Handbook of Structural Stability_ Ed by Column Research Comitee of JAPAN-1971

A;ii HANDBOOK OF STRUCTURAL STABILITY

Quarterly of Applied MathematicsReport of the Shipbuilding Research Association of JapanReport of Ship Research InstitlteSAE JournalSAE TransactionsSchiffbau ForschungStructural EngineerThe Aeroaautical Journal (of the Royal Aeronautical

Society)The Aeronautical QuarterlyTransaction of Institution of Engineers and Shipbuilders in

Scotland

Transaction of North East Coast Institute of Engineers andShipbuilders

Transaction of Royal Institute of Naval Architecture

Transactions of the American Society of MechanicalEngineers

Transactions of the Architectural Institutes of JapanTransactions of the Japan Society for Aeronautical and

Space Sciences

Transactions of the Japan Society of Civil EngineersTransactions of the Jaran Society of Mechanical Engineers

Transactions of the Society of Naval Architects and MarineEngineers

rr'elding JournalZeitschrift fr Angervantlte Mathematik und MechanikZeitschrift fr Angervandte lvlathematik und PhysikZeitschrif t f r F'lugrvissenschaftenNational Advisory Committee for Aeronautics Technical

NoteNational Advisory Committee for Aeronautics Technical

Report

Quart. Appl. Math.Rcp. Ship. Res. Ass. JapanRep. Ship Res. Inst.SAE Jour.SAE Trans.Schiff. Forsch.Struct. Eng.

Aero. Jour.Aero. Quart.

Trans. Inst. Eng. Ship.Scotland

Trans. North East CoastInst. Eng. Ship.

Trans. Roy. Inst. NavalArch.

Trans. American Soc. Mech.ng.

Trans. Arch. Inst. Japan

Trans. Japan Soc. Aero.Space Sci.

Trans. Japan Soc. Civil Eng-Trans. Japan Soc. Mech.

Eng.

Trans. Soc. Naval Arch.Mar. Eng.

Weld. Jour.ZAMMZAMPzFw

NACA TNNACA TR

Part I.

CONTENTS

STRAIGHT MEMBERS

A. FLEXURAL BUCKLING... ..................r-.31. Axial compression............... ..............r-3

l.l Prismatic bar, concentric ioading........'... ..................t-31.2 Prismatic bar, cccentric loading and columns with initial curvature .....-..-...-..-...--..-.1-Zs1.3 Prismatic bar on the elastic foundation..................... ....----.........-.-..1-271.4 Columns with variabie section ........'... ............-......r-3t1.5 Composite members anC built-up members ........--...1-s9

2. Combined bending and axial compression .............r-7i2.1 Load-deflection re1ation............ .-.---.1-742.2 Ultimate strength of beam-columns................'..'. ............................I-802.3 Special cases -........1-Iog

B. LATERAL BUCKLING OF BEAMS .....................1-rr11. Elastic lateral buckling ..................1-trl

1.1 Narrow beams .....-I-1II7.2 l-beams and wide flange ems............... ...............1-ItB1.3 Tee-beams ............t-r2q1.4 Narrow beams with varying cross sections ............1-I2s1.5 Restrained beams..'............ --..-......-I-I2g1.6 Others ............. ......r-2

2. Design formula for iateral buckling strength ......r-rll3. Inelastic lateral buckling ................... ....................-3t4. Others .......... ...........r-144

C. FLEXURAL TORSIONAL BUCKLING1. Elastic buckling under axial compression.... ...........r-146

1.1 Concentric Ioading .... ....... ............1,1487.2 Eccentric 1oading............. ..............1-tS61.3 Restrained beams and columns ......1-Igo

2, Elastic buckling under axial compression and end moments ......1-t6J3. Inelastic buckling ...........................-7J4. Others .......... ...........r-r78

D.

E.

Part II. FRAMES AND CURVED MEMBERSA. COhITINUOUS BARS

........................2_J

1. Flexural buckling ..............................2_J

l'1 Continuous bars on rigid supports........... ...................p_J1.2 Continuous bars on elastic supports .........................2_11

2. Flexural torsional buckling .............2_16

B. RIGID FRAMES ................ ...............2_zJ

1. Buckling of frames under vertical loads .............. ..........................z-zr1.1 In-plane buckling of frames

..........2_zg1.2 Spatial buckling of frames

.......,.....2_so

2. Ultimate strength o{ frames under combined loading ...................p_ssC. TRUSSES

...................2_s7

1. Plane truss............... .........................2_s7

l.l In-plane buckling........... .................2_57

1.2 Out-of-plane buckling ...................2_sg

1.3 Lateral buckling ............................2_sg

1.4 Snap-through.... ......2_66

2. Space truss ................2_67

D. ARCHES AND CURVED BEAMS .........................2_721. Parabolic arch ...............

l.l Uniformly distributed vertical load ...................-.....2-721.2 Nonuniformly distributed vertical load ...................2-80

2. Circular arch................ .....................2-82

2.1 Uniformly distributed radial load ............................2_gz2.2 Uniformly distributed verrical load .........................z_s62'3 Nonuniformly distributed load ............... ..---..........2_rol2.4 Pure bending....

.....z_tls2'5 Horizntal forces at supports .........2_IOs2-6 Concentrated load at the crown..............

--..............2_Iog2.7 Impulsive loadings, step loading and rectangular loading.. ..............-..-..-................2_ItJ2.8 Static hydraulic pressure

v ...........--.....2_116

3, Tapered circular arches connected at the crown............. ..............2-116

CONTENTS i

3.1 Uniformly distributed radial load ...........................2-1164. Sinusoidal shallow arch ............. ...........................2-rrs

4.1 Uniformly distributed vertical load ........................2-IIg4.? Sinusoidaly distributed vertical load .....................2-II94.3 Sinusoidaly disributed radial load ........................2-Izt4.4 Concentrated vertical load ............2-I2z4.5 Horizontal thrust...........'.. .....-.-.....2-1234.6 Stochastic vertical load .............--..-2-124

5. Arch with the axis of arbitrary shape..........'.... ....2-r2i5.1 Concentrated load at the crown..........'... ..-.............2-124

6. Circular ring................ ....................2-rzs6.1 Uniformly distributed external pressure........... .......2-I2s6.2 Uniformly distributed pressure and circumferential force ......--.......2-I3o6.3 Circular ring in a cavity...........'.. .........................2-rJI

8I8LIOGR4PHY.................. .....................2-rrs

'ii HANDBOOK OF STRUCTURAL STABILITY

4.4 Other loading.... ......J-765. Postbuckling behaviour'of plates ............................s-77

5.1 Compression ..........g-775.2 Shear ..........'.. ......s-sl5.3 Combined 1oading...........'. ...............5-945.4 Thermal loading ............ ................J-96

6. Plastic buckling of plates ...............r-roo6.1 Compression ........-J-IOO6.2 Shear ..........'.. .....s-to,6.3 Comprcssion and bending ..--.-.........3-Io,

B. POLYGONAL PLATES ..................3-r1o1. Triangular p|ates...................................:.. ................s-tto

l.l Compression .........3-1Io1,2 Compression and shear ..................J-II7

2. Parallelogram. ...........J-tts2.1 Compression .........9-IIs2.2 Shear ............. .....3-tI6

3. Trapezoidal plates ...'......... ..............9-rr73.1 Compression .........3-117

4. Other regular polygons ......-......-...'s-12t4.1 Compression '........3-1214.2 Shear ..'....'..'.. .--.-3-122

C. CIRCULAR PLATES .-...'....'..'......'..'s-I2s1. Circular plates '............ ..............'.....3-123

l.l Uniform external pressure """ ""'3-1231.2 Partial external pressure """""""'3-126

2. Circular plates with variable thickness ..............'...3-1272.1 Uniform external pressure '.'..".'..3-127

3. Annular plates......'...... ..'....'............3-1293.1 Uniform external pressur "".....'.'3-12!)3.2 Uniform internal pressure .........'..'....... """"""""3-1313.3 Uniform external and internal pressure -.-...---..-'.....3-132

4. Sectoral plates .......'...'. .....'......... ."3'1314'l Bending and other loading ...'.....".3-134

5. Postbuckling and finite deformation behaviour of circular plates........................3-1375.1 Uniform external pressure ....'.".."'..,v " """""'3'1371.2 Fi:i: deformation behaviour under thermal loading..'..'.. .... ..-.....-..3-139

CONTENTS iii

D. STIFFENED PLATES ......'..............r-r13l. Stiffened p1ates........ ..... '.'..'...'.........3-143

l. I Compression ....-..-.3-1431.2 Shear .....'....... .....3-581.3 Bending .......... ..-.'3-1671.4 Comprcssion and bending ---.-..-.......3-1711.5 Compression and shear .'..'...'.........3-.f83

2. Plastic buckling of stiffened plates ............. ........'..3-t832.1 Compression .'"".' 3-83

3. Ultimate load of stiffened plates .......'.... ...........'...3-1853.1 Compression """'3-185

E. PLATE STRUCTURES... ... ....'..'...... .......'...............3-rerl. Plate structures........ '..'..............-.....-3-191

:l i::ffi;'",;;;: :: ::: : :: :: ::'';:2. Tension eld of plate structures """" """"""""'3-202

2.1 Tension eld """"'3-2023. Ultimate load of plate structures " " """"""""""3-208

3.1 Compression " """3-2083.2 Bending .'........ ""'3-2Io3.3 Shear ............. .....3-214

F' SANDWICH PLATES """"""""""'3-2171. Isotropic sandwich plates ......."..' """""""""""""'3-217

I.l Compression (Overall type) """"""3-211.2 Compression (lryrinkling type) """"""" """"""""3-2211.3 Shear.'........... ""'3-223

2. Sandwich plates with isotropic faces and orthotropic cores """"""""""""""""3-2242.1 Compression """"'3-2242.2 Compression and bending """"""""3-2292.3 Compression and lateral pressure """"""""""""""3-231

3. Orthotropic sandwich plates.. " """ """""""""""3-2333.f Compression """""3-2333.2 Shear ... "'. " """3'237

4. Sandwich plates with corrugated cores...."""""' """""""""" """3-2394.1 Co-nression """"'3-2394'2 S and comoression """"""""'3-243\t4.3 Bending .....'.... ""'3-215

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',i HANDBOOK OF STRUCTURAL STABILITY

2.4 Edge moment.... .....1-3Os2.5 External pressure and concentrated load......'........... ........................4-gO42.6 External pressure and edgc moment.......'..... .-..-..-..-4-3062.7 Dynamic buckling ......................-.1-3072-8 Creep buckling ............1-31s

3. Isotropic trunci'tc(l s'h1 ri;l shclls .............. ...---....4-g163.1 Axial comprcssion3.2 Tension . ... ... ................1-3173-3 ltS own rveight .....4-3t73.4 Torsion .... .. . .. . 1-318

4. Stiffened spherical she11s............... ..........................'4-s2o4.1 External prcssurc............ ........ ......4-3204.2 Half-side cxternal pressure '.. ...... .4-323

5. Sandwich srherical shells ............. ..........................4-Jzs5.1 Extcrnal pressure..'........' ...............1-325

E. SHELLS IffITH MISCELLANEOUS SHAPE .........4-3281. Elliptic cylindrical shells (Including stiffened elliptic cylindrical shells and

D-section cylinders) ........................4-szgl.l Axial compression .'...'....'....... . ....4-328l-2 Bending .....,,.....,.,4-3301.3 Torsion .......... ... '1-333

2. Other cylindrical shells ......'....'....'.4-3352.1 Non-circular cylindrical shell-Axial compression ...........................4-3352.2 Cylindrical shell with circular-arc section-Bending '...'...................1-335

3. Spheroidal shells.'....'....'.. ----............1-3363.1 External pressure............ .. ... '......'4-3363.2 Internal pressure..................... ..'...1-339

4. Toroidal shells and toroidal shell segments .......'.. .1-s4o4.1 External pressure............ ............,..4-3404.2 Axial tension.... .....1-343

5. Other axi-symmetrical shells '.'...'...... ....................4-3115.1 Cylindrical shell with hemispherical ends-External hydrostatic pressure.'............... 1-3445.2 Torispherical bulkhead and elliptical bulkhead-internal pressure ........'................'.4-3155.3 Logarithmic shell-Torsion ...-........1-318

6. Hyperbolic paraboloidal shells....'...'..... ....-..--........"4-3496.1 Its own weight. ..-.-4-340

8I8LIOGR4PHY.................. .....................4-350

Plnr ISTRAIGHT MBMBERS

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.

ifiir:ifci{: . ' .{6. r ".' ..,l:',t

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ls:b":D:d:E:

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hi

SYMBOLS USED IN PART I

area of flangeare of webwidth of rectangular cross section' width of plateeffective plate widthflexural rigidity of a plate per unit widthdepth of sectionstress-strain modulus of elasticitystrain-hardening modulus (initial)tangent modulusdistance from centroid of girder cross section to shear center (positive

if shear center lies between centroid and compression flange' other-wise negative), distance from shear center to the middle plane of achannel web, eccentricity of end load in a beam-column

compressive stress due to bending momentdistance from shear center of girder to point of application of transverse

load (positive when load is below shear center, other-\'vise negative)depth of a rectangular cross section' clear depth of plate girder web

between flange comPonentsdistance to compression-flange centroid from centroid of sectiondistance to tension flange centroid from centroid of sectiondepth of webmoment-of-inertia of section

I" z moment-of-inertia of compression flange bout y axisI : moment-of-inertia of a flange about 3t axisIo : polar moment-of-inertia of cross sectionf. : moment-of-inertia of stiffener about web-face axisIt i moment-of-inertia of tension flange about I axis

1,,1" :, moment-of-inertia of cross section, and g denoting the coordinateaxes

J z torsion constantj , lateral-torsional buckling constant' number of panels in a stifiened PlateK : effective or equivalent length factor

L,t "

length of member, particularly a laterally unbraced lengthM a moment

Ma,Mo: end moments acting on a beam-column at ends and ' respectivelyM^o, i maximum bending moment

:g:

h, Ihtah*:Iz

1_2 I. STRAIGHT MEMBERS

Mo : plastic bending momentMr,My,Mr: moment.about coordinate axes ,y, and z, respectively

M, I yield momentN ; nominal axial loadP : column axial load

P", : critical loadEuler buckling loadmaximum column loadtangent-modulus column loadcolumn axial load at full-yield conditiontransverse shear in centrally loaded columnradius-of gyration of memberpolar radius-of-gyration of the cross section about its shear center

radius-of-gyration of one chord in a battened columnradius-of-gyration about the certroidal axis - (strong axis)radius-of-gyration about the centroidal axis g-y (weak axis)thickness of tension flangethickness of web plates of box-section beam, thickness of webdisplacement in the directiondisplacement in the y directiondisplacement in the directioncoordinate axiscoordinate axis, particularly a principat axis, distancedistance between the shear center and the centroid in the direction

the . axiscoordinate axiscoodinate axis, particularly a principal axisdistance from centroidal axis - to face of tee flangedistance between the shear center and the centroid in the y axiscoordinate axisangle of twist of cross sectionstrain at initial strain hardeningelastic strain at yield stressPoisson's rationormal str*sEuler buckling stressyield stressangle of rotation, curvatureangle of rotationshear stress

A. FLEXURAT BUCKTING1. Axial compression1.1 Prismatic bar, concentric loadinga) Elastic buckling

cond itions

At one end, rotation isrestrained, free to srvaY,another end is fixed.

lPrr\

,I

One encl hinged, another6xecl

Upper end is elasticallysuprorted with springconstant r. lorver end is6xed.

P,P,,P,P, :D.ta.

Pt:Pvi@:r:

ro I

rrzrvltt It-:u'.iu'.

X-X,r-x :uiIo '.

ref.

of

Y-Y,v-v :yilc"loi2a:

sr 3

tiv3o'.

oE,d :dv i:0:

P"=2'946I'EIvalue of the coefficient is 2.04574

P-=t:+The coefficient ro is obtained from

*"*--.Q-{ff)

1_30 I. STRAIGHT MEMBERS

conditions

Column on a elastic foun-dationUniformlY distributedlateral loads ge and axial

Foundation characteris-tics

p=sinh (zru)

t = reaction of founda-tion Per unit length

Buckling of microbersInfinitely long 6ber inin6nitely large volume ofelastic matrixFiber has round crosssection.Compression is aPPlied to

the fiber.Two possible bucklingmodes are given in thefigure.

Fiber buckled into srnecurve in the r, z-Planewith orientational linesperpendicular to (a) thez-axis and (b) the bentshape of the central'line of the ber.

(a) F=0

ref. l'4 Columns with varible sectioncond itions

Both ends hingedThe section is varYinglinearly.

ta\.r=r_*\7)Approximate formula :

Equivalent length l-_k__t- I -

where,

A. FLEXURAI- IJUCKI.ING I_3I

resul t6 ref.30

n",=otl{,-Tble 1

32

33

is obtained as,2tt

/-l;"={ 7;

Teble 2

Approximate formula :

/r=depth of the section

Upper end freeLower end fixedMoment of inertiavarying linearly,

Ir = Io(l -)

Critical axial force is given bYN,,=2'/81"""h6

in which=uniform lateral displacement of beam caused by a

uniform lateral load qo at the neutral state ofequilibrium

e'2 E'I-, tr_

ro2

et and relative contraction of the fiber' * are given

in the 6gures.

0_

Fig. 1. GraPhs of erand + versusE ID for t=O.4and for P=0'

0.1

Fig. 2. GraPhs of crand * vcrsusf,,tlE tor v=O.4and for =1.

100E IE

v=Poisson's ratio of matrixE'=modulus of elasticitY of 6berE=modulus of elasticitY of matrixro=radius of fiber v

thrust NBro is,"n qL =0

@*l;T;l o'f I o a I oo I r o, l" *.i, rr.l, *q''*tl'=*i'-t'*ti,-*F*tit*til *E'*t

(b) =l

1_32 I. STRAIGHT MEMBERS

-

C2(K2 -l) tt2 EIoIi,=- Xzttz .- F

(o is given by the first root of the next equation,J'(K Y (K()

-Jr (K() Yr (() = 0where

Kr=L, */;;

(02(K2 _ l)u=- 4yz-

P"=l'AP

--l- t. h ;il ;; I '.tl ; I .-o, I i.;

--j-Ei@lel 'i; :l : "i l"rlrs

P"--oo'#

-,_l Q-9,\'pn'=\- 2 -- Iwhere Co is the rst oot of the next equation,

J"- '() = 01-

'll+r'"iial;

F

conditions

Both ends hingedDistribution o{ thement of inertia is,

Upper end freeLower end fixedDistribution of the mo-ment of inertia is,,=,"(i),01n12

Both ends hingedDistribution of the mo-ment of inertia is,

,=i,{r-(f)'}'aL Illa"ll/l\i I I \.iIIJ]

lP ,"

resulte

A. FLEXURAL BUCKLING I-3.?

condition.q results ref.

59Upper end : Rotation isrestrained but side swayis free.Lo'er end 6xedCircular section

w

-

tr72a2b2E EI"t;=_;,p =p Fwhere I is the 6rst root of the next equation,

t'---.;':t-. ftlt ,,. 1 =,,,.{ ( r -)o r I.b_atdblA--T - tan A

p

t

0 0.2 0.{

-

a/b

35

60

Both ends xedCircular section

o

",LtlzI l,1l I

'#

-

I "E EIn ahl'(t=-nt'tt'F=11 I; , ^=-;-where is the rst root of the next equation,

l, - f t : -!=, :'-l*,r . -, { !!fl r }^=1t*a-t>u:1"-;.{__

A. FLEXURAL BUCKLING I-35

I is given by the first root of the ncxt equation'

*,]r9=;=), * -, (r't) = -+ll is presented in the diagram'

]ztr / a-b\z EIoP,,= so'o\ t-)=P t"given by the first root of the next equation'

'W^a/ i ^)

v " -

v,(/ i )t,

is presented in the diagram'

4 olb

T_34 I. STRAIGHT MEMBERS

conditions

Circular section-

ttb'to_ 4

P 2o]lifi

results

r+f r""(L:xr)e;aF;r cot (Kl)=|

-- b-a

t,

ref.

62Upper end freeLower end fixedCircular section

, -obn,o_ 4

ililLW

nlzab2 / a_b\2 EIoP,'= a,,;\ t )=o t"is the rst root of the next equation,

a"le--s)fo-:) ^

) - *, (* = *where3?

63Tapared round columnBoth ends hinged

l--'/,Ii '[ 1,

#r

n2qbz / a_b\2 EIor",=-tr" \- t )=o F 37

conditions

Rcctangular cross sectionrvith constant thickness,f and varying breadthBoth ends hinged

Rectangular cross sections'ith constant thickness,f and varying breadthOne end hinged,another end 6xed

p is presented in the diagram'

ilnto- A

2r;t3 / a_b\2 EIoP"= "b\ t )=o FI is given by the 6rst root of the next equation'

,,(/:i ^)

y,e) -

y,(r/ !br)"r, ro _ .

^ ^! (r

-

a) (b -

a)- ,-j:--) /- i--=-tanl *r- l;( + ^) v" - v,(/ i lt "

1_36 I. STRAIGHT MEMBERS

- _

'P!_(.b-a\2_. EIo,1,= qlr"b\-- ) --p t -

I is given by the 6rst root of the next equation,t"V i^)v - v"(/ ; t)r"a>

.-_ r(r_a)(1),lW=tanl 2;b ^l

p is presented in the diagram.

o -

zDtt ( .b-a\2- .. EIs.r"= a1',b\ t )--u r

I is given by the rst root of the next equation,when a I,

t (/ i t) v "> - v'(/ vu":y =, ^ ^!

tL--e e-,,(/ I ^)",,^) - r,(/

:i-t-;*^,=^"lT

"F ^l

when a=1,

',(/:^) :,,'(/l^) %(r)

,r is presented in the diagram.

A. FI-EXURAI- BII(-KLING 1 ..?7

),zEcb / a_lt\2 EIn= 48'. \ / -)=o t"

I is given by the rst root of the next equation'

ref.conditions

Rectngular cross sectionwith constant breaclthd varying thicknessBoth ends hinged

Rectangular cross sectionrvith constant breadthand varying thicknessUpper end hingedLower end fixed

t,(/ * ^) Y - v,(/'L t)t "tl

Q 1^,on",/ !\t,o,>r is presented in the diagram.

2 (0-a)(t-a)1=T+cotl 2;; I

;-t--11-r'-- ; t-_ to

rp-o I I o.zszl/lo | 0.4671 1.5r/s I t.oezl l.orl4 | t.tul t.zznl/3 | z. o I a. aoertz I e.ozsl o.oos2t3 | s.zl 2.s993/4

_l 6.483 | & 542

t

conditions

Rectangular cross sectionwith constant thickness,f and varying breadthBoth ends 6xedUpper end free to swaY

Rectangular cross sectionwith constant thickness,t and varying breadthUpper end freeLower end fixed

>-O a

o -

72Ecb ( a-\'- .. EIoY"= 4B,.,\ t )=oV'is given by the rst root of the next equation,

ff"'-r-',V+t*t,(/

* ^) v,

-

v,(r/ 1 t)"r , *ffi'"\fi?)^l

=T- b- -

(o-d)(b4)ldb ^-n\- 2;b l

l is presented in the diagram.

u20

r/r0u5rl41/3u2213311

02.tu4.510s.6857.6

11. 340M.7m16.346

tt20t8t6llt2t086420

+

lt-

1-38 I. STRAIGHT MEMBERSconditione

Rectangular cross sectionwith constant breadthand varying thicknessUpper end freeLower end fixed

Rectangular cross section,thickness of which beingin reverse proportion tobreadth in the upper re-gionUpper end hingedLower end fixed

ref.

A. FLEXURAL BUCKLING /-39ref.

o _ 2Ecb_( g-\'?_ .. EIor*= 4B'"-\--)--P t,I is given by the first root of the next equation,

3938

c

----9- ,o.r-THfrdH',"fl

conditions

Rectangular cross section,thickness of which beingin reverse proportion tobreadth in the upper re-gionBoth ends fixed

Rectangular cross section,thickness of which beingin reverse proportion tobreadth in the upper re-gion

P",=+^'t!-=r+zr is given by the next equations,/__ ryX=y' r-n\-t_",,l is given by the 6rst root of the next equation.

{,() +'{,() } s;,,r, (} r"s ) : z tlz / i - " "*r, (} r 1) }where

,=;fu"^l--::=ur=*l-!:-z(t-lb--ur=,|

, (^) = -b# ./ | -F "^l(l-?:? u -t I*!#*"1#u=rl(t-2a)(b-a) f(l-a)(-),-

,\t)=-cos\ -i , 'yL-^'I

-lrio-t-!#tu= ,1x"nl\>i!=>'/;trl

wn"" f;> |, r uu"o*"s imaginary.r is presented in the diagram.

'!1 * !!1(/! t) n -

z -, ^^!, - a)rs:-4 r

1/ |^rt^>-r,Vnr ^)t*='-'^"1 zt- ^p is presented in the diagram.-s.g&-\l ' I't"o I lo

u20 | | 0.646l/r0 | 0.32r1 r.ro3l/s | 0.5971 1.618rl4 | 0.7281 r.7721/3 | 0.e351 r.953rl2 | r.3371 2.174213 | r.?2rl 2.3314 | r.9721 2.359

I

ca

0 0.2 0.1_ olb

n",--$-'ff-=r+zz is given by the next equations,

t----7-mT/ulffi)=tI is given by the first root of the next equation,

I cotr, (j ros l)+ t = vr-=L+#tr=Y.*"12#ur-l

"^ ffi a=r-."1('-?#") tt=' Iwhen ffi< ], I b""o-"s imaginary.

0.2 0.1* lb

-

I "Ecb EIorc,=Em-- z =lt F

ze is given by the next equations,t___7 ndy

/ t-4\)=^,l is given by the first root of the next equation.

1-40 I. STRAIGHT N'IEMBERSconditions

Upper end freeLower end fixed

Rectangular cross section,thickness of which beingin reverse ProPortion tobreadth in the uPPer re-gionBoth ends hinged

-r-

conditions

Symmetrical column withstraight code membersBoth ends hinged

/ denotes height of crosssection.

r=r.(r-|)'

Column with varing crosssectionUpper end freeLower end xed

,='.('- l)'

Column with varYingcross sectionBoth ends hinged

r()=r^-(r^-u>(z;" ,n - lrr=t

A. FLEXURAL BUCKLING 1 41

n2 EIP",=p:_;!,

-;i: l_r=l';t--l ;;-u I o.lol I onae I o610'l-----l--

t"th^ | ou I ot I tot I o.zg | _088t l--r'oo-

Approximation on condition, holh^20'2 "

t'=0.34+o-66v/*

-- ; '-l ',, _l__l--l '

-

-,* | l*?_l _'ot I o1u _

, EI^ EIopc?=ttn-T=paV.p^ arrd p are presented in the diagram'

n='I^

ref.

33

r-tr,(jr.sl)+r="/r4"1e4:Feut*l

*n n ffi'|' I b""ot"" imaginarY'p is Presented in the diagram'

2.62.12.22.0

| - Ecb EIoPc,=Tm'-tl= ttVz is given bY the next equations,

/----7-ru/ r-4\b4 ) =^

I is given by the first root of the next equation'

r*,r,(|r.sl)+rt{r_a)!_3)upl

= _y'r _t, cott ' 2 ob ,

w"" #>T, I becomes imasinarv'g is presented in the diagram'

-

alb

I. STRAIGHT MEMBERSconditions

ends hinged

{;

Both ends hinged

/ x\nr=r,\ul),03r3h2L

en(t-$l.--+(,-/+)

Both ends xed

,rrftrr,

)/t-2

r=r.(t-ft*=-i,t,

A. FLEXURAL BUCKLING 1-43

zlo.2 I 4.430.4 I o. zs0.6 I a.

r"'=PJ!)='(2t1',where

p=qo"z_"gffIf r=9, p and (oare given Y fuUl" t'If z=1, (o is given bY Table 2'

Table In 113 312 s/3 ?14

tpr_l Il(o5.762.40 3.00 3.83 5.14 6.38

0.8

0.6

p(ItCo

7.022.t2

s.062.70

4.253.30

3.214.30

2.495.20

8. 4lt.74 2.t5 2.50 3.00 3. 30

1.80nnl;" 9.30ltt r.45 r.55 1.70Table 2

>t'l o.r o.2 0.4 0.6 0.8l.o | 2.800.8 I

".oo0.6 l t.*0.4 I r. 30

3.2.742.61.30

4.663. 502.451.50

7.324.702.90r.60

7.003.60r.85

Both ends hinged

EINP,,=P-F'

lr-'

A. FLEXURAI. BUCKLING T-45

conditions

Both ends hinged

ref

Both ends hingedr'-- ' Lllo /--------\ t.H

,=,,(i).0!rS12

At the ends, 1=/o

,=,,(i),0lxl12

At the ends, f=/o

] ,44 I. STRAIGHT MEMBERS

p,,=ptrL

l is presented in the table.

0.4 o.6 0.8

8.688.087.U7.

8.918.68.494.42

9.559.489. 469. 45

9.279.249.239.23

9. 549.509.509. 49

9.639.449.399.38

9.749. 639.629.62

r I 6.48 I ?.s82 | s.4o | 6.623l 5.011 6.324| 4.sr| 6.nt I r.ot-l-ise2l 6.371 7.493 I 6.14 | ?.3r4l 6.02l7.zo

I ll 8.611 e.r2ou I 3 I 3:3 I 3:31i_i I 101 I

-n ol

f",-pE*p is presented in the table.H,, l,,

"ltlI234

I34

20.29

19_ 60

zlt22.39

22.3621.2520.8820.7r

.0027.67n.24n.03

It4

27.8027.3527.20n.

a.8.5228.4028.33

33.0832. s932.4432.35

tt34

32.2032.0231.9631.94

32.9232.7732.7232.69

35.m35.6435.6035.56

t_I ra.art_I ra. ra

I 37.84I sz. alI sz.I az.ao

36.36 I 36- 8436.34 I 36.8036.32 I 36.m.32 | yt.78

36.0035.9735.96ara_

l I 35.40 |2 I 35.35 Is I gs.gz Irl.rel

conditions

Both ends 6xed

P,,=pIrLr is presented in the tables.

Table 2

;_-r-,i{;l Fi f ;; li ; 1 oJ

,-1tl o lo.orl o.r I o.z I o.r I o.o I o.ao li.l I s.ra lo.s lJ.o' ll ls.oi ls.z?0.2 lz.ulr.tt I z. | t.so I a.ss I n. t, I n t.0.4 I B.3s I s.4o I s.63 I 8.e0 I g.rg I g.ss I s.680.6 l r. I g.oo I g.ou I s.t. I g.to I g.zo I s.r,0.8 ls.eole.aoln.azls.rrls.*ls.ruln.*

Tble 3

Elll-l-1 1 [:I I j1[;T'" "o I z.ss I s.or I o. r+l z.sz e.so I s.z=3 o.z I a.or I o.sz I z.ar I a. | s.oz I s.so0.4 | s.4zl z-u I 8.4e1 e.rol s.aol s.os0.6 I z.sel s.rl s.ssl s.oei s.zl s.s

o.B I s.oal s.ttl e.atl saol s.asj s.

Strut with constant thickness plate

n,. = | o +.r' ), ?1,. M)i,l'Strut with two solid cones

P",=la"f:"]r.ffi]"Stut with four angles connected by lattices

r-r-l,=,,(i).01r S 12

At the ends, l=/o

1-46 STRAIGHT MEMBERS

IF;

conditions results ref.

Buckling in r-7 plane r",--\ffila\l'-,J':,i"2It

d=: lo

42

44

(1)86Tapered struts of lengthIBoth ends hingedCompressive force

o l,

,",m'

fl4r"\l/_lP!It

Symmetrical with resPectto the central section

(1) ,=r"(:)^(2) I- In*''rt,

o 1r 1ll2

where

For m=1,

For m=4,

Fot m=2,

(2)

,(r-\)o-',/^) -y zEroP"=l-r:;v^:w- l--fI, / b\^

,=7;=\;)n",=lo+tr"o"'o

.-- .2EI"p,,={ a- lz -

n-n(fr;\ *Er-'-1 k2 r2EIo-:-

' ct- 4 (l_e-k/2)2 12

87(1) Tapered column

---+Iti:l-'

l-.''=n['-;('-*)Jn--l:. weak-axis buck-

ling of a flat plate hav-ing uniformly varYingwidth and constantthickness

n=2: open web crosssection with constant

(1)

I=kzo

5

4

3

2

0

(a)h lbo

Buckling loadscotumns (hinged

for tapered'cd ends).

Itr-

conditions

flange area and taPer-ing depth (strong axisbuckling), or a taPeredtower section havingconstant reas concen-trated at the corners

z=3: strong axis buck-ling of a at Platc hav-ing uniformlY varYingwidth and constantthickness, or weak axisbuckling of a ll-sectionwith uniformlY varYingflange width

n-4: a solid truncatedcone or PYramid

A. FLEXURAL BUCKLING I-47ref.

o"==,'39.18

bt/bo(b) Buckling loads for tapcred

columus (fixed-fixed ends).

--L------1o[Mo'a I=t z'_ EI"L, = t l,2-u"ffi=,, 2.6

1.81.6t.4I

2.tt

00

1.0

0.0.2

0

hl bo(c) Buckling loads for taPered

columns (xed-hinged ends).

h/bo(d) Buckling loads for tapered

columns (xed-frce ends).

conditons

(2) Cantilevered taPere