analytical solutions in elasto-plastic bending of beams with.pdf

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Analytical solutions in elasto-plastic bending of beams withrectangular cross section

Boris tok *, Miroslav HalilovicFaculty of Mechanical Engineering, University of Ljubljana; Akerceva 6, Ljubljana, Slovenia

a r t i c l e i n f o

Article history:Received 31 August 2007Received in revised form 14 March 2008Accepted 25 March 2008Available online 4 April 2008

Keywords:BeamsBendingElasticplastic materialFinite deflectionsAnalytic functionsPlastic collapse

a b s t r a c t

Deflection analysis of beams with rectangular cross section is considered under specificloading conditions, resulting in at most quadratic bending moment distribution, andassuming elasto-plastic behaviour with no hardening. Within the framework of smallstrain and small displacement approach analytical solutions are derived, which enableelasto-plastic analyses of beams to be performed in a closed analytical form. In conse-quence, clear tracing of the elasto-plastic response evolution with a propagation of theplastic zone through the volume, i.e. its spreading along the beams longitudinal axis aswell as its penetration through the cross section, is enabled as loads increase, from theappearance of a first plastic yielding in a structure till its collapse. With the derivation ofthe general solution, listed explicitly by Eqs. (21)(23), which was never presented inany article or book before, the presented article fills the gap in the analytical non-linearmechanics of beams.

2008 Elsevier Inc. All rights reserved.

1. Introduction

Bending of beam-like structures, being rather frequently addressed in technical practice, has been adequately and thor-oughly analysed, considering even more rigorous approaches, especially for elastic problems [1,2]. Elastic bending of beamsis nowadays still studied theoretically[3], mainly in purpose of dynamic behaviour[4,5]and in studying new materials, likecomposites[5], laminates[6], etc. Elasto-plastic analyses of beam-like structures, where by assumed formation of plastichinges limit fully plastic loads are evaluated, causing a structure to collapse, are treated within a framework of the limit anal-ysis theory[79]. Plastic hinge occurence can be predicted also using refined plastic-hinge theory, where evolution of plasticzone is replaced with gradual stiffness degradation up to zero at fully plastic section[10,11], or with introduction of singu-larities in stiffness at the location of plastic hinge[12]. Based on that, numerical applications for prediction of collapse ofbeams and frames are developed. However, considering a statement from[13]. . .in the presented elasticplastic hinge meth-

od, the material is assumed to be either perfectly elastic or fully plastic. . ., evolution of elasto-plastic behaviour cannot beappropriately modelled using variations of limit analysis.

Since in general the governing equation of the elasto-plastic beam bending problem is not solvable analytically, there areabove all numerical solutions and experimental results that are met in literature[1416]. In[17]deflection of elasto-plasticbeams is related to moment-curvature relationship. Elasto-plastic deflection is investigated with the development and test-ing of new materials: polymers, composites, laminates, reinforced materials, etc. Commonly, researches are focused into par-ticular technical problems, what from a theoretical point of view means that they are dealing with a particular boundaryproblem: simply supported beam[16,1821], pure bending of beam[22,23], bending of thread[24]. In classical books

S0307-904X/$ - see front matter 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2008.03.011

* Corresponding author. Tel.: +386 41 694 534; fax: +386 1 2518 567.E-mail address:[email protected](B. tok).

Applied Mathematical Modelling 33 (2009) 17491760

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describing plasticity of beams[7,2528]major attention is paid to stress state evolution in a particular cross section, but verylittle is usually written about deflection line of plastically deformed beams. Analytical solution is described at most for a con-stant bending moment distribution, while for a quadratic bending moment distribution it is derived for a particular problemonly[9,26]. In [29]authors among others calculate deflection line of a beam with rectangular cross section for a particularcase, in which two plastic zones occur at the same time. But the solution is found considering At the corresponding value ofthe load. . ., the presence of two fully plastic sections. . .causes an indefinitely large deflection of the beam . . .Since this is an extre-mely difficult problem, it is circumvented by the assumption that the beam is made of an idealized I section. . .. Still recently, nogeneral solution of this problem can be found. Namely, in Ref. [9]deflection lines for several cases with rectangular crosssectional beams were computed: cantilever beam with point load and with uniform load, simply supported beam with uni-form load, propped cantilever. . .and in all cases only one plastic zone occurs. All those cases are calculated one-by-one fromthe beginning, considering respective specifics of the analysed case, and no general equations applicable to all boundaryproblems are presented.

In this paper we concentrate, by assuming elasto-plastic material with no hardening, on the investigation of elasto-plasticbending of beams with rectangular cross section area. The functional solutions derived under the assumption of at most qua-dratic bending moment distribution enable fully analytical tracing of the elasto-plastic state evolution in structural compo-nents by monotonic and proportional application of loads to a beam structure. After presentation of the derived generalequations for elasto-plastic deflection lines the elaborated example will show, that determination of a deflection in the case,where two plastic zones coexist, is simple. Furthermore, because the solution method is general, there is no limitationregarding either the number of plastic zones or possible static indeterminacy.

2. Formulation of elasto-plastic beam bending problem

2.1. Basic assumptions, governing equations in general

Let us consider a straight beam of a cross sectional area A(x), (Fig. 1a), subject to elasto-plastic bending in the (x,z) plane,wherexandzare respectively, the longitudinal axis of the beam and a corresponding principal axis of the cross section. Forbrevity, let the area moment of inertia with respect to the principal axisybe denoted byI(x). While the material behaviour isassumed to obey Hookes law in the elastic region, no restrictions on the nature of irreversible inelastic response are posedfor the moment. Also, the BernoulliNavier assumptions on the cross section planarity and respective perpendicularity to theneutral axis are respected. The established stress state rij(x,y,z) is characterized, in accordance with the nature of the con-sidered problem, by the following resultants being zero:

ZAxrxy dATx 0; ZAx

rxzy rxyz dAMxx 0; ZAx

rxxydA Mzx 0: 1

Furthermore, it is assumed, that external loads are in accordance withZAx

rxxdANx 0; 2

while the remaining two stress resultantsZAx

rxxzdAMyx;Z

AxrxzdATzx dMyxdx 3

are non-zero, in general. By considering the zero stress components (ryy(x,y,z) =rzz(x,y,z) =ryz(x,y,z) = 0) the BernoulliNavier assumptions lead to a linear distribution of the axial strain exxacross the cross section height, and consequently, in

Fig. 1. Elastic and elasto-plastic stress distribution in a beam bending problem.

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case of elastic response (Fig. 1a) to a linear stressrxx(x,y,z) distribution, too. By introducing simplified notations rxx=randMy=Mthis relationship can be written, in accordance with Eq.(3), as

rx;z MxIx z: 4

In case of elasto-plastic response (Fig. 1b), which manifests when stresses exceed the yield stress r0(rxxP r0(x)), therespective stress distribution is governed by the nature of the actual response

rx;z signMx z apxqpx

r0x . . . jz apxj 6 qpx;

signz apxrpepx;z . . . jz apxj > qpx:

8>: 5

Here, the equivalent plastic strain epis taken as absolute value of the axial plastic strain exx=e, while rp(ep) denotes astress-plastic strain relationship, as determined from uniaxial tensile test. The range of the zcoordinate in Eqs. (4) and(5),z2 [h1(x), +h2(x)], is defined by the shape of the cross sectionA(x). While thexaxis performs actually as a neutral axisexperiencing r(x,z= 0) = 0 within the elastic response, Eqs.(2) and (4), the neutral axis is subject, in general, to a displace-ment from the elastic position in case of elasto-plastic response. The magnitude jap(x)j of the respective displacement in thezdirection is such that Eq.(2)is fulfilled. A part of the cross sectional area,z2 [ap(x) qp(x),ap(x) +qp(x)] \ [h1(x), +h2(x)],spreading centrally around the actual neutral axis is the area of pure elastic response. Provided that bending moment M(x) isgreater thanMe(x), i.e.jM(x)j >Me(x) > 0, whereMe(x) denotes the limit value of bending moment still causing only elasticdeformation in the cross section at a longitudinal position x, the two parameters, qp(x) andap(x), that describe the degree

of plastification and its impact on the displacement of the neutral axis, are obtained as a solution of the system of Eqs.(2) and (3), when considering the non-linear stress distribution(5). The respective equations take the following formZ

Apxsignz apxrpepx;zdA

ZAex

r0z apx

qpx

!dA0;

ZApx

signz apxrpepx;zzdA Z

Aexr0

z apxqpx

!zdA jMxj;

6

where the division of the cross section areaA(x) into elastic and plastic region,Ae(x) andAp(x), respectively is made in accor-dance with

Aex fx;z :jrx;zj 6 r0xg;Apx fx;z:jrx;zj > r0xg:

7

From the structure of the above system, Eq. (6), it follows that at a certain level of the externally applied loads bothparameters, the displacement of the neutral axisap(x) as well as the degree of the plastification of the cross section qp(x),are directly related to the actual intensity of the stress resultant M(x) and to the geometry of the considered cross section

A(x).When considering a beam structure that occupies a domain X,x 2 X, the stress distribution through a cross section A(x) is

subject, specific cases excepted, to a variation along the structure. The reason is either a variation of the stress resultant M(x)or a variation of the cross section A(x), or even a variation of both of them at a time. The onset of the first plastic yielding inthe structure, as well as subsequent spreading of the plastic zone, both through the cross section and along the structure, areprimarily dealing with the increasing of external loads, of course. In this context there will be, after a certain level of theapplied load is exceeded, a split of the initially elastic domain X Xeinto two parts, Xeand Xp, respectively. The subdomaincontaining those cross sectionsA(x) where the yield stress is not exceeded will be called elastic domain and denoted as Xe,while the remaining part of the domain X, called elasto-plastic domain and denoted as Xp, will comprise cross sections thatare characterized by the presence of plastic deformations (Fig. 2). In accordance with the given physical interpretation the

Fig. 2. Elastic and elasto-plastic domain decomposition.

B. tok, M. Halilovic/ Applied Mathematical Modelling 33 (2009) 17491760 1751


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domain decompositionX Xe [ Xpwill depend definitely on the actual load level, and will be performed on the basis of theestablished stress distribution r(x,y,z) in the structure, fulfilling the following relations:

Xe fx:jrx;y;zj < r0x . . . y;z2Ax ^x2Xg;Xp fx:jrx;y;zjP r0x . . . y;z2Ax ^x2Xg:

8

With regard to the limit elastic momentMe(x), which is actually only a function of the cross sectionA(x), the above rela-tions can be set, considering the established bending moment distributionM(x) in the structure, also as

Xe fx:jMxj 6 Mex . . . x2Xg;Xp fx:jMxj > Mex . . . x2Xg:

9

Finally, considering the BernoulliNavier assumptions and the existing relationship between curvature of the neutral axis,given by the respective radiusR(x)(Fig. 3), and its deflection w(x) =w(x,z=ap(x)) in thezdirection

ex;z z apxRx z apx

d2wxdx2

1 dwxdx

2" #32 z apx d

2wxdx2

10

differential equations governing the two possible cases, i.e. that of pure elastic response and that of evolved plastic strains inthe cross sectionA(x), can be deduced

d2wx

dx2 Mx

Ex

Ix

. . . x2Xe

signMx r0xExqpx

. . . x2Xp:8>>>>>: 11

2.2. Governing equations in case of rectangular cross section and material with no hardening

In order to escape from a stack of difficulties, which no doubt arise in solving of differential equation(11) in case of elasto-plastic response and arbitrary cross sectional areaA(x), we will limit ourselves in the sequel on the consideration of beams ofrectangular cross section, defined by width band height 2h, constant along thexaxis (A(x) = 2bh,I(x) = 2/3bh3). The materialproperties will be likewise assumed constant (E(x) =E,r0(x) =r0) and isotropic, also regarding hardening in tension and com-pression. In consequence, considering theyaxis is also a symmetry axis (h1(x) =h2(x) =h), the neutral axis will remain iden-tical to the center of gravity line, i.e. thex axis, regardless of the undergone plastic deformation (ap(x) = 0).

Further simplification, obtained by assuming no plastic hardening (rp(ep) =r0), yields stresses in a partially plastically de-formed cross section distributed in accordance with

rx;z signMxz

qpxr0 . . . jzj 6 qpx;

signzr0 . . . qpx Me)

qpx hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3 1 jMxjbh2r0

!vuut : 13

Fig. 3. Bending deformation of a beam.



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Considering respectivelyqp=hand qp= 0 the limit valuesMeand Mpof the bending moment, as shown inFig. 4, are ob-tained from this equation. Thus, with jM(x)j =Methe conditions for initiation of plastic deformation are established, whereasfulfilment ofjM(x)j =Mp, the latter being termed the fully plastic moment, causes the plastic zone to spread over the wholecross section

Me23 bh2

r0; Mpbh2r0)MeMp 23: 14

With the assumptions specified in this subsection the governing equation(11)of the bending problem, which is definedover a domain X (X=Xe [ Xp, Xe \ Xp= 0), transforms to

d2wxdx2

3Mx2Ebh3 . . . x2Xe;Kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

MpjMxjp . . . x2Xp;

80 x2Xp; 23a

K

2ffiffiffiffiffiffiffiffi

jm2j3p Txarsh TxjDj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 T2x

q ; signMxm2 >>>>: 23

Above,T(x) is the shear force distribution which is evaluated from Eq.(3)considering(19), andDis a constant obtainedfrom the second order polynomial coefficients as

D2 B2 4AC; 24

the considered polynomial being defined by the relationf(x) =Mp jM(x)j =Ax2 +Bx+C.Regarding the correct selection of the deflection line solution wP(x) in case of parabolic moment distribution M(x), Eq.

(23a,b), attention is to be paid to the established functional behaviour of the functionf(x) in the elasto-plastic domain Xp.This behaviour is characterized, as seen inFig. 5, by two physically different situations which appear alternatively, and lead,in consequence, to duality of the solution wP(x). Mathematically, the moment distributionM(x) given, these situations areuniquely defined by the product of the leading polynomial coefficient m2and the sign of the moment M(x). From the dis-played graphs of all possible moment distributionsM(x) it follows that the distributions (a) and (b), which are otherwisecharacterized by sign (M(x)) m2> 0, are equivalent with respect to the functional behaviour of the functionf(x), representedby curvature d2w(x)/dx2 < 0. Similar equivalence, yielding however curvature d2w(x)/dx2 > 0, can be attributed to the distri-butions (c) and (d) that are characterized by sign(M(x)) m2< 0.

4. On mechanical response properties by elasto-plastic loading

In terms of the given assumptions a closed form analytical solution for any boundary problem can be obtained by con-sidering the derived differential equations of the deflection curve, i.e. Eq.(15a)and(15b), respectively for elastic and elasto-

Fig. 5. Possible parabolic moment distributions causing plastic yielding.



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plastic response. It should be emphasized that with the stress resultants, i.e. internal forces, known the plastic domain Xpaswell as spreading of the plastic zone through the beam height(13)are uniquely defined. It is also characteristic for staticallydeterminate beam structures (Fig. 6a) that neither distribution nor magnitude of the internal forces along the structure areinfluenced by the presence of plastic strains in the structure. Regardless of the material behaviour and non-linear stress dis-tribution across a cross section the internal forces are increased proportionally by a proportional increase of the appliedloads. On the contrary, the described property is not valid for statically indeterminate structures (Fig. 6b). With ongoingof the plastic deformation the bending stiffness is actually subject to a continuous variation, affecting thus reaction forcesto vary in a non-proportional way in spite of a proportional application of the external loads. In consequence, the internalforces are changed non-proportionally as well. Regarding application of the derived differential equation of the deflectioncurve in case of demonstrated elasto-plastic response(15b)in a solution of the bending problem it should be noted thatits use in the analysis of statically determinate structures is needed only if deflections are investigated, whereas in staticallyindeterminate problems the solution cannot be achieved at all without its use.

Let give parameter k (0 < k 6 1) the role of loading parameter which defines the level of the applied loading measuredwith respect to the limit loading (k= 1), by which loss of structural functionality with transition to a mechanism occurs. Fur-ther, let kedenote the loading level corresponding to elastic limit loading, which is still characterized by absence of plasticstrain in the structure. Application of the loads that result in a linear elastic response ( k6 ke) is characterised by proportion-ality, regardless of the static (in)determinacy. Therefore

Mkx kke

Mex; 0 < k 6 ke; 25

whereMe(x) denotes functional dependence of the bending moment at the load level k=ke, andMk(x) is the respective mo-ment dependence corresponding to a load level k. On the other side, respecting the properties stated above, loading beyondelastic limit (k>ke) is characterised by

Mkx kke

Mex kMpx; ke


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where the obtained solution is explicitly derived and exact, is highly efficient and simple to use, what will be demonstratedby considering a numerical example.

5.2. Numerical example

Mechanical response of a simply supported overhanging beam, subject to a concentrated forceF(FP 0) at its free end andto a uniformly distributed loadq(qP 0) between the supports (Fig. 6a), will be considered in the sequel. Of importance forthe evolution of plastic strains there is, along with increase of the loading parameter k, also the ratio established between thetwo loads. It is actually this ratio that defines, by fixing a location of the maximum bending moment, the position of the firstplastic yielding. Let denote the considered load ratio by a coefficient w

w FqL1

: 27

Since by assumed proportional loading the ratio between the loads q and Fremains fixed for any load level k, the loadnotations may be enlarged in order to complete the information by adding index w, i.e.q ? qwand F? Fw. Application ofthe loads (dk> 0) is thus characterised by

Fw kFpw^ qw kqpw; 0 < k 6 1; 28whereqpwand F

pwdenote magnitudes of the loads causing collapse of the structure, which actually happens at the occurrence

of a plastic hinge.

Considering the bending moment functionM

(x

) (06x

=x

16 L

1^ 06x

=x

26 L

2) expressed in terms of non-dimensionalcoordinates n (nL1=x1) and g (gL2=x2), where obviously 0 6 n, g 6 1

M1n FwL2 m2w 1 n 1

nqwL2112

1 n wm

n;

M2g FwL2g 1 qwL21w

mg 1; mL1

L2;

29

three possible cases (Fig. 7) can be exposed in relation to the load ratio w, regarding the elasto-plastic state evolution. Whileprevailing of the loadq(0 6 w6 w1) results in plastification of the beam between the supports ( Fig. 7a), prevailing of the loadF(w3 6 w< 1) leads to the beam plastification at the right support (Fig. 7d), both regardless of the load level (ke


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w1 2 ffiffiffiffiffiffi

15p

2

!m; w2

32

ffiffiffi

2p

m; w3 7 2

ffiffiffiffiffiffi10

p

6

!m: 30

It is worth mentioning here, that magnitudes of the collapse loads qpwandFpware dependent on the load ratiowand geom-

etry aspect ratiom=L1/L2, the respective variation of the load capacity of the structure being depicted inFig. 8. Because ofthe established relationFpwwqpwL1the two diagrams, expressed in either of the two loads,qpwor Fpw, are actually equivalentand showing the collapse of the structure at a given load ratio and geometry parameters. The impact of each of the twoparameters,wandm, respectively, on the load capacity of the structure can be retrieved from the displayed graphs. Perform-ing an in-depth study it can be demonstrated, that it is mostly due to repositioning of the internal loads within the structurethat the load capacity is affected. In fact the maximal load carrying capacity, evidenced as a peak in the diagram, is obtainedat specific load ratio w=w2, which is characterized by occurrence of two plastic hinges theoretically at a same time, thusresulting in equivalence of the bending moment distributions ofFigs. 7b and c. At lower values of the load ratio w , w


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particular geometry of the structure,L1= 2L2, and three characteristic values of the load ratiowthere are plotted inFig. 9a therespective plastic strain evolutions, as the load level is increased within the interval ke


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rwx wxwex 1

100%;

rux ux e ux euex e uex e 1

100%;

rd

xw

xew 1 100%:

31

For a given ratio wit is reasonable in the investigated example to observe, beside the rdquantity, the following quantities:rw(xM) at the position of the maximum bending moment between the supports and rw(xC) at the free end,r/(xM) at the posi-tion of the maximum bending moment between the supports and r/(xB) at the right support, while taking e = 10 mm. Theobserved quantities, obtained by the analysis of load cases with the characteristic values of the load ratiow , are presentedinTable 1.

The tabulated quantities show that proportional increase of the considered loading from k =keto k ? 1, the tabulationbeing made for greater evidence on the basis of equidistant load step incrementation (Dk= const.), leads to growing influenceof the non-linearity. As a matter of fact this is quite expected. Namely, with intensification of the plastic propagation andconsecutive weakening of the structural elastic resistance the departure from linearity becomes more and more apparentas loads approach k? 1. Near the singularity point, where the ultimate load levelk= 1 is reached, all the observed quantities,except the rd quantity, exhibit rapid increase, with their values asymptotically approaching infinity. At k= 1 the moststressed cross section becomes fully plastified, and the structure, the problem being a statically determinate one, transformsinto a mechanism due to the occurrence of a plastic hinge. This happening the originally designed functionality and load car-rying capacity of the structure are lost, which is equivalent to a collapse.

6. Conclusion

In the article we considered bending of a straight beam with rectangular cross section, exhibiting elasto-plastic behaviourwith no hardening. In the respective deflection curve analysis following the classical beam bending formulation the explicitdeflection line functions for partially plastified domains of the beam under characteristic loading cases, including applicationof concentrated forces and moments as well as application of uniformly distributed loads, were derived. With respect to theevidenced bending moment distribution constant, linear or quadratic, three distinct functions were obtained for therespective particular solutions. They are listed in the Eqs.(21)(23), thus forming a firm mathematical framework for elas-to-plastic analyses of the considered kind of beam structures.

From a computational point of view it can be concluded, that the derivation of general form elasto-plastic solutions isquite an achievement, which is clearly demonstrated and fully exploited in the investigation of the considered numerical

Table 1

Deviations from the linear response; L1= 2 m, L2= 1 m, h= 20 mm, Dk= 1/12

w ki ru(xM) [%] ru(xB) [%] rw(xM) [%] rw(xC) [%] rd[%]

0 8/12 0.0 0.0 0.0 0.0 0.009/12 2.6 0.0 1.0 0.7 0.0010/12 13.1 0.0 6.0 4.3 0.0011/12 45.4 0.0 21.2 15.4 0.001109 24163.6 0.0 827.3 529.6 0.00

w1 8/12 0.0 0.0 0.0 0.0 0.009/12 2.6 0.0 0.9 1.6 0.1310/12 13.1 0.0 5.7 10.4 0.5611/12 45.4 0.0 20.4 37.6 1.481109 21580.4 0.0 783.0 1292.8 7.49

w2 8/12 0.0 0.0 0.0 0.0 0.009/12 2.6 2.1 0.9 6.8 0.1610/12 13.1 11.4 5.7 36.0 0.6711/12 45.4 39.3 20.4 121.5 1.781109 17144.8 995.0 679.2 6743.0 9.39

w3 8/12 0.0 0.0 0.0 0.0 0.009/12 0.0 2.2 0.0 0.5 0.0010/12 0.0 11.7 0.0 3.8 0.0011/12 0.0 40.1 0.04 14.6 0.021109 0.0 524.1

0.16 85.5

0.08

1 8/12 / 0.0 / 0.0 0.009/12 / 2.5 / 0.3 0.0110/12 / 12.5 / 2.1 0.1011/12 / 43.0 / 8.1 0.491109 / 826.8 / 48.1 1.83

B. tok, M. Halilovic/ Applied Mathematical Modelling 33 (2009) 17491760 1759


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example. Namely, by taking the derived explicit solutions into account the procedure needed to solve an elasto-plastic bend-ing beam problem becomes very simple, irrespectively of the exhibited non-linear material behaviour. Considering the com-plete problem solution is obtained by solving a relatively small system of linear equations, this procedure is also efficient. Inthe article this was demonstrated by considering a statically determinate problem due to simplicity and clearness of the pre-sentation, but the use of the derived equations is similar (simple and efficient) also for statically indeterminate problems.Though explicitly derived for a rectangular cross section the contribution presented in this article may be considered, pri-marily due to its explicit solvability and, in consequence, by given ability of very clear tracing of the elasto-plastic responseevolution, as relevant.

Considering the derived general elasto-plastic solutions, listed explicitly by the Eqs.(21)(23),were never presented inany article or book before, we can state in conclusion that the presented article fills the gap in the analytical non-linearmechanics.

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