A BEM formulation for analysing the coupled stretching bending problem of plates reinforced by rectangular beams with columns defined in the domain

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1 Comput Meh 21 45: OI 1.17/s ORIGINAL PAPR A BM formulation for analysing the oupled strething bending problem of plates reinfored by retangular beams with olumns defined in the domain Gabriela Rezende Fernandes Guido J. enipotti anilo H. Konda Reeived: 18 Marh 29 / Aepted: 9 eember 29 / Published online: 16 February 21 Springer-Verlag 21 Abstrat In this work, a boundary element formulation to analyse plates reinfored by retangular beams, with olumns defined in the domain is proposed. The model is based on Kirhhoff hypothesis and the beams are not required to be displayed over the plate surfae, therefore eentriity effets are taken into aount. The presented boundary element method formulation is derived by applying the reiproity theorem to zoned plates, where beams are treated as thin sub-regions with larger rigidities. The integral representations derived for this omplex strutural element onsider the bending and strething effets of both strutural elements working together. The standard equilibrium and ompatibility onditions along interfae are naturally imposed, being the bending trations eliminated along interfaes. The in-plane trations and the bending and in-plane displaements are approximated along the beam width, reduing the number of degrees of freedom. The olumns are introdued into the formulation by onsidering domain points where trations an be presribed. Some examples are then shown G. R. Fernandes B Civil ngineering epartment, Federal University of Goiás UFG, CAC Campus Catalão, Av. r. Lamartine Pinto de Avelar, 112, Setor Universitário, CP 757- Catalão, GO, Brazil grezfernandes@itelefonia.om.br G. J. enipotti Civil ngineering epartment of São Paulo State University, UNSP, Al. Bahia, 55, Ilha Solteira, Brazil guido.denipotti@imesul.ind.br. H. Konda letri ngineering epartment, Federal Institute of duation, Siene and Tehnology IFT, Campus Vitória da Conquista, Av. Amazonas, 315, Bairro Zabelê, CP Vitória da Conquista, BA, Brazil dhkonda@gmail.om to illustrate the auray of the formulation, omparing the obtained results with other numerial solutions. Keywords Plate bending Boundary elements Building floor strutures 1 Introdution The boundary element method BM has already proved to be a suitable numerial tool to deal with plate bending problems. The method is partiularly reommended to evaluate internal fore onentrations due to loads distributed over small regions that very often appear in pratial problems. Moreover, the same order of errors is expeted when omputing defletions, slopes, moments and shear fores. Shear fores, for instane, are muh better evaluated when ompared with other numerial methods. They are not obtained by differentiating approximation funtion as for other numerial tehniques. The BM is partiularly reommended for the analysis of building floor strutures where the ombinations of slab, beam and olumn elements an be more aurately represented, onsidering that the method is very aurate to ompute the effets of onentrated in fat loads distributed over small areas and line loads, as well to evaluate high gradient values as bending and twisting moments, and shear fores. The diret BM formulation applied to Kirhhoff s plates has appeared in the seventies 1 3]. These works, as well as several other more reent publiations, have demonstrated that the method is a robust numerial tehnique to deal with plates in bending, taking into aount its auray and onfidene. Reently Chen et al. 4], presented a novel BM for plate bending where singularity analysis and treatment

2 524 Comput Meh 21 45: of the disontinued orner point are not needed at all. In this ontext, it is worth mentioning two edited books 5,6] ontaining formulations of the BM applied to plate bending showing several important appliations in the engineering ontext. Bezine 7] apparently was the first to use a boundary element to analyse building floor strutures by onsidering plates with internal point supports. The works published by Song and Mukherjee 8] and Hartmann and Zotemantel 9] have shown interesting BM approahes to deal with building frame floors, where displaement restritions at internal points and the use of hermitian interpolations are disussed in details. More reently, several woks oupling BM with FM have been presented to deal with this kind of strutures 1 12], where the slabs are modelled by boundary elements, while beams and olumns are represented by finite elements. As usual, they are ombined together by enforing equilibrium and ompatibility onditions along the interfaes. Although, for omplex floor strutures the number of degrees of freedom inreases rapidly and the auray of the solution diminishes. In 13 16] are proposed BM formulations for analysing the bending problem of beam-stiffened elasti plates. An alternative sheme to deal with zoned domains without dividing them into sub-regions has been proposed by Venturini 17,18]. In Chaves et al. 19] the authors presented a plate bending BM formulation to deal with the general ase of varying thikness problem. Paiva and Aliabadi in 2] presented a BM formulation to analyse building floors strutures whih are modelled by a zoned plate with different thikness. In 21] the same authors proposed a formulation for zoned plates where the boundary integral equations of urvatures of points loated at the zone s interfaes are dedued in a very easy way allowing getting the bending moments at these points easily. By using also a sub-region tehnique Leite et al. 22] have presented a BM formulation for two-dimensional solids reinfored by thin bars, where the number of degrees of freedom is dereased by approximating the displaements along the bars ross setion. Reently Fernandes and Venturini 23,24] have proposed two numerial models to perform linear bending analysis of plates reinfored by retangular beams using only a BM formulation based on Kirhhoff s hypothesis. In these work the building floor is modelled by a zoned plate where the beams are onsidered as narrow sub-regions with larger thikness for whih some kinemati approximations were assumed to redue the number of degrees of freedom. The bending trations are eliminated along the interfaes, reduing therefore the total number of unknowns. This omposed struture is treated as a single body, being the equilibrium onditions automatially satisfied. In 23] the authors present a formulation to perform simple bending analysis of building floor strutures. Then this formulation is extended in 24] to onsider the plate and beam elements not neessarily defined in the same plane, so that the bending and strething problems have to be oupled. In this ontext is interesting mentioning the works 25 27] in whih the eentriity effets are also onsidered and the ross setion of the reinfored beams is not restrited to a retangular one. In these works are presented formulations using the analog equation method AM whih is a BM based method. Besides, the warping influene arising from both shear fores and twisting moments is taken into aount. In 26,27] the authors have improved the model presented in 25] to onsider a nonuniform distribution of the interfae transverse shear fores and the nonuniform torsional response of the beams. Also leading with simple bending analysis of plates reinfored by retangular beams, Fernandes and Konda 28] presented a BM formulation based on Reissner s hypothesis, where some approximations are adopted for both trations and displaements, along the beams width, to redue the number of degrees of freedom. Moreover, in 29] Fernandes and Venturini have extended the BM formulation developed in 24] to perform non-linear analysis of stiffened plates. In this paper the linear formulation proposed in 24] is extended to define olumns in the stiffened plate domain. Initially domain points where bending and in-plane trations an be presribed are introdued into the formulation. Then, the olumns reations over the plate are onsidered as presribed trations in the entral point of the olumn-plate interfae. Some numerial examples are presented to illustrate the auray of the results and the apability of the formulation to analyse omplex building floor strutures. 2 Basi equations Without loss of generality, let us onsider the plate depited in Fig. 1a, where t 1, t 2 and t 3 are the thikness of the sub-regions 1, 2 and 3, whose external boundaries are Ɣ 1,Ɣ 2 and Ɣ 3, respetively. The total external boundary is given by Ɣ while Ɣ jk represents the interfae between the adjaent sub-regions j and k. In the simple bending analysis all sub-regions are represented by their middle surfae, as shown in Fig. 1, while for the oupled strething bending problem the Cartesian system of o-ordinates axes x 1, x 2 and x 3 is defined on a hosen referene surfae see Fig. 1b, whose distane to the sub-regions middle surfaes are given by 1, 2 and 3. As in Fig. 1b the referene surfae is adopted oinident to 2 middle surfae one has 2 =. For a point plaed at any of those plate sub-regions, the following basi relationships are defined for the bending problem:

3 Comput Meh 21 45: Fig. 1 a General zoned plate domain, b referene surfae view, and plate middle surfae view a Ω 1 Γ 1 Γ 12 Γ 21 Ω 2 Γ 2 Γ 23 Ω 3 Γ 32 Γ 3 b X3 Referene surfae x 3 Middle surfae C 1 t 1/2 X2 X1 t 2/2 C 3 t 3/2 t 1 t 2 t 3 x 2 x1 quilibrium equations in terms of internal fores: m ij, j q i = 1 q i,i g = 2 where g is the distributed load ating on the plate middle surfae, m ij are bending and twisting moments and q i represents shear fores, with subsripts taken in the range i, j ={1, 2}. The plate bending differential equation, m ij,ij g = 3 or w, iijj = g/ i, j = 1, 2 4 where = t 3 /121 ν 2 is the flexural rigidity and w, iijj = 4 w, being 4 the bi-harmoni operator. The generalised internal fore displaement relations, m ij = νδ ij w,kk 1 νw,ij 5 q i =w, jji 6 The effetive shear fore, V n = q n m ns / s 7 where n, s are the loal o-ordinate system, with n and s referred to the plate boundary normal and tangential diretions, respetively. Similarly, for the strething problem the in-plane equilibrium an be expressed by: N ij, j b i = 8 where N ij is the membrane internal fore and b i represents the orresponding body fores distributed over the plate middle surfae. By assuming plane stress onditions, membrane internal fore displaement relations now read: ] 2ν N ij = Gt 1 ν u k,kδ ij u i, j u j,i The in-plane equilibrium an also be written in terms of displaements by replaing q. 9 into q. 8 asfollows: 1 ν 1 ν u j,ij u i, jj b i /Gt = 1 The problem definition is then ompleted by assuming the following boundary onditions over Ɣ : u i = u i on Ɣ 1 generalised displaements: defletions, rotations and in-plane displaement omponents and p i = p i on Ɣ 2 generalised trations: normal bending moment, effetive shear fores and in-plane trations, where Ɣ 1 Ɣ 2 = Ɣ. 3 Integral representations In this setion, initially we are going to derive the integral equations for the general ase of zoned domain plate problems whih will be partiularized to the ase of plates reinfored by retangular beams. Finally, domain points where trations an be presribed will be introdued into the formulation in order to define olumns in the stiffened plate domain. The plate thikness may vary from one sub-region to another whose middle surfaes are not neessarily the same. The equations will be derived by applying the reiproity theorem to eah sub-region and summing them to obtain the reiproity relations for the whole body. Complementary domain integral terms will be inserted in the reiproity relation to take into aount the effets of the relative position of the sub-region middle surfaes. 9

4 526 Comput Meh 21 45: Let us initially onsider a single sub-region,forwhih the following reiproity relations an be written: V m σ jk m εm ijk dv = V m σijk m εm jkdv 11 where εijk m m and σijk are fundamental solutions with the unit load ating in the diretion x i, m is the Young s modulus and V m the orresponding volume; no summation is implied on m. For the oupled bending and strething problem, the fundamental strain and stress omponents are divided into two parts as follows: εijk m = εms ijk εijk mb σijk m = σ ijk ms σijk mb 12a 12b where B and S refer, respetively to bending and strething problems; εijk mb =x3 mw,m ijk with w, ij being the fundamental plate surfae urvature. Considering 12 and arrying out the integrals of q. 11 aross the thikness t m one obtains: εijk Sm N m jk d = Nijk m εsm jk d w, m jk mm jk d m m jk w,m jk d 13 quation 13 an be used separately to obtain the wellknown strething and bending reiproity relations for any sub-region. In this work, the Poisson s ration is adopted the same for all sub-regions, so that we an say that trations fundamental m m jk = m jk values are the same for all sub-regions, i.e. Nijk m = N, while the in-plane strain and urvature fundamental solutions εijk m values εijk,w, jk ijk and and w,m ijk an be written in terms of the, and referred to the sub-region where the load point is plaed as follow: ε Sm ijk w, m jk = ε S ijk / m = / m] w, jk 14a 14b where m = m t m, being m the Young s modulus in the sub-region. Considering 14 and summing all sub-regions equations the following reiproity relations an be derived for the entire body. For the strething and bending problems one obtains, respetively: εijk s N N s jkd = w, jk m jkd = N s m m ε sm jk N ijk d w, m jk m jk 15a d 15b where N s is the number of sub-regions. Now to derive the reiproity relations in whih strething and bending effets are oupled, we have to write strain and moment values of sub-region in terms of the referene surfae values, as follows: ε Sm jk = ε jk m w, jk 16a m m jk = m jk m N jk 16b where ε jk and m jk are the strain and moment values referred to the referene surfae. It is worth noting that the internal normal fores and the urvatures do not depend on the plate surfae position and therefore the loal values are replaed by the global ones, i.e., N m jk = N jk and w, m jk = w, jk. Replaing q. 16 into q. 15 gives: εijk N jkd = N s m w, jk m jkd N s = m N ijk ε jkd m w, jk m jk d m N ijk w, jkd w, jk N jkd 17a 17b quations 17 an be integrated by parts to give the oupled representations of in-plane displaements and defletions. For the strething problem, the integral representation of displaements is: sub K w,i w, i K ui u i = Sub m m b a m p in w, n pis w, un pin u s pis un pin u s pis

5 Comput Meh 21 45: Ɣ b b a a u in p n uis p p in w, n pis w, b u in b n uis b s d 18 where pik = N ik, with k = n, s, is the usual tration fundamental solutions for the strething problem; no summation is implied on n and s that are loal normal and shear diretion o-ordinates; the subsripts b and a refers, respetively, to the beam sub-region and its adjaent sub-region, N int is the number of interfaes; the free term values are given by: K w,i = R and K ui = 1 for an internal point; K w,i = R /2 and K ui = 1/2 for boundary points; K wi = 2 1 R at a a t and K ui = at a t for interfae points, being R the distane of the olloation point sub-region to the referene surfae. For the bending problem, the integral representation of defletions is: K q w q Sub = N 1 m N 2 N 3 j R j w j Vn w w M n j a b a Ɣ R j w j Vn w w M n N R j wj V n w w M n gw d g sub sub m b a m pn w, n p sw, ] pn w, n p sw, ] bn w, n b sw, s d 19 where 1, 2 and 3 are different kinds of orners for their definitions and their orresponding free term values see Fernandes and Venturini 23]; g is the plate loaded area; K q = 1, K Q =.5 and K Q =.51 a /, respetively, for internal, boundary and interfae points, N is the total number of orners. In qs. 18 and 19 all values are related to the referene surfae, being defined on the external boundary eight values: w, w, n ; u n, u s, p n, p s M n and V n. Along interfaes, the bending trations have been eliminated remaining four generalized displaements, w, w, n ; u n and u s and two in-plane trations, p n and p s as unknown values. The rotation w, s is onveniently replaed by numerial derivatives of w, therefore leading to six unknowns at eah interfae node. quations 18 and 19 are the exat representations of in-plane displaements and defletions of a zoned plate bending-membrane problem. To partiularize the formulation to deal with plates reinfored by beams, we assumed that beams are modelled by narrow sub-regions. Moreover, in order to redue further the number of degrees of freedom we have assumed some kinemati hypothesis along the beam ross setion. Linear approximation has been adopted for both defletions and in-plane displaements, while the rotation w, ij n has been onsidered onstant along the beam width in the skeleton normal diretion n see 24] for more details. Besides these displaements approximations we have also assumed linear distribution of stresses aross the beam setion, being the trations pk divided into two parts see 24]. The first part of the trations pk is referred to onstant stress field aross the beam, therefore an be written in terms of displaement derivatives using Hooke s law, as follow: ] pk = Gt 2ν 1 ν u l, l n k u k, n u n, k k =n, s 2 where the n is the beam axis outward vetor. The seond part p k refers to the linear stress distribution aross the beam setion and represents new independent values, i.e., new degrees of freedom. Note that in q. 2 the displaements derivatives u n,s and u s,s with respet to beam axis, diretion s, are replaed by numerial derivatives of u n and u s, respetively. Thus, by adopting those approximations the interfae displaement vetor defletion, derivative defletions and in-plane displaement omponents as well as the interfae trations are translated to the skeleton line, remaining eight values at eah beam skeleton node: three displaements w, u n and u s three rotations w, n ; u n,n and u s,n and two distributed fores p n and p s. Note that for external beams, only the interfae trations are approximated, as the boundary trations represent the atual boundary values. But in the external beams the interfae values p k an be onveniently eliminated, therefore they do not represent new independent values see 24]. It is important to stress that the integrals are still performed along the interfaes. Thus, no singular or hyper-singular term is found when transforming the integrals representations into algebrai ones. Moreover, as the beam widths are very small,

6 528 Comput Meh 21 45: P C P C Ω C ΩC P C P C Ω C Ω C body fores uniformly distributed over lines defined inside the sub-region see Fig. 4b. Thus, inluding into the formulation the distributed load σ as well as the line loads b x and b y, the displaements representations for strething and bending problems are given, respetively, by: Fig. 2 Plate with olumns defined in the domain N x M x N y M x R M y R N y M y N x Ω Fig. 3 Columns reations over the plate a Ω C L x σ C L y Plate middle surfae Column Fig. 4 a istributed load over. b Line loads inside the sub-region to perform the resulting quasi-singular integrals we have used an appropriate sub-element integration see 23]. Let us now onsider the inlusion of olumns into the formulation developed previously see Fig. 2 where the subregion represents a olumn ross setion in ontat with the plate and P C the entral point of. The olumn bending reations moments M x and M y and normal fore R over the plate see Fig. 3 where y and x indiate the olumn prinipal diretions will be transformed into the normal stress σ uniformly distributed over the olumn ross setion. Then, the stress σ will be onsidered as additional distributed load ating on the plate sub-region in onnetion with the olumn see Fig. 4a. Following the same idea, the olumn in-plane reations N x and N y will be transformed into line loads b x and b y whih will be assumed as additional L y b y b x Ω L x b y x sub K w,i w, i K ui u i = Sub Ɣ b N ol m m b a m p in w, n pis w, b b a a un pin u s pis u in p n uis p u in b n uis b s d Ɣ y b x uix ] N ol yj un pin u s pis p in w, n pis w, Ɣ x where N ol is the olumns number. K q w q Sub = N 1 m j R j w j Vn w w M n b a Ɣ N 2 N 3 ] b y u i y xj 21 j a R j w j Vn w w M n N R j wj V n w w M n gw d g sub m b a pn w, n p sw, ] pn w, n p sw, ]

7 Comput Meh 21 45: sub N ol m j Ɣ y bn w, n b sw, N ol s d b x w, ] N ol x yj j Ɣ x j σ j w d j ] b y w, y xj 22 Let us now write the loads σ, b x and b y in terms of the olumns reations. The olumn bending reations an be written in terms of the normal stress σ uniformly distributed over and the olumns in-plane reations are transformed into the line loads b x and b y as follow: σ = M y I x y M x I y x R A 23 b x =N x L y b y =N y L x 24a 24b where L y and L x are the length of the olumn ross setion sides see Fig. 4, A is the ross setion area, I x and I y are the inertia moment with respet to diretions x and y, respetively. Finally, in order to have the final set of equations given in terms of the olumn generalized displaements we have to write relations 23 and 24 in terms of displaements. Considering the olumn stiffness matrix, we an define the following relations for the bending reations: M y = a I x L M x = a I y L w y w x b I x L 2 u y 25 b I y L 2 u x 26 R = A L w 27 In qs. 25, 26 and 27 and L are, respetively, the Young s modulus and the length of the olumn; w, u x, u y, w x and w y are generalized displaements in the olumn ross setion; the values a and b depend on the olumn boundary onditions: a = 4 and b = 6 for fixed olumns while a = 3 and b = for simply supported olumns. For the in-plane reations we an define the following trations displaements relations: N x = d I y L 2 N y = d I x L 2 w x w y e I y L 3 u x e I x L 3 u y 28a 28b where d = and e = 3 for simply supported olumns; d = 6 and e = 12 for fixed olumns. Replaing now 25, 26 and 27 into23, one obtains the normal stress σ in terms of displaements: σ = a w b L x L 2 u x x a w b L y L 2 u y y w 29 L The line loads in terms of displaements is obtained by replaing qs. 28 into24: b x = 1 d I y L y L 2 b y = 1 d I x L x L 2 w x w y e ] I y L 3 u x e ] I x L 3 u y 3a 3b Considering that the displaements related to the olumn must be equal to the ones defined on the plate sub-region C and adopting onstant approximation for the displaements w, w, x,w, y, u x and u y over C, these displaements will be defined at the entral point P C of the sub-region C see Fig. 2. Replaing 29 and 3 into qs. 21 and 22, one obtains the final integral representations of displaements for the strething and bending problems whih are given, respetively, by: sub K w,i w, i K ui u i = Sub Ɣ m m b a m p in w, n pis w, b b a a un pin u s pis u in p n uis p un pin u s pis p in w, n pis w, b u in b n uis b s d

8 53 Comput Meh 21 45: N ol 1 L y j uix yj Ɣ y d j j I xj L 2 j K q w q Sub = N 1 m d j j I yj L 2 j 1 L x j w y j j R j w j w x j e j j I xj L 3 u y j j Ɣx Vn w w M n b a Ɣ N 2 N 3 e j j I yj L 3 u x j j j a ui y xj R j w j Vn w w M n N R j wj V n w w M n sub sub N ol m b a m a j j L j pi N ol pn w, n p sw, ] pn w, n p sw, ] bn w, n b sw, s d a j j L j yw d j j 31 gw d g w b j j x 2 u x j xw d j j L j pi w b j j y 2 u y j j L j j w j L j pi 1 d j j I y j L y j L 2 j e j j I y j L 3 u x j w, x y j j Ɣ yj w d j w x j 1 d j j I x j L x j L 2 j w y j e j j I x j L 3 u y j j w, y x j 32 Ɣ x where w, w, x,w, y, u x and u y are defined at the entral point P C of the sub-region C see Fig. 2. Considering this sheme four equations must be written for eah boundary olloation point, one equation for eah orner node and for eah external and internal beams axis nodes, six and eight relations are required, respetively see Set. 4 for more details. Besides these equations, when we onsider the inlusion of olumns into the formulation, five new values remain as unknowns on the olumn-plate interfae: w, w, x,w, y, u x and u y. Thus, to omplete the neessary number of equations to solve the problem we write the orresponding five algebrai equations of displaements at the entral point P C of the olumn ross setion C see Fig. 2. Note that the integral representation of w, m an be easily obtained by differentiating equations 32: w, m q Sub = N 1 m j N g sub N ol V n m w M n m R m w j b a w R j m Ɣ w N sub g m d b a m N 2 N 3 w j a V n m w M n m R m w j w w V n m M w n m m w, n p n m p w, ] s m w, n p n m p w, ] s m w, n b n m b w, s s d m a j j L j w b j j x 2 u x j j L j

9 Comput Meh 21 45: pi pi N ol w x m d j a j j L j w y m d j j w j L j j pi 1 d j j I y j L y j e j j I y j L 3 u x j j Ɣ yj 1 d j j I x j L x j L 2 j Ɣ x L 2 j w b j j y 2 u y j j L j w m d j w x j w, x m y j w y j e j j I x j L 3 u y j j w, y m x j 33 To obtain the integral representation of u n,n or u s,n one has to differentiate q. 31 and the urvature integral representation is derived by differentiating one more q. 32. Then, bending and twisting moment integral representations are obtained by simply applying the definition given in q. 5. Furthermore, one has to remember that the derivatives with respet to beam axis o-ordinate s are replaed by differenes. To obtain the shear fore integral representation, ompleting the internal fore values at internal points, one an differentiate the urvature equation one to apply the definition given in 6. 4 Algebrai equations As usual for any BM formulation, the integral representations for instane, qs. 31, 32 and 33 have to be transformed into algebrai expressions after disretizing the boundary and interfaes. For the present ase, the boundary and the interfaes were disretized into geometrially linear elements, while quadrati shape funtions were adopted to approximate the variables along the boundary and beam axes. With this approximation we an write algebrai representations of displaements in-plane and defletion omponents and their derivatives for olloation points taken inside the domain, along the boundary, along beam axes and out of the plate. After seleting an appropriate number of olloation points with the orresponding algebrai equations, one an assemble a onvenient set of relations to solve the problem in terms of boundary, beam axis and olumns values. The orresponding boundary nodal values remained in the algebrai system are the standard ones usually adopted in BM formulations: two in-plane displaements u n and u s for the strething problem; the defletion w and its normal derivative w, n for the bending problem. The ounterpart values are respetively: in-plane trations p n and p s forthe strething problem; moment M n normal to the boundary and the effetive shear fore V n for the bending problem. Therefore, four equations must be written for eah boundary node beause there are four unknowns per node. In eah orner the defletion and the orner reation are defined, requiring therefore one relation in this ase. The skeleton nodal values maintained in the algebrai system are: two in-plane displaements u n and u s, two displaement derivatives with respet to the skeleton line normal diretion, u n / and u s / for the strething problem; the defletion w and one defletion derivative w, n with respet to the skeleton line normal diretion for the bending problem. The bending ounterpart values along interfaes were all eliminated, remaining the in-plane trations p n and p s as unknowns in the internal beams. Thus, for eah external and internal beams axis node, six and eight relations are required, respetively. In the entral points of the olumn-plate interfaes are defined the following generalized displaements: w, w, x, w, y, u x and u y. Thus we have to write five additional relations in eah one of these points to omplete the neessary number of equations to solve the problem. We have hosen to write the orresponding equations of displaements. For eah boundary node we define two olloation points: the first one is the node itself or another point plaed along the adjaent element when boundary value disontinuity is assumed; the seond olloation is an external point very near the boundary. For the olloation defined along the boundary, we write three displaements algebrai relations: two in-plane displaement relations obtained from qs. 31 and one defletion relation from q. 32. The last algebrai relation is also obtained from q. 32, but written for the external olloation. For eah beam skeleton node we write two in-plane displaement relations obtained from q. 31; one defletion relation from q. 32; two in-plane displaement derivative relations; and one slope relation from q. 33. Besides, for internal beams two in-plane tration relations have to be added. For the beam skeleton nodes, all olloations are defined along the skeleton line. They are oinident with the node when variable ontinuity is assumed or defined at skeleton element internal point when variable disontinuity is required. After seleting the reommended olloation points and writing the orresponding algebrai relation for all of them, one obtains the following set of equations:

10 532 Comput Meh 21 45: ] H] B H] C H H s B ] ol H 2 ]ol ] H ] H] B S ] H 2 ] ol {U} B {w} C H ] B H ] ] H C ] H S B H ] {U} S ol 2 ol {U ] B } ol H ] H ] ] {U S } ol ] B S H 2 ol G] B G] C G ]S {T } B ] ] G] S {P} = G ] B G ] B {T } S G ] {R} C { C {P} S S T } B ] ] G ] { T } S S 34 In q. 34 the subsripts B, C, S and ol indiate, respetively, bending, orner, strething and olumn. The H and G matrix are divided into four parts: the first blok of equations refers to the bending algebrai equations 32 or 33 written for boundary, external and beam axis nodes. The seond blok of equations are related to the strething problem q. 31 and the equations of the in-plane displaements derivatives and in-plane trations written for boundary and beam axis nodes. The last two bloks refer, respetively, to the bending and strething equations written in olumns points. quation 34 an be written as the standard set of equations given as usual by: is based on Kirhhoff s theory and the beam element is represented by the usual stiffness method relations, whih are introdued into the boundary element system onsidering the equilibrium and displaement ompatibility onditions. To onsider the inlusion of olumns into the plate formulation we have adopted in this paper the same proedure desribed in Paiva 3]. However, in Pavia 3] is onsidered only the simple bending analysis while in the proposed model the membrane effets are also taken into aount, so that new values arise in the formulation. It is important to stress that the strutural systems modelled by ANSYS, by the model presented in 3] and by the proposed formulation are not exatly the same and therefore the results an be only similar. For the ANSYS analysis finite solid elements solid brik 8 node 45 have been used to disretize the slabs, beams and olumns. In the proposed model we have used plate elements and we have treated the whole body as a solid, therefore without splitting the plate and the beams; beams are inlusions in the whole body. Observe that in the proposed model the elements plaed at external beams ends, along the beam width, are automatially generated by the ode, so that there is no need of defining them. For all examples the onvergene performane of the proposed formulation was verified by solving the problem using several meshes. In the examples will be presented only the results of one mesh, whose solution was onsidered aurate enough. HU = GP T 35 where U ontains the generalized displaement nodal values defined in the olumns, along the boundary and along skeleton lines, P ontains boundary nodal trations and in-plane trations on the internal beam axis; T is the independent vetor due to the applied loads. 5 Numerial appliations Three examples are now shown to demonstrate the performane of the proposed formulation: a simple plate with four olumns defined in the domain, a plate reinfored by external beams supported by four olumns defined on the orners and a more omplex building floor struture ombining many beams and slabs and also ontaining four olumns on the orners. The results are ompared either to a well-known finite element ode ANSYS, version 9 or to a numerial model proposed by Paiva 3] where the beams and olumns are modelled by finite elements and the plate by boundary elements. In the model presented in 3] the plate formulation 5.1 Plate with olumns defined in the domain This example onsists of a simply supported square plate with a distributed load g of 1 kn/m 2 applied on the whole plate surfae and with four olumns defined in the domain as depited in Fig. 5. The plate side has been onsidered equal to 12 m while its thikness, Young s modulus and Poisson s ration were assumed, respetively, as: t p =.15 m, p = kn/m 2 and ν =.167. The Young s modulus and the length adopted for the olumns are: = kn/m 2 and L = 3 m. Besides, the olumns have been assumed fixed on their base and one has adopted square ross setion with the partiular value of L y = L x = 1 m for their sides to give exat solution. The adopted disretization has 1 elements in eah plate side, resulting into 4 elements and 84 nodes, as depited in Fig. 6. The numerial results are ompared to the exat solution given in 3]. For the A A axis the defletion, the moment Mx and the moment My values are displayed, respetively, in Figs. 7, 8 and 9. The same values along the B B axis are shown in Figs. 1, 11 and 12. As an be observed the results ompare very well with the solution presented in Paiva 3].

11 Comput Meh 21 45: Fig. 5 View of the plate with four olumns defined in the domain Mx 1 knm/m PAIVA 28] xm Fig. 6 Plate disretization Fig. 8 Moments Mx along the A A axis w m x m Fig. 7 efletions along the A A axis PAIVA 28] My 1kNm/m PAIVA 28] 5.2 Plate reinfored by external beams with olumns defined on the orners Let us now onsider the plate shown in Fig. 13 reinfored by external beams along all plate boundaries and supported by four olumns defined on the orners. The Young s modulus, the Poisson s ratio, the plate and beams thiknesses adopted to analyse this struture are, respetively: = kn/m 2,ν=.2, t p =.1m and t b =.3 m A distributed load of 2 kn/m 2 is applied x m Fig. 9 Moments My along the A A axis on the whole surfae of the struture and all external beams axes have been assumed free. For the olumns, whih are assumed fixed on their bases, the following data are adopted: length L = 3 m and square ross setion with sides equal to L y = L x =.2 m. In the ANSYS disretization for

12 534 Comput Meh 21 45: w m X m Fig. 1 isplaements along the B B axis Paiva 28] 2.2m Mx 1kNm/m X m Fig. 11 Moments Mx along the B B axis My 1kNm/m X m Fig. 12 Moments My along the B B axis PAIVA 28] PAIVA 28] Fig. 13 View of the plate reinfored by external beams in Figs. 18 and 19. As an be observed the results for the simple bending analysis obtained using the proposed formulation, are smaller than those related to the ommerial pak ANSYS, but they are similar. On the other hand, for the oupled strething bending problem they ompare very well. As expeted, due to the membrane effets, the displaements related to the oupled strething bending analysis are smaller than the ones referred to the simple bending analysis. It is important to note that for almost all analysis we have already obtained enough aurate results for a mesh with 4 elements see Figs. 2, 21. Only for moments along the beam axis we had to onsider the finer mesh with 14 elements to get the results onvergene see Figs. 22, Building floor struture the simple bending analysis we have used solid elements whose sides have been adopted equal to 1 m see Fig. 14a while for the oupled strething bending analysis we have defined free disretization resulting into 19,813 elements see Fig. 14b. For the proposed model the finer disretization used to solve this problem, shown in Fig. 15, ontains 22 nodes with 14 quadrati elements along the beam axes inluding 24 nodes and 8 elements used for the beam intersetions whih are automatially generated by the ode. For the oupled strething bending analysis the referene surfae was adopted oinident to the slab middle surfae. Figures 16 and 17 show, respetively, the displaements omputed along the beam axis and the A A axis, onsidering simple bending analysis denoted by SB and the oupled strething bending analysis denoted by CP. The bending moments in the diretion of the respetive axis are displayed This example onsists of a simple bending analysis of a square plate, whose length side between external beam axis is adopted equal to 9 m. The plate is reinfored by several internal and external beams and supported on four olumns, as depited in Figs. 24 and 25. The distane between beam axis is adopted equal to 3 m and a distributed load g = 1 kn/m 2 is applied over all stiffened plate surfae whose external beams axis are onsidered all free. The olumns ross setion are adopted square with sides equal to L y = L x =.3 m and their length has been assumed as L = 4 m. The plate thikness t p was adopted equal to.8 m while for the external and internal beams we have onsidered t B =.8 m. For the slabs and beams we have adopted elasti modulus = kn/m 2 and Poisson s ratio ν = 1/6, while for the olumns = kn/m 2 has been onsidered. To analyse the onvergene results we have onsidered three meshes. The poorest one had 252 nodes and 18

13 Comput Meh 21 45: Fig. 14 a ANSYS disretization for simple bending analysis. b ANSYS disretization for the oupled strething bending problem elements defined along the beams axis 4 elements on eah side where the beam has an interfae with a slab and 1 element where one beam ross another one, as shown in Fig. 26. Some neessary elements at beams ends are not shown in the disretization, beause they are automatially generated by the ode. In the seond mesh 24 elements have been defined along the beam axis, resulting into 444 nodes. For the finer mesh we have adopted 3 elements giving the total amount of 636 nodes. Regarding the displaement w along the axis A and B see Figs. 27, 28 there was no signifiant differene between the three meshes. In Figs. 27 and 28 the results are ompared with both a finite element analysis presented in Paiva 3] and the model proposed by Paiva 3], all of them related to the simple bending analysis. As an be observed in Fig. 28, the results along the internal beam axis axis B ompare very well with the finite element analysis and the model proposed by Paiva 3]. On the other hand, along the middle axis A A Fig. 27 the slab displaements obtained with the proposed model are bigger than the ones obtained with the other two models, although they ompare very well along the internal beam. In the proposed model, near to the internal beam the displaements derease strongly see Fig. 27, evidening the inreasing of rigidity due to the beam what does not happen in the FM analysis. Note that in the proposed Fig. 15 isretization for the proposed model model we have modelled the plate and the beams as a single body without splitting the plate and the beams. As an be observed, we have obtained bigger urvatures along the plate whih has flexure rigidity muh smaller than the beams. The model behaves as the slabs were partially supported on the beams. In the model presented by Paiva 3], the beams are onsidered as inlusions into the plate formulation. Note that the plate and the beams present the same urvature; it seems that the beams inrease the plate flexure rigidity dereasing the displaements along the plate. 5.4 Stiffened plate omposed by beams with different thikness The stiffened plate depited in Fig. 29 is reinfored by internal and external beams with different thikness and supported on four olumns whose ross setion have been adopted square with sides equal to L y = L x =.1 m. A distributed load g =.5kN/m 2 is applied over all stiffened plate surfae whose external beams axis are onsidered all free. The olumns length has been assumed as L = 3 m and they have been onsidered fixed at their extremity. The plate thikness t p was adopted equal to.8 m while for the external and internal beams we have onsidered different thikness: t B1 = t B2.18 m; t B3 = t B8 = t B9 = t B1 = t B5 = wm Beam axis m Fig. 16 isplaements along the beam axis - SB ANSYS - SB - CP ANSYS - CP

14 536 Comput Meh 21 45: w m x m Fig. 17 isplaements along the axis A - SB ANSYS - SB - CP ANSYS - CP w m xm model CP -4 elems Fig. 21 Convergene of displaements along the axis A, for the oupled analysis Mss knm/m Beam axis m - SB ANSYS - SB - CP ANSYS - CP Fig. 18 Bending moment along the beam axis, in the tangential diretion s MxkNm/m A-A axism Fig. 19 Bending moment along the axis A MxkNm/m xm - SB ANSYS - SB - CP ANSYS - CP model SB - 4 elems model SB - 14 elems Fig. 2 Convergene of moments along the axis A, for simple bending analysis t B7 =.25 m; t B4 = t B6 =.3 m. For the slabs, beams and olumns we have adopted elasti modulus = kn/m 2 and the Poisson s ratio ν =.16 has been assumed for the slabs and beams. In this example we have onsidered only the oupled strething bending analysis, being the numerial results ompared to ANSYS. The slabs middle surfae has been assumed as the referene surfae and the adopted mesh for the proposed model has 293 elements Mss knm/m Beam axis m model SB - 4 elems model SB - 14 elems Fig. 22 Convergene of bending moments along the beam axis, for simple bending analysis Mss knm/m Beam axis m CP - 4 elems model CP - 14 elems Fig. 23 Convergene of bending moments along the beam axis, for oupled analysis defined along the beam axis, resulting into 63 nodes as depited in Fig. 3. Moreover, when one external beam ross with another external beam are defined two more elements whih are not shown in the disretization, beause they are automatially generated by the ode. For the ANSYS disretization shown in Fig. 31 we have used solids elements. In Figs. 32, 33 and 34 are shown the numerial results for displaements whih ompare very well with the ANSYS analysis mainly at the entre of the stiffened plate. At the extremities the displaement obtained with the proposed model are a little bit different, maybe due to the approximations for displaements adopted along the beam width.

15 Comput Meh 21 45: m B w m FM PAIVA 28] y 4.5m A.2m 9m x m axis A x Fig. 27 isplaement w along the axis A Fig. 24 Plate middle surfae view 3m 3m 3m w m FM PAIVA 28] x m axis B Fig. 28 isplaement w along the axis B B 6.15m Fig. 25 Building floor geometry B 7 S 4 S 3 B 1 B 4 B 5 3.m.2m X S 1 S 2 B B 8 1.5m 9 B 3 B 1 B 2.1m 3.m.2m 3.m.1m 3.m.15m Fig. 29 Stiffened plate omposed by beams with different thikness Conlusions Fig. 26 Building floor disretization A BM formulation for analysing the bending problem of plates reinfored with retangular beams has been extended to define olumns inside the stiffened plate domain. Beam rigidity is taken into aount by assuming narrow subregions, without dividing the reinfored plate into beam and plate elements. As the membrane effets are onsidered, the beams are not required to be displayed over the plate surfae. The olumns are introdued into the formulation by onsidering domain points where trations an be presribed. In this formulation the equilibrium and ompatibility onditions are

16 538 Comput Meh 21 45: Fig. 3 BM disretization y x wm xm ANSYS - CP CP Fig. 33 isplaements along axis x Fig. 31 ANSYS disretization wm xm Fig. 32 isplaements along B9 and B1 beams axis ANSYS - CP CP wm xm Fig. 34 isplaements along B4 beam axis ANSYS - CP - CP obtained by using other numerial models as the well-known finite element ode ANSYS. automatially guaranteed by the global integral equations. Besides, it avoids unneessary approximations usually present when treating this problem with the standard sub-region tehnique, reduing very muh the number of unknowns and inreasing the auray of the results. The performane of the proposed formulation has been onfirmed by omparing the results with analytial solutions and also numerial solutions Referenes 1. Bezine GP 1978 Boundary integral formulation for plate flexure with arbitrary boundary onditions. Meh Res Comm 54: Stern MA 1979 A general boundary integral formulation for the numerial solution of plate bending problems. Int J Solids Strut 15:

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