 5950-1:2000 gives three methods for checking the in-plane stability ofsingle-storey frames:

The way-check method

TheAmplifiedMoment method

Second-order analysis

Themetho ds apply to portal frames designed either by elastic design (seeClause 5.5.2 of BS 5950-1) o r by plastic design (see Clause 5.5 .3 ofBS 5950-1) .

It will almost alway s be preferable to perform these checks by software. It ispossible to perfo rm the checks by hand, but he results will almost invariablybe less econo mical. The only benefitof the hand me thod of second-orderanalysis is to gain a greater unde rstanding of he response of he frame to thesecond-order (P-delta) effects andhe loss of stiffness resulting from theformation of plastic hinges.

1.2 The metho ds n brief1.2.1 The Sway-check method

Range of application

The Sway -check metho d may be used for portals that are not ied portals andwhich satisfy the following geo metrical limitations:

Spad heigh t to eaves isnot more than5 .

Rise of apexabovecolum n tops s not mo re than span /4 for symmetricalspans or a value given by a formula for asymmetric rafters.

Either the notional sway deflection from notional forces (calculated by first-order analysis) is not more thanh/1000, or the spanldepth ratio of herafters is within a limit given by a formula. The stiffnessof the cladding isnot to be considered in calculating the notional sway deflection forpredominantly gravity load cases (e.g. Combination1).

Advantages and disadvantages

The Swa y-check method is the simplest method and gives econom ical designsifthe frame is sufficiently stiff tosatisfy eithe r theh/1000 check or the formula

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c h e c k b e c a u s e o f t h e s e c t i o n s i z e s s e l e c t e d e i to r t o s a t i s f y t h e S e r v i c e a b i l i t y L i m i t S t a t e (w i l l o f t e n g i v e t h e m o s t e c o n o m i c a l d e s i g n s fb e r e l a t i v e l y s t i f f . E c o n o m y i s a c h i e v e d b e c as t r e n g t h f o r t h e g r a v i t y l o a d c a s e s ( L o a d c o mC r a n e c o m b i n a t i o n 1 o f C l a u s e 2 . 4 . 1 . 3 ) t h a tl o a d c a s e s .M a n y m u l t i - s p a n f r a m e s w i l l n o t sr e q u i r e m e n t s w i t h o u t i n c r e a s i n g t h e s i z e o f t h ef o r s t r e n g t h o r f o r S L S r e q u i r e m e n t s . W h e n us t e e l s t r e n g t h ( e . g . S 2 7 5 o r S 3 5 5 ) h a s n o c a l c u l a t i o n .

T h e d e s i g n s t e p s f o r t h i s m e t h o d a n d f u r t h e r S e c t i o n 3 .

1 . 2 . 2T h e A m p l i f i e d M o m e n t m e t h o dR a n g e o f a p p l i c a t i o nT h eA m p l i f i e d M o m e n t m e t h o d i s a m e t h o d t h ad o e s n o t m e e t t h e l i m i t a t i o n s o f t h e s w a y - cp o r t a l s t h a t a r e n o t t i e d p o r t a l s a n d w h i c h h a v2 c r ,n o tl e s s t h a n 4 . 6 . T h e e l a s t i c c r i t i c a l b u c k l ii s d e s c r i b e d i nS e c t i o n 2 . 3

A d v a n t a g e sa n d d i s a d v a n t a g e sT h eA m p l i f i e d M o m e n t m e t h o d i s a s i m p l e m e t2 c ri s k n o w n . I f e a s y - t o - u s e s o f t w a r e i s a v a i l a bW h e n s o f t w a r e i s n o t a v a i l a b l e , t h e n t h e f o r mt h e y a r e c o m p l e x a n d s e v e r a l f o r m u l a e n e ef r a m e . T h e m e t h o d g i v e s r e a s o n a b l y e c o nr e l a t i v e l y s t i f f b e c a u s e o f t h e s e c t i o n s i z e s r es t r e n g t h o r t o s a t i s f y t h e S L S r e q u i r e m e n t s . I n1 0 ,t h e r ei s n o r e d u c t i o n i n f r a m e s t r e n g t h . T h u s , t hd e s i g n s f o r m o s t s i n g l e s p a n p o r t a l s b e c a u s ew i l l a l s o g i v e r e a s o n a b l y e c o n o m i c a l d e s i g nr e l a t i v e l y s t i f f .H o w e v e r , m a n y m u l t i - s p a n f r a mr e q u i r e m e n t t h a t 2 c r4 . 6 , u n l e s s t h e s i z e o f t h e m e m b e r ss i z e r e q u i r e d f o r s t r e n g t h o r S L S r e q u i r e m e n t

i m p r o v e m e n t i n i n - p l a n e s t a b i l i t y o f t h e f r a ms t r e n g t h s t e e l ( g r a d e S 3 5 5 s t e e l ) . T h i s i m p r o vn o t f r o m 2 c r ,w h i c hi s i n d e p e n d e n t o f t h e c h a n g e o f s t e

T h e d e s i g n s t e p s f o r t h i s m e t h o d a n d f u r t h e r dS e c t i o n 4 .

1 . 2 . 3S e c o n d - o r d e r a n a l y s i sR a n g e o f a p p l i c a t i o nS e c o n d - o r d e ra n a l y s i s i s a n o t h e r a l t e r n a t i v e m e t h o dm e e t t h e l i m i t a t i o n s o f t h e S w a y - c h e c k m e t hi n c l u d i n g t i e d p o r t a l s .T i e d p o r t a l s m u s t b e d e s i g n e d a n a l y s i s . F o r t i e d p o r t a l s , t h e a n a l y s i s m e t h o d n o n - l i n e a r b e h a v i o u r o f t h e a p e x d r o p , a c a p aa l l p a c k a g e s t h a t d e s c r i b e t h e m s e l v e s a s ' s e c o n

2

check because of the section sizes selected either to give the necessary strengthor o satisfy the Serviceability Limit State(SLS) requirements. This methodwill o ften give the most econom ical designs for single span portals that tend tobe relatively stiff. Econom y is achieved because there is no reduction in framestrength for the gravity load cases (Load combinationl of Clause 2 . 4 . 1 . 2andCrane combination 1 of Clause 2 .4 .1 .3 ) that are generally the critical designloadases. Many multi-span frames will not satisfy the notionalwayrequireme nts without increasing the size of the memb ers above the size requiredfor strength or forSLS requirements. Wh en using he Sway-check method, thesteel strengthe.g. S275 or S355) has no effect on then-planetabilitycalculation.

The design steps for this method and further detailsof the method are g iven inSection 3.

1.2.2 The Ampl i f i ed Moment me thod


The Amplified Moment method isa method that may be used where the framedoesnot meet the limitations of he sway-check method. It maybeused forportals that are not tied portals and which havean elastic critical buckling ratio,A,,, not less than 4 . 6 . The elastic critical buckling ratio,IC,,, s described inSection 2 .3


The Am plified Mom ent method is a simple method to apply when the value ofA,, is known. Ifeasy-to-use software is available, the metho d is easy touse.

W hen software is not available, then the formulae in Section4 may be used, butthey are complex and several formulae need to beapplied for a multi-spanframe.The method gives reasonably economical designs if the frame isrelatively stiff because of he section sizes required either o give he requiredstrength o r to satisfy theSLS requirements. In particular, whe reA,, 2 10, thereis o reduction in frame trength.Thus, themethodwill ive economicaldesigns for most single span portals because they end oberelatively stiff. Itwill also give reasonably economical designs for multi-span frames that arerelatively stiff.ow eve r, many multi-span frames will not satisfyherequirement thatAcr2 4 . 6 , unless the size of the mem bers is increased above thesize required for strengthor SLS requirements. The method does recognise the

improvement in n-plane stability of he frame resulting from the use of higherstrength steel (gradeS355 steel). This improvement comes from an increase in,l,,, not from A,,, which is independentof the change of steel grade.

Th e design steps for this method and fu rther details of this method are given inSection 4 .

1.2.3 Second-order nalysi s


Second-order analysis is another alternative method where the frame doesnotmeet the limitations of he Sway-check method.It maybeused for all portalsincluding tied portals. Tied portals must beesignedsing second-orderanalysis. For tied portals, the analysis method must also be ableto calculate thenon-linear behaviour of the apex dro p, a capability that may not be included inall packages that describe themselves as second-order.

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A d v a n t a g e s a n d d i s a d v a n t a g e sS e c o n d - o r d e ra n a l y s i s i s s i m p l e t o a p p l y i f t h e r ea v a i l a b l e .I t w i l l g i v e t h e m o s t e c o n o m i c a l d e s i gs u c h a s m u l t i - s p a n f r a m e s . I t m a y g i v e l e s s

m e t h o d s f o r s t i f f e r f r a m e s b e c a u s e i t w i l l a l ws t r e n g t h f r o m s e c o n d - o r d e r ( P - d e l t a ) e f f e c t s .s t i f f n e s s v a l u e s a b o v e w h i c h t h e s t r e n g t h im e t h o d d o e s r e c o g n i s e t h e i m p r o v e m e n t ir e s u l t i n g f r o m t h e u s e o f h i g h e r s t r e n g t h s t e e l

F u r t h e r d e t a i l s o f t h i s m e t h o d a r e g i v e n i n S e c

1 . 3 S e l e c t i n g m e t h o d s f o r d i f ff r a m e s

1 . 3 . 1 S i n g l e - s p a n f r a m e s ( n o t t i e d p oS i n g l e - s p a nf r a m e s m a y b e d e s i g n e d b y a n y o f t ha b o v e .W h e r e t h e f r a m e s a r e w i t h i n t h e g e oS w a y - c h e c k m e t h o d a n d p a s s e i t h e r t h e h I l 0 0S e c t i o n 1 . 2 . 1 a b o v e ) , t h e m e t h o d d o e s n o t g if o r t h e g r a v i t y l o a d c a s e s . W h e r e t h e f r a ml i m i t a t i o n s o f t h e S w a y - c h e c k m e t h o d o r f a i lb e u s e d . F o r f r a m e s s l i g h t l y o u t s i d e t h e g ew o r t h m a k i n g m i n o r a l t e r a t i o n s t o t h e s c h e ma n i n c r e a s e i n s t i f f n e s s o f t h e f r a m e t o s a t i s

t h e b a s e s d e e p e r t o s u i t t h e s p a n t o h e i g h t r a tW h e r e t h e S w a y - c h e c k m e t h o d i s n o t s a t i s f im e t h o d o r S e c o n d - o r d e r a n a l y s i s s h o u l d b e u s

1 . 3 . 2 M u l t i - s p a n f r a m e s ( n o t t i e d p o r tM u l t i - s p a nf r a m e s o f t e n h a v e r e l a t i v e l y l o w s t iA l t h o u g h s o mm u l t i - s p a n f r a m e s m i g h t b e s u f f i c i e n t l y s t i f f w i l l n o t . W h e r e t h e f r a m e s a r e t o o f l e x i b l e at h e m o s t e f f i c i e n t w a y t o i m p r o v e t h e f r a m et h e i n t e r n a l c o l u m n s t i f f n e s s .

T h e a m p l i f i e d m o m e n t m e t h o d m a y g i v e a n eh a s a v a l u e o f 2 , 4 . 6 .W h e r e t h e v a l u e o f 2 c r1 0 , t h e r e i s n o r e d ud e s i g n s t r e n g t h i n t h i s m e t h o d . H o w e v e r , m av a l u e o f , ,l e s st h a n 4 . 6 , s o t h i s m e t h o d c a n n o t b e c h o i c e b e t w e e n s t i f f e n i n g t h e f r a m e a n d u s i n g

1 . 3 . 3T i e d p o r t a l sT i e dp o r t a l s s h o u l d a l w a y s b e d e s i g n e d u s i ns o l u t i o n m e t h o d f o r t h i s a n a l y s i s i s n o t s p e c i ft o c h o o s e a s u i t a b l e r o u t i n e .I t s h o u l d b e n o t e d t h a t f o r t i er o o f s l o p e s , t h e r e i s a n i m p o r t a n t n o n - l i n e aa r i s e s b e c a u s e t h e c o m p r e s s i o n o f t h e r a f t e r at h e h e i g h t o f t h e a p e x , w h i c h r e d u c e s t h e v e r tT o m a i n t a i n e q u i l i b r i u m , a n i n c r e a s e d r a f t e r t h e a p e x d e f l e c t i o n u n t i l e i t h e r e q u i l i b r i u m i s

3


Second-order analysis is simple to apply if there is easy-to-use softwareavailable. It will give themost econom ical designs for more flexible framessuch as multi-span frames. It may give less econo mical designs than the othermetho ds for stiffer frames because it will alw ays calculate a reduction of fram estrength from second-order (P-delta) effects. The other method s have thresholdstiffness values abovewhich the strength is not reduced. TheSecond-ordermetho d oes recognise the improv emen tn in-plane stability ofhe frameresulting fro m the use of higher strength steel (gradeS355 steel).

Further details of this method are given in Section5 .

1.3 Selecting methods for diff erent ypes offrames

1.3.1 Single-span rames not ied por ta l s )

Single-span frames may be designed by any of he three methods describedabove. W here the framesre within the geom etrical limitations ofheSway-check method and pass either theh/1000 check or the formula check (seeSection 1.2.1 abo ve), the method does not give any reduction of frame strengthfor the gravity load cases. W here the frames re outside the geometricallimitations of he Swa y-check metho d or fail the checks, another method mustbe used. For rames slightly outside the geometrical limitations, it may beworth mak ing mino r alterationsto the scheme to fit into the limitations, such asan increase in stiffnessof the frame to satisfy the deflection check,or setting

the bases deeper to suit the span to height ratio or a change of rafter geometry.W here the Sw ay-check method is not satisfied, either the Am plifiedMomentmethod or Second-order analysis should be used.

1. 3 . 2 Mul t i - span rames not ied por ta l s )

Multi-span frames often have relatively low stiffness. Althou gh somemulti-span frames migh t be sufficiently stiff for the Sw ay-check metho d, manywill not. W here the frames are too flexible and have slender internal colum ns,the most efficient way o imp rove the frame stiffness will often be to increasethe internal column stiffness.

Th e amplified mo men t method may give an econom ical frame whe re the framehas a value of A,, 2 4.6. Where the value ofAcr2 10, there is no reduction ofdesign strength in his metho d.However,many multi-span frames will have avalue of Acr less than 4.6, so this metho d cannot be applied. This leaves thechoice betw een stiffening the frame and using second-order analysis.

1.3 .3 Tied por ta l s

Tied portals should alway s be designed using second-order analysis. Thesolution method for this analysis is not specified inBS 5950-1, leaving freedomto choo se a suitable routine. It should be noted that for tied portals with owroof slopes, there is an impo rtant non-linea rity in he apex deflections. Thisarises because the compressionof the rafter and the stretching of the tie reducethe height of the apex, w hich reduces the vertical comp onent of the rafter force.To maintain equilibrium, an increased rafter force is required, which increasesthe apex deflection until either equilibrium is reached or the apex snaps through.

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T h e r e f o r e , w h a t e v e r r o u t i n e i s s e l e c t e d , i t mb e h a v i o u r o f t h e r a f t e r a n d t i e s y s t e m , w h i c hi t e r a t i v e p r o c e d u r e .

T i e d p o r t a l s o f e c o n o m i c a l p r o p o r t i o n s w i

f o r c e s i n t h e r a f t e r s . T h e s e f o r c e s o f t e n c a us t a b i l i t y o f t h e f r a m e .T h e r e f o r e , r a f t e r s w i l l o f t e ns i g n i f i c a n t l y s t i f f e r t h a n t h e s e c t i o n t h a t w o u l d

1 . 3 . 4 S t a b i l i t y p o r t a l s o r ' w i n d p o r t a lS t a b i l i t yp o r t a l s a r e o u t s i d e t h e s c o p e o f t h i s d op o r t a l s u s e d t o s t a b i l i s e s t r u c t u r e s w h e r e c r of r a m e s h a v e l i t t l e v e r t i c a l l o a d i n g d i s t r i b u t e ds m a l l a x i a l l o a d s i n t h e b e a m . T h e d o m iS e c o n d - o r d e r a n a l y s i s , t h e A m p l i f i e d M o mm e t h o d ( l a t e r a l l o a d c a s e ) w o u l d b e a p p r o p rb u t t h e g r a v i t y l o a d c a s e o f t h e S w a y - c h eA l t e r n a t i v e l y , w h e r e t h e a x i a l f o r c e i n t h e b ed e s i g n s u c h f r a m e s a c c o r d i n g t o t h e r u l e s f ot h a n t h e r u l e s f o r o r d i n a r y p o r t a l f r a m e s .

1 . 4 R e q u i r e d l o a d f a c t o r , A rB S5 9 5 0 - 1 C l a u s e s 5 . 5 . 2 a n d 5 . 5 . 3 i n t r o d u c e ti s a f a c t o r t o a l l o w f o r P - d e l t a e f f e c t s w h e r e t h e g l o b a l a n a l y s i s . F o r e l a s t i c d e s i g n o f p

f i r s t - o r d e r g l o b a l a n a l y s i s w i t h U L S l o a d s mm e m b e r r e s i s t a n c e s a r e c h e c k e d . F o r p l a s t i c A , c a l c u l a t e d b y f i r s t - o r d e r g l o b a l a n a l y s i s t h a n 2 . M e m b e r s t r e n g t h a n d s t a b i l i t y c a l c u l ar a t h e r t h a n 1 . 0 x U L S .

1 . 5 B a s e s t i f f n e s sB S5 9 5 0 - 1 C l a u s e 5 . 1 . 3 g i v e s g u i d a n c e o n a s s u m e d i n d e s i g n . T h i s m a y b e s u m m a r i sf r e q u e n t l y o c c u r r i n g i n p o r t a l f r a m e d e s i g n .

B a s ew i t h a p i n o r r o c k e rT h eb a s e s t i f f n e s s s h o u l d b e t a k e n a s z e r o

N o m i n a l l yp i n n e d b a s eI ft h e b a s e m o m e n t i s a s s u m e d t o b e z e r o , t hp i n n e d i n t h e g l o b a l a n a l y s i s u s e d t o c a l c u l a tt h e f r a m e . H o w e v e r , t h e b a s e s t i f f n e s s m a y t h e c o l u m n s t i f f n e s s w h e n c h e c k i n g f r a m e e f f e c t i v e l e n g t h s , w h i c h f o r m p a r t o f t he l a s t i c - p l a s t i c d e s i g n , a n a p p r o p r i a t e b a s e c a p aF o r c a l c u l a t i n g d e f l e c t i o n s a t S L S , t h e b a s e s t io f t h e c o l u m n s t i f f n e s s , b u t t h i s s h o u l d n o t b e

4

The refore, whatever routine s selected,it must ake account of henon-linearbehav iour of he rafter and ie system, whichwill almost certainly involve aniterative procedure.

Tied portals of econo mical prop ortions will ormally ave very high xial

forces in he rafte rs. These forces often cause a significant reduction inhestability ofhe frame.here fore, rafters willfteneedoeadesignificantly stiffer than the section that would satisfy a first-order analysis.

1.3.4 Stability portals or 'wind portals '

Stability portals are outside the scope of this documen t. Stability portals areportals used ostabilise structures where cross-bracing isnot acceptable. Suchframes have little vertical loading distributed along he beam elem ent,so havesmall axial loads inhe beam. The dominant failure mode isy sway.Second-order analysis, the Amplified Moment method or the Sway-checkmethod (lateral oad case) wouldbe approp riate for check ing stability frames,

but the gravity load case ofhe Sway-check method should not e used.Alternative ly, where the axial force in the beam is very low ,i t is reasonable todesign such frames according to the rules for multi-storey sway-frames ratherthan the rules for ordinary portal frames.

1.4 Required load acto r, hBS 5950-1 Clauses5.5.2 and 5.5.3 introduce the required load factorA. Thisis a factor to allow for P-delta effects where these have not been calculated inthe global analysis. Fo r elastic design of portal frames, the outpu t from a

first-orde r global analysis with UL S loads must be multiplied by/i, before themem ber resistances are checke d. Fo r plastic design, the plastic collapse factor,4, calculated by first-order global analysis withULS oads must not be essthan 4. Me mbe r strength and stability calculations should be made at/2, x ULSrather than 1.0x ULS.

1.5 Base stiffnessBS 5950 -1 C lause 5 .1 .3 gives guidance on the asetiffnesshatmay eassum ed in design. This may be summarised as follows for the cases mostfrequently occurring in portal frame design.

Base w it h a pin or rocker

The base stiffness should be taken as zero

Nominally pinned base

If he base mom ent is assumed to be zero, thebase should be assumed to bepinned in he global analysis used to calculate the moments and forces aroundthe frame. Ho wev er, thebase stiffness maybeassumed obe equal to 10%ofthe column stiffness when checking frame stability or determining in-planeeffective lengths, which formart ofheLS process. Whensing

elastic-plastic design, an appropriate base capacity must also be specified.For calculating deflections atSLS, the base stiffness may be assumed to be20of the column stiffness, bu t this should not be used for in-plane stability checks.

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O t h e r t y p e s o f b a s eB S5 9 5 0 - 1 C l a u s e 5 . 1 . 3 a l s o g i v e s g u i d a n c e f oa n d n o m i n a l s e m i - r i g i d b a s e s .

A p p l i c a t i o n o f t h e s e p r o v i s i o n s f o r b a s e s t ic h e c k i n g f r a m e s i s g i v e n i n S e c t i o n 3 . 3 . 4S e c t i o n 4 . 3 . 5 f o r t h e A m p l i f i e d M o m e n t S e c o n d - o r d e r m e t h o d s . T h e a p p l i c a t i o n t o tc a l c u l a t i o n si s g i v e n i n A p p e n d i x A . 2 . 4 f o r A p p e n d i x B . 2 . 4 f o r t i e d p o r t a l s .

1 . 6 N o t i o n a l h o r i z o n t a l f o r c e1 . 6 . 1 G e n e r a lB S 5 9 5 0 - 1 u s e s n o t i o n a l h o r i z o n t a l f o r c e sf a c t o r e d v e r t i c a l d e a d a n d i m p o s e d l o a d s . Tt h e c o l u m n s f o r s i m p l i c i t y , o r a t t h e p o i n t oF i g u r e 1 . 1 .

0 . 0 5 ( R 3 - C 1 )0 . 0 5

0 . 0 5 ( R 4 - C 2 )- * . 0 . 0 5C 2

t R 6F i g u r e 1 . 1 N o t i o n a l h o r i z o n t a l f o r c e s ( f o r

1 . 6 . 2 )

0 . 0 5 R 1 0 . 0 5 A 2-

C 1 C 2

0 . 0 5 ( A -

0 . 0 5 - + - 0 . 0 5 R

0 . 0 5 ( R T Q 2 )

0 . 0 5 Q 2

A 5

5

Other typ es of base

BS 5950-1 Clause 5.1.3 also gives guidance for the use of nominally rigid basesand nominal semi-rigid bases.

Applicationof these provisions or base stiffness to he differentmethods ofcheckingramess given in Section .3.4or the Swa y-checkmethod, inSection4.3.5 or the Am plified Mom ent method and in Section5.3.4 orSecond-ordermethods.Theapplication o the hand method of second -ordercalculationss given in Appendix A.2.4orommonortals andAppendix B.2.4 for tied portals.

1.6 Notional horizontal orces1.6.1 General

BS 5950-1uses notional horizontal orces, which are taken as0.5% of hefactored vertical dead and imposed loads. They may be applied at the tops ofthe column s for simplicity, or at the point of application of load, as show n inFigure 1.1.

0.05 (R 5- Q 1)*

0.05 Q 1-b

Figure 1.1 Notional horizontal forces (for mezzanines etc, see Section1.6.2)

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T h e s e n o t i o n a l h o r i z o n t a l f o r c e s a r e u s e d f o r t w

( i ) F o r c h e c k i n g f r a m e s t r e n g t hT h e n o t i o n a l h o r i z o n t a l f o r c e s a r e a p p l i e da l l o w f o r t h e e f f e c t s o f p r a c t i c a l i m p e r f e c t i

a s g i v e n i n C l a u s e 2 . 4 . 2 . 4 . T h e n o t i o n a l hL o a d c o m b i n a t i o n 1 o f C l a u s e 2 . 4 . 1 . 2 , w hp l u s i m p o s e d l o a d s ( g r a v i t y l o a d s ) .

( i i ) F o r c h e c k i n g f r a m e s t i f f n e s sT h e n o t i o n a l h o r i z o n t a l f o r c e s a r e a p p l i e d ac h e c k o f f r a m e s s u c h a s i n C l a u s e 5 . 5 . 4n o t i o n a l h o r i z o n t a l f o r c e s a r e a p p l i e d t ol o a d i n g t o a s s e s s t h e s t i f f n e s s o f t h e f r a md e f l e c t i o n s o f t h e c o l u m n t o p s a s s u m iC l a u s e 5 . 5 . 4 . 2 . 1 s a y s t h a t t h e f o r c e s s hv e r t i c a l r e a c t i o n a t t h e b a s e o f t h e r e s p e c tt h e c o l u m n r e a c t i o n s a r e k n o w n e x a c t l y f o r c e s a r e d e f i n e d . I n p r a c t i c e , t h e d e f l ed i s t r i b u t i o n o f t h e n o t i o n a l h o r i z o n t a l f o r cm a y b e m a d e i n t h e d i s t r i b u t i o n o f t h e s e li s t h a t n o t i o n a l h o r i z o n t a l f o r c e s m u s t b e l o a d s o n t h e b u i l d i n g a n d t h i s i s m o s tc o n s i d e r i n g t h e v e r t i c a l r e a c t i o n s o f t h e c o l u

A l t h o u g h t h e m a g n i t u d e o f t h e f o r c e s i n b o t h0 . 5 % o f f a c t o r e d l o a d s , t h e r e i s a n i m p o r ta p p l i e d i n t h e c a s e o f c r a n e l o a d s . I n C l a u s e c r a n e l o a d s n e e d n o t b e i n c l u d e d w h e n c a l c u l af o r c h e c k i n g f r a m e s t r e n g t h . B y c o n t r a s t , aw h e n c h e c k i n g t h e f r a m e s t i f f n e s s , h e n c e i nh o r i z o n t a l f o r c e s m u s t i n c l u d e 0 . 5 % o f t h e vi n - p l a n e s t a b i l i t y o f t h e f r a m e i s n o t a f f e cn o t i o n a l h o r i z o n t a l f o r c e s h o u l d b e t a k e n a sw i t h o u t d y n a m i c o r i m p a c t e f f e c t s .

1 . 6 . 2M e z z a n i n e s a n d o t h e r c o n n e c t e d s tW h e r ea m e z z a n i n e f l o o r o r o t h e r s t r u c t u r e i s c o ns t a b i l i t y o f t h e c o n n e c t e d s t r u c t u r e m u s t b e c o

s t r e n g t h a n d t h e s t i f f n e s s o f t h e p o r t a l . W h e r eo w n s t a b i l i t y s y s t e m ( e . g . c r o s s - b r a c i n g , sc o n n e c t i o n s ) t h a t m a k e s t h e c o n n e c t e d s t r u cf r a m e , t h e n t h e p o r t a l n e e d n o t r e s i s t n o t i oc o n n e c t e d s t r u c t u r e . W h e r e t h e c o n n e c t e d ss t a b i l i t y s y s t e m , t h e s u m o f t h e n o t i o n a l h o r is t r u c t u r e m u s t b e a p p l i e d t o t h e p o r t a l f r a mw h e r e t h e c o n n e c t e d s t r u c t u r e p r o v i d e s s o m ep o r t a l f r a m e , t h e n o t i o n a l h o r i z o n t a l f o r c e s f r os h a r e d .

T h e s t i f f n e s s o f t h e c o n n e c t e d s t r u c t u r e a n d t hi n t e r m s o f t h e s l o p e o f t h e c o l u m n s i n d u c e d A l t e r n a t i v e l y , i t m a y b e c a l c u l a t e d i n t e r m s op o i n t s i n d u c e d b y t h e n o t i o n a l h o r i z o n t a l f o r

6

These notional horizontal forces are used for two completely different purposes:

(i) For checking frame strength.

The notional horizontal forces are applied as a design horizontal load toallow for the effects of practical imperfection such as a lack of verticality,

as given in Clause 2.4 .2 .4. The notional horizontal forces are applied inLoad combination 1 of Clause 2.4 .1.2 , which s combination of dead loadplus imposed loads (gravity loads).

(ii) For checking frame stiffnessTh e notional horizontal forces are applied as the loading used in a stiffnesscheck of frames such as in Clause 5.5 .4.2 .1. Inhis application , thenotional horizontal forces re applied tohe frame without any otherloading to assess the stiffness of the frame by calculating the horizontaldeflections ofhe colum n tops assuming linear elastic behaviour.Clause 5.5 .4.2 .1 says thathe forces should be equal to 0 .5 % ofhevertical reaction at the baseof he respective colum n. Thi s assumes thatthe colum n reactions are known exactly before the otional horizontalforces are defined. In practice, the deflections are not sensitive to thedistribution of henotional horizontal forces. Thu s, some approximationmaybe made in he distribution of hese loads. The most important pointis that notional horizontal forces must becalculated from all heverticalloads on theuilding and this is most conveniently calculated byconsidering the vertical reactions of the columns.

Althou gh the mag nitude of he forces in both (i) and (ii) above is he sam e, at0.5% of factored loads, there is an important difference in the loads to beapplied in the case of crane loads. In Clause 2.4.2 .4, it is clear that the verticalcrane loads need not be included when calculating the notional horizontal forcesfor checking frame strength. By contrast,all vertical oads must be appliedwhen checking the frame stiffness, hence in the stiffness check, thenotionalhorizontal forces must nclude 0.5% of the vertical crane loads. How ever, thein-planetability fhe frames ot affected by dynamic loading,so thenotional horizontal force should be takenas 0.5% of the factored crane loadwithout dynamic or impact effects.

1.6.2 Mezzanines and other connected s t ruc tures

W here a m ezzanine floor or other structure is connected to the portal frame, thestability of the connected structure mu st be considered w hen checking both the

strength and the stiffness of the portal. W here a connected structure contains itsown stability system (e.g. cross-bracing, stability portal or rigid momentconnections) that makes the connected structure at least as stiff as the portalframe, then the portal need not resist notional horizontal forcesrom theconnected structure. W here the connected structure is not restrained by nystability system, the sum of the notional horizontal forces from the connectedstructuremu st be applied to he portal frame. In he intermediate condition,where the conn ected struc ture provides som e stability but is not as stiff as theportal fra me , the notional horizontal forces from the connected structure may beshared.

Th e stiffness o f the connected structure a nd the portal frame may be calculatedin terms of the slope of the colum ns induced by the notional horizontal fo rces.Alternatively , it may be calculated in terms of the deflection at the connectionpoints induced by the notional horizontal forces. It is rare o find heseslopes

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o r d e f l e c t i o n s u n i f o r m t h r o u g h o u t a s t r u c t uc a l c u l a t e d v a l u e s m a y b e u s e d .

1 . 7 L o c a l c o n c e n t r a t e d l a t e r a l l oB u i l d i n gs t r u c t u r e s a r e o f t e n s u b j e c t t o l o c a l c o nl o a d s . W h e r e t h e s e c a u s e s w a y d e f l e c t i o n s (h o r i z o n t a l f o r c e s ) , t h e s e l o a d s m a y b e s hb u i l d i n g s w i t h m e t a l r o o f s h e e t i n g o r w i t h c o n

7

or deflectionsuniform hrough out a structure,so the mean or median of hecalculated values may be used.

l .7 Local concentrated ateral oads n buildingsBuilding structures are often subject o local concen trated oads, such as craneloads.W here these cause sway deflections e.g.cranesurge loads or notionalhorizontalorces), these loads may be shared by the adjacentrames inbuildings with metal roof sheeting or with continuous bracing.

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2 I N T R O D U C T I O N T O IS T A B I L I T Y

2 . 1 W h y a r e t h e r e i n - p l a n e s t aA l ls l e n d e r m e m b e r s r e s i s t i n g a x i a l c o m p r e s s ia x i a l f o r c e w e r e l a r g e e n o u g h .S t a b i l i t y c h e c k s c a l c u l a t i o nr e s i s t a n c e t o b u c k l i n g i s g r e a t e r t h a n t h e a p ps t a b i l i t y o f a c o l u m n , t h e b u c k l i n g r e s i s t a n c eb o t h t h e m a j o r a x i s a n d t h e m i n o r a x i s .

I n f r a m e s , t h e s t a b i l i t y c h e c k s m u s t a l s o v e rr e s i s t a n c e a b o u t b o t h t h e m a j o r a x i s a n d t hI n n o r m a l p o r tf r a m e s , b u c k l i n g o u t o f t h e p l a n e o f t h e f r a mf o r a n y o t h e r b e a m - c o l u m n , c o n s i d e r i n g b u c kb e t w e e n t o r s i o n a l r e s t r a i n t s p r o v i d e d b y b r a c ie f f e c t i v e l e n g t h s o f e a c h e l e m e n t e a s i l y i d e n tp l a n e o f t h e f r a m e i s m o r e c o m p l i c a t e d t h a n iT h i s i s b e c a u s e t h e r e i s n o r m a l l y n o b r a c i n g it h e r e s t r a i n t t o a n y c o l u m n d e p e n d s o n t h e s t ic o l u m n s . E q u a l l y , t h e r e s t r a i n t t o a n y r a f t e rc o l u m n s a n d t h e o t h e r r a f t e r s . T h e r e f o r e , c h em u s t c o n s i d e r t h e e n t i r e f r a m e s t i f f n e s s . A l t h oc h e c k i n g t h e b u c k l i n g r e s i s t a n c e o f c o l u m n s u sl e n g t h s o f p o r t a l f r a m e s c a n o n l y b e d e f i n ee n t i r e f r a m e i s c o n s i d e r e d .T h e i n - p l a n e s t a b i l i t y c h e c k s f o r p o r t a l f r a m ef o r b e a m a n d c o l u m n b u i l d i n g s . T h i s i s b e c a uh a v e a m u c h g r e a t e r e f f e c t o n t h e s t a b i l i t y o f m i g h t o c c u r i n t h e b e a m s o f c o m m o n b e a m a n

2 . 2 A x i a l c o m p r e s s i v e f o r c e s i n 2 . 2 . 1 G e n e r a l

I n - p l a n es t a b i l i t y d e p e n d s o n t h e m a g n i t u d e o f tm e m b e r s , s o i t i s i m p o r t a n t t o u n d e r s t a n d t h e i n t h e r a f t e r s a n d c o l u m n s .

M o s t f r a m e s h a v e a x i a l c o m p r e s s i v e f o r c ed i s t r i b u t i o n o f f o r c e s d e p e n d s n o t o n l y o n t hs t r u c t u r a l f o r m o f t h e f r a m e a n d t h e b e n d i n gT h e m a g n i t u d e o f t h e s e c o n d - o r d e r b u c k l i n gm a g n i t u d e o f t h e f o r c e , b u t a l s o o n t h e e l a sm e m b e r s a n d t h e e l a s t i c c r i t i c a l b u c k l i n g ld i s c u s s e d i n S e c t i o n 2 . 4 . 2 a n d S e c t i o n 2 . 4 .b u c k l i n g l o a d s , t h e g r e a t e r w i l l b e t h e s e c o n dc o m p r e s s i v e f o r c e .

W h e r e t h e r e i s a x i a l t e n s i o n i n t h e m e m b e r s , t h e s t i f f n e s s o f t h e f r a m e , s o n o r e d u c t i o n i n f r

8

2 INTRODUCTION O IN-PLANESTABILITY

2.1 Why are there in-plane stability checks?All slender members resistingaxial com pression wouldbuckle if heappliedaxial force were large eno ugh . Stability checks calculations verifyhatheresistance to buckling isgreater than heapplied forces. Whenchecking hestabilityof a column , the buckling resistance is calculated for bucklingaboutboth the major axis and the minor axis.

In frames, the stability checks must also verify he adequacy of hebucklingresistance about both the major axis and the mino r axis. In normal portalframes, buckling out of he plane of he frame is checked in he same way asfor any other beam-column, considering buckling between lateral restraints andbetween torsional restraints provided by bracings etc. These bracings make heeffective lengths of each elemen t easily identifiable. Ho we ver, buckling n heplane of the frame is more complicated than in normal beam colum n elements.This is because there is normally no bracing in the plane of the frame, and thusthe restraint to any colum n depends on the stiffness of the rafters and the othercolum ns. Equally, the restraint toany rafter depends on the tiffnessof hecolum ns and the other rafters. Therefore, checks for thestabilityof he framemust consider the entire frame stiffness. Although engineers are accustomed ocheck ing the buckling resistance of colum ns using effective lengths, the effectivelengths of portal framescan onlybedefined correc tly if thestiffnessof he

entire frame is considered.

The in-plane stability checks for portal frames inBS 5950-1 differ from thosefor beam and colum n buildings. This is because the axial loads in portal raftershave a much grea ter effect o n the stability of the frame than the axial loads thatmight occur in the beams of common beam and column buildings.

2.2 Axial compressive forces in rames2.2.1 General

In-plane stability dependson themagnitudeof he axial compression in hemembers, so it is imp ortan t to understand the relative magnitude of these forcesin the rafters and column s.

Most frames have axial compressive forces in some of the mem bers. Thedistribution of forces depends not only on the applied loads, but also on thestructural form of the frame and thebending mom ents throughout the frame.The magnitude of the second-order buckling effects depends notonlyon hemagnitude of the force, but also on he elastic critical buckling oadof hemembers and the elastic critical bucklingoad of the entire frame. This isdiscussed in Section 2.4.2 nd Section 2.4 .3.Th e low er the lastic riticalbuckling loads, the greater willbe he second-order effects from a givenaxialcompressive force.

W here there s axial tension in the memb ers, the second-order effects increasethe stiffness of the fram e,so no reduction in frame capacity need be considered.

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2 . 2 . 2 O r d i n a r y p o r t a l sA t y p i c a l b e n d i n g m o m e n t d i a g r a m f o r a n o r dl o a d i n g i s s h o w n i n F i g u r e 2 . 1 . T h e r e i s a ht h e c o l u m n s t o m a i n t a i n e q u i l i b r i u m w i t h t h eT o m a i n t a i n t h e h o r i z o n t a l e q u i l i b r i u m o f t h ec a r r y a n a x i a l c o m p r e s s i o n a s s h o w n i n F i gf o r c e s a r e n o t l a r g e i n m a g n i t u d e , b u t t h e y mt h e e l a s t i c c r i t i c a l b u c k l i n g l o a d o f t h e r a f t e r sl o n g . T h i s e f f e c t i s c o n s i d e r e d i n S e c t i o n 2 . 4

T h e a x i a l c o m p r e s s i v e f o r c e i n t h e r a f t e r i s s ep o r t a l s p a n t o t h e c o l u m n h e i g h t . T h i s i s b ec o l u m n t o p d e p e n d s o n t h e s p a n a n d t h e h o r id e p e n d s o n t h e m o m e n t a t t h e c o l u m n t o p am o m e n t a t t h e c o l u m n t o p i s g i v e n a p p r o x i m a

2C o l u m n t o p m o m e n t , M

w h e r e :1 2

w i st h e d i s t r i b u t e d l o a d o n t h e r a f t e rL i s t h e s p a n o f t h e p o r t a l .

T h e h o r i z o n t a l r e a c t i o n f o r a p i i m e d b a s e i s t h

H = - L - - - -H 1 2 hw h e r e :

h i s t h e h e i g h t o f t h e c o l u m n .T h e r e f o r e , f o r a g i v e n l o a d i n g a n d s p a n , t h e l e s s f o r a h i g h p o r t a l f r a m e t h a n f o r a l o w f r a

T h e a x i a l c o m p r e s s i o n i n t h e r a f t e r s p r o d u c e sw h i c h r e d u c e s t h e i n - p l a n e s t a b i l i t y o f t h e f r ae f f e c t s f r o m t h e a x i a l c o m p r e s s i o n i n t h e c o l u

9

F i g u r e 2 . 1 B e n d i n g m o m e n t s i n a t y p i c a l

2.2.2 Ordinary ortalsA typical bending mom ent diagram for an ordinary portal frame unde r verticalloading s shown in Figure2.1. There s a horizontal reaction at the bases ofthe colum ns to maintain equilibrium with the bending mo ments in the column s.T o maintain the horizontal equilibrium of these horizontal reactions, the rafterscarryanaxialcompression as shown in Figure2.2. These axial compressiveforces are not large in mag nitude, but they maybe significant com pared withthe e lastic critical buckling load of the rafters, b ecause the rafters are relativelylong. This effect is considered in Section2.4.2

The axial com pressive force in the rafter is seriously affected by the ratio of theportalspan o the columnheight.This sbecause the bendin g mom ent at thecolum n top depends on the span and the horizontal reaction at the colum n basedepen ds on the moment at the column top and the height of he colum n. Themoment at the column top is given approximately by:

WLColumn top m oment, M =12

where:

W is the distributed load on the rafter

L is the span of the p ortal.

The horizontal reaction for a pinned base is then given by:

M wL2

H 12hH = - = -

where:

h is the height of the column.

Th erefore , for a give n load ing and sp an, the axial com pression in the rafters isless for a high portal frame than for a low frame.

The axial compression in the rafters produc es second -order effects in the rafters,which red uces the in-plane stability of the frame in addition to the secon d-ordereffects from the axial compression in the columns.

4 kFigure 2.1 Bending moments in a typical frame

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F i g u r e 2 . 2 H o r i z o n t a lr e a c t i o n s a n d r a f t e r a x i a l f o r c e

2 . 2 . 3 T i e d p o r t a l sT i e dp o r t a l s i n w h i c h t h e t i e i s n e a r t h e e a v e s l e vo r d i n a r y p o r t a l s . T h e s t r u c t u r a l b e h a v i o u r i s t r u s s o n p o s t s . T h e a x i a l c o m p r e s s i v e f o r c e s i

i n o r d i n a r y p o r t a l s , e s p e c i a l l y f o r p o r t a l s w i t h T h e b e n d i n g m o m e n t s f o r a t i e d - p o r t a l a r eT h eb e n d i n g m o m e n t d i a g r a m i s s i m i l a r t o a p a i r os p a n f r o m e a v e s t o a p e x . T h e r e f o r e , t h e b e na d i n t h e c o l u m n s a r e a p p r o x i m a t e l y a q u a r to r d i n a r y p o r t a l . T h i s r e d u c t i o n i n t h e b e n dr a f t e r w i t h a m u c h s m a l l e r b e n d i n g r e s i s t a n ca r e a c o n s e q u e n c e o f t h e t r u s s a c t i o n o f t h es h o w n i n F i g u r e 2 . 4 .

7 - 0 -

F i g u r e 2 . 4C o l u m n s h e a r s a n d r a f t e r a n d t i e a x i

T h e h i g h a x i a l l o a d s o n r a f t e r s t h a t r e q u i rr e s i s t a n c e m e a n s t h a t t h e r a f t e r s a r e s e r i o u s l y F o r t h i s r e a s o n , i t i s r e c o m m e n d e d t h a t t i e d

F i g u r e 2 . 3 B e n d i n g m o m e n t s i n a t i e d p o

a

1 0

Figure 2.2 Horizontal reactions and rafter axial force

2.2.3 Tied portals

Tied portals in which the tie is near the eaves level behave very differently fromordinary portals. The structural behaviour is more like that of a rigidly-jointedtruss on pos ts. The axial com pressive forces in the rafters are much higher than

in ordinary portals, especially for portals with low roof slopes.The bendingmoments for tied-portal arellustrated in Figure 2.3. Thebending moment diagram is similar to a pairof fixed ended beams, each with aspanfromeaves oapex.Therefore, the bendingmomentsboth in the raftersad n the columnsareapproximatelyaqua rter of the bendingmom ents in anordinaryportal.Th is reduction in the bendingmom ent allow s the use of arafter with amuchsmaller bending resistance. The reducedbendingmomentsarea consequence of the truss action of the tied portal.The axial loads areshown in Figure2.4.

Figure 2.3 Bending moments in a tied portal

Figure 2.4 Column shears and rafter and t ie axial forces

The igh axial loads onafters that require only relatively small bendingresistance means that the rafters are seriously affected by second -order effects.For this reason, it is recommended that tied-portals are always heckedby

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' i c r

w h e r e :

c i )C - )

0U -

P y

F i g u r e 2 . 5

L 2

1 1

s e c o n d - o r d e r a n a l y s i s . H o w e v e r , i f t h i s i s t ol e s s r i g o r o u s t h a n t h e i n - p l a n e c h e c k s o nI n a d d i t i o n , tc a l c u l a t i o n s m u s t a l l o w f o r t h e i n c r e a s e i n a xi n t h e h e i g h t b e t w e e n t h e a p e x a n d t h e t i ec o n s e q u e n c e o f t h e s t r a i n s i n t h e r a f t e r s aa v o i d i n g t h i s r e d u c t i o n i n h e i g h t i s t o i n s t a l l t o m a i n t a i n a c o n s t a n t h e i g h t b e t w e e n t h e T h i s m u s t bp r o p e r l y r e s t r a i n e d a g a i n s t o u t - o f - p l a n e d i se n d s .

2 . 3 E l a s t i c c r i t i c a l b u c k l i n g o f fS t r u t sh a v e a t h e o r e t i c a l e l a s t i c c r i t i c a l b u c k l ic o u l d o n l y b e r e a c h e d i f t h e s t r u t h a s a n i n f il o a d , o r E u l e r l o a d , f o r a p i n - e n d e d s t r u t i s g i

= n ; 2 E 1

E i s t h e Y o u n g ' s m o d u l u sI i s t h e i n e r t i a o f t h e s t r u tL i s t h e l e n g t h o f t h e s t r u t .

T h e c r i t i c a l b u c k l i n g l o a d i s a t h e o r e t i c a l l ol o a d o f a r e a l s t r u t a s s h o w n i n F i g u r e 2 . 5s q u a s h l o a d P , ( =A r e ax y i e l d s t r e s s ) a r e s h o w n .

c r

r ' f a i l

S l e n d e r n e s s

E l a s t i c c r i t i c a l b u c k l i n g l o a d o f a s t

second -order analysis. How ever, if thisis to be d one , the check s should be noless rigorous than the in-plane ch ecks nrussafter. In addition, thecalculations must allow for the increase in axial forces arising from a reductionin the height betweenhe pex and the tie. This reduction in heightsconsequence f the strains in the rafters and tie.A convenient method ofavoiding this reduction in height is to install a strut between the apex and the tieto maintain a onstant height between the apex and the tie.Thismust beproperly restrained againstout-of-plane displacements of the frames at bothends.

2.3 Elastic critical buckling of f r a m e sStruts have a theoretical elastic critical buckling load, o r Euler oad, whichcou ld only be reached if the strut has an infinitely high strength. The bucklingload, or Euler load, for a pin-ended strut is given by:

where:

E is the Youngsmodulus

Z is the inertia of the strut

L is the length of the strut.

Thecritical buckling load isa theoretical load and exceed s the actual failureload of a real strut as shown in Figure2.5. In the figure, both P,, and thesquash loadP,, ( = Area x yield stress) are shown.

Slenderness

Figure 2.5 Elastic critical buckling load of a s trut

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S i m i l a r l y , f r a m e s h a v e a t h e o r e t i c a l e l a s t i c c ro n l y b e r e a c h e d i f t h e f r a m e h a s a n i n f i n i t eT h i s w i l l b er e f e r r e d t o i n t h i s d o c u m e n t a s V c r .T h i si s c o m m o n l y e x p r e s s ec a l l e d ' l a m b d a c r i t ' , 2 c r ,w h i c hi s d e f i n e d a s :

v c rc r

V U L S

w h e r e :

V c r i s t h e e l a s t i c c r i t i c a l b u c k l i n g l o a dV U L Si st h e a p p l i e d l o a d i n g a t U L S .

T h e v a l u e o f V c rd e p e n d so n t h e d i s t r i b u t i o n o f l o a d o n t h eb e c a l c u l a t e d f r o m v a l u e s o f V c r a n d V U L S td i s t r i b u t i o n o f l o a d .

T h e v a l u e o fv a r i e s a c c o r d i n g t o t h e m a g n i t u d e o f V U L S .A l a r g e v a l u e o f 2 , i n d i c a t e s t h a t t h e l o a d i nt h e b u c k l i n g r e s i s t a n c e . A v a l u e o f 2 c r j u s t a bi s n e a r t o i t s f a i l u r e l o a d . I t m u s t b e r e m e m b ew e l l b e l o w V c r d u e t o b e n d i n g s t r e s s e s i n t h e ff i n i t e v a l u e o f y i e l d s t r e s s . H o w e v e r , , l . i si n d i c a t o r o f t h e s e n s i t i v i t y o f t h e f r a m e ta m p l i f i c a t i o n f a c t o r s .

2 . 4 S e c o n d o r d e r ( P - d e l t a ) e f f e c t2 . 4 . 1 G e n e r a lT h es t r e n g t h c h e c k s f o r a n y s t r u c t u r e a r e v a l i d oa g o o d r e p r e s e n t a t i o n o f t h e b e h a v i o u r o f t h e a

W h e n a n y f r a m e i s l o a d e d , i t d e f l e c t s a n d i t s st h e u n d e f o r m e d s h a p e . T h e d e f l e c t i o n c a u s e s a c t a l o n g d i f f e r e n t l i n e s f r o m t h o s e a s s u md i a g r a m m a t i c a l l y i n F i g u r e 2 . 6 a n d F i g u r e 2 . 7c o n s e q u e n c e s a r e v e r y s m a l l a n d a f i r s t - o r d e r

t h e d e f l e c t e d s h a p e ) i s s u f f i c i e n t l y a c c u r a t es u c h t h a t t h e e f f e c t s o f t h e a x i a l l o a d o n t h e d ec a u s e s i g n i f i c a n t f u r t h e r d e f l e c t i o n , t h e fs e c o n d - o r d e r e f f e c t s . T h e s e s e c o n d - o r d e r es u f f i c i e n t t o r e d u c e t h e r e s i s t a n c e o f t h e f r a m e .

S e c o n d - o r d e r e f f e c t s a r e g e o m e t r i c a l e f f e c t sm a t e r i a l - n o n - l i n e a r i t y .

T h e r e a r e t w o c a t e g o r i e s o f s e c o n d o r d e r e f f e c

( i ) E f f e c t s o f d e f l e c t i o n s w i t h i n t h e l e n g t h o f

( P - l i t t l e d e l t a ) e f f e c t s .( i i ) E f f e c t s o f d i s p l a c e m e n t s o f t h e i n t e r s e c t i o

P . 4 ( P - b i g d e l t a ) e f f e c t s .

1 2

Similarly,frames have a theoretical elastic critical buckling load, whichcouldonly be reached ifhe frame has an infinitely igh strength. Th is will ereferred o in this document asV,,.

in-plane stability of portal frames

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