Boundary Element Analysis of Building Slabs

Transcription

1 , July 1-3, 15, London, U.K. Boundary Element Analyi of Building Sla Charle Jater de Oliveira, João Batita de Paiva, Ângelo Vieira Mendonça Atract In thi work a oundary element formulation for the analyi of plate-eam interaction i ued in the analyi of practical uilding la and waffle la. Thi formulation ue a oundary element with three degree of freedom per node and the eam element i replaced y their action on the plate, that i, a ditriuted load and end of element force. From the olution of the differential equation of a eam with linearly ditriuted load the plate-eam interaction traction can e written a a function of the nodal value of the eam. With thi tranformation a final ytem of equation in the nodal value of diplacement of plate oundary and eam node i otained and from it, all unknown of the plate-eam ytem are otained. The reult how an excellent agreement with thoe from the a finite element analyi. Index Term Boundary element method, plate in ending, eam, tiffner T I. INTRODUCTION HE oundary element method wa firt applied to the analyi of uilding la y BÉZINE [1]) that analyzed the prolem of plate with internal upport that could e ued to imulate a plate upported on rigid column. Since then, everal author have developed formulation for the analyi of uilding la via the Boundary Element Method,, [,3,4,5,6,7]. In the uual formulation of the oundary element method for plate in ending the nodal parameter of the oundary element are w the diplacement w and it derivative. A each eam n node ha three nodal parameter the compatiility of diplacement and rotation of node elonging to the eam and the oundary of the plate are hard to manage, requiring a re-organization of the final matrix of the ytem of equation generating pecial line at the end of thi ytem for the nodal parameter that do not coincide with thoe of the contour of the plate. To olve thi prolem a formulation with three nodal parameter wa propoed,[8,9] ut keeping Kirchhoff thin plate hypothei [1]. With thi formulation coupling the plate with eam and column i much impler. However in thi cae the connection of the plate with the eam i made excluively y mean of vertical force at the node of the finite element. Due to thi punctual force the ending moment at internal connecting plate-eam node are infinite, repreented the olution of the differential equation of plate. In thi work, a new oundary element formulation for the analyi of the plate-eam interaction i preented, in which the plate i modeled y the formulation referred to aove and the eam i replaced y it action on the plate, a ditriuted load and force at it end [11,1]. In thi formulation each eam element ha three node, each with two nodal value, w and w/, and the tranvere diplacement of the eam i approximated y a fifth degree polynomial that repreent the analytical differential equation olution for a eam under tranvere loading with linear variation. A the interaction force etween the eam and the plate can e written a a function of tranvere diplacement and a it i written a a function of the nodal parameter, integral equation for the plateeam coupling can e written in term of the nodal diplacement of the plate and the eam. By impoing the oundary condition and olving the ytem of equation, the diplacement and traction on the eam and plate can readily e calculated. Thi formulation wa then ued in the analyi of aic plate tiffened with eam and the reult were excellent. However the formulation needed to e teted for more complex prolem, uch a plate with eam not parallel to it ide and alo in uual uilding tructure, uch a waffle la. Thi work preent the integral equation for the plate and the olution of the eam and it coupling and then how reult otained in the analyi of plate with nonparallel eam and alo for a typical uilding floor deigned a waffle la. The reult are compared with thoe otained y the finite element method demontrating excellent agreement and confirmed that thi tool can e effectively ued in the analyi of uilding floor la. II. INTEGRAL EQUATIONS Manucript received January th, 15 revied March nd, 15. Thi work wa upported y the reearch council CNPq. C. J. Oliveira i with Poitivo Univerity, Curitia, PR,818-33, Brazil, chjater@hotmail.com J. B. Paiva i with the School of Engineering of São Carlo- Univerity of São Paulo, São Carlo, SP, , Brazil, paiva@c.up.r A. V. Mandonça i with Federal Univerity of Paraia, Civil Engineering Department, João Peoa, PB, 5839, Brazil, mendonca@ct.ufp.r). For a plate in ending with concentric eam, Figure 1, the following oundary integral equation can e written, employing the alternative formulation of Boundary Element Method () with three nodal diplacement parameter [8,9] in which the eam i replaced y the plateeam interaction traction(fig. 1 (II)):

2 , July 1-3, 15, London, U.K. Fig. 1. Plate and eam: coordinate ytem and interface traction w K ( S) w( S) qn ( S, Q) w( Q) mn ( S, Q) ( Q) n w mn ( S, Q) ( Q) d( Q) Vn ( Q) w ( S, Q) N c w n ci ci n i 1 g 3 g w i Vw i i M i Vk wk m ( Q) ( S, Q) d( Q) R ( Q) w ( S, Q) M k g ( q) w d ( q) P ( q) w ds ( q) w k (1) where w, m n and V n are, repectively, the tranvere diplacement, the ending moment and the equivalent hear force along the oundary; g(q) and Ω g are the tranvere load and the urface where it i applied; p 3 (q), V i, V k, M i and M k are traction at the plate-eam interface and S i the coordinate along each eam element axi. The ymol i ued here to indicate fundamental olution. From equation (1) the integral repreentation of the derivative of the diplacement with repect to a direction m, of a ytem of coordinate (m, u ), can e derived a follow: qn w w ( SQwQ, ) ( ) K1(S) (S)+ K(S) (S)+ m ms u S mn w mn w ( SQ, ) ( Q) ( SQ, ) ( Q) ms n ms w w V n(q) (S,Q) mn(q) (S,Q) d (Q) ms ms n N c wci ( S, Q) w Rci ( Q) g( q) m dg ( q) m i1 g i w w P3 ( q) ds Vi m m w i wk w k i k k m m m M ( Sq, ) V + M ( Sq, ) () in aove equation, K() = 1 for internal point ; K(S) = / for a point S at a oundary corner, with internal angle ; K(S) = ½ for a point S on a mooth oundary; R ci mn mn i the corner reaction; 1 K 1(S) inin 8 (3) 1 K (S) co co 8 (4) where i the angle etween the coordinate ytem (n,), at the diplacement point, and (m, u ), at the ource point (Figure 1). The integral equation [1,] are now written to oundary point and the plate-eam internal connecting point. Thu, the oundary of the plate i divided into egment called oundary element with node at their end. The rotation w/ n and the ending moment m n are approximated in each oundary element y linear function, and the equivalent hear force, V n, i approximated y concentrated reaction R k applied to the element node, a decried previouly in [13]. A the corner reaction act on the ame node, their value are alo repreented y the reaction R k. The tranvere diplacement in each oundary element i approximated y a cuic polynomial () and written a a function of the nodal parameter, w and w/, at the end node of the element. Thu, the tranvere diplacement and it directional derivative can e expreed: where: w( ) = [ ]{ e } w ' ' ' ' ( )=[ ]{ e } 1 () = ( )/4 1() = ( )/4 () = ( )L/8 () = ( )L/8 3 () = ( )/4 3() = (3-3 )/4 4 () = ( )L/8 4() = ( )L/8 where L i the length of the oundary element (5) (6)

3 , July 1-3, 15, London, U.K. III. BEAM REPRESENTATION The eam element adopted in thi formulation i ujected to a tranvere loading linearly ditriuted along it length, a hown in Figure 1. The differential equation which repreent the diplacement field of thi element i: dw 4 p (7) d 3 4 The eam element have a node at each end and one at the midpoint (ee Fig. 1 (II)). At thi tage of the analyi, eam torion ha not een included, and thu only nodal parameter related to ending are ued: the vertical diplacement and it directional derivative along the eam axi (w and w/ ). The olution of thi differential equation i a fifth degree polynomial, a follow: w( ) = [ 1]{ i } (8) where { i } i a vector of generalized contant. With the w impoition of oundary condition wi e ( ) i at the node of the element the olution of the differential equation (7) can e otained a a function of the nodal parameter: e w( ) = [ ( ) ( ) ( ) ( ) ( ) ( )]{ }...(9) Where { e } i the vector of nodal variale of the eam: T dw dw i j dwk { e} [ wi wj wk ]...(1) d d d The hape function i ( ) are given y: ( ) ( ) L ( ) ( ) ( ) L ( ) ( ) (11) L 6 ( ) (4 8 5 ) In thee expreion, L i the length of the eam element and ξ=s /L i a dimenionle coordinate along the eam axi (S), with origin at node i. From the differential equation olution of the eam can e otained traction and moment at the interface etween the plate and eam, viz. the ditriuted load, p 3 (q), and the traction on the end, V i, V k, M i and M k. The ditriuted load at the plate-eam interface i otained y utituting the hape function for w (9) in the equation (7) reulting in: e p ( ) [ ( ) ( ) ( ) ( ) ( ) ( )]{ }...(1) where: ( ) (88 163) 1 4 ( ) (48 88) ( ) 3 4 ( ) (19 96)...(13) 4 3 ( ) (48 19) 5 4 ( ) (48 19) 6 3 The ending moment and hear force at the initial (i) and terminal (k) node are otained from the expreion 3 dw dw M and V, putting and d d 3 1 repectively: Mi [ ]{ } L e M k [ ]{ } L e Vi [ ]{ } e L L Vk [ ]{ } e.(14) L L Sutituting thee expreion for the plate-eam interface traction, in term of the diplacement of the eam node, into equation (1) and () give integral equation written in term of the diplacement and traction at the plate oundary node and the eam element node diplacement. IV PLATE-BEAM COUPLING SYSTEM OF EQUATIONS By writing the oundary equation for the diplacement and their derivative in the normal and tangential direction for all node on the oundary and y performing numerically all the integration, the following et of linear equation can e otained: w H H = [G]{V }+{p}...(15) w where {w Ω } contain the diplacement and their derivative for all eam node in the plate domain. Thi new et of linear equation ha more unknown than equation and thu, to alance the unknown and equation, further oundary equation are written for the diplacement and their derivative at all eam node in the domain of the plate, reulting in the following et of linear equation: w H H = [G ]{V }+{ p } w Equation (14) and (15) can e comined to give:...(16)

4 , July 1-3, 15, London, U.K. H H w G p = { V } + H H w G p...(17) Figure 4 and 5 how the diplacement along the eam B and B4.Once again, the agreement etween the two et of reult i excellent. After applying the oundary condition, equation (17) ecome: [ A ] { X }= { B }...(18) in which {X} i a vector compoed of the unknown. After olving thi ytem of equation (18), diplacement and curvature at any point on the plate can e computed from equation (1), with K(S) = 1. Boundary diplacement w (cm).6.4. V NUMERICAL RESULTS The firt example to how the performance of the formulation on la analyi with parallel eam i the uilding floor ketched in Figure with a contant thickne of 1cm, upported at it corner and ujected to a uniform loading of 6.8kN/m. For the concrete ued, Young modulu E = kn/cm and Poion ratio = E x(cm) Fig. 4. Tranvere diplacement at point along eam B..5E-1.E-1 Diplacement w(cm) 1.5E-1 1.E-1 5.E- Fig.. Building floor upported at ix point on the oundary In thi analyi la oundary wa divided into 4 element, the horizontal eam into 4 element and the vertical into element. In Finite Element analyi the la wa mehed with 876 DKT finite element [14]. Figure 3 how the vertical diplacement along the axi of ymmetry (x) and oth, and reult are the ame..e y(cm) Fig. 5. Tranvere diplacement at point along eam B4. The next example i the la with eam inclined with repect to their edge hown in Figure 6. Loading data, upport and dimenion are the ame a the previou example..6.5 Diplacement w(cm) x(m) Fig. 3. Vertical diplacement along the axi of ymmetry (x). Fig. 6. Sla with inclined eam The plate oundary wa divided into 48 oundary element and 4 eam element; the internal eam into 4 element each. Figure 7 how the vertical diplacement along the inclined eam otained with the propoed formulation and the finite element method. Both reult are practically the ame.

5 , July 1-3, 15, London, U.K..8.6 aumed are: Young' modulu E = 38kN/cm Poion' ratio =.. and w(cm) S(cm) Fig. 7. Vertical diplacement along eam B6 Figure 8 how the vertical diplacement along eam B7 and Figure 9 how the reult along coordinate. Thee reult how an excellent concordance among a analyi. w(cm) W(cm) S Fig. 8. Vertical diplacement along eam B S 3 Fig. 9. Vertical diplacement along S Te next example i a waffle la upported at the four corner, a preented in figure 9. The plate ha a contant thickne of 5 cm, and i ujected to a uniform load of 7.5kN / m². The oundary eam (B1) cro ection i cmx6cm and the internal eam (B) cro ection i 8cmx4cm. In the numerical analyi the concrete data Fig. 1. Waffle la Tale 1 how the reult otained in the analyi of thi la with the propoed formulation and thoe otained with finite element method. For the analyi a meh with 4 oundary element, 6 eam element and 1 node were ued. For the analyi were adopted two mehe, one of 4 and another of 16 finite element. Thee reult how excellent agreement among oth formulation. TABLE I WAFFLE SLAB ANALYSIS RESULTS node Coordinate Diplacement w(cm) x y

6 , July 1-3, 15, London, U.K [13] Paiva, J.B. "Boundary element formulation for plate analyi with pecial ditriution of reaction along the oundary". Advance in Engineering Software and Worktation 13: , July 1991 [14] M.N. De Rezende, J. Batita de Paiva, "A Quadrilateral Dicrete Kirchhoff Finite Element for Building Sla Analyi", in M. Papadrakaki, B.H.V. Topping, (Editor), "Advance in Finite Element Technique", Civil-Comp Pre, Edinurgh, UK, pp 5-31, VI CONCLUSION In thi work practical example of uilding floor are analyzed with a comination oundary element method, to repreent the plate, with the olution of the differential equation of eam to repreent the interaction traction among thee two tructural element. Adopting a linear ditriution for the traction etween plate and eam the olution of the differential equation i a fifth degree polynomial which i then written a a function of the nodal parameter adopted for the eam. So the integral equation for the plate are written excluively in function of the nodal diplacement of the oundary of the plate and eam. The reult were compared with thoe of the finite element method howing excellent agreement. REFERENCES [1] BÉZINE, G. A oundary integral equation method for plate flexure with condition inide the domain. Int. J. for Numerical Method in Engineering, 17: , [] PAIVA, J.B.; VENTURINI, W.S. Boundary element algorithm for uilding floor la analyi. Boundary Element Technology Conference, Adelaide (Autr.) Nov., [3] PAIVA, J.B.; VENTURINI, W.S. Analyi of uilding tructure conidering plate-eam-column interaction. International Conference on Boundary Element Technology- Rio de Janeiro- Jun., [4] HARTLEY, G.A.; ABDEL-AKHER A.; CHEN P. Boundary element analyi of thin plate internally ounded y rigid patche. Int. J. for Numerical Method in Engineering, 35: , 199 [5] HU, C.; HARTLEY, G.A. Elatic analyi of thin plate with eam upport. Engineering Analyi with Boundary Element 13, p. 9 38, 1994 [6] FERNANDES, G.R.; VENTURINI, W.S. Stiffened plate ending analyi y oundary element method. Computational Mechanic, 8, p ,. [7] FERNANDES, G.R.; VENTURINI, W.S. Building floor analyi y the oundary element method. Computational mechanic, 35, pp , 5. [8] OLIVRA NETO, L.; PAIVA, J.B. A pecial for elatotatic analyi of uilding floor la on column. Computer & Structure, v.81, n.6, p , March, 3. (ISSN: ) [9] OLIVRA NETO, L.; PAIVA, J.B. Cuic approximation for the tranvere diplacement in for elatic plate analyi. Engineering Analyi with Boundary Element, v.8, p , 4. (ISSN: ) [1] KIRCHHOFF, G. Üer da Gleichgewicht und die Bewegung einer elatichen Scheie. J. Math., v.4, p.51-58, 185. [11] MENDONÇA, A. V.; PAIVA, J.B. Boundary Element Analyi of Plate-Beam Interaction. In: XXXII Jornada Sud Americana de Ingenieria Etructural, Santiago, 8. [1] PAIVA, J. B. ; MENDOÇA, A. V. A coupled oundary element/differential equation method formulation for plate eam interaction analyi. Engineering Analyi with Boundary Element, v. 34, p , 1.