USF Libraries
USF Digital Collections

A study of the critical condition of a battened column and a frame by classical methods

MISSING IMAGE

Material Information

Title:
A study of the critical condition of a battened column and a frame by classical methods
Physical Description:
Book
Language:
English
Creator:
Bekdache, Jamal A.H
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
Publication Date:

Subjects

Subjects / Keywords:
bending
shear
compression
buckling
deflection
Dissertations, Academic -- Civil Engineering -- Masters -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
ABSTRACT: Knowledge of structural stability theory is of paramount importance to the practicing structural engineer. In many instances, buckling is the primary consideration in the design of various structural configurations. The first chapter introduces a simplified treatment of the elastic stability of a battened column using classical methods without getting involved with lengthy and complicated mathematical operations. In chapter two, a treatment of the elastic stability of a frame is presented, including effects of elastic restraints. In this study, a theoretical treatment is given which although approximate, is believed to constitute a satisfactory solution of the structure.
Thesis:
Thesis (M.S.C.E.)--University of South Florida, 2003.
Bibliography:
Includes bibliographical references.
System Details:
System requirements: World Wide Web browser and PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Jamal A.H. Bekdache.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 29 pages.

Record Information

Source Institution:
University of South Florida Library
Holding Location:
University of South Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001430597
oclc - 53171155
notis - AJL4058
usfldc doi - E14-SFE0000118
usfldc handle - e14.118
System ID:
SFS0024814:00001


This item is only available as the following downloads:


Full Text
xml version 1.0 encoding UTF-8 standalone no
record xmlns http:www.loc.govMARC21slim xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.loc.govstandardsmarcxmlschemaMARC21slim.xsd
leader nam Ka
controlfield tag 001 001430597
003 fts
006 m||||e|||d||||||||
007 cr mnu|||uuuuu
008 031007s2003 flua sbm s000|0 eng d
datafield ind1 8 ind2 024
subfield code a E14-SFE0000118
035
(OCoLC)53171155
9
AJL4058
b SE
SFE0000118
040
FHM
c FHM
049
FHME
090
TA145
1 100
Bekdache, Jamal A.H.
2 245
A study of the critical condition of a battened column and a frame by classical methods
h [electronic resource] /
by Jamal A.H. Bekdache.
260
[Tampa, Fla.] :
University of South Florida,
2003.
502
Thesis (M.S.C.E.)--University of South Florida, 2003.
504
Includes bibliographical references.
516
Text (Electronic thesis) in PDF format.
538
System requirements: World Wide Web browser and PDF reader.
Mode of access: World Wide Web.
500
Title from PDF of title page.
Document formatted into pages; contains 29 pages.
3 520
ABSTRACT: Knowledge of structural stability theory is of paramount importance to the practicing structural engineer. In many instances, buckling is the primary consideration in the design of various structural configurations. The first chapter introduces a simplified treatment of the elastic stability of a battened column using classical methods without getting involved with lengthy and complicated mathematical operations. In chapter two, a treatment of the elastic stability of a frame is presented, including effects of elastic restraints. In this study, a theoretical treatment is given which although approximate, is believed to constitute a satisfactory solution of the structure.
590
Adviser: Carpenter, William
653
bending.
shear.
compression.
buckling.
deflection.
0 690
Dissertations, Academic
z USF
x Civil Engineering
Masters.
773
t USF Electronic Theses and Dissertations.
4 856
u http://digital.lib.usf.edu/?e14.118



PAGE 1

A S t u d y o f t h e C r i t i c a l C o n d i t i o n o f a B a t t e n e d C o l u m n a n d a F r a m e b y C l a s s i c a l M e t h o d s b y J a m a l A H B e k d a c h e A t h e s i s s u b m i t t e d i n p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e m e n t s f o r t h e d e g r e e o f M a s t e r o f S c i e n c e i n C i v i l E n g i n e e r i n g D e p a r t m e n t o f C i v i l & E n v i r o n m e n t a l E n g i n e e r i n g C o l l e g e o f E n g i n e e r i n g U n i v e r s i t y o f S o u t h F l o r i d a M a j o r P r o f e s s o r : W i l l i a m C a r p e n t e r P h D S t a n l e y K r a n c P h D R a j a n S e n P h D D a t e o f A p p r o v a l : J u l y 8 2 0 0 3 K e y w o r d s : b u c k l i n g c o m p r e s s i o n s h e a r b e n d i n g d e f l e c t i o n C o p y r i g h t 2 0 0 3 J a m a l A H B e k d a c h e

PAGE 2

i T a b l e o f C o n t e n t s L i s t o f T a b l e s i i L i s t o f F i g u r e s i i i A b s t r a c t i v C h a p t e r O n e : C l a s s i c a l S o l u t i o n o f t h e B a t t e n e d C o l u m n 1 I n t r o d u c t i o n 1 A n a l y s i s 2 C h a p t e r T w o : B u c k l i n g o f C l a m p e d F r a m e 8 P u r p o s e o f S t u d y 8 C a s e S t u d y 8 A n a l y s i s 1 0 C h a p t e r T h r e e : F i n d i n g s 1 4 D e s c r i p t i o n o f F i n d i n g s 1 4 R e f e r e n c e s 2 4

PAGE 3

i i L i s t o f T a b l e s T a b l e 1 C r i t i c a l p a r a m e t e r s c o l u m n A B 1 7 T a b l e 2 C r i t i c a l p a r a m e t e r s c o l u m n C D 1 7 T a b l e 3 C r i t i c a l l o a d f o r f r a m e 1 7

PAGE 4

i i i L i s t o f F i g u r e s F i g u r e 1 B a t t e n e d c o l u m n 1 F i g u r e 2 U n i t e l e m e n t o f w e b 2 F i g u r e 3 B e n d i n g m o m e n t s o n c o l u m n 3 F i g u r e 4 D i s p l a c e d c o l u m n 3 F i g u r e 5 P / Q v / s s t i f f n e s s r a t i o : L / b = 4 6 F i g u r e 6 B e n d i n g m o m e n t f o r v a r i o u s s t i f f n e s s r a t i o s 7 F i g u r e 7 F r a m e u n d e r l o a d s PA a n d PC 8 F i g u r e 8 B u c k l i n g o f f r a m e 9 F i g u r e 9 B a r u n d e r c o m p r e s s i v e f o r c e s 1 0 F i g u r e 1 0 B e n d i n g o f t o p h a l f o f c o l u m n A B 1 0 F i g u r e 1 1 B e n d i n g o f b o t t o m h a l f o f c o l u m n A B 1 1 F i g u r e 1 2 R e a c t i v e s h e a r a t m i d s p a n 1 3 F i g u r e 1 3 C r i t i c a l c o n d i t i o n s f o r c o l u m n A B 1 5 F i g u r e 1 4 C r i t i c a l c o n d i t i o n s f o r c o l u m n C D 1 5 F i g u r e 1 5 T o t a l c r i t i c a l l o a d 1 6 F i g u r e 1 6 F i r s t m o d e c r i t i c a l p a r a m e t e r v / s s t i f f n e s s r a t i o f o r c o l u m n A B 1 8 F i g u r e 1 7 C r i t i c a l p a r a m e t e r f o r c o l u m n A B f i r s t m o d e s h a p e 1 9 F i g u r e 1 8 C r i t i c a l l o a d f o r c o l u m n A B f i r s t m o d e s h a p e 1 9 F i g u r e 1 9 C r i t i c a l p a r a m e t e r f o r c o l u m n C D f i r s t m o d e s h a p e 2 0 F i g u r e 2 0 C r i t i c a l l o a d f o r c o l u m n C D f i r s t m o d e s h a p e 2 0 F i g u r e 2 1 T o t a l c r i t i c a l l o a d f i r s t m o d e s h a p e 2 1 F i g u r e 2 2 D e f l e c t e d f r a m e u n d e r c r i t i c a l l o a d s f i r s t m o d e s h a p e 2 2 F i g u r e 2 3 D e f l e c t e d f r a m e u n d e r c r i t i c a l l o a d s s e c o n d m o d e s h a p e 2 3

PAGE 5

i v A S t u d y o f t h e C r i t i c a l C o n d i t i o n o f a B a t t e n e d C o l u m n a n d a F r a m e b y C l a s s i c a l M e t h o d s J a m a l A H B e k d a c h e A B S T R A C T K n o w l e d g e o f s t r u c t u r a l s t a b i l i t y t h e o r y i s o f p a r a m o u n t i m p o r t a n c e t o t h e p r a c t i c i n g s t r u c t u r a l e n g i n e e r I n m a n y i n s t a n c e s b u c k l i n g i s t h e p r i m a r y c o n s i d e r a t i o n i n t h e d e s i g n o f v a r i o u s s t r u c t u r a l c o n f i g u r a t i o n s T h e f i r s t c h a p t e r i n t r o d u c e s a s i m p l i f i e d t r e a t m e n t o f t h e e l a s t i c s t a b i l i t y o f a b a t t e n e d c o l u m n u s i n g c l a s s i c a l m e t h o d s w i t h o u t g e t t i n g i n v o l v e d w i t h l e n g t h y a n d c o m p l i c a t e d m a t h e m a t i c a l o p e r a t i o n s I n c h a p t e r t w o a t r e a t m e n t o f t h e e l a s t i c s t a b i l i t y o f a f r a m e i s p r e s e n t e d i n c l u d i n g e f f e c t s o f e l a s t i c r e s t r a i n t s I n t h i s s t u d y a t h e o r e t i c a l t r e a t m e n t i s g i v e n w h i c h a l t h o u g h a p p r o x i m a t e i s b e l i e v e d t o c o n s t i t u t e a s a t i s f a c t o r y s o l u t i o n o f t h e s t r u c t u r e

PAGE 6

1 F i g u r e 1 B a t t e n e d c o l u m n C h a p t e r O n e : C l a s s i c a l S o l u t i o n o f t h e B a t t e n e d C o l u m n I n t r o d u c t i o n O n e o f t h e c o m m o n e s t o p e n p a n e l s t r u c t u r e s i s t h e b a t t e n e d c o l u m n t h a t c o n s i s t s o f t w o f l a n g e s c o n n e c t e d b y e v e n l y s p a c e d f l a t b a t t e n p l a t e s F i g 1 s h o w s d i a g r a m m a t i c a l l y t h e c o l u m n t o b e s t u d i e d T w o e q u a l f l a n g e m e m b e r s a r e c o n n e c t e d b y f l a t b a t t e n p l a t e s i n s u c h a w a y t h a t t h e j o i n t s a r e r i g i d i e t h e r e w i l l b e n o r e l a t i v e m o v e m e n t s b e t w e e n p l a t e s a n d f l a n g e s a t t h e i r j u n c t i o n s I t w i l l b e a s s u m e d t h a t t h e g e n e r a l d e s i g n i s s u c h a s t o e l i m i n a t e t h e p o s s i b i l i t y o f a n y s e c o n d a r y f a i l u r e t h e b a t t e n p l a t e s m i g h t f a i l i n b e n d i n g i f t h e y w e r e m a d e t h i n o r i n s u f f i c i e n t l y w i d e I t w i l l a l s o b e a s s u m e d t h a t t h e d i s t a n c e b e t w e e n b a t t e n p l a t e s i s m u c h s m a l l e r t h a n t h e l e n g t h o f t h e c o l u m n a n d t h a t t h e f l a n g e s i n s t e a d o f b e i n g j o i n e d b y a f i n i t e n u m b e r o f b a t t e n p l a t e s a r e c o n n e c t e d b y a w e b m e d i u m w h i c h c a n a p p l y f l e x u r a l r e s t r a i n t t o t h e f l a n g e s T h e f l e x u r a l r i g i d i t y o f t h i s w e b w i l l b e t a k e n a s I E p e r u n i t l e n g t h o f c o l u m n L / 2 X Y a L / 2 P P P P b d L

PAGE 7

2 A n a l y s i s L e t P b e t h e c r i t i c a l a x i a l c o m p r e s s i o n o n t h e p i n e n d e d c o l u m n i e t h e c o m p r e s s i o n w h i c h w o u l d p r o d u c e i n s t a b i l i t y o f t h e c o l u m n a s a w h o l e A t h e a r e a o f e a c h f l a n g e I t h e s e c o n d m o m e n t o f a r e a o f e a c h f l a n g e I c t h e s e c o n d m o m e n t o f a r e a o f b a t t e n p l a t e s I¢, t h e s e c o n d m o m e n t o f a r e a o f t h e w e b m e d i u m a b o u t t h e a x i s o f b e n d i n g p e r u n i t l e n g t h L t h e l e n g t h o f t h e c o l u m n b e t w e e n p i n s b t h e l e n g t h o f t h e b a t t e n p l a t e s d t h e d i s t a n c e b e t w e e n b a t t e n p l a t e s n t h e n u m b e r o f b a t t e n p l a t e s n¢, t h e s t i f f n e s s r a t i o d L E I I EC C Q t h e c r i t i c a l l o a d f o r o n e f l a n g e 2 2 L E Ip = W h e n f l e x u r e o c c u r s t h e c o n d i t i o n s a t o n e o f t h e b a t t e n p l a t e s a r e s h o w n i n F i g 2 T h e p l a t e i s c o n s i d e r e d t o b e a n e l e m e n t o f t h e w e b o f l e n g t h x d W h e n t h e c o l u m n i s d e f l e c t e d f r o m i t s i n i t i a l p o s i t i o n t h e e l e m e n t i s d i s t o r t e d a s s h o w n a n d a p p l i e s c o u p l e s o f m a g n i t u d e x m d t o e a c h f l a n g e a n d a l s o r e a c t i v e f o r c e s b x m 2 d T h e r e a c t i o n s i n c r e a s e t h e l e n g t h o f t h e l e f t f l a n g e a n d d e c r e a s e t h e l e n g t h o f t h e r i g h t f l a n g e T h e s l o p e o f t h e f l a n g e s a t a n y p o i n t i s t h e r e f o r e t h e s u m o f t w o c o m p o n e n t s ; f c a u s e d b y t h e s e l o n g i t u d i n a l s t r a i n s a n d q d u e t o t h e f l e x u r e o f t h e b a t t e n p l a t e s I t w i l l b e a s s u m e d t h a t t h e c o n t r i b u t i o n o f f i s v e r y s m a l l c o m p a r e d w i t h t h a t d u e t o t h e c r o s s b e a m s a n d i t w i l l t h e r e f o r e b e n e g l e c t e d T o d e t e r m i n e q l e t Q R i n F i g 2 r e p r e s e n t a u n i t e l e m e n t o f t h e w e b c a r r y i n g c o u p l e s m a t e a c h e n d a n d e q u i l i b r a t i n g f o r c e s b m 2 a s s h o w n A t a d i s t a n c e x f r o m Q F i g u r e 2 U n i t e l e m e n t o f w e b

PAGE 8

3 b m x 2 m d x y d I E2 2 C C+ = A b m x m x d x d y I E2 C C+ + = B A x b 3 m x 2 m x y I E3 2 C C+ + + = T h e s e c o n d c o n s t a n t o f i n t e g r a t i o n B b e i n g 0 s i n c e y i s 0 a t Q F r o m t h e s y m m e t r y o f t h e l o a d i n g t h e r e c a n b e n o d e f l e c t i o n a t t h e m i d p o i n t o f t h e e l e m e n t i e y = 0 w h e n x = b / 2 a n d s o 6 m b A=. T h e r e f o r e w h e n x = 0 t h e s l o p e a t t h e e n d o f t h e e l e m e n t i s C CI E 6 m b d x d y = T h i s s l o p e i s t h e s a m e a s t h a t o f t h e f l a n g e s a t Q a n d R I f b e n d i n g m o m e n t s r e q u i r e d t o p r o d u c e c u r v a t u r e a s s h o w n i n F i g 3 a r e t a k e n t o b e p o s i t i v e t h e s l o p e a t a n y p o i n t d x d y i s p o s i t i v e A s s u m i n g t h a t L d £ a n d i f t h e r e a r e n e q u a l b a t t e n p l a t e s i n a l e n g t h L o f t h e s t r u c t u r e t h e t o t a l f l e x u r a l r i g i d i t y i s C CI n E a n d t h e f l e x u r a l r i g i d i t y o f t h e m e d i u m p e r u n i t l e n g t h i s L / I E nC C, w h i c h w i l l b e d e n o t e d b y E ¢ I ¢ A n d s o i f t h e e q u a t i o n f o r t h e s l o p e d e r i v e d f o r a b a t t e n p l a t e b e r e w r i t t e n i n t e r m s o f t h e c o n t i n u o u s m e d i u m w e h a v e f o r e i t h e r f l a n g e I E 6 E I m b d x d y E I ¢ ¢ = . . . . . . . . . . . . . . . ( 1 ) F i g u r e 3 B e n d i n g m o m e n t s o n c o l u m n F i g u r e 4 D i s p l a c e d c o l u m n

PAGE 9

4 I f t h e c o l u m n i s d i s p l a c e d f r o m i t s n o r m a l p o s i t i o n w h e n c a r r y i n g a l o a d l e s s t h a n t h e c r i t i c a l v a l u e i t w i l l r e c o v e r i t s s t r a i g h t n e s s w h e n t h e d i s p l a c i n g f o r c e i s r e m o v e d W h e n i t c a r r i e s t h e c r i t i c a l l o a d i t w i l l r e m a i n i n i t s d i s p l a c e d p o s i t i o n F i g 4 s h o w s t h e c o l u m n i n t h i s s t a t e h a v i n g b e e n d i s p l a c e d a n a r b i t r a r y a m o u n t a a t t h e c e n t e r S i n c e t h e f l a n g e s a r e a s s u m e d t o b e s i m i l a r t h e l o a d i s e v e n l y d i v i d e d b e t w e e n t h e m a n d e a c h c a r r i e s P / 2 I f a n o r i g i n b e t a k e n a t t h e c e n t e r o f t h e d i s p l a c e d c o l u m n a n d x a n d y m e a s u r e d i n t h e d i r e c t i o n s h o w n t h e n t h e m o m e n t i n e a c h f l a n g e a t x f r o m t h e o r i g i n d u e t o t h e w e b r e a c t i o n s a n d t o t h e a p p l i e d l o a d P / 2 i s =2 L / x a y 2 L / x 2 2d x d x d y d x b m 2 d x m y ) ( a 2 P d x y d E I F r o m ( 1 ) b I E 6 d x d y m ¢ ¢ = t h e r e f o r e t h e t h i r d t e r m o n r i g h t h a n d s i d e o f t h e a b o v e e q u a t i o n c a n b e n e g l e c t e d a s b e i n g a s m a l l q u a n t i t y o f t h e s e c o n d o r d e r s o =2 L / x 2 2x d m y ) ( a 2 P d x y d E I a n d m d x d y 2 P d x y d E I3 3+ = h e n c e d x d y 2 P d x y d E I m3 3+ = . . . . . . . . . . . . . . . . . ( 2 ) F r o m ( 1 ) d x d y b I E 6 m ¢ ¢ = T h i s o n s u b s t i t u t i o n i n ( 2 ) g i v e s 0 d x d y c d x y d1 3 3= + W h e r e 2 / 1 1E I 6 I E 6 E I 2 P ¢ ¢ = c T h e a p p r o p r i a t e s o l u t i o n o f t h e e q u a t i o n t a k i n g a c c o u n t o f s y m m e t r y i s B x c c o s A y1+ = . . . . . . . . . . . . . . . . . ( 3 ) W h e n x = 0 y = 0 ; a n d w h e n x = L / 2 y = a U s i n g t h e s e c o n d i t i o n s t h e c o n s t a n t s a r e

PAGE 10

5 n r = 1 ) 2 / L c ( c o s a A1 n r = 1 2 ) L / ( c c o s a1B A l s o 0 d x y d2 2= w h e n x = L / 2 a n d i f t h i s c o n d i t i o n i s u s e d i n e q u a t i o n ( 3 ) 0 ) 2 / L c ( c o s A c1 2 1= o r 0 2 ) L / c o s ( c 1 2 ) L / ( c c o s c a1 1 2 1= n r T h i s m u s t b e s a t i s f i e d f o r a n y v a l u e o f a w h i c h i s a n a r b i t r a r y d i s p l a c e m e n t a n d s o c o s ( c1L / 2 ) = 0 H e n c e t h e c o n d i t i o n t h a t d e t e r m i n e s t h e v a l u e o f t h e c r i t i c a l l o a d P i s c1L = p . . . . . . . . . . . . . . . . . ( 4 ) s u b s t i t u t i o n o f t h i s i n t h e c o n s t a n t s o f i n t e g r a t i o n g i v e s A = – a a n d B = a a n d t h e d e f l e c t e d f o r m o f t h e c o l u m n i s f r o m ( 3 ) ) L x c o s 1 ( a y p = . . . . . . . . . . . . . . . . . ( 5 ) T h e n f r o m ( 2 ) L x s i n ) Q 2 P ( L a m p p = . . . . . . . . . . . . . . . . . ( 6 ) a n d f r o m ( 4 ) 2 2 2 1 L c = S u b s t i t u t i o n o f t h e v a l u e o f c1 i n t h e a b o v e e x p r e s s i o n y i e l d s 2 2 L b E I I E 6 E I 2 P = ¢ ¢ o r b I E 1 2 L E I 2 P2 2¢ ¢ + p = . . . . . . . . . . . . . . . . . . ( 7 )

PAGE 11

6 a n d E I b L I E 1 2 2 Q P2 2¢ ¢ + = . . . . . . . . . . . . . . . . . . ( 8 ) b y s u b s t i t u t i o n a b o v e e x p r e s s i o n c a n a l s o b e r e w r i t t e n a s : E I b L I n E 1 2 2 Q P2 C Cp + = o r b L n 1 2 2 Q P2p ¢ + = w h e r e E I I E n nC C= ¢ F i g 5 i s a p l o t o f Q P V / S n ¢ w i t h b L t a k e n e q u a l t o 4 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 0 5 1 0 1 5 n 'P / QL / b = 4 F i g u r e 5 P / Q v / s s t i f f n e s s r a t i o n ; L / b = 4

PAGE 12

7 0 5 1 0 1 5 2 0 2 5 3 0 0 0 1 0 2 0 3 0 4 0 5 x / L m ( L3/ p3E I a ) # i n / i nn = 2 n = 5 n = 8 n = 1 0 F i g u r e 6 B e n d i n g m o m e n t f o r v a r i o u s s t i f f n e s s r a t i o s F i g u r e 6 i s t h e p l o t o f t h e c o u p l e m a t d i f f e r e n t l o c a t i o n s a l o n g t h e c o l u m n a n d f o r v a r i o u s v a l u e s o f t h e s t i f f n e s s r a t i o n

PAGE 13

8 C h a p t e r T w o : B u c k l i n g o f C l a m p e d F r a m e P u r p o s e o f S t u d y F r a m e s o f v a r i o u s t y p e s a r e u s e d i n s t r u c t u r a l c o n f i g u r a t i o n s s u c h a s b u i l d i n g s a n d b r i d g e s T h e s e f r a m e s a r e s u b j e c t e d t o c o n c e n t r a t e d a n d d i s t r i b u t e d l o a d s w h i c h i n m a n y c a s e s m a y c a u s e b u c k l i n g o f a n e l e m e n t o r g r o u p o f e l e m e n t s o f t h e f r a m e B e c a u s e t h e m e m b e r s a r e r i g i d l y c o n n e c t e d t o o t h e r m e m b e r s f l e x u r a l d e f o r m a t i o n s i n o n e e l e m e n t c a u s e d e f o r m a t i o n s i n t h e n e i g h b o r i n g e l e m e n t s K n o w l e d g e o f t h e c r i t i c a l c o n d i t i o n i s e s s e n t i a l i n t h e d e s i g n o f b o t h s i m p l e a n d c o m p l e x f r a m e s N e g l e c t i n g l o n g i t u d i n a l s t r a i n s t h e f o r e g o i n g a n a l y s i s p r e s e n t s a s i m p l i f i e d m e t h o d t h a t c a n b e s u c c e s s f u l l y u s e d t o a r r i v e a t t h e c r i t i c a l c o n d i t i o n a n d t h e e q u a t i o n o f t h e c o r r e s p o n d i n g d e f l e c t e d s h a p e C a s e S t u d y L e t u s c o n s i d e r t h e f r a m e s h o w n i n F i g 7 W e a r e i n t e r e s t e d i n f i n d i n g t h e s m a l l e s t p o s s i b l e l o a d s PA a n d PC t h a t w i l l c a u s e b o t h c o l u m n s A B a n d C D o f t h e f r a m e t o b u c k l e T o a c c o m p l i s h t h i s w e m u s t c o n s i d e r a l l p o s s i b l e m o d e s o f b u c k l i n g a n d e s t a b l i s h t h r o u g h a c o m p a r i s o n t h e c o r r e s p o n d i n g b u c k l i n g m o d e T h e f r a m e u n d e r s t u d y i s s y m m e t r i c a n d t h e d i f f e r e n t b u c k l i n g m o d e s a r e s h o w n i n f i g 8 N o t i n g t h a t t h e r e i s n o p o s s i b i l i t y o f a s w a y b u c k l i n g m o d e a t t h e m i d s p a n w h e n t h e h o r i z o n t a l b a r b u c k l e s s y m m e t r i c a l l y o n l y a n t i s y m m e t r i c s w a y b u c k l i n g m o d e s w i l l b e f u r t h e r d i s c u s s e d F i g u r e 7 F r a m e u n d e r l o a d s PA a n d PC

PAGE 14

9 F i g u r e 8 B u c k l i n g o f f r a m e ( a ) s y m m e t r i c a l – n o s w a y ( b ) a n t i s y m m e t r i c a l – n o s w a y ( c ) a n t i s y m m e t r i c a l – s w a y a l l o w e d

PAGE 15

1 0 F i g u r e 1 0 B e n d i n g o f t o p h a l f o f c o l u m n A B A n a l y s i s W h e n a b a r s u c h a s A B i n f i g 9 i s i n i t i a l l y s t r a i g h t a n d o f p e r f e c t g e o m e t r y a n d i t i s s u b j e c t e d t o t h e a c t i o n o f a c o m p r e s s i v e f o r c e w i t h o u t e c c e n t r i c i t y t h e c o l u m n i s c o m p r e s s e d b u t r e m a i n s s t r a i g h t W e t h e n n e e d t o k n o w i f t h e c o l u m n w i l l r e m a i n s t r a i g h t n o m a t t e r w h a t t h e l e v e l o f t h e a p p l i e d f o r c e i s T o d e t e r m i n e t h i s w e s e e k n o n t r i v i a l s o l u t i o n s ( w 0 ) f o r t h e e q u a t i o n s g o v e r n i n g t h e b e n d i n g o f t h i s c o l u m n u n d e r a n a x i a l c o m p r e s s i v e l o a d P a n d s u b j e c t t o t h e s i n g u l a r i t i e s a t m i d s p a n a n d t o t h e p a r t i c u l a r s e t o f b o u n d a r y c o n d i t i o n s w h e r e : E I t h e b e n d i n g s t i f f n e s s o f e a c h c o l u m n E¢l¢, t h e b e n d i n g s t i f f n e s s o f t h e h o r i z o n t a l b e a m L t h e l e n g t h o f e a c h c o l u m n b t h e l e n g t h o f t h e h o r i z o n t a l b e a m R t h e s t i f f n e s s r a t i o 2 2 b L E I I E¢ ¢ = PA c r, t h e c r i t i c a l l o a d f o r c o l u m n A B PC c r, t h e c r i t i c a l l o a d f o r c o l u m n C D Pc r, t h e t o t a l c r i t i c a l l o a d f o r t h e f r a m e N o t e t h a t i n d e r i v i n g t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n s i t w a s a s s u m e d t h a t t h e a p p l i e d c o m p r e s s i v e l o a d s r e m a i n e d p a r a l l e l t o t h e i r o r i g i n a l d i r e c t i o n a n d t h e r e w a s n o e c c e n t r i c i t y i n e i t h e r t h e g e o m e t r y o r t h e a p p l i e d l o a d s T h e m a t h e m a t i c a l f o r m u t a t i o n o f t h i s p r o b l e m i s g i v e n b e l o w T h e e q u a t i o n g o v e r n i n g t h e b e n d i n g o f t h e t o p h a l f o f c o l u m n A B ( S e e F i g 1 0 ) i s g i v e n b y D E 1 A 1y P y E I = ¢ ¢ ) 2 / L x 0 ( £ £ B C 0 ) 0 ( y1= ¢ ¢ y1 ( 0 ) = 0 A s s u m i n g t h a t t h e b e n d i n g s t i f f n e s s E I o f t h e c o l u m n i s c o n s t a n t a n d i n t r o d u c i n g t h e p a r a m e t e r E I / P KA 2 1= a l l o w s u s t o w r i t e t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n i n F i g u r e 9 B a r u n d e r c o m p r e s s i v e f o r c e s

PAGE 16

1 1 t h e f o l l o w i n g f o r m : 0 y K y1 2 1 1= + ¢ ¢ T h e g e n e r a l s o l u t i o n o f t h i s e q u a t i o n i s g i v e n b y y1 = A1s i n K1x + B1c o s K1x T h i s s o l u t i o n m u s t s a t i s f y t h e p r e s c r i b e d b o u n d a r y c o n d i t i o n s T h i s r e q u i r e m e n t l e a d s t o t w o l i n e a r h o m o g e n e o u s a l g e b r a i c e q u a t i o n s i n t h e t w o c o n s t a n t s A1 a n d B1 0 ) 0 c o s ( K B ) 0 s i n ( K A2 1 1 2 1 1= A1s i n ( 0 ) + B1c o s ( 0 ) = 0 o r B1 = 0 a n d y1( x ) = A1s i n K1x T h e m a t h e m a t i c a l f o r m u l a t i o n o f t h e b o t t o m h a l f o f t h e c o l u m n i s g i v e n b e l o w ( s e e F i g 1 1 ) D E ) ] 2 / L ( y y [ s m y P y E I2 2 2 A 2+ + = ¢ ¢ L x 2 / L £ £ B C 0 ) L ( y2= ¢ i n t r o d u c i n g t h e p a r a m e t e r E I s P KA 22= a l l o w s u s t o w r i t e t h e g o v e r n i n g d i f f e r e n t i a l e q u a t i o n i n t h e f o l l o w i n g f o r m E I ) 2 / L ( y s m y K y2 2 22 2= + ¢ ¢ w h e r e ) 2 / L ( y b I E 6 m2¢ ¢ ¢ = a n d b m 2 s= o r ) 2 / L ( y b I E 1 2 s2 2¢ ¢ ¢ = T h e g e n e r a l s o l u t i o n o f t h i s e q u a t i o n i s g i v e n b y 22 2 2 2 2 2 2E I K ) 2 / L ( y s m x K c o s B x K s i n A ) x ( y + + = T h i s s o l u t i o n m u s t s a t i s f y c o m p a t i b i l i t y a t m i d s p a n i e F i g u r e 1 1 B e n d i n g o f b o t t o m h a l f o f c o l u m n A B

PAGE 17

12 EI m )2/L(y)2/L(y2 1= )2/L(y)2/L(y2 1 = and )2/L(y)2/L(y2 1= The first two conditions above lead to two linear homogenous algebraic equations in the two constants A2 and B2 EI m 2/LKcosKB2/LKsinKA2/LKsinKA2 2 22 2 2 22 1 2 11-= and A1K1 cos K1L/2 = A2K2 cos K2L/2 B2K2 sin K2L/2 Solution of the above equations yields 2 2 2 2 1 2 1 2 1 2 1 12EIK 2/LKsinm 2/LKcos2/LKcos2/LKsin2/LKsin K K K K AA + = 2 2 2 2 1 2 1 2 1 2 1 12EIK 2/LKcosm 2/LKsin2/LKcos2/LKcos2/LKsin K K K K AB = hence, xKsin EIK 2/LKsinm 2/LKcos2/LKcos2/LKsin2/LKsin K K K K A)x(y2 2 2 2 2 1 2 1 2 1 2 1 1 2b t n f r + = xKcos EIK 2/LKcosm 2/LKsin2/LKcos2/LKcos2/LKsin K K K K A2 2 2 2 2 1 2 1 2 1 2 1 1b t n f r + 2 2 2EIK )2/L(ysm + Where )2/L(y b IE6 )2/L(y b IE6 m1 2 = = )2/L(y b IE12 )2/L(y b IE12 s1 2 2 2 = = y2 (L/2) = y1 (L/2) Substitution yields: f r + = 2/LKcos2/LKcos2/LKsin2/LKsin K K K K A)x(y2 1 2 1 2 1 2 1 1 2

PAGE 18

13 Figure 12. Reactive shear at midspan xKsin2/LKsin2/LKcos EIKb KIE62 2 1 2 2 1 + 2/LKsin2/LKcos2/LKcos2/LKsin K K K K2 1 2 1 2 1 2 1 xKcos2/LKcos2/LKcos EIKb KIE62 2 1 2 2 1 b t n+ )2/LKsinA b 2 1(2/LKcos EIKb KIE611 1 2 2 1 The necessary boundary condition at the fixed-end leads to the characteristic equation: 2/LKcos2/LKcos K K 2/LKsin2/LKsin K K 02 1 2 1 2 1 2 2 2 1+ -= b t n + 2/LKsin2/LKcos EIKb KIE62 1 2 2 1 but EI sP KA 2 2= and EI P KA 2 1= \ 2/1 1 1 2 1 2 12)2/LKcosA EIb KIE12 K(K -= This upon substitution in the above characteristic equation yields the critical parameter K1 and hence the critical load PA for column AB. Identical work was done for member CD of the structure noting however that the reactive shear s assumes an opposite direction in this case (See Fig. 12).

PAGE 19

1 4 C h a p t e r T h r e e : F i n d i n g s D e s c r i p t i o n o f F i n d i n g s S o l u t i o n o f t h e c h a r a c t e r i s t i c e q u a t i o n f o r c o l u m n A B i s p l o t t e d i n F i g 1 3 A n a r b i t r a r y d e f l e c t i o n a t m i d s p a n r e l a t i v e t o t h e f r e e e n d a t p o i n t A w a s t a k e n e q u a l t o 0 1 L T h e s t i f f n e s s r a t i o 2 2 b L E I I E R¢ ¢ = w a s t a k e n e q u a l t o 1 0 F i g 1 4 i s t h e s o l u t i o n o f t h e c h a r a c t e r i s t i c e q u a t i o n f o r c o l u m n C D c o n s i d e r i n g t h a t a t m i d s p a n a n d d u e t o t h e b u c k l i n g s y m m e t r y o f t h e h o r i z o n t a l b a r b o t h m e m b e r s A B a n d C D h a v e e q u a l s l o p e s U p o n s u b s t i t u t i o n w h e r e Pc r = ( K1L )2 E I / L2, v a l u e s o f b o t h c r i t i c a l l o a d s PA c r a n d PC c r f o r c o l u m n s A B a n d C D r e s p e c t i v e l y c a n b e d e r i v e d F i g 1 5 i s a p l o t o f t h e c h a r a c t e r i s t i c e q u a t i o n s o l u t i o n v / s t h e t o t a l c r i t i c a l l o a d Pc r = PA c r + PC c r. T h e c r i t i c a l v a l u e s f o r t h e f i r s t a n d s e c o n d m o d e s h a p e s a r e s u m m a r i z e d i n T a b l e 1 F i g 1 6 p l o t s t h e v a l u e s o f t h e c r i t i c a l p a r a m e t e r K1 f o r c o l u m n A B a g a i n s t t h e s t i f f n e s s r a t i o R W e c a n s e e t h a t w h e n R = 0 t h e n K1 a s s u m e s t h e v a l u e p/ 2 t h e s a m e a s f o r a c o l u m n w i t h a f i x e d e n d a t s u p p o r t I n a d d i t i o n w h e n R a p p r o a c h e s i n f i n i t y t h e h o r i z o n t a l b a r a c t s l i k e a f i x e d s u p p o r t a t m i d s p a n a n d K1 = p. F i g 1 7 a n d 1 8 a r e f i r s t m o d e p l o t s o f t h e c r i t i c a l p a r a m e t e r K1 a n d c r i t i c a l l o a d PA c r f o r c o l u m n A B v / s s t i f f n e s s r a t i o R A l o g a r i t h m i c s c a l e w a s u s e d t o e n a b l e u s c o v e r a w i d e r a n g e o f t h e s t i f f n e s s r a t i o S a m e i s p l o t t e d i n F i g 1 9 a n d 2 0 f o r c o l u m n C D V a l u e s o f t h e t o t a l c r i t i c a l l o a d f o r t h e F r a m e i n t h e f i r s t m o d e s h a p e Pc r = PA c r + PC c r, a r e p l o t t e d a g a i n s t t h e s t i f f n e s s r a t i o R i n F i g 2 1 a l s o u s i n g a l o g a r i t h m i c s c a l e F i g 2 2 a n d 2 3 s h o w t h e d e f l e c t e d s h a p e o f t h e s t r u c t u r e f o r t h e f i r s t a n d s e c o n d m o d e s s h a p e s w i t h R t a k e n e q u a l t o 1 0

PAGE 20

1 5 3 0 0 2 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 0 1 2 3 4 5 6 7 8 9 1 0s l o p e X = L F i g u r e 1 3 C r i t i c a l c o d i t i o n s f o r c o l u m n A B K 1 L 3 0 0 2 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0s l o p e X = L F i g u r e 1 4 C r i t i c a l c o n d i t i o n s f o r c o l u m n C D k1L

PAGE 21

1 6 3 0 0 2 0 0 1 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0PC R L2/ E I = ( PA C R+ PC C R) L2/ E Is l o p e X = L F i g u r e 1 5 T o t a l c r i t i c a l l o a d

PAGE 22

1 7 R M O D E K1L K2L C R I T I C A L L O A D R E A C T I O N E R R O R S H A P E PA C R L2/ E I A T S U P P O R T 0 1 1 5 7 1 3 8 3 1 5 7 0 7 8 3 2 4 6 9 2 4 3 1 6 6 2 4 6 7 3 5 8 6 1 1 5 4 4 E 0 9 0 2 4 7 1 2 1 8 5 4 7 1 2 7 8 4 2 2 2 0 4 6 8 2 9 6 2 2 2 1 0 3 3 6 4 2 3 8 1 E 0 9 1 0 1 2 8 0 6 9 8 3 1 4 8 0 0 4 2 7 8 7 9 1 5 5 9 7 6 2 1 9 0 5 2 4 8 8 4 1 0 6 E 0 7 1 0 2 3 9 0 0 8 1 1 5 8 2 1 6 4 2 1 5 2 1 6 3 2 6 8 5 3 3 8 9 1 5 2 1 1 8 7 6 8 E 0 8 1 0 0 0 1 3 1 3 7 4 1 4 1 4 0 6 1 9 1 9 8 4 3 3 6 7 0 0 3 1 9 7 7 3 7 3 2 8 6 8 5 4 E 0 8 1 0 0 0 2 3 1 5 7 1 2 8 6 2 7 6 6 0 1 9 9 6 7 4 5 5 2 9 5 3 9 3 9 5 7 1 6 6 4 1 7 1 E 0 7 R M O D E K1L K2L C R I T I C A L L O A D R E A C T I O N E R R O R S H A P E PC C R L2/ E I A T S U P P O R T 0 1 1 5 7 1 1 6 5 1 5 7 1 7 6 4 2 4 6 8 5 5 8 5 4 6 2 4 7 0 4 4 3 1 9 9 4 E 0 9 0 2 1 5 7 1 6 1 5 6 9 8 0 1 2 4 6 9 9 2 7 6 1 9 2 4 6 4 2 7 4 1 5 3 1 1 7 E 0 9 1 0 1 2 7 5 4 7 7 6 3 6 4 3 8 1 9 7 5 8 8 7 8 8 7 7 7 1 3 2 7 7 4 1 9 8 7 1 1 2 E 0 8 1 0 2 6 9 6 4 2 3 8 5 4 6 1 2 6 6 4 8 5 0 0 6 1 5 3 4 2 9 8 2 5 4 2 1 0 1 2 7 1 E 0 7 1 0 0 0 1 3 1 3 7 4 0 3 4 2 0 8 2 4 1 9 8 4 3 2 9 4 5 4 7 1 7 7 0 9 2 8 8 2 6 1 4 8 E 0 7 1 0 0 0 2 8 2 9 4 6 1 8 6 2 7 4 7 4 5 6 8 8 0 0 6 9 1 4 7 3 9 3 7 2 4 3 0 1 2 5 9 3 E 0 8 R M O D E T O T A L C R I T I C A L L O A D S H A P E PC R L2/ E I = ( PA C R+ PC C R ) L2/ E I 0 1 4 9 3 7 8 0 2 0 2 2 4 6 7 4 6 1 1 0 1 1 5 4 6 7 9 4 1 0 2 6 3 7 1 6 9 4 1 0 0 0 1 1 9 6 8 6 6 6 1 0 0 0 2 7 8 7 6 8 1 5 T a b l e 1 C r i t i c a l p a r a m e t e r s c o l u m n A B T a b l e 2 C r i t i c a l p a r a m e t e r s c o l u m n C D T a b l e 3 C r i t i c a l l o a d f o r f r a m e

PAGE 23

1 8 0 1 2 3 4 5 6 7 8 9 1 0K1LRp p / 2 F i g u r e 1 6 F i r s t m o d e c r i t i c a l p a r a m e t e r v / s s t i f f n e s s r a t i o f o r c o l u m n A B

PAGE 24

1 9 p p / 2 0 1 2 3 4 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0RK1L F i g u r e 1 7 C r i t i c a l p a r a m e t e r f o r c o l u m n A B f i r s t m o d e s h a p e 0 1 2 3 4 5 6 7 8 9 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0RPA C R E I / L2p2/ 4p2 F i g u r e 1 8 C r i t i c a l l o a d f o r c o l u m n A B f i r s t m o d e s h a p e

PAGE 25

2 0 pp / 2 0 1 2 3 4 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0 RK1L F i g u r e 1 9 C r i t i c a l p a r a m e t e r f o r c o l u m n C D f i r s t m o d e s h a p e 0 1 2 3 4 5 6 7 8 9 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0RPC C RL2/ E Ip2/ 4p2 F i g u r e 2 0 C r i t i c a l l o a d f o r c o l u m n C D f i r s t m o d e s h a p e

PAGE 26

2 1 0 5 1 0 1 5 2 0 2 5 0 0 1 0 1 1 1 1 0 1 0 0 1 0 0 0 1 0 0 0 0RPC R L2/ E Ip2/ 22 p2 F i g u r e 2 1 T o t a l c r i t i c a l l o a d f i r s t m o d e s h a p e

PAGE 27

2 2 F i g u r e 2 2 D e f l e c t e d f r a m e u n d e r c r i t i c a l l o a d s f i r s t m o d e s h a p e

PAGE 28

2 3 F i g u r e 2 3 D e f l e c t e d f r a m e u n d e r c r i t i c a l l o a d s s e c o n d m o d e s h a p e

PAGE 29

2 4 R e f e r e n c e s P i p p a r d A J S ( 1 9 4 8 ) P h i l o s o p h i c a l m a g a z i n e ( s e r i e s 7 ) T h e c r i t i c a l l o a d o f a b a t t e n e d c o l u m n L o n d o n : I m p e r i a l C o l l e g e S i m i t s e s G e o r g e J A n i n t r o d u c t i o n t o t h e e l a s t i c s t a b i l i t y o f s t r u c t u r e s N e w J e r s e y : P r e n t i c e H a l l