Plastic Buckling of Columns: Development of a Simplified Model of Analysis

Transcription

1 Plastic Bucklig of Colums: Developmet of a Simplified Model of Aalysis Filipe Pereira filipe.s.pereira@tecico.ulisboa.pt Istituto Superior Técico, October, 016 Abstract Colum bucklig is a pheomeo usually associated with sleder colums, that buckle i the elastic rage. However, colums made of elasto plastic materials of lower slederess may also buckle i the plastic rage. For these colums, the maximum load bearig capacity is greater tha the bucklig load. The difficulty behid the aalysis of the colum bucklig i the plastic rage lies i the fact that, right after bucklig occurs, fibres i the covex side of the colum start to elastically uload, so, for every step alog the aalysis of the post bucklig behaviour of the colums, each cross sectio poit may be i oe of three states: plastic loadig i compressio, elastic uloadig or plastic loadig i tesio. This dissertatio aims to study the plastic bucklig of colums. A simplified model is proposed based o the assumptio that exists a direct relatio betwee the curvature ad the trasverse displacemet of the mid-spa sectio, makig it possible to obtai the complete post-bucklig equilibrium path. This simplified model is the geeralized, i order to allow for a more accurate descriptio of the colums, by aalysig the equilibrium of multiple cross sectios. These models are the evaluated by comparig their results to those of fiite elemet models. At last, the proposed model is used to study a set of colums of differet geometrical ad material characteristics. 1 Itroductio I structures, colums are elemets which are desiged to trasmit loads, both vertical ad horizotal, to the foudatios. I most structures, durig its life cycle, the vertical loads are predomiat ad, cosequetly, the study of colums uder compressive loads has bee a subject of iterest for ceturies. The simplicity associated with the geometrically liear aalysis of these elemets is goe i the problem whe the colum s bucklig has to be studied. Havig to combie the geometric ad material o-liearities whe studyig the bucklig behaviour of colums with elasto-plastic materials makes this problem much harder to aalyse. The first approach to describe the pheomeo of bucklig was made by Euler, whe he determied the critical load which causes a elastic colum to buckle uder a compressive force P cr = P E = π EI L e ( 1 ) where E is the elastic modulus of the material, I is the momet of iertia of the cross-sectio about the flexural axis ad L e is the effective legth of the colums, which is simply the legth of the colum for simply supported colums. Studies made sice the 19 th cetury, ad throughout the last cetury have cocluded that the bucklig load of colums i the plastic rage may be determied by the same expressio, but i which the elastic modulus is replaced by a taget modulus E t, which is the modulus at which the stress of poits o the yield surface evolve i the plastic rage. P cr = P t = π E t I L e ( ) This expressio was first proposed by Egesser i 1889 [1], but whe he first did it, he made so uder wrog assumptios, by ot takig ito accout the elastic uloadig that happes i the covex side of the colum. As a result, i 1895 [] he corrected his origial theory by creatig the reduced modulus theory P cr = P R = π E R I L e ( 3 ) The reduced modulus, E R takes ito accout the existece of both plastic ad elastic domais i the cross-sectio, therefore the coditio E t E R E is always true. Vo Karma supported this ew perspective ad determied expressios for the reduced modulus [3]. However, later studies established that P t does, ideed, predict the bucklig load for the situatio of plastic bucklig, because at the istat whe bucklig occurs, every poit i the cross-sectio is uder plastic compressio, ad thus have their behaviour cotrolled by 1

2 E t. Oe of the authors who cotributed to this coclusio, Shaley [4], also determied P t to be the lowest possible bucklig load, sice loads ragig from P t to P cr are all possible bucklig loads. This load, P t, is the oe which coditios the behaviour of the imperfect colums The post bucklig behaviour of the colums was, however, hard to determie. I the decade of 1970, Hutchiso [5], with the study of his cotiuous model, explored the importace of the elastic uloadig i the post-bucklig behaviour of the colums. Nevertheless, the post-bucklig equilibrium path was t accurately reproduced, ofte because of icorrect modellig of tesio yieldig at the covex side. This dissertatio aims to develop a simplified umerical model of aalysis of the plastic bucklig, adoptig a simplified hypothesis which makes it possible to establish the equilibrium of the colum mid-spa cross-sectio ad get the complete post-bucklig behaviour of the colums, correctly takig ito accout the effects of plasticity, elastic uloadig ad tesio yieldig. It also presets a more sophisticated approach of the model, by geeralizig it i order to model the behaviour of multiple sectios throughout the colums legth. These models are the validated ad used i parametric studies. The models study simply supported colums, ad elastoplastic materials with plastic hardeig. The model is implemeted usig MATLAB. Plastic Bucklig.1 Goverig Equatios A deformed colum of legth equal to L, uder a compressive load P, ad i a geeric post-buckled positio, is represeted i Figure 1. At ay time of the aalysis, the equilibrium betwee the iteral ad exteral forces must be satisfied, F it F ext = 0 ( 4 ) For situatios like the oe show i the previous figure, the exteral forces ca be represeted as N = P ( 5 ) M = P(w + w 0 ) Where w is the trasverse displacemet ad w 0 represets the geometrical imperfectios which may be preset. O the other had, the iteral forces are etirely depedet o the stresses, σ 33, which develop i the cross-sectio as a result of its deformatio profile, ε 33 (x ) = ε(x ) = ε g + χ x ( 6 ) where ε g is the axial strai of the cross-sectio ad χ is its curvature. So, for the elastic case, the stresses i a cross-sectio are writte as σ 33 = E ε 33 ( 7 ) For the plastic case, this expressio becomes, for each poit i the cross-sectio σ = E (ε ε p ) ( 8 ) where ε p the plastic strai. The relatio betwee this variable ad ε depeds o the loadig history of the cross-sectio poits. I ay case, the iteral forces of a cross-sectio may be writte as N = σ 33 da A M = σ 33 x da A This allows for the equilibrium to be writte as N N = σ 33 da + P = 0 A M M = σ 33 x da P(w + w 0 ) = 0 A. Elastic Behaviour ( 9 ) ( 10 ) Takig i cosideratio the equatio of the equilibrium of momets, it is possible to determie the expressio which gives the post-bucklig displacemets of a colum with elastic behaviour. From M M = 0 we ca get, for a perfect colum EI χ(x 3 ) P w(x 3 ) = 0 ( 11 ) From the hypothesis of small displacemets ad Euler- Beroulli s theory of thi beams, the curvature is equal to so ( 11 ) becomes χ = d w(x 3 ) dx 3 ( 1 ) Figure 1: Deformed colum EI d w(x 3 ) dx 3 P w(x 3 ) = 0 ( 13 ) The solutio of this equatio [6] is the oe which allows the descriptio of the colum bucklig displacemets. It results i

3 .3 Plastic Behaviour w(x 3 ) = B si ( πx 3 L ) ( 14 ) The expressio ( 14 ) is o loger valid whe the colum starts to experiece plasticity. The colum is o loger respodig accordig to the elastic modulus, E, i every poit of its cross-sectios ad as a result, the way it deforms will ot be simply described by a sie wave. Besides the correct formulatio of the trasversal displacemets, perhaps the biggest challege, as it was previously stated, is the correct evaluatio of the stresses alog poits of the cross-sectio. The evaluatio of ( 8 ) requires the defiitio of ε p at each step of the post-bucklig aalysis. The plastic strai evolves for poits beig loaded ad i yield. For these poits, it ca be writte f = σ σ y (ε p ) = 0 ( 15 ) where σ y (ε p ) deotes the strai hardeig law, which is the size of the yieldig surface ad is depedet of ε p, the accumulated equivalet plastic strai ad the costitutive law beig applied. Note that the variatio of ε p ad ε p relate by ε p = ε p. A icremetal evaluatio allows to determie the chage of all these properties. Assume ε0 p ad ε 0 p at the begiig of a icremet. If f trial = E(ε ε 0 p ) σ y (ε ) 0 p > 0 ( 16 ) the values of ε0 p ad ε 0 p must be updated accordig to ε p = ε 0 p + Δε p ε p = ε 0 p + Δεp ( 17 ) For the poit to be withi the yieldig surface this leads to ad Δε p = f trial E + H dσ dε = E E E + H ( 18 ) ( 19 ) where H is the plastic modulus which depeds of the chose costitutive law. Three such laws were cosidered i this dissertatio: the biliear, the triliear ad a oliear law. Figure 3: No-liear costitutive law By evaluatig ( 10 ) it ca be oted that the iteral forces are solely determied by the deformatio profile, ad, therefore, ε g ad χ. So these expressios ca be writte as N(ε g, χ) + P = 0 ( 0 ) M(ε g, χ) P(w + w 0 ) = 0 The relatio betwee χ ad w is the basis behid the developed simplified model. Assumig the bucklig mode ( 14 ) remais valid i the plastic case, we have So χ(x 3 ) = d w(x 3 ) = B π dw L si (πx 3 L ) ( 1 ) χ(x 3 ) w(x 3 ) = π L χ(x 3 ) = π L w(x 3) ( ) This relatio turs the system of equatios defied by ( 0 ) ito a solvable icremetal successio of systems of two o-liear equatios. 3 Developmet of the Simplified Model 3.1 Simplified model The simplified model is about extedig the validity of the relatio preseted i ( ) to the plastic bucklig aalysis. So, at the mid-spa sectio, startig icremets are made to w, startig from 0, chagig the value of χ at the same time. This results i a situatio where the equilibrium, ( 14 ), is ot satisfied. R N 0 = N(ε g, χ) + P 0 R M 0 = M(ε g, χ) P(w + w 0 ) 0 ( 3 ) Therefore, a iterative process must be made to fid the values of ε g ad P which re-establish the equilibrium For the re-establishmet of the equilibrium, the Newto- Raphso method is used, ad may be described as Figure : Biliear ad triliear costitutive laws R = R 0 + dr dδx ΔX + ( ) = 0 ( 4 ) i which R 0 groups the equilibrium errors at the begiig of the icremet ( 3 ), ΔX groups the variatios of the depedet variables of the problem, Δε g ad ΔP, ad dr dδx 3

4 is the taget matrix, which dictates the way R will chage i each iteratio, util the situatio of R = 0 is reached. The defiitio of dr dδx is as follows R N R N dr dδx = ε g P ( 5 ) R M R M [ ε g P ] After the defiitio of this matrix. the depedet variables i the ext iteratio is X = X 0 dr R dδx 0 ( 6 ) The updated value for P is used to defie the ew exteral forces 1 N = P M = P (w + w 0 ) ( 7 ) whereas the updated value for ε g is used to defie the ew deformatio profile ε 33 = ε g + χ x ( 8 ) The stress σ 33 is re-calculated, which results i the updated values of the iteral axial force ad bedig momet, N ad M. It is also i this step that the taget matrix is updated. After these steps are take, the values of R N ad R M are evaluated ad if they are ot reasoably close to zero, the iterative process is repeated. Otherwise, a ew icremet of w takes place, ad the calculatio of the post-bucklig trajectory cotiues. 3. Geeralized Model The geeralizatio of the simplified model cosists i expadig the aalysis by cosiderig multiple crosssectios throughout the colum legth. For sectios aalysed, sie fuctios must be cosidered, each correspodig to higher levels of bucklig modes. They are defied as w j = B j si ( kπx 3 L ) ( 9 ) where j = 1,,, ad k = j 1 correspod to the odd, symmetrical, bucklig modes (see Figure 4). Because of the symmetry of the problem, the sectios aalysed are all i oe half of the colum, ad they divide the colum i equal legths. Figure 4:Represetatio of 3 of the bucklig modes which ca be cosidered For this model, the icremetal variable is B 1. It s the chage i this variable which leads to the o-satisfactio of the equilibrium equatios which are ow N i (ε gi, χ i ) + P = 0 M i (ε gi, χ i ) P (w i + w 0i ) = 0 I which i deotes the aalysed cross-sectio. ( 30 ) By havig B 1 as a idepedet variable, this problem becomes depedet of variables: the axial strai at each sectio aalysed, ε gi, the remaiig factors for the sie fuctios B,, B ad the load P. The variables χ i = χ(x 3 = x i ) ad w i = w(x 3 = x i ) are t direct variables of the problem because they are solely depedet o the B j variables. They are obtaied as w i = B j si ( kπx i L ) j=1 χ i = k π j=1 L si ( kπx i L ) B j ( 31 ) ( 3 ) where w 0i defies the value of the iitial geometrical imperfectio at sectio i w 0i = w 0 (x i ) = B 0 si ( πx i L ) ( 33 ) Fially, we ca write similarly to what was previously doe R 0 Ni = N i (ε gi, χ i ) + P 0 R 0 Mi = M (ε gi, χ i ) P(w i + w 0 ) 0 ( 34 ) I this problem, the, we have a system of equatios to be solved by chagig the variables metioed before. Oce more, the re-establishmet of the equilibrium lies i the applicatio of the Newto-Raphso method ( 4 ), the differece beig that the umber of equatios ad variables is differet. X groups ε gi (for i = 1,, ), B j (for j =,, ) ad P, so ΔX groups their variatios. The taget matrix is for this case dr dδx = [ R Ni ε gi R Mi ε gi R Ni B j R Mi B j R Ni P R Mi P ] ( 35 ) 4

5 P(kN) with i = 1,, ad j =,,. The process that follows is the same which has already bee explaied for the simplified model. The variables are updated accordig to ( 6 ), however, sice the values of B,, B chaged, the trasverse displacemet ad the curvature i each sectio also has to be updated, accordig to ( 31 ) ad ( 3 ), resultig i ew values at each iteratio, which we ll call w i ad χ i. The exteral forces are updated as i ( 7 ), with N = P i every sectio aalysed, ad M = P (w i + w 0i ). The deformatio profile is also updated through ε 33i = ε gi + χ i x ( 36 ) The procedure for the calculatio of the iteral forces for each sectio is the same as it was previously explaied. Oce more, after these steps are take, the values of R N ad R M are evaluated ad if they are ot reasoably close to zero, the iterative process is repeated. Otherwise, a ew icremet of B 1 takes place, ad the calculatio of the post-bucklig trajectory cotiues. 3.3 Numerical Implemetatio The models described were implemeted i MATLAB, where fuctios were developed for the aalysis of simply supported colums with rectagular ad I cross-sectios (Figure 5 ad Figure 6). Figure 7: Discretizatio of the cross-sectios Its most iterestig outputs are the complete loaddisplacemet trajectory of the colums, ad the possibility of savig for each icremet of the fuctio, the evolutio of the stresses i each of the itegratio poits. 3.4 Comparig the Simplified ad the Geeralized Models I order to evaluate how the simplified model compares to the more sophisticated geeralized model, a compariso was made, aalysig the resultig trajectories. For this compariso, it was chose a colum with the followig properties: b = 0,00 m (the dimesio i the directio of bucklig), h = 0,00 m, L =,5 m. It was chose a biliear law, with E = 10 GPa ad E t = 63 GPa (E t E = 0,30), ad a iitial yieldig stress f y = 35 MPa, with a discretizatio of the cross-sectio i 00 areas of itegratio alog b (dx = 1 mm) The results were the followig (see Figure 8): Figure 5: Simply supported colum ,0 0,04 0,06 0,08 0,1 0,1 Figure 8: Load-displacemet trajectories for the colum, for several cross-sectios. The dashed lie is the simplified model. Figure 6: Cosidered cross-sectios The fuctios allow for the choice of 3 differet costitutive laws: biliear, triliear ad o-liear. Geometric ad material imperfectios may be take ito accout. The program s iputs are the geometrical (the dimesios of the colum ad the cross-sectios as show above) ad the material properties (the parameters which defie the costitutive laws. The discretizatio of the cross-sectio, i.e., the dimesio of the areas of itegratio, may also be chose, by choosig the values of dx ad dy. The dashed curve, which stads out, is the result of the applicatio of the simplified models. The other is a superpositio of curves resultig from the use of the geeralized model, for, 3, 4, 5, 7 ad 10 sectios. At the scale preseted they are all idistiguishable. The aalysis made of the results has show that the differece betwee the results of the 5 ad 10 curves are miimal, so the model with 5 sectios was take as referece. Regardig the simplified model, the resultig curve is clearly differet, but still a very good result, cosiderig the simplicity of the model. Also, evaluatig the relative error, it results that, although the error i the displacemet 5

6 P(kN) at the poit of maximum load is aroud 16%, the error i the estimatio of the load itself was oly 0,93%, relative to the result of the 5 sectios curve. 3.5 Ifluece of the Number of Itegratio Areas I the last sectio, it was revealed that, although the umber of cross-sectios cosidered alog the spa of the colum ifluece the results, they are t largely differet. I this sectio, for the same colum ad for the simplified model, it was studied the ifluece, i the results, of usig differet umbers of itegratio areas. The umber of areas cosidered were, 4, 8, 16, 5, 50, 100, 00 ad 000. The results led to the coclusio that, for this colum, the examples with less tha 16 areas do t come close to reproducig the post-bucklig trajectory of the colum as see i Figure 9: Load-displacemet trajectories for the colum, aalysed with differet umbers of itegratio areas. Bottom curve: areas; top curve: 16 areas., where the results for the use of, 4, 8 ad 16 itegratio areas are show Figure 10: Real stress profile (dashed lie) versus the stresses i the itegratio poits, xi(1) ad xi() The use of a higher umber of itegratio poits gradually lead to better results. The curve with 000 areas precisely reproduces the bifurcatio load estimated by Egesser s expressio, P t, for this colum. For all the subsequet aalysis, 00 itegratio areas were cosidered. 4 Model Validatio Usig a Fiite Elemet Aalysis 4.1 Plae Stress Model The same colum aalysed i sectio Figure 8 was modelled ad aalysed usig ADINA [7], which is able to perform physical ad geometrical o-liear aalyses. The results give by ADINA i the plae stress aalysis are here compared to the oes of MATLAB where 5 sectios were cosidered. The plae stress model defied i ADINA is show i ,0 0,04 0,06 0,08 0,1 0,1 Figure 9: Load-displacemet trajectories for the colum, aalysed with differet umbers of itegratio areas. Bottom curve: areas; top curve: 16 areas. Figure 11: Defied plae stress model The imperfect model was made by offsettig P upwards, so that the middle plae of the colum had a triagular shape, like show i The trajectories are much more iflueced by the lack of discretizatio alog the cross-sectio tha alog the colum s legth. This is because the lack of itegratio poits does t allow for the correct evaluatio of the stresses alog the cross-sectio. The pheomeo of elastic uloadig, which was prove to be of great importace, eve datig back to the studies of Hutchiso, is oly very roughly reproduced by the models with few itegratio poits. That idea is exemplified i Figure 10, where itegratio poits were used. Figure 1: implemetatio of the geometric imperfectio Sice the imperfectios i MATLAB were beig implemeted through a sie wave, i order to get a perfect fit betwee the results give by ADINA ad the oes give by the MATLAB model, chages were made so that the implemetatio of the imperfectios i the developed model also resulted i a triagular shape. To do that, the imperfectios were implemeted i the form of a Fourier series with a triagular shape, usig the first 5 terms. 8 w 0i = B 0 ( 1)( 1)/ si ( πx i π L ) =1,3,5, ( 37 ) 6

7 P(kN) P(kN) 4. Compariso of Results The results for the perfect colum ad for a imperfect colum with w 0 = 0,01 m are represeted i Figure 13 ad i Figure 14 (the MATLAB results are the red, dashed, curves, while ADINA s are the blue curves). Perfect Colum (w 0 = 0) Figure 13: Compariso of the results for the perfect colum. MATLAB red; ADINA blue Imperfect Colum (w 0 = 0, 01 m) ,0 0,04 0,06 0,08 0,1 0, ,00 0,0 0,04 0,06 0,08 0,10 0,1 Figure 14: Compariso of the results for the imperfect colum. MATLAB red; ADINA blue The graphics show that the geeralized model developed i MATLAB matches the results give by a sophisticated, ad harder to use, fiite elemets program. Although the results from the simplified model differ from these, it still offers a very good estimate of the maximum load. 5 Parametric Studies Havig established that the models created are able to reproduce colum s behaviour, i this sectio parametric studies are performed i order to better uderstad the post-bucklig behaviour for differet situatios. These studies were performed for colums of differet slederess ratios, λ = L i, but the colums were t directly picked by their slederess ratios, but rather for their bucklig/yieldig load relatios, P bif /P y. The effect of geometric (w 0 = 0,001 ad 0,01 m) ad material imperfectios was also studied for the biliear law. 5.1 Rectagular Cross-sectio with Biliear ad Triliear Laws The cross-sectio dimesios are b = 0,00 m ad h = 0,100 m. The parameters which defie the costitutive laws are σ y0 = 35 MPa, σ y1 = 450 MPa, E = 10 GPa, E t = E t1 = 63 GPa ad E t = 0. For this colum, we have P y = 4700 kn. The colum here represeted are: a) P bif = P t = P y (L =,10 m); b) P bif = P y = 1,5P t (L = 3,43 m); c) P bif = P y = P t (L = 4,0 m); d) P bif = P E = 0,5P y (L = 7,67 m). The resultig post bucklig curves for these colums were as show i Figure 15. The biliear law is represeted, i the results, with a black lie while the triliear is the red lie. P/P y 3,5 1,5 1 0, ,05 0,1 Figure 15: Resultig post-bucklig curves for differet colums with the two laws. Biliear-black; triliear-red It ca be see that i the d) curve, both laws are coicidetal, i the rage of results represeted. This happes because the secod yieldig level, for the triliear law, is t reached, which is the reaso why, i curves b) ad c), the results diverge. I curve a) the triliear law ca t reach the load of P bif = P t = P y, because that would correspod to a level of stresses applied i the a) b) c) d) 7

8 sectio which does t comply with the limit of σ y1 = 450 MPa. P y leads to a applied load of P = 9400 kn, ad the triliear law ca oly support a load of P = 9000 kn, so as it reaches this value it buckles ad immediately starts losig load bearig capacity. I geeral, it ca be said that the results are more adversely iflueced with ρ > 0. These results seem to be cosistet with the oes obtaied by Ritto Corrêa [8], i his study of the cotiuous model of Hutchiso. 5. Effect of the Imperfectios for the Biliear Law ρ = 0, 5 ρ = 0 ρ = +0, 5 Below, i Figure 17, it is show, for the same colums from the last sectio, the effects of geometric ad material imperfectios. The material imperfectios aalysed have the profile, alog b, as see i Figure 16. Figure 16: Material imperfectio, i the rectagular cross-sectio, alog b where the log dashed lies represet a geometric imperfectio of w 0 = 0,001 m ad the smaller dashed lies a imperfectio of w 0 = 0,01 m. It ca be see from the figure that: I a geeral way, the presece of geometric imperfectios lowers the maximum load; The colums of lowest slederess (L =,10 m) are particularly sesible to the geometrical imperfectios. It s show that eve for the smallest geometric imperfectio there s a big deviatio from the trajectory of the perfect colum; These colums are t very sesible to the material imperfectios; For the secod set of colums, the presece of material imperfectios results i a bigger bifurcatio load, for both ρ = 0,5. For ρ = +0,5 the bifurcatio load remais the same, but the iitial slope of the curve is bigger; For the third set of colums the bifurcatio load is raised for ρ = 0,5 ad lowers for ρ = +0,5; For the slederest colums, the bifurcatio is elastic. ad for the situatio with o material imperfectios, the bifurcatio follows a straight, horizotal lie, util the poit yieldig is reached. For the versio with material imperfectios, we ca see that they both buckle for the same value of P P y = 0,5, but the versio with ρ = 0,5 has a iitial break right after bucklig, ad a secod break aroud w = 0,075 m, whereas the versio with ρ = +0,5 seems to follow a steeper, cotiuous, descedig path after bucklig; Figure 17: Effects, i post-bucklig behaviour, of geometric ad material imperfectios 6 Coclusios I this paper, a simplified model for the study of the bucklig ad post-bucklig behaviour of elasto-plastic colums was itroduced, based i the applicatio, for this situatio, of the simple relatio that there is i the elastic case betwee the trasverse displacemet of the colum ad the curvature of its sectios. This allowed for the loaddisplacemet trajectory of the colums to be obtaied through the aalysis of the equilibrium, i the mid-spa cross-sectio, of the iteral ad exteral forces, through a icremetal/iterative method, with the equilibrium equatios beig checked for every value of trasverse 8

9 displacemet, w, cosidered, relyig o the variatio of oly depedet variables, ε g ad P. Whe cosidered the compariso with more sophisticated fiite elemets models, it was show that the simplified reproduces with, a good level of accuracy, the postbucklig behaviour of colums, which is, i itself, a very iterestig ad satisfyig result to be take. 7 Refereces [1] F. Egesser, Ueber die Kickfestigkeit gerader Stäbe, Zeitschrift für Architektur ud Igeieurwese, [] F. Egesser, Uber Kickfrage, Schweuzerische Bauzeitug, [3] T. Vo Karma, Utersuchuge uber Kickfestigkeit, Mitteiluge über Forschugsarbeite auf dem Gebiete des Igeieurweses, [4] F. Shaley, Ielastic Colum Theory, Joural of Aeroautic Sciece, vol. 14, [5] J. W. Hutchiso, Plastic Bucklig, Advaces i Applied Mechaics, [6] F. Virtuoso, Estabilidade de Estruturas. Coluas e Vigas-colua, em Folhas da Disciplia de Estruturas Metálicas - IST, 013. [7] I. ADINA R&D, Theory ad Modellig Guide, 015. [8] M. Ritto-Corrêa, Estabilidade Elastoplástica de Coluas: Estudo do Modelo Cotíuo de Hutchiso,