beam column behaviour

8/15/2019 BEAM COLUMN BEHAVIOUR

1/15


2/15


3/15


4/15

15 Beam Columns Ch 3 Art 3 3 Beam Column with Distributed Lateral oi

which, in view of (3.8) and (3.13), can be written as

Ql PQ13M m a x

1

4 8 E I I - PIP,,)

Simplifying the tcrm inside the bracket givcs

from which

Th e 1eft:hand fact in (3.16) is the maxim um mom ent that would exist ifnoaxial force were present. Thus., letting

IFig 3 4 Beam co lumnwi th laterally : L .----Adistr ibuted load.

solving the governing clifl 'esential 'eq uat ion . T o illustrate an altcmcthod ol'nnalysis, wc shall now ubsc thc K;~ylcigh-Ritzmcthod.- -.. . -.. - .,. I

/ I n Leam column, bending and axlaT ompression usually 1_isimultaneously. However, the bending deformations can be assumeindependent of the axial deformations as long as deformations inremain small. Th e analysis of a beam c olum n by the energy metho difore similar to the analysis of an axially loaded m emb er. Tha t is, the eraxial compression is omitted and only bending energy is considelArticle 2.2).

Th e strain energy that is stored in the mem ber as it bends is

and th e potent ial energy of the external loads is-t he max imum moment in the beam co lumn can be g iven as

I O.ISP/P,,)- M m x = o (3.18)- PlPC,) Thu s the total energy In thc system isEqua tion (3.18) shows tha t the effect- - f axial- ompression- .- . on the.b e n d ~ n g-..

imom ent is very similar t o the eiTEst that an axial load has on the cicfleclio~i.----.- - --- - 0 1 / =Like the deflection, the mo me nt that exists in the absence of axiaicompressiol)is amplified by the presence of a n axial load. Tt is also interesting to notc the

a similar ity between the amplif icat ion factor for nloment and theco rres pond ~ngi

T o satisl) the boundary cond itions, Ihc deflectionJJ is assumed to beamplification fac tor fo r deflection.

4 -- 6 sin nx

3.3 BEAM COLUMN WITH DISTRIBUTEDLATERAL LOAD where i b the midspan deflection. Substrtution of this expression in

(3.19) givesLet us now consider the case of a s imply supported member bent bya i nuniformly distributed lateral load w and a set of axial forcesP s shown in

3 Fig. 3-4. As before, we assume that the m ater ial obeys Hooke's law, thatdeformations remain small , and tha t the mem beris restrained against lateralbuckling. In A rticle 3.2 the investigation was carried out by setting up arid

iIi


5/15

152 Beam Colunrns CI?. Art 3 3 Beam Column with Distributed Lateral oad 75.

To evaluate (3.21), we m ake use of tile follow ing definite integrals:

{ : s i n 2 y d . y -

Thu s Eq. (3.21) becomes

Equat ion (3.26) gives the m aximum deflection of a simply supported beamtha t is bent simultaneously bya dis t r~bu tedransverse load 14 and axial forces

P Sincet%.med shape for y was n ot exact, the deflection given by (3.26)i s o 3 ~ n ximatyon . However, i t Ks Te en shown by Timoshenko andb e r e (Ref. 1.2), who solved thc problcm rigorously, that the approximatesolutio n differs from th e exact answer only slightly 3w

~ / j r J 7 -The maximum mo ment in the member is- - \ - ---- /-=--:- ,&+yv l L k pM (3.27)

E16 'n4 211~51 P6?n"[I . I = I - - --4 1 ' n 41 (3 .22)

2 E t P J y l c ~ j d k i a r / w y I n ;iew of (3.25) an d (3.26) this cxpression can be written asF o r l ie system to be in equil ibrium the derivative ofU 4 Y with respcct to ? '

f Cv 2 ~PIYI(

6 must vanish. Tha t is, n x = 38481 (PIPc,)

d(U -1- V ) E16n4 ?II / P n2$8 21' 7~ 21

from which I .

If the num erator an d den omin ator of (3.23).are multiplied by 5/3S4EI, oncobtains

whicli reduces to

y or very nearly to

The i e f t-hand fac to r in th i s r e l a t io ~~s the detlcc tiol~ h;\t would csist if thelateral load 1: were acting by itself. Thus we let

an d rewrite (3.24) as

Simplifying the term inside the brackets, one obta ins

from which

The term outside the brackets is the ma ximum m ornent that would exist if noaxial force were present.If one lets

(3.25) can b e written in th e form

Th e maximum deflection of the beam column given byEq. (3.26) and themaximum m omen t given byEq. (3.30) are thus bot h e&al-tXhe Froduct oftwo terms, the maximu m d eflk tion o r moment tha t would exist if only laterallo rd were present a w m p l i f i c a t i o n f a ~ a c c o u n t sor th e effect ofth e axial load. ~ L a ts perhaps most significant in these relations is their

similarity to the corresponding expressions for deflectio~l and mom entobtained previously for a c ancentratcd lateral load. It is at least partially due

;


6/15

n a<

:g g

. 0J l

E l pg :

9O

S IV .

z5

=, a

: E

V

z;

E gVa

.

z FY O

352VV

z


7/15

- 7 5 0 Beam Columns Ch 3 Art 3 4 Effect of Axial oad on Bending Stiffness 157Substitution of these results in Eq. (3.34) gives In a s imr~armanner, th e ~-otatio n t endB of the member can be obtained.

Setting x , - 1 n Eq (3.37) gives

Mn6 , = $f ( l - - k / c s ~ k l ) + ~ ( l l c o t k l) . J

from w hichfrom which

M1 cos k l cos kx M ~ ' 1 - l csc kl) + A ( 1 l co t kl) ,= kZEIl '

s in k l(3.36 .

or A4o,, := 9 . x 9n 4 (3.43)

I Making use of the identity where K / , and 4 are defined in (3.40)Icos(a - P ) = sin a sin P - cos a cos/?

and multiplying the numeratorand denominator of the second t e rmi n

c:\cilparenthesis of (3.36) byI. one obta ins

kl cos k(1 x ] -,,- E D , l cos k x ) (3.37)sin kl - PI sin kl

/ :. .

Solving Eqs. (3.39) and (3.43) forM Aand M,, one obtains

and

I The end rotat ion atA is obtairied by setting = 0. Thus LettingM M

8 = z4 l l cot k l ) -i- -*( - l csc I


8/15


9/15

160 Beam Columns Ch 3

Fig. 3 7 Variation of bentling stilfness with ratio of axirrl loatf lo c:ills:ilload.

if 8, and are set equal to zero in Eq. (3.56). Thu sa is proportional to themoment M A hat is needed to maintain a rotation 8, when 0 A : 0. Inother words, a is a measure of the bending stiffiless of the member. f (hequantrly kl is rewritten in the form

Art 3 5 Failure of Beam Columns 161

greater than 4.49, a is negative, which means that the moment and therotation are oppositely directed. Values ofkl in excess of 4.49 correspond tomembers th at ar e elastically restrained by other members at the end to whichthe rotation is applied. Ano ther way of putting th e same idea is to say thatIcl < 4.49 corresponds to cases where 8, is induced by the adjacent memberand kl > 4.49 to cases where 8, is resisted by the adjacent member.

3 . 5 FAILURE OF BEAM COLUMNS.Up to this point in our study of beam columns, we did not concern ourselveswith th e suhiect of failure, and it was therefore possible to limit the analysis toelastic behavior. Now, however, we are specifically interestedi n determiningthe failure load, and, since failure involves yielding, it becomes necessary tointroduce the complexities of inelastic behavior into the investigation. Whenstudying the behavior of columns i n Chapter I it was pointed out that prob-lems which involve inelastic bending do not possess closed-form solutions.They must either be solved numerically, which entails lengthy and tirne-consuming calculations, o r approximate answers mustbe sought by makingsimplifying assumptions, In this article we shall study the failure of beamcolumns using the latter of these two approaches to the problem.

Let us consider the simply supported, symmetrically loaded membershown in Fig. 3-8a. The member is simultaneously bent and compressed by

where Pt is ihe Euler load, it beconies evident thatkl is a measure of the ratioof the axial load to the E uler load. T h k u r v e in Fig. 3-7 thus gives the vari3-tion of the bend ing stiffn ess with the ratio of axial load to critical load.

tWhen kl = 0 that is, when there is no axial load, a; = 4. This value of h -

a, is used in routi ne stru ctll ral analysis whe re the eA :ct of axial conzpress ion 7on the bending stiffness s neslected. Betweel? Icl = 0 and kl -- 4.49, a 1 ~ i uyl ro indecreases as kl incrcascs. T he hending s[ili ncss is tllus rccluccdby a n incrcnsc Fig. 3 8 ldcalized hccun c o l u m ~ lin the magnitude of thc axial load.A t I; : 4.41 or = : 2.04Pc a . 0 1 1 1 ~ Jezek. b) c)reason th e bending stiffness vanisllcs IL this loadis that the n~em her hich wehave considered up t o this point isi n elyect hinged a t one su ppo rt andfixed equal end couples M and axial forces P. It has been demonstrated by Jezekat the, other, and therefore has a critical load o f P-- 2.04Pt. For values ofkl (Refs. 3.3 and 3.4) that a closed-form solu tion for the load-?rf:ct.m c h m -


10/15

A r t . 3 5 Failure of Beam olumrrs 1m62 Beam Columns Ch.

terist ics , beyond thep roport ion al l imit , can be obtained provided t l~ef ol lo ~vin gassumptions are introduced.

the curvature can be wri t ten a s

d 2 v SnZZ sin -; xa = (3.61)I The cross sect ion of the m ember is rectangular (Fig. 3-8b .2. Th e material is an ideal elastic-plastic material (Fig. 3-8c).3. The bending deflection of the member takesthc form 01 L Ir;~lf-

sinewave.

. Substitution of (3.61) into (3.54) givcs

STIZ TIXM Py = EI- s nZ IThe reason inelastic bending is difficult to analyze is -that the relation of

stress to strain varies in a complicated manner both along the 111elnberandacross the section, once the proportional limit has been exceeded.t is withthis problem in mind that Jezek introduces the foregoing assumptions.Assumption 3 makes i t possible to predict the behavior of the ent iremem berfrom a consideration o f the stresses at only a single cross section, and assump-tions 1 and 2 greatly simplify the manner in which stress and strain vary atthat one section. In addition to these major idealizations, the followingassumptions are made

which reduces to

E16n21. p J ~ 6I 3.62)

at midspan.Assuming that M is propo rtiona l to P, we introduce the notation

and rewrile Eq. (3.62) in thc form

SEIz26 - 24. Defc>r~l~ationsrc f ini te but s ti l l small e~ lo ug l~o tha t thc cu~-v ;~ iu rc:111

bc approxirnatcd the second derivative.5. T he memb er is initially straight.6. Bending takes place about the major principal axis. o Re 1 6 == SP 3.64)

:: the coord inate axes are taken as indicatedin Fig. 3-8a, t l ~ c xtcrnnlbending moment a t a distances rom the origin is

where P = n2E 12 s the Eule r load o f the member. If both sides of (3.64)are divided by th e depthh and th e terms rearranged, on e obtains

This expression isYalid rega rdless of whe the r the elastic limit of th e n1;iteriaIhas been exceeded o r not. Th e characteristics of the internal resisting mom entdo, however, depend o n the state of s tress in the member. A s long as Flooke slaw remains valid, the inte rnal mo men t is given by the well-known relation

where 0 = P,/bh is the Euler stress and a = Plblz is the a vera ge axial stress.As long as stresses remain elastic, Eq. (3.65) gives the correct load-

deflection relationship for the me mber. To determ ine the load at which Eq.(3.65) becomes invalid, one m ust conside r the m aximum stressin theme mber.Tha t lat ter is

To determine the relat ion between load and deflect ion up t o the propor-tional limit, we equ ate (3.58) t o (3.57). Thus

or, substituting Pc for ll iind o or Plbh,f the deflection is now assumed t o be of the f or n~

n xy = 6 sin


11/15

I . , :164 Beam Columns

Th e elastic load-deflection relation given by (3.65) becomes inviilid wlicna . as given by (3.67) equals the yield stress.

Of the three expressions, (3.57), (3.58), and (3.60), used to determi ne Lllcelastic load-deflection relat ion, only (3.58), the moment-curvature relation,must be revised when the elastic limit is exceeded. To determ ine tlie inelastic

moment-curvature expression tliat is used in place of (3.58) in the illelasticrange, let us consider the stress distributions d epicted in F ig. 3-9. As indicatcd,

Stress distributionfor small e=M/P

0)

Stress distributionfor large e = M/P

. b)

Art 3 5 Failure of Beam Columns 165

The distances c d and are defined in Fig. 3-9a; and a and a re, respec-tively, the yield stress acting at the extreme fiber on the concave side of themember and the tensile stress acting at the extreme fiber on the convex side.

The internal moment is obtained by taking the moment of all the forcesabout the cent ro ida l axi s. ~ h 6 s

Noting that c + h Eqs. (3.68) an d (3.69) can be solved fo; c. Aftersome, fairly involved algebraic manipulation s, which are not reproducedhere, one obtains

~ r o n ; ig. 3-921 it is evident that

where p is the radius of curvature.

Fig. 3-9 Stress distribu tion for beam colunin in inelastic range. Adapt ed . Ifrom Ref. 1.12.)

Thus

two different distributions of stress are possible. If the ratioe MIP is

relatively small, yielding occurs onlyor1 the concave side of t he inember pr iorto failure . This case is depicted in Fig. 3-9a. On the o ther ha nd, ife is relativelylarge, both the convex as well as the cponcave side of the member will liavestarted to yield before the maximum load is reached (see Fig. 3-9b). Tosimplify the analysis, we shall restrict g,urselves to small values of and thuslimit our concern to the stress distribution in Fig. 3-9a.

Equilibrium of forces in the s irection gives

.whi ch, r..?ter dividing b oth sides by bli can be written as

from which

Finally, s ubstituting the expression fo r cgiven in (3.70) into (3.73) leads to

Thi s is the inelastic moment-curvature relation tha t must be used in place ofEq. (3.58) once the stresses have excqeded the proportional linit.

In view of(3.57), (3.61), and (3.63), the curvature and mom ent at m idspanare given by

67

qq7

a n d P e 8


12/15

.

166 Beam Columns Clr. 3 A r t . 3 . 5 F a il u re o f ~ e a m o lu mn s 7 7

Substitution of these expressions in to (3.74) leads to Table 3 1 Load dellcction data-for beam column

. urnax

(ksi) 8 11 (ksi (ksi) 61h

2 0.036 6.4 8.0 0.21Since the Euler stress can be expressed as 4 0.080 14 8.5 0.24

6 0.137 23 9.0 ,0.30a2EI x2EhZ 8 0.212 34 9.1 0.35us= =AI 1212 d 10 0.314 - 9.0 0.40

12 0.46314 0.7 10 -Eq. (3.75) can also be written in th e form b) Inelastic Range Eq. 3.76)16 1.150 -

6 e Z 3) ] - I ) (3.76)[ a, 54 a a,

.i:>; ;..

Equatio n (3.76) gives the load-deflection relation in t he inelastic range. It can

be used from the on set ofyi eldi; lg up to failure, provided failur e occurs beforeyielding commences on the convex side of the member.

With the aid of Eqs. (3.65) a n d (3.76) it is possiblc lo obtiiin lllc snli1.cload-deflection curve, from thcbeginning ol lo ;~d ingo failur c, Sorilnymcm-ber that falls within the limit;ltio~ls utlincd at thc start of thc ntlalysis.

As an example, let us consider a simply supported, rectangular, steelbeam column with the following dilnensions and properties:

length I = 120 in.radius ofgyrat ion r = 1 in.ratio of moment to axial load e = 1.15 in.yield stress a = 34 ksimodulus of elasticity E = 30 x lo3 ksi

Based on this. dat a

and h = 2 E .- 0.6 ksi(J/r)2

The load-deflection data fo r the elastic range, obtained usingEq. (3.65),are given in columns I a n d 2 of Table 3.1~1.Corresponding to each set ofvalues for a, and d/lr listed in the table, the maximum stress has been deter-

mined using Eq. (3.67). Th e latter is given in colum n3 of the table. It is evidentfrom these data that the i~~a xirn ulntl.ess i n the 111embcr rcnchcs 34 ksi lhc

(a) Elastic range Eqs. 3.65) nd 3.67)

yield Stress, at approximatelya, = 8 ksi., As a result, Eq. (3.76): the inelastic

load-deflection relatio n,mu st be u sed to obtain deflections for axial stressesin excess of 8 ksi. The load-deflection data f or the inelastic range, obtainedusing Eq. (3.76), a re listed in Ta blc 3- b..

Th e entire load-deflection curvc, including both the elastic and theinelastic portions, is plotted i n Fig 3-10. The solid line represents the actualbehavior of the member. A dashed line denoting the invalid part of theelasticcurve is also included for comparison. U p toa, = 8 ksi, the material obeysHooke s law and the deformations are relatively small. However, as soon asyielding spreads beyonda, == 8 ksi, there occurs a noticeable decrease in thestiffness of the mem ber.T his decrease builds up fairly rapidly until at approxi-mately a, = 9.1 ksi th e mem ber is no longer ab le to resist an increase in load.In othe r words, a, = 9.1 ksi represents the maximum load that the membercan support .

The results obtained here for a rectangular section and for a perfectelastic-plastic material are typical of the behavior exhibitedby other shapesand oth er materials. However, the determination of the maximum load formost-other sections andm ateria ls involves considerably more effort than wasrequired to analyze th e rectangular section with t h e perfect elastic-plasticmaterial. In the majority of instances, closed-form solutions of the typepresented here are o ut of the question, and numerical meth ods are the onlymeans available for obtaining the m aximum load. On e such numerical solu-tion for the maximum load of a structural-steel1 beam is given by Galambos

.

and Ketter (Ref. 3.5).

In view of the fact that thedetermination of the maximum load of a beamcolumn is invariably cot~lplcx nd time consuming, the laada t which yielding


13/15

168 Beam Columns Ch. 3 A r t . 3 6 Design of Be ah Columns-Interaction Equation 169

member, the calculat ions would h nic been evc nm ore coinplex and fAr moretime consl~ rning. he impracticability of obtainin g the collapse load of a beamcolunln by purely theoretical procedures an d the need for an empirical designformula are th us self-evident.

W h e n a member is subject to a combined loading, such as bending and

axial compression, an interaction equation provides a convenient way ofapproximating th e ul t imate s trength (Ref.3.6). Knowing th e strength of themember in both pure compression and pure bend ing and knowing thatthe

less compression a nd less bending when both o f these loadsif.either is acting by itself, one con estimate how much

bendhig a nd com pression can be resisted-if both a re present. Such an approxi-mation can then be verified experimentally.

To develop a n interaction equatio n fo r combined 'bending ,and axialcompressiorl, let us introduce the ratiosPIP a n d MJM here

P = axial load act ing on the me mber at fai lure when both axial com-. pression and bending a re present

. h = ultimate load of the member when only axial compression ispresent, that is, the buckling load of the member

M = niaxi~nurn primary bending mom ent act ing on the -member at. f i~i lurewhen bo th bcncling end ax ial compression ex.ist; this

0 0.2 0 4 6 0.8 10cxcludcs thc amplification in thc morncnt duc to prcscncc of theaxial loatl

Fig. 3 10 Load-deflection curve for beam columnM = ulti~nate ending morncnt when only bending exists, that is, the

plastic moment of the sectionbegins has often been used in place of the maximu m lo ad as the limitofstructura l usefulness. The load corre spond ing t o initial yielding is an attrac- Let us now calculate the above ratios for the rectangular beam column

tive design criteria. because it is relatively eilsy to obtain and it givesa ilnalyzed in Article 3 5. For tha t m ember the axial stress at failure was found

conser :dtive estinlate of the actua l collapse loa d. H owever, it does have the to be0 --

9.1 ksi and the Eulcr stress isa :=

20.6 ksi. .Thusdisadvantage of being often too conservative. Fortunately, there has been

P 9.1developed a n alternative semiempirical design criterion that is both ;~cc.tirate -- - 0.44P a 2 0 . 6 -an d relatively easy to use. This design iterion, th e interaction equallo n, isconsidered in th e following article. The rat io of the niaxi~num rimary momcnt a t failure to the plast ic mom ent

of the section can bc written n the form

3.6 DESIGN OF BEAM COLUMNSNTERACTION EQUATION

In Article 3.5 the col lapse load of a beam column was calculatcd. To sin~plifythe a nalysis as mu ch a s possible, a. very idealized mem ber was chosen, arectangular sect ion mad e out o fa perfect elastic-plastic material. Nevcrtiicless,fairly lengthy and c omplicated calculations were neededt o obtain the desiredresult. Ha d we attem pted to determ ine the m axiin un~ oad for sc)nlc olliel-

Subst i tut ion of lh .33 a and a = 3 4 gives

m . .


14/15

..

. ..

170 ' Beam Columfrs Ch 3 Clt 3 References 77

The rat ios PIP, = 0.44 and cl4/M = 0.35 give the ~l iaximum ?i~lucsl' lineI

P and h.1 that t l ie recta~igularbeam column with e / h == 0.33 a~?slyz ccl nArticle 3.5 can resist. In Fig. 3-1 this result s shown plotted as point /I O I I ;I -- -1 , 1.0

u A I,,(3.77)

Fig 3-1 Interaction equation for beam column.

depicted by th e dashed line in Fig. 3-1 I. All the theoretically and ex pe rh en -tally o btaine d failure loads included in th e figure fall below this curve. It can

therefore be concluded th at Eq. (3.77) gives an unconservat ivtest imate of th eI maxim um stre ngth of beam colum ns and is not a satisfactory design criterion., .

The reason for the discrepallcy betweenEq, (3.77) and.the actual failure: loads is that M in the equation, is only the primary part o fth e total momenti

tha t sits on the mem ber. In other words,M does n ot includ e-the secondarynloment p roduc ed by th e product o f thc axial load and t he lateral deflection.It was shown in Article 3.2 that the preseocc of an axial load amplifies theprimary bendin g mo ment roughly by the rat io 1/[1 (PIP,)]. If this factor isincorporated into Eq. (3.77), one ob tains

giaph whose ordinate is PIP and whosc abscissa is MIM . C:~lcul:itio~i::similar to those leading to point 4 II;I\T bccn cnrricd out by .Ic;r.ck 1tcl.s. 3.3and 3.4 for rectangular beam columns with various values ofe l;, :ind tlicresults thus obtained i r e tabulated by Bleich (Ref. 1.12). Points13 and C in

t h e f igure have been plot ted using these da ta.I'n addition to these results,depicted by squa re points, the figure includ es tllree circular points giving ihcmaxiniuni loads for structural-stecl I beams and two tr i: i~igular po in ~s hat

correspond to aluminum-alloy tubes. T he data used t o plot the circular pointswere obtained by Galambos and Ketter (Ref. 3.5) using a numerical Integra-tion nietllod, and th e failure loads represented by the triangular points wereobtained experimentally by Clark (Ref. 3.7);

The lack of scatter exhibited by the failure loads plotted in Fig. 3-1indicates that in all probability a single analytical expression can be foundwhich will predict the ma xim um loa d fo r a variety of different beam columns..Such a relation, which is also simple enough to be useful in routine engineer-ing, will now be developed.

It isfairly obvious th at PIP, = O when MlM, = 0 and that MIM, == 1.05-hq P /P. = 0 The desired CUNe must therefore pass through the points61, 2:): 0, 1). The simplest expression tha t satisfies this criterion is the straight

Th is relation is shown plotted as a solid line in Fig. 3-1I It is evident that Eq.(3.78) agrees much better with the actual failure loads than did the stiaightl ine and tha t Eq. (3.78) appears to offer n satispdctory design criterion.

Although agreement has been shown to exist betweenEq. (3.78) and onlya limited num ber o f cascs, Eq. (3.78) is actui~ lly ble to predict the ultimateload for n largc varicly o situations. Thc ccluation is applicable toI beams aswcll as rectangular sections. and to aluminun i as wcll as steel. F urthe rmore , itmakes no difference whether the primary nlomcnt is due to eccentric axialloading or to transverse loads or to a combination of the two. The onlyrestriction is that the maximum momcnl occur at or near the center of thebeam. Equation (3.78) is still applicable if this condition is not satisfied.

However, a suitable factor must be introduced in the moment term of theequa tion (Rcf. 3.8). Jn view of the fact thatEq. (3.78) is both simple to applyand renlarkably ac curate for a large numb er of different situations, it is usedextensively asa design criterion for beam columns.

eferences

3.1 J. I. PARCEL nd R. B. B. MOORMAN,nalysis of Sfalically indererminateStrrrctrrres (New Yo rk: John Wiley Sons, Inc.,1955).

3.2 G . WINTER,. T. HSU,B. KOO, nd M. H. LOH, Buckling of Trusses andRigid Frames, Cornell Unive rsity Ellginrering E-rperimental Station BulletinNo. 36 Ithaca, N.Y., 1948.


15/15

172 Beam Columnsa

Ch 3

3.3 K. JEZEK N a h e r ~ ~ ~ ~ g s b e r e c h n ~ ~ ~ l ger Tragkraft exzentrisch gcdriickter~ t ah l s tbbe , Dr r Sialrlborr Vol. 8, 1935.

3.4 K. JEZEK Die Tragfiihigkeit axial gedriicktel. und n~l f iegi~ ng conspruchIcr-Stahlstabe, Der Stal:lbatr Vol. 9 1936.

Ch. 3 Problems

P3-3 is given by the implicit relation

3.5 T. V. GALAMBOS nd R. L. KETTER Columns Under Combined Bending and Noting thatThrust, Traiisactioiis ASCE Vol. 120 1955. P P( -t u m , x c

3.6 F. R . SHANLEN: nd E. I. R Y D E R Stress Ratios. Avio/iorr ~01. '36, o . 6, 1937.uy umax I

3.7 J W C L A R K ,Ec~c~i t r ic :~IIyoiided Columns, ~ c r t r s a c / i o ~ r s S C E Vol. 120 derive the relation.1955. Th e terms used in the relations are defined as follows:

3 8 W MCGUIRE, te el S ~ I - I I C I I ~ ~ ~ SEnglewood Cliffs, N.J.: Prentice-Hall, Inc.. d = cross-sectional area

1968). c = distance from neutral axis to extreme fiberr = radius of gyration

a = yield stressProblems = maximum stress

3 1 Obtain expressions for the niaxin~unl eflection and maximum monicnt ofabeam colu mn whose ends arc built in an d that is loaded witha concentratedload a t midspan as shown n Fig. P3-1.

3.2 ~ e t e r r n i n che niaxinium moment for ;I beiilii column that is bent inn reversecurve as show n.in Fig. P3-2, when

(a) PIP = 0 2b) PIP = 0 8

where P = nZEIIL2. n view of the foregoing results, what can be concludedregarding the maximum moment in a beam column with reverse curvature?

3.3 The load P a t which yielding commences in th e bean1 column s hownin Fig.

I1 Fig. P3-3

beam column behaviour

Documents