Design for Buckling Columns and Plates

7/26/2019 Design for Buckling Columns and Plates

1/20

A12 - Design for Column and Plate Buckling 1

Design for Column and Plate Buckling

The critical buckling load for a long slender column waspreviousl obtained !see A1" and A11# b solving the governing

differential e$uation of e$uilibrium and is given b%2

2cr

EIP c

L

=

where c is a constant depending upon the end conditions%

clamped-free% c&"'2(

pinned-pinned% c&1

clamped-pinned% c&2clamped-clamped% c&)

*$uation can be written as a critical buckling stress+ and can also

be put in terms of a non-dimensional ratio called slenderness ratio

as follows' The critical buckling stress is simpl%


2/20


2 2

2 2 ! , #

crcr

P EI Ec c

A L A L A I

= = =

The term !A,# is related to the radius of gyrationdefined b

,I A= !units of length#

*$uation becomes2

2 2!1, #cr

Ec

L

= ' .o finall we write the

Euler critical buckling stressas%

2

2! , #E

EcL

=

The term ,L is non-dimensional and is known as the

slenderness ratioof the column'


3/20

A12 - Design for Column and Plate Buckling /

0hen *ulers e$uation is compared to eperimental results+ it

found that the slenderness ratio must be 3large3 in order to obtain

acceptable correlation' 0hat is large will be considered shortl'

4or columns that have a cross-section such that the moments of

inertia are different about the two aes+ the minimum moment of

inertia must be used' 4or eample+

suppose we have an aluminum 0)"'1(cross-section' This is a cross-section that

is )3 deep and has a web that is "'1(3

thick' The top and bottom caps are "'2/3

thick and the shear web is /'()3 long' 0e

have the following section properties%

2 ) )1'56( + ('62 + 1'")xx yyA in I in I in= = = Conse$uentl+ thecolumn will buckle so that bending occurs about the -ais !

)min 1'")I in= #'

x 4

0.23

0.23

3.54

0.15


4/20

A12 - Design for Column and Plate Buckling )

Example' Consider an aluminum column ! 61"') 1"E x psi= # withthe cross-section above that is pinned on each end !c&1# and

7&1""3' The radius of gration is min , "'828I A= = 3 and theslenderness ratio is e$ual to , 1""3, "'8283 1/8'6L = = ' Thebuckling stress becomes%

2 2 6

2 2

!1"') 1" #1 (+ )2(

! , # !1""3, "'8283#

E

E x psic psi

L

= = =

4or a tpical aluminum+ we note that the ield stress is around)"+"""y psi = !or greater#' 9ence+ buckling will occur well

before the ield stress is reached+ and buckling for long+ slender

columns !large ,L # is thus geometricall dominated+ not materialielding dominated'

4or ver short columns !small ,L #+ the column will not buckle

but simpl compress+ and a simple ,P A= model is sufficient'

4ailure will then be due to ielding of the material'


5/20

A12 - Design for Column and Plate Buckling (

There is an intermediate range of ,L where neither *ulers model

nor a P,A model matches eperimental results' Johnson's

solutionis often used in the intermediate range and is given b2

2

! , #1

)

yJ y

L

c E

=

:ote that ;ohnsons e$uationis *ulers solution inverted

and offset b a constant ! y

=yield stress#' f one graphs

e$uations and


6/20


specific slenderness ratio' :ote that *ulers method goes to

infinit when the slenderness ratio goes to >ero+ whereas ;ohnsons

solution is e$ual to y

for an slenderness ratio of >ero' Thetangent point can be found b setting the two solutions e$ual to

each other%22

2 2

! , #1

! , # )

yy

LEc

L c E

=

7etting 2! , #a L = + the above can be written as the $uadratice$uation%

2 2 2 2 ) 2! # !) # ) "y ya c E a c E + =

which has the solution 22 ! , #ya c E = ' :ow ( ),L a = '9ence+ the slenderness ratio at the e$ual point is given b

2 , y

equal

LcE

=


7/20


4rom eperimental observation+ one finds that the *uler solution is

good for slenderness ratios greater then this value+ while the

;ohnson solution is good for slenderness ratios smaller than thisvalue' 4or the case of c&1 !pinned-pinned# and aluminum with1"')E Mpsi= and )"y ksi = + we have the following plot with

the e$ual point at! , # 81'6)equalL = '

:ote that this plot+ and the

resultant slenderness ratio,L where the *uler and

;ohnson models are e$ual+ is

a function of column end

conditions !c# and the

material being used !Eand

y #' 9ence+ the

determination of which model to use !*uler or ;ohnson# must be

determined for each problem' 4or this material !tpical aluminum#

?se ;ohnson ?se *uler

Slenderness ratio


8/20

A12 - Design for Column and Plate Buckling @

and end condition !c&1#+ we see the following% for values of! , # 81'6)L < + *ulers solution will over estimate the critical

stress' 4or! , # 81'6)L >

+ ;ohnsons solution will under estimatethe critical stress'

Example% Consider the case of a column 2"3 long !7&2"3# with

the same 0)"'1( cross-section ! "'8283= # and aluminum

material as before !1"')E Mpsi

= and)"

y ksi =

#' Theslenderness ratio for the column is e$ual to%, 2"3, "'8283 28'(2L = = ' The transition point on the two curves

is given b 2 , 81'6)ytransition

LcE

= =

' 9ence+ this

indicates that one should use the ;ohnson solution since28'(281'6)' ;ohnsons solution gives the critical buckling stress

as%

2

2

! , #1 /8+ ")5

)

yJ y

Lpsi

c E

= =


9/20


Buckling of Flat Plates

n the notes b Prof' Pollock !see A11#+ the buckling of flat plateswas briefl discussed' This included flat plates subected to in-

plane compression or shear' Also+ due to bending loads+ but note

that the bending moment was about an ais perpendicular to the

plate not the usual plate bending discussed in A"( where the

bending moment is about an or ais which lies in the plane of

the plate'

The buckling load for a flat plate is obtained b starting with the

governing differential e$uation for displacements for a plate as was

derived in A"( but modified so as to include the coupling between

in-plane and out-of-plane displacements !as was done for the

column in A1"#' 4or a particular set of edge boundar conditions+

a series solution of sine and cosine functions is assumed that

satisf the governing differential e$uation' As for the column+ the

result is an eigenvalue problem that must be solved to determine


10/20

A12 - Design for Column and Plate Buckling 1"

the critical load under consideration !compression+ shear or

bending moment#' uch of the earl work on the subect was

done b Eerard and Becker and is reported inHandbook of

Structural Stability+ :ACA T: /8@1+ 15(8+ and also in

Introduction to Structural Stability Teory+ Eerard+ cEraw-9ill+

1562'

The result of their work is still utili>ed toda' Although the finite

element method ma be used to predict bucking and collapse of

plates and more comple structures+ it is $uite computer intensive

and not done often in practice !because it re$uires the solution of

nonlinear e$uations of e$uilibrium#'


11/20


4lat Plates in Compression

Consider a flat plate of thickness t3+ dimensions a and b+ and

subected to in-plane compression as shown below'

a

b

:ote that 3b3 is width of the plate !edge where the load is applied#+

and 3a3 is the length of the plate' Eerards solution for a flat plate

in compression with various edge boundaries can be summari>ed

with the following e$uation% 22

212!1 #

ccr

k E t

b

=

The constant ck is compressive buckling coefficient and is a

function of the edge boundar conditions and !a,b#'


12/20


The value of ck can be plotted as follows%


13/20

A12 - Design for Column and Plate Buckling 1/

:ote that there are ( edge condition cases presented for the

unloaded edges !length of 3a3# and for each of these cases a curve

for the loaded edges !width of 3b3# being either clamped or simpl

supported' :otation is% c&clamped+ ss&simpl supported+ f&free'

*ach one of the 3scalloped3 portions of a curve in 4ig' C('2 is the

solution for a particular buckling mode% n&1 !half sine wave#+ n&2

!full sine wave#+ etc' 4or clamped+ would be cosine waves'

n&1 n&2 n&/

4or the top curve !Case A+ loaded edges clamped#+ ou can identif

up to n&8' Thus for !a,b#&2+ the plate will buckle with n&/+ i'e'+sin!/ , #x a where is the coordinate ais in the direction of load

application !a direction#'


14/20

A12 - Design for Column and Plate Buckling 1)

Example% Consider the problem outlined in Pollocks notes !A11#'

A 5"36"3 flat plate with s$uare tube stiffeners as shown below is

to withstand an in-plane load of )" lbf,in' All plate edges are

assumed to be full clamped' The material for both the plate and

tubes is aluminum with a

Foungs modulus of 1"')

psi+ Poissons ratio of

"'/+ ield stress of )" ksi

and specific weight of

"'"5@ lbf,inG/'

The design parameters are

the plate thickness !t#+ the

number of added stiffeners

running parallel to the 5"3 edge and si>e of the stiffeners' The

stiffeners are s$uare tubes that have a wall thickness e$ual to that

chosen for the plate' As man stiffeners as desired ma be used+ so

long as the total cross-sectional area of the stiffeners does not

90

60


15/20

A12 - Design for Column and Plate Buckling 1(

eceed /"H of the area of plate !area over which the load is

applied - on one end#'

The added stiffeners will relieve some of the load from the plate'

The amount of load carried b the s$uare tubes depends on the

cross-sectional area of each tube and that of the plate' Fou ma

reduce the amount of the edge loading on the plate+ accordingl'

.imilarl+ the addition of stiffeners breaks the plate into two or

more smaller plates that are constrained along all four edges'

.mall tubes !less than 1'(3 1'(3# can be taken act as simpl

supported constraints for the plate !on edges parallel to tubes#'

Tubes larger than this si>e act as clamped constraints for the plate'

simply supported between stiffeners clamped between stiffeners

small stiffeners lare stiffeners

Design the plate for minimum weight'


16/20


0e could start the design in one of two was% 1# Assume the

stiffener spacing and solve for plate thickness t+ or 2# assume the

plate thickness t and solve for the stiffener spacing'

.uppose we start the design with a 23 23 stiffener ever 2"3 !total

of 2 stiffeners#' This will mean that the plate si>e between

stiffeners is b&2"3 !and the length is a&5"3#'

b!20

2

t2t

The cross-sectional area of the plate is !2"3#pA t= ' The area ofthe tubes within the 2"3 length is 2!13 23 13# @3TA t t= + + = !sameas area of one tube#' The total area is 2@t' 0e assume that the load

carried of the plate and tubes will be in the ratio of their areas'

9ence the load carried b the plate is


17/20


)" , !2" , 2@# 2@'(8 ,plate! lbf in lbf in= =And the load carried b the tubes is

)" , !@ , 2@# 11')/ ,tubes! lbf in lbf in= =The problem stated that the edges are clamped where the loads areapplied' .ince we have chosen 2323 tubes+ we assume these are

large enough so that the provide a clamped edge for the plate

along their length' 9ence the 5"3 2"3 is assumed to be clamped

on all edges' 4rom Eerards plot+ we choose Case A and thedashed curve' 4or

5"3)'(

2"3

a

b= = + we find that 8'2ck ' Eerards

e$uation for plates with in-plane compression is22

212!1 #

c

cr

k E t

b

= ' The plate must carr 2@'(8 lbf,in of loadhence the allowable stress is , 2@'(8! , # ,cr p! t lbf in t = = '

.ubstituting into Eerards e$uation gives%


18/20

A12 - Design for Column and Plate Buckling 1@

22 6

2

2@'(8 , !8'2#!1"') 1" #

2"312!1 !"'/# #

lbf in x psi t

t

=

.olving for t gives% "'"((3t=

Check to see if the stiffener !a column# will buckle under the

compressive load that it must carr !neglecting that it is attached to

the plate#'Area of tube% 2@3!'"((3# "'))tA in=; !using nominal dimensions

onl#

oment of inertia !about centroid and ais parallel to plate#%

{ }/ / 2 )

2 '"((3!23# ,12 23!'"((3# ,12 !23 '"((3#!13# "'25/I x in= + + =Iadius of gration% ) 2, "'25/ , "')) "'@163I A in in= = =

.lenderness ratio%5"3

11""'@163

L

= =

:ow determine which column e$uation to use% *uler or ;ohnson'


19/20


The transition point between the e$uations is at

62 , 2!)#!1"') 1" # ,!)" # 1)/ytransition

LcE x psi ksi

= = =

.ince the slenderness ratio for the tube is 11"+ which is less than

1)/+ then the ;ohnson e$uation should be used' ;ohnsons e$uation

gives the buckling stress as

2 2

2 2 6

! , # )" !11"#1 )" 1 2@+ 2""

) )!)# !1"') 1" #

yJ y

L ksiksi psi

c E x psi

= = = The load carried b the column is 11')/ , !@3# 51')P lbf in lbf= = and the compressive load is 2, 51') , "')) 2"@P A lbf in psi= = =

:ote% 2"@ 2@+ 212Jpsi psi = < = and 2"@ )"ypsi ksi = < = '


20/20

A12 - Design for Column and Plate Buckling 2"

9ence+ the tube stiffener is not even close to buckling or ielding'

0ith this design+ when the plate buckles+ the stiffened plate will

still carr significantl more load !via the tube stiffeners#'

0eight of the stiffened plate as designed%

Plate onl% /!6"3#!5"3#!"'"((3#!"'"5@ , # 25'11lbf in lb=Tubes !2 of them at 2"3 spacing+ each 23 s$uare#%

2 /

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Design for Buckling Columns and Plates