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A new model for slab and beam structures––comparison with other models
Computers and Structures 80 (2002) 459–470 www.elsevier.com/locate/compstruc
A new model for slab and beam structures––comparison with other models Evangelos J. Sapountzakis *, John T. Katsikadelis Department of Civil Engineering, Institute of Structural Analysis, National Technical University of Athens, Zografou Campus, GR-157 73 Athens, Greece Received 2 June 2001; accepted 9 January 2002
Abstract In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons, while a three-dimensional elasticity model is also employed for the verification of the accuracy of the results of the examined models. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Model of elastic stiffened plate; Reinforced plate with beams; Bending; Ribbed plate; Analog equation method
1. Introduction Slab and beam structures are widely used in the construction of long river or valley bridges, long span slabs or floors of aircraft carriers. Stiffened plate structures are efficient, economical, functional and readily constructed of most common materials. The use of structural plate systems stiffened by beams improves their strength and performance as it minimizes their dead weight. The extensive use of the aforementioned plate structures necessitates a rigorous analysis.
* Corresponding author. Tel.: +301-772-1718; fax: +301-7721720. E-mail address: [email protected] (E.J. Sapountzakis).
The analysis of the behaviour of the aforementioned structural plate systems has been approximated by converting this system to an equivalent homogeneous slab of constant thickness using the stiffness properties of the beams and applying the orthotropic plate theory [1–6]. This approximation may be applicable only when the stiffened plate satisfies two limitations. The first one is that ratios of spacing between two consecutive stiffeners to slab boundary dimensions are small enough to ensure approximate homogeneity of stiffness. The second limitation is that the ratio of stiffener rigidity to the slab rigidity must not become so large that the beam action is predominant. Moreover approximate methods using energy principles [7,8] as well as numerical methods such as finite element method (FEM) and BEM [9–15] have been employed for the analysis of stiffened plate systems. In
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 2 0 - 2
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the employed models for the development of these methods the inplane forces and deformations due to the shear forces at the interfaces have been neglected. This assumption results in discrepancies from the actual response of the plate-beams system. Moreover, it does not allow the establishment of these forces, which are necessary for the study of prefabricated or composite structures. In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system [16]. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems, which are non-linearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons. Finally, a three-dimensional (3-D)
elasticity model is also employed for the verification of the accuracy of the results of the examined models. The analysis of the stiffened structure using the last five models is achieved employing the FEM. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations.
2. Statement of the problem Consider a thin elastic plate having constant thickness hp , occupying the domain X of the x, y plane and stiffened by a set of parallel beams. The plate may have J holes while its boundary C ¼ [Jj¼0 Cj may be piecewise smooth (Fig. 1). For the sake of convenience the x-axis is taken parallel to the beams. The plate is subjected to the lateral load g ¼ gðxÞ, x : fx; yg and is supported on its boundary, whereas the beams may have point supports. According to the proposed model the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, while tractions at the fictitious interfaces are taken into account (Fig. 2). These tractions result in the loading of the beam as well as the additional loading of the plate. Their distribution is unknown and is established by imposing displacement continuity conditions at the interfaces. The integration of the tractions along the width of the beam result in line forces per unit length which are denoted by qx , qy and qz . Taking into account that the
Fig. 1. Two-dimensional region X occupied by the plate.
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
461
Fig. 2. Thin elastic plate stiffened by beams (a) and isolation of the beams from the plate (b).
torsional stiffness of the beam is small, the traction component qy , in the direction normal to the beam axis is ignored. The other two components qx and qz produce the following loadings along the trace of each beam.
On the base of the above considerations the response of the plate and of the beams may be described by the following boundary value problems. 2.3. For the plate
2.1. In the plate (i) A lateral line load qz at the interface. (ii) A lateral line load oMp =ox due to the eccentricity of the component qx from the middle surface of the plate. Mp ¼ qx hp =2 is the bending moment. (iii) An inplane line body force qx at the middle surface of the plate.
The plate undergoes transverse deflection and inplane deformation. Thus, for the transverse deflection we have o2 wp o2 wp o2 wp Dr4 wp Nx 2 þ 2Nxy þ Ny ox ox oy oy 2 ! K X oMpðkÞ dð y yk Þ in X ð1Þ qðkÞ ¼g z þ ox k¼1 a1 wp þ a2 Vn ¼ a3
2.2. In each beam (i) A transverse load qz . (ii) A transverse load oMb =ox due to the eccentricity of qx from the neutral axis of the beam cross-section. (iii) An inplane axial force qx . The structural models of the plate and the beams are shown in Fig. 3.
on C b1
owp on
where wp ¼ wp ðxÞ is the transverse deflection of the plate; D ¼ Ep h3p =12ð1 m2 Þ is its flexural rigidity with Ep being the elastic modulus and m the Poisson ratio; Nx ¼ Nx ðxÞ, Ny ¼ Ny ðxÞ, Nxy ¼ Nxy ðxÞ are the membrane forces per unit length of the plate cross-section; dðy yk Þ is the Dirac’s delta function in the y direction; Mn and Vn are the bending moment normal to the boundary and the effective reaction along it, respectively, and they are given as 2 o wp o2 wp Mn ¼ D þ m ð3Þ on2 ot2
o 2 o o2 wp Vn ¼ D r wp ðm 1Þ on os on ot
Fig. 3. Structural model of the plate and the beams.
ð2Þ
þ b2 Mn ¼ b3
ð4Þ
Finally, ai , bi ði ¼ 1; 2; 3Þ are functions specified on the boundary C. The boundary conditions (2) are the most general linear boundary conditions for the plate problem including also the elastic support. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived form these equations by specifying appropriately
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the functions ai and bi (e.g. for a clamped edge it is a1 ¼ b1 ¼ 1, a2 ¼ a3 ¼ b2 ¼ b3 ¼ 0). Since linear plate bending theory is considered, the components of the membrane forces Nx , Ny , Nxy do not depend on the deflection wp . They are given as oup ovp þm ð5aÞ Nx ¼ C ox oy oup ovp Ny ¼ C m þ ox oy 1m Nxy ¼ C 2
oup ovp þ oy ox
ð5bÞ
ventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from Eq. (9) by specifying appropriately the coefficients ai , bi (e.g. for a simply supported end it is a1 ¼ b2 ¼ 1, a2 ¼ a3 ¼ b1 ¼ b3 ¼ 0). Since linear beam bending theory is considered, the axial force of the beam does not depend on the deflection wb . The axial deformation of the beam is described by solving independently the boundary value problem i.e. Eb Ab
o2 ub ¼ qx ox2
in Lk ; k ¼ 1; 2; . . . ; K
ð10Þ
at the beam ends x ¼ 0; l
ð11Þ
ð5cÞ c1 ub þ c2 N ¼ c3
where C ¼ Ep =ð1 m2 Þ; up ¼ up ðxÞ and vp ¼ vp ðxÞ are the displacement components of the middle surface of the plate and are established by solving the plane stress problem, which is described by the following boundary value problem (Navier’s equations of equilibrium) 1 þ m o oup ovp 1 r2 up þ qx dð y yk Þ ¼ 0 þ þ 1 m ox ox Gp oy in X 1 þ m o oup ovp r 2 vp þ þ ¼0 oy 1 m oy ox ð6Þ
where N is the axial reaction at the beam ends. Eqs. (1), (6), (8) and (10) constitute a set of five coupled partial differential equations including seven unknowns, namely wp , up , vp , wb , ub , qx , qz . Two additional equations are required, which result from the continuity conditions of the displacements in the direction of z and x axes at the interfaces between the plate and the beams. These conditions can be expressed as wp ¼ wb up
c1 un þ c2 Nn ¼ c3
ð12Þ
hp owp hb owb ¼ ub þ 2 ox 2 ox
ð13Þ
ð7Þ
on C d1 ut þ d2 Nt ¼ d3
in which Gp ¼ Ep =2ð1 þ mÞ is the shear modulus of the plate; Nn , Nt and un , ut are the boundary membrane forces and displacements in the normal and tangential directions to the boundary, respectively; ci , di ði ¼ 1; 2; 3Þ are functions specified on C.
3. Numerical solution
2.4. For each beam
3.1. The analog equation method for the plate equation
Each beam undergoes transverse deflection and axial deformation. Thus, for the transverse deflection we have
Let wp be the sought solution of the boundary value problem (1) and (2). If the biharmonic operator is applied to this function we have
d4 wb o2 wb oMb Nb 2 ¼ qz 4 dx ox ox k ¼ 1; 2; . . . ; K
Eb Ib
in Lk ;
r4 wp ¼ qp
at the beam ends x ¼ 0; l
in X
ð14Þ
ð8Þ
a1 wb þ a2 V ¼ a3 owb b1 þ b2 M ¼ b3 ox
The differential equations are solved numerically. More specifically, Eqs. (1), (6), (8) and (10) are solved using the AEM.
ð9Þ
where wb ¼ wb ðxÞ is the transverse deflection of the beam; Eb Ib is its flexural rigidity; Nb ¼ Nb ðxÞ is the axial force at the neutral axis; V, M are the reaction and the bending moment at the beam ends, respectively and ai , bi ði ¼ 1; 2; 3Þ are coefficients specified at the boundary of the beam. It is apparent that all types of the con-
Thus, the problem is converted to that of establishing the unknown the fictitious load distribution qp ðxÞ. This is accomplished using BEM as follows. The solution of Eq. (14) is given in integral form as Z Z ow p Mn ewp ðxÞ ¼ w p qp dX w p Vn on X C owp
M ds ð15Þ þ wp Vn on n where e ¼ 1, 1/2 or 0 depending on whether the point x is inside the domain X, on the boundary C, or outside
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
X, respectively and w p is the fundamental solution, which is given as w p ¼
1 2 r ln r 8p
ð16Þ
with r ¼ ½ðn xÞ2 þ ðg yÞ2 1=2 , x, y 2 X [ C, n, g 2 C. The quantities Mn and Vn are obtained from Eqs. (3) and (4), respectively, by replacing wp with w p . Eq. (15) and its normal derivative for x 2 C yield the following boundary integral equations Z Z ow p 1 Mn w p qp dX w p Vn wp ðxÞ ¼ on 2 X C owp
M ds ð17Þ þ wp Vn on n
1 owp ðxÞ ¼ 2 onx
Z
ow p qp dX onx
Z
ow p o2 w p Mn Vn onx onx on X C ! o owp o
ð18Þ þ wp V M ds onx n on onx n
Eqs. (17) and (18) together with Eq. (2) can be employed to express the unknown boundary quantities wp , owp =on, Mn , Vn in terms of qp . This is accomplished numerically. The boundary integrals are evaluated using constant boundary elements, while the domain integrals using constant cells. If N is the number of boundary nodal points and M that of the domain nodal points, this procedure yields the following set of linear equations 2 38 w 9 > ½A11 ½0 ½0 ½A14 > > >n p o> > 6 ½0 ½A22 ½A23 ½0 7< owp = 6 7 on 4 ½A31 ½A32 ½A33 ½A34 5> > > > fMn g > > ½A41 ½A42 ½A43 ½A44 : fV g ; n 8 9 2 3 fa3 g > ½0 > > > < = 6 ½0 7 fb3 g 7 qp ð19Þ þ6 ¼ 4 5 0 ½ C f g > > 3 > > : ; ½C4 f0g in which [A11 ], [A14 ], [A22 ], [A23 ] are diagonal N N matrices including the nodal values of the functions a1 , a2 , b1 , b2 of Eq. (2) and [Aij ], ði ¼ 3; 4; j ¼ 1; 2; 3; 4Þ are square N N known coefficient matrices originating from the integration of the kernels on the boundary elements; {a3 }, {b3 } are N 1 known vectors including the N nodal values of the functions a3 , b3 in Eq. (2) and ½Ci ði ¼ 3; 4Þ are N M rectangular known matrices originating from the integration of the kernels on the domain cells. Finally, fwp g, fowp =ong, fMn g, fVn g are vectors including the N unknown nodal values of the respective boundary quantities and {qp } is a vector including the M unknown nodal values of the fictitious load.
463
The discretized counterpart of Eq. (15) when applied to all nodal points in X yields owp wp ¼ ½C qp ½A1 wp þ ½A2 on ð20Þ þ ½A3 fMn g þ ½A4 fVn g where [C] is an M M known matrix and [Ai ], ði ¼ 1; 2; 3; 4Þ are M N also known matrices. Elimination of the boundary quantities from Eq. (20) using Eq. (19) yields wp ¼ ½ P qp ð21Þ where [P] is an M M matrix. Further, differentiating Eq. (15) for x 2 X, using the same boundary and domain discretization and eliminating the boundary quantities by means of Eq. (20) we obtain wp;x ¼ ½Px qp ð22aÞ wp;xx ¼ ½Pxx qp
ð22bÞ
wp;yy ¼ Pyy qp
ð22cÞ
wp;xy ¼ Pxy qp
ð22dÞ
where [Px ], [Pxx ], [Pyy ], [Pxy ] are known M M coefficient matrices. The final step of AEM is to apply Eq. (1) to the M nodal points inside X. This yields D qp ½Nx ½Pxx þ 2 Nxy Pxy þ Ny Pyy qp ¼ fgg ½Z fqz g ½ X fqx g
ð23Þ
where [Nx ], [Nxy ] and [Ny ] are unknown diagonal M M matrices including the values of the inplane forces; {qz } and {qx } are vectors with L elements; L is the total number of the nodal points at the interfaces; [Z] is a position vector which converts the vector {qz } into a vector with length M. The matrix [X] results after approximating the derivative of Mp with finite differences. Its dimensions are also M L. 3.2. The analog equation method for the plane stress problem Following the procedure presented in Katsikadelis and Kandilas [17] and using the same boundary and domain discretization the membrane forces for homogeneous boundary conditions (7) (c3 ¼ d3 ¼ 0) are expressed as follows
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fNx g ¼ ½Fx fqx g Nxy ¼ Fxy fqx g Ny ¼ Fy fqx g
ð24Þ
where [Fx ], [Fxy ] and [Fy ] are known matrices with dimensions M L. 3.3. The analog equation method for the deflection of the beams Let wb be the sought solution of the boundary value problem described by Eqs. (8) and (9). Differentiating this function four times yields 4
d wb ¼ qb dx4
ð25Þ
Eq. (25) indicates that the solution of the original problem can be obtained as the deflection of a beam with unit flexural rigidity subjected only to a transverse fictitious load qb under the same boundary conditions. The fictitious load is unknown. However, it can be established using BEM as follows. The solution of Eq. (25) is given in integral form as l Z l ow
owb
M wb ðxÞ ¼ w b qb ds þ w b V b M þ wb V ox ox 0 0 ð26Þ
where w b is the fundamental solution which is given as w b ¼
1 3 l ð2 þ jqj3 3jqj2 Þ 12
ð27Þ
with q ¼ r=l, r ¼ n x, x 2 Lk , k ¼ 1; 2; . . . ; K, n at the beam ends x ¼ 0; l and the quantities M and V are given as M ¼ 12lð1 jqjÞ
ð28aÞ
V ¼ 12sgn q
ð28bÞ
Eq. (26) and its derivative for x ¼ 0 and x ¼ l yield wb ¼
Z
l 0
l ow
owb
M w b qb ds þ w b V b M þ wb V ox ox 0 ð29Þ
owb ¼ oxb
Z
l
ow b ow b o2 w b M qb ds þ V ox ox ox b b b ox 0 l o owb o þ wb V M
ox oxb oxb 0
ð30Þ
Eqs. (29) and (30) together with Eq. (9) can be employed to express the unknown boundary quantities wb , owb =on, M, V in terms of qb . This is accomplished numerically. If
k is the number of parallel beams and L that of the nodal points at the interfaces this procedure yields the following set of linear equations 9 2 38 ½E11 ½0 ½0 ½E14 > fwb g > > > 6 ½0 ½E22 ½E23 ½0 7< owb = on 6 7 4 ½E31 ½E32 ½E33 ½E34 5> f M g > > > : ; ½E41 ½E42 ½E43 ½E44 fV g 8 9 2 3 ½0 fa3 g > > > > < = 6 fb3 g ½0 7 7fq g 6 ð31Þ þ4 ¼ ½ f 0 g F3 5 b > > > > : ; ½F4 f0g in which [E11 ], [E14 ], [E22 ], [E23 ] are diagonal 2k 2k matrices including the nodal values of the functions a1 , a2 , b1 , b2 of Eq. (9) and [Eij ], ði ¼ 3; 4; j ¼ 1; 2; 3; 4Þ are square 2k 2k known coefficient matrices; {a3 }, {b3 } are 2k 1 known column matrices including the boundary values of the functions a3 , b3 in Eq. (9) and [Fi ] ði ¼ 3; 4Þ are 2k L rectangular known matrices originating from the integration of the kernels on the axis of the beams. Finally, fwb g, fowb =ong, fMg, fV g are vectors including the 2k unknown nodal values of the respective boundary quantities and {qb } is a vector including the L unknown nodal values of the fictitious load. The discretized counterpart of Eq. (26) when applied to all nodal points at the interfaces yields owb fwb g ¼ ½ F fqb g ½E1 fwb g þ ½E2 on ð32Þ þ ½E3 f M g þ ½E4 fV g where [F] is an L L known matrix and [Ei ], ði ¼ 1; 2; 3; 4Þ are L 2k also known matrices. Elimination of the boundary quantities from Eq. (32) using Eq. (31) for homogeneous boundary conditions (9) (a3 ¼ b3 ¼ 0) yields fwb g ¼ ½ B fqb g
ð33Þ
where [B] is an L L matrix. Further, differentiating Eq. (26) for x 2 Lk and eliminating the boundary quantities by means of Eq. (31) we obtain wb;x ¼ ½Bx fqb g ð34aÞ
wb;xx ¼ ½Bxx fqb g
ð34bÞ
where [Bx ], [Bxx ] are known L L coefficient matrices. The final step of AEM is to apply Eq. (8) to the L nodal points at the interfaces. This yields Eb Ib fqb g ½Nb ½Bxx fqb g ¼ fqz g ½Q fqx g
ð35Þ
where [Nb ] is unknown diagonal L L matrix including the values of the inplane forces; {qz } and {qx } are vectors with L elements. The matrix [Q] results after approxi-
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
mating the derivative of Mb with finite differences. Its dimensions are also L L. 3.4. The analog equation method for the axial deformation of the beams The solution of Eq. (10) is given in integral form as l Z l dub du b ub u b qx ds þ u b ð36Þ ub ðxÞ ¼ dx dx 0 0 where u b ¼
1 ðl jrjÞ 2Eb Ab
ð37Þ
Following the same procedure as for the plane stress problem the axial force at the neutral axis of the beam for homogeneous boundary conditions (11) (c3 ¼ 0) can be expressed as follows fNb g ¼ ½Fb fqx g
ð38Þ
where [Fb ] is known matrix with dimensions L L. Eqs. (23) and (35) after elimination of the quantities Nx , Ny , Nxy , Nb using Eqs. (24) and (38) together with continuity conditions (12) and (13) which after discretization at the L nodal points at the interfaces are written as wp ¼ fwb g ð39Þ hp hb wp;x ¼ fub g þ wb;x up 2 2 constitute a non-linear system of equations with respect to qz , qx , qp , qb . This system is solved using iterative numerical methods. Subsequently, the deflection at any interior point x 2 X of the stiffened plate is established using the discretized counterpart of Eq. (15) when applied to this point. For the solution of the non-linear system of equations the two-term acceleration method [18] has been employed. According to this method, an initial vector, say qð0Þ x ¼ 0 is assumed. Using this vector and Eqs. (35) ð0Þ ð0Þ and (39) the values of the vectors qð0Þ are z , qp , qb computed. Introducing these values into Eq. (23) a 0Þ vector qð1 is obtained. To accelerate convergence the x new initial vector is given as qð1Þ x
¼
aqð0Þ x
þ
0Þ bqð1 x
465
where [Fp ], [Fb ] are known L L flexibility coefficient matrices the elements of which are evaluated numerically following the steps of the AEM. 4. Presentation of additional models In Fig. 4 the structural model of ‘‘model I’’ is presented. According to this the stiffened plate is approximated by a simply supported beam with a cross-section consisting of a part of the plate corresponding to the effective breadth and the beam cross-section. The analysis of the model is achieved by solving the beam bending governing differential equation EI
d4 w ¼ q in the interior of the beam dx4
ð42Þ
satisfying the boundary conditions a1 w þ a2 V ¼ a3 ow þ b2 M ¼ b3 b1 ox
at the beam ends
ð43Þ
where w ¼ wðxÞ is the transverse deflection of the stiffened plate along the axis of the stiffening beam; EI is the flexural rigidity of the approximating beam; V, M are the reaction and the bending moment at the beam ends, respectively and ai , bi (i ¼ 1; 2; 3) are coefficients specified at the boundary of the beam. In Figs. 5 and 6 a typically discretized stiffened plate according to ‘‘model II’’ and ‘‘model III’’, respectively, are presented. According to these the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, while only qz tractions at the
Fig. 4. Cross-section and structural system of model I.
ð40Þ
where a, b appropriately chosen weight factors with a þ b ¼ 1. It is worth here noting that in Eq. (39) the column matrices {up }, {ub } are evaluated from the relations ð41aÞ up ¼ Fp fqx g fub g ¼ ½Fb fqx g
ð41bÞ
Fig. 5. Discretized stiffened plate according to model II.
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Fig. 8. Discretized stiffened plate according to model V. Fig. 6. Discretized stiffened plate according to model III.
fictitious interfaces are taken into account (Fig. 2). These tractions result in the loading of the beam as well as the additional loading of the plate. The difference between the two models is due to the eccentricity of the structural model of the beam with respect to that of the plate, equal with the distance between the center of gravity axis of the beam and the middle surface of the plate, which is taken into account in model III. The analysis of the plate is accomplished solving the plate bending governing differential equation Dr4 wp ¼ g
ð44Þ
in the interior of the plate, satisfying the boundary conditions (2) at its boundary, where wp is the transverse deflection of the plate. Moreover, the beam bending problem is governed from Eq. (42) and has to satisfy the boundary conditions (43). Both the plate and beam problems are solved using the FEM [19], while the distribution of the qz interface forces is established by imposing the displacement continuity condition (12) at the interfaces. In Fig. 7 a typically discretized stiffened plate according to ‘‘model IV’’ is presented. The difference between this model and model III is restricted in the analysis of the beam, which is accomplished by solving the plane stress problem in terms of displacements, given as 2 o u o2 w lr2 u þ ðk þ lÞ þ þ bx ¼ 0 ox2 ox oz ð45Þ 2 o u o2 w lr2 w þ ðk þ lÞ þ 2 þ bz ¼ 0 ox oz oz
and applying the continuity conditions (12) and (13) at the interfaces. In Eq. (45) u is the displacement along the beam axis; w is the deflection of the beam; k, l are Lame constants and bx , bz are body forces. Both the plate and beam problems are solved using the FEM [19]. Till now, in all the presented models of this section the inplane forces and deformations due to the shear forces at the interfaces have been neglected. Finally in Fig. 8 a typically discretized stiffened plate according to ‘‘model V’’ is presented, which takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. The analysis of this model is accomplished by solving for both the plate and the beams the 3-D elasticity problem in terms of displacements, given as 2 ou o2 v o2 w lr2 u þ ðk þ lÞ þ ð46aÞ þ þ bx ¼ 0 ox2 ox oy ox oz lr2 v þ ðk þ lÞ
lr2 w þ ðk þ lÞ
o2 u o2 v o2 w þ 2þ ox oy oy oy oz
o2 u o2 v o2 w þ þ 2 ox oz oy oz oz
þ by ¼ 0
ð46bÞ
þ bz ¼ 0
ð46cÞ
and satisfying prescribed displacement uðx; y; zÞ ¼ u0
ð47aÞ
vðx; y; zÞ ¼ v0
ð47bÞ
wðx; y; zÞ ¼ w0
ð47cÞ
or traction
Fig. 7. Discretized stiffened plate according to model IV.
tx ðx; y; zÞ ¼ tx0
ð48aÞ
ty ðx; y; zÞ ¼ ty0
ð48bÞ
tz ðx; y; zÞ ¼ tz0
ð48cÞ
boundary conditions of the stiffened structure. The analysis of this model and the solution of the aforementioned problem is achieved using the FEM [19]. It is worth here noting that the continuity conditions at
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
467
the fictitious interfaces are automatically satisfied for this model, as the stiffened structure is examined as a whole. It is obvious that model V approximates better the behaviour of the stiffened plate. The disadvantage of this model compared with the proposed one is focalized in its inability to estimate the interface forces, the knowledge of which is very important in the design of composite or prefabricated ribbed plates.
5. Numerical examples On the basis of the analytical and numerical procedures presented in the previous sections, representative examples have been studied to verify the accuracy, the efficiency and the range of applications of the proposed model. Example 1: A rectangular plate with dimensions ap bp ¼ 18:0 m 9:0 m and thickness hp ¼ 0:20 m subjected to a uniform load g ¼ 10 kN/m2 and stiffened by a beam of width 1.0 m through its centerline has been studied (Ep ¼ Eb ¼ 3:0 107 kN/m2 , m ¼ 0:154). The plate is simply supported along its small edges, while the other two edges are free according to the transverse boundary conditions and Nx ¼ Nxy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nyx ¼ 0 along the edges y ¼ bp =2 and y ¼ bp =2 according to the inplane boundary conditions. In Table 1 the deflections of the plate at its center for different beam heights hb are shown as compared with those obtained from the five additional presented models. Moreover, in Fig. 9 the distributions of the deflection w along the axis of symmetry of the interface, according to the proposed model and model V, are presented for various values of the beam height hb . The discrepancy of the results of the proposed model (AEM results) from the corresponding ones of models I, II, III and IV is obvious. The results of model V, especially for high beams, are somewhat similar with those of the proposed model. The observed discrepancy between the proposed model and model V is attributed to the different implementation of the boundary conditions between the two models. Example 2: The same plate as in Example 1, with all edges simply supported according to the transverse
Fig. 9. Deflections along the axis of the beam of the stiffened plate of Example 1 for various values of the beam height hb .
boundary conditions and ux ¼ 0, Nxy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nyx ¼ 0 along the edges y ¼ bp =2, y ¼ bp =2 according to the inplane boundary conditions has been studied. In Table 2 the deflections of the plate at its center for different beam heights hb are shown as compared with those obtained from the five additional presented models. Moreover, in Fig. 10 the distributions of the deflection w along the axis of symmetry of the interface, according to the proposed model and model V, are presented for various values of the beam height hb . The same conclusions as in Example 1 can easily be drawn. Example 3: A rectangular concrete plate (Ep ¼ 3:0 107 kN/m2 , m ¼ 0:154) with dimensions a b ¼ 18:0 m 9:0 m subjected to a uniform load g ¼ 10 kN/ m2 and stiffened by an I-section steel beam (Eb ¼ 3:0 108 kN/m2 ) through its centerline has been studied (Fig. 11). The plate is simply supported along its small edges, while the other two edges are free according to the transverse boundary conditions and ux ¼ uy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nxy ¼ 0 along the edges y ¼ bp =2 and y ¼ bp =2 according to the
Table 1 Deflections w (mm) at the center of the stiffened plate of Example 1 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.60 1.00 1.25 2.00
33.44 9.55 5.30 1.49
42.72 21.91 7.37 2.25
174.61 46.07 24.31 6.08
38.30 10.79 6.02 1.70
44.79 12.80 7.12 1.56
24.85 7.76 4.57 1.48
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Table 2 Deflections w (mm) at the center of the stiffened plate of Example 2 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.60 1.00 1.25 2.00
7.86 2.18 1.22 0.35
42.72 21.91 7.37 2.25
27.38 16.10 10.71 3.42
14.40 5.63 3.37 1.01
7.90 4.11 2.73 0.97
4.58 1.61 1.02 0.39
6. Concluding remarks In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models previously used. The adopted model contrary to the already used models takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. The main conclusions that can be drawn from this investigation are
Fig. 10. Deflections along the axis of the beam of the stiffened plate of Example 2 for various values of the beam height hb .
inplane boundary conditions. In Tables 3 and 4 the deflections of the plate at its center and at the middle of the free edge, respectively, for different beam heights hb are shown as compared with those obtained from the five additional presented models. The conclusions of Example 1 are once again verified. Finally, in Fig. 12 the distributions of the interface forces qx are presented for various values of the height hb of the beam.
(a) The discrepancy of the results of the proposed model from the corresponding ones of models I, II, III and IV, which ignore the inplane forces and deformations due to the shear forces at the interfaces, is remarkable. This discrepancy necessitates the consideration of the inplane forces and deformations. (b) The results of model V, which analyzes the stiffened plate solving a 3-D elasticity problem, are somewhat similar with those of the proposed model. The observed discrepancy between the proposed model and model V is attributed to the different implementation of the boundary conditions between the 3-D elasticity model and the plate-beam model, which is more intensive in flexible structures. (c) The adopted model permits the evaluation of the inplane shear forces at the interface between the plate and the beams, the knowledge of which is very important in the design of composite or prefabricated plate-beams structures (estimation of bondage, shear connectors or welding).
Fig. 11. Plan (a) and cross-section (b) of the stiffened plate of Example 3.
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469
Table 3 Deflections w (mm) at the center of the stiffened plate of Example 3 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.50 1.00 1.50 2.00
9.00 2.873 1.342 0.756
51.819 18.091 9.244 5.711
117.16 35.532 16.848 9.881
46.271 16.168 8.310 5.165
18.240 7.181 4.154 2.851
13.898 5.871 3.490 2.434
Table 4 Deflections w (mm) at the middle of the free edge of the stiffened plate of Example 3 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.50 1.00 1.50 2.00
28.56 22.58 21.09 20.51
1197.2 679.2 472.7 363.3
138.673 56.983 38.291 31.324
67.616 37.568 29.730 26.594
34.070 23.155 20.148 18.846
29.180 21.662 19.427 18.433
Fig. 12. Distribution of the qx forces along the beam axis of the stiffened plate of Example 1.
Acknowledgements The author wishes to thank his postgraduate student, Diplom Civil Engineer Mr. A. Kontizas for his assistance in obtaining the numerical results.
References [1] Huber MT. Die Theorie des Kreuzweise Bewehrten Eisenbetonplatten. Der Bauingenieur 1923;4:354–92. [2] Guyon Y. Calcul des Ponts Larges a Poutres Multiples Solidarisees por les Entretoises. Ann Ponts Chausees 1946;116(5):553–612.
[3] Massonet C. Method of calculations for bridges with several longitudinal beams taking into account their torsional resistance. Int Assoc Bridges Struct Eng 1950;10:147–82. [4] Cornelius W. Die Berechnung der Ebener Fl€achentragwerke mit Hilfe der Theorie der Orthogonal Anisotropen Platten. Der Stahlbau 1952;2:21–6. [5] Pama RP, Cusens AR. Edge beam stiffening of multibeam bridges. J Struct Div 1967;93(ST2):141–61. [6] Powell GH, Ogden DW. Analysis of orthotropic steel plate bridge decks. J Struct Div 1969;95(ST5):909–21. [7] Kukreti AR, Rajapaksa Y. Analysis procedure for ribbed and grid plate systems used for bridge decks. J Struct Eng 1990;116(2):372–91. [8] Kukreti AR, Cheraghi E. Analysis procedure for stiffened plate systems using an energy approach. Comput Struct 1993;46(4):649–57. [9] Cheung YK. Finite strip method in analysis of elastic plates with two opposite sides simply supported ends. Proc Inst Civil Eng 1968;40:1–7. [10] King IP, Zienkienwicz OC. Slab bridges with arbitrary shape and support conditions: a general method of analysis based on finite element method. Proc Inst Civil Eng 1968;40:9–36. [11] Tanaka M, Bercin AN. A boundary element method applied to the elastic bending problem of stiffened plates. Boundary Elem Method 1997;XIX:203–12. [12] Hu C, Hartley GA. Elastic analysis of thin plates with beam supports. Eng Anal Boundary Elem 1994;13:229– 38. [13] de Paiva JB. Boundary element formulation of building slabs. Eng Anal Boundary Elem 1996;17:105–10. [14] Ng SF, Cheung MS, Xu T. A combined boundary element and finite element solution of slab and slab-on-girder bridges. Comput Struct 1990;37:1069–75. [15] Cheung MS, Akhras G, Li W. Combined boundary element/finite strip analysis of bridges. J Struct Eng 1994;120:716–27.
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[16] Sapountzakis EJ, Katsikadelis JT. Analysis of plates reinforced with beams. Computat Mech 2000;26:66– 74. [17] Katsikadelis JT, Kandilas CB. Solving the plane elastostatic problem by the analog equation method. Comput Struct 1994;64:305–12.
[18] Sapountzakis EJ, Katsikadelis JT. Unilaterally supported plates on elastic foundations by the boundary element method. J Appl Mech 1992;59:580–6. [19] Sofistik, GmbH, Software f€ ur Statik und Konstruktion Gesellschaft mit beschr€ankter Haftung, Ringstrasse 29a, D-8042, Oberschleißheim, 1995.
A new model for slab and beam structures––comparison with other models Evangelos J. Sapountzakis *, John T. Katsikadelis Department of Civil Engineering, Institute of Structural Analysis, National Technical University of Athens, Zografou Campus, GR-157 73 Athens, Greece Received 2 June 2001; accepted 9 January 2002
Abstract In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons, while a three-dimensional elasticity model is also employed for the verification of the accuracy of the results of the examined models. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Model of elastic stiffened plate; Reinforced plate with beams; Bending; Ribbed plate; Analog equation method
1. Introduction Slab and beam structures are widely used in the construction of long river or valley bridges, long span slabs or floors of aircraft carriers. Stiffened plate structures are efficient, economical, functional and readily constructed of most common materials. The use of structural plate systems stiffened by beams improves their strength and performance as it minimizes their dead weight. The extensive use of the aforementioned plate structures necessitates a rigorous analysis.
* Corresponding author. Tel.: +301-772-1718; fax: +301-7721720. E-mail address: [email protected] (E.J. Sapountzakis).
The analysis of the behaviour of the aforementioned structural plate systems has been approximated by converting this system to an equivalent homogeneous slab of constant thickness using the stiffness properties of the beams and applying the orthotropic plate theory [1–6]. This approximation may be applicable only when the stiffened plate satisfies two limitations. The first one is that ratios of spacing between two consecutive stiffeners to slab boundary dimensions are small enough to ensure approximate homogeneity of stiffness. The second limitation is that the ratio of stiffener rigidity to the slab rigidity must not become so large that the beam action is predominant. Moreover approximate methods using energy principles [7,8] as well as numerical methods such as finite element method (FEM) and BEM [9–15] have been employed for the analysis of stiffened plate systems. In
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 2 0 - 2
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the employed models for the development of these methods the inplane forces and deformations due to the shear forces at the interfaces have been neglected. This assumption results in discrepancies from the actual response of the plate-beams system. Moreover, it does not allow the establishment of these forces, which are necessary for the study of prefabricated or composite structures. In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models used by various researchers. The adopted model contrary to the models used previously takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system [16]. According to this model the stiffening beams of the structure are isolated from the plate by sections parallel to the lower outer surface of the plate. The forces at the interface, which produce lateral deflection and inplane deformation to the plate and lateral deflection and axial deformation to the beam, are established using continuity conditions at the interface. The solution of the arising plate and beam problems, which are non-linearly coupled, is achieved using the analog equation method (AEM). The adopted model describes better the actual response of the plate-beams system and permits the evaluation of the shear forces at the interface, the knowledge of which is very important in the design of composite or prefabricated ribbed plates. Four additional models neglecting the shear forces at the interfaces are presented and used for comparison reasons. Finally, a three-dimensional (3-D)
elasticity model is also employed for the verification of the accuracy of the results of the examined models. The analysis of the stiffened structure using the last five models is achieved employing the FEM. The findings from this investigation, using the adopted model, which approximates better the actual response of the plate-beams system, necessitate the consideration of the inplane forces and deformations.
2. Statement of the problem Consider a thin elastic plate having constant thickness hp , occupying the domain X of the x, y plane and stiffened by a set of parallel beams. The plate may have J holes while its boundary C ¼ [Jj¼0 Cj may be piecewise smooth (Fig. 1). For the sake of convenience the x-axis is taken parallel to the beams. The plate is subjected to the lateral load g ¼ gðxÞ, x : fx; yg and is supported on its boundary, whereas the beams may have point supports. According to the proposed model the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, while tractions at the fictitious interfaces are taken into account (Fig. 2). These tractions result in the loading of the beam as well as the additional loading of the plate. Their distribution is unknown and is established by imposing displacement continuity conditions at the interfaces. The integration of the tractions along the width of the beam result in line forces per unit length which are denoted by qx , qy and qz . Taking into account that the
Fig. 1. Two-dimensional region X occupied by the plate.
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
461
Fig. 2. Thin elastic plate stiffened by beams (a) and isolation of the beams from the plate (b).
torsional stiffness of the beam is small, the traction component qy , in the direction normal to the beam axis is ignored. The other two components qx and qz produce the following loadings along the trace of each beam.
On the base of the above considerations the response of the plate and of the beams may be described by the following boundary value problems. 2.3. For the plate
2.1. In the plate (i) A lateral line load qz at the interface. (ii) A lateral line load oMp =ox due to the eccentricity of the component qx from the middle surface of the plate. Mp ¼ qx hp =2 is the bending moment. (iii) An inplane line body force qx at the middle surface of the plate.
The plate undergoes transverse deflection and inplane deformation. Thus, for the transverse deflection we have o2 wp o2 wp o2 wp Dr4 wp Nx 2 þ 2Nxy þ Ny ox ox oy oy 2 ! K X oMpðkÞ dð y yk Þ in X ð1Þ qðkÞ ¼g z þ ox k¼1 a1 wp þ a2 Vn ¼ a3
2.2. In each beam (i) A transverse load qz . (ii) A transverse load oMb =ox due to the eccentricity of qx from the neutral axis of the beam cross-section. (iii) An inplane axial force qx . The structural models of the plate and the beams are shown in Fig. 3.
on C b1
owp on
where wp ¼ wp ðxÞ is the transverse deflection of the plate; D ¼ Ep h3p =12ð1 m2 Þ is its flexural rigidity with Ep being the elastic modulus and m the Poisson ratio; Nx ¼ Nx ðxÞ, Ny ¼ Ny ðxÞ, Nxy ¼ Nxy ðxÞ are the membrane forces per unit length of the plate cross-section; dðy yk Þ is the Dirac’s delta function in the y direction; Mn and Vn are the bending moment normal to the boundary and the effective reaction along it, respectively, and they are given as 2 o wp o2 wp Mn ¼ D þ m ð3Þ on2 ot2
o 2 o o2 wp Vn ¼ D r wp ðm 1Þ on os on ot
Fig. 3. Structural model of the plate and the beams.
ð2Þ
þ b2 Mn ¼ b3
ð4Þ
Finally, ai , bi ði ¼ 1; 2; 3Þ are functions specified on the boundary C. The boundary conditions (2) are the most general linear boundary conditions for the plate problem including also the elastic support. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived form these equations by specifying appropriately
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the functions ai and bi (e.g. for a clamped edge it is a1 ¼ b1 ¼ 1, a2 ¼ a3 ¼ b2 ¼ b3 ¼ 0). Since linear plate bending theory is considered, the components of the membrane forces Nx , Ny , Nxy do not depend on the deflection wp . They are given as oup ovp þm ð5aÞ Nx ¼ C ox oy oup ovp Ny ¼ C m þ ox oy 1m Nxy ¼ C 2
oup ovp þ oy ox
ð5bÞ
ventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from Eq. (9) by specifying appropriately the coefficients ai , bi (e.g. for a simply supported end it is a1 ¼ b2 ¼ 1, a2 ¼ a3 ¼ b1 ¼ b3 ¼ 0). Since linear beam bending theory is considered, the axial force of the beam does not depend on the deflection wb . The axial deformation of the beam is described by solving independently the boundary value problem i.e. Eb Ab
o2 ub ¼ qx ox2
in Lk ; k ¼ 1; 2; . . . ; K
ð10Þ
at the beam ends x ¼ 0; l
ð11Þ
ð5cÞ c1 ub þ c2 N ¼ c3
where C ¼ Ep =ð1 m2 Þ; up ¼ up ðxÞ and vp ¼ vp ðxÞ are the displacement components of the middle surface of the plate and are established by solving the plane stress problem, which is described by the following boundary value problem (Navier’s equations of equilibrium) 1 þ m o oup ovp 1 r2 up þ qx dð y yk Þ ¼ 0 þ þ 1 m ox ox Gp oy in X 1 þ m o oup ovp r 2 vp þ þ ¼0 oy 1 m oy ox ð6Þ
where N is the axial reaction at the beam ends. Eqs. (1), (6), (8) and (10) constitute a set of five coupled partial differential equations including seven unknowns, namely wp , up , vp , wb , ub , qx , qz . Two additional equations are required, which result from the continuity conditions of the displacements in the direction of z and x axes at the interfaces between the plate and the beams. These conditions can be expressed as wp ¼ wb up
c1 un þ c2 Nn ¼ c3
ð12Þ
hp owp hb owb ¼ ub þ 2 ox 2 ox
ð13Þ
ð7Þ
on C d1 ut þ d2 Nt ¼ d3
in which Gp ¼ Ep =2ð1 þ mÞ is the shear modulus of the plate; Nn , Nt and un , ut are the boundary membrane forces and displacements in the normal and tangential directions to the boundary, respectively; ci , di ði ¼ 1; 2; 3Þ are functions specified on C.
3. Numerical solution
2.4. For each beam
3.1. The analog equation method for the plate equation
Each beam undergoes transverse deflection and axial deformation. Thus, for the transverse deflection we have
Let wp be the sought solution of the boundary value problem (1) and (2). If the biharmonic operator is applied to this function we have
d4 wb o2 wb oMb Nb 2 ¼ qz 4 dx ox ox k ¼ 1; 2; . . . ; K
Eb Ib
in Lk ;
r4 wp ¼ qp
at the beam ends x ¼ 0; l
in X
ð14Þ
ð8Þ
a1 wb þ a2 V ¼ a3 owb b1 þ b2 M ¼ b3 ox
The differential equations are solved numerically. More specifically, Eqs. (1), (6), (8) and (10) are solved using the AEM.
ð9Þ
where wb ¼ wb ðxÞ is the transverse deflection of the beam; Eb Ib is its flexural rigidity; Nb ¼ Nb ðxÞ is the axial force at the neutral axis; V, M are the reaction and the bending moment at the beam ends, respectively and ai , bi ði ¼ 1; 2; 3Þ are coefficients specified at the boundary of the beam. It is apparent that all types of the con-
Thus, the problem is converted to that of establishing the unknown the fictitious load distribution qp ðxÞ. This is accomplished using BEM as follows. The solution of Eq. (14) is given in integral form as Z Z ow p Mn ewp ðxÞ ¼ w p qp dX w p Vn on X C owp
M ds ð15Þ þ wp Vn on n where e ¼ 1, 1/2 or 0 depending on whether the point x is inside the domain X, on the boundary C, or outside
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X, respectively and w p is the fundamental solution, which is given as w p ¼
1 2 r ln r 8p
ð16Þ
with r ¼ ½ðn xÞ2 þ ðg yÞ2 1=2 , x, y 2 X [ C, n, g 2 C. The quantities Mn and Vn are obtained from Eqs. (3) and (4), respectively, by replacing wp with w p . Eq. (15) and its normal derivative for x 2 C yield the following boundary integral equations Z Z ow p 1 Mn w p qp dX w p Vn wp ðxÞ ¼ on 2 X C owp
M ds ð17Þ þ wp Vn on n
1 owp ðxÞ ¼ 2 onx
Z
ow p qp dX onx
Z
ow p o2 w p Mn Vn onx onx on X C ! o owp o
ð18Þ þ wp V M ds onx n on onx n
Eqs. (17) and (18) together with Eq. (2) can be employed to express the unknown boundary quantities wp , owp =on, Mn , Vn in terms of qp . This is accomplished numerically. The boundary integrals are evaluated using constant boundary elements, while the domain integrals using constant cells. If N is the number of boundary nodal points and M that of the domain nodal points, this procedure yields the following set of linear equations 2 38 w 9 > ½A11 ½0 ½0 ½A14 > > >n p o> > 6 ½0 ½A22 ½A23 ½0 7< owp = 6 7 on 4 ½A31 ½A32 ½A33 ½A34 5> > > > fMn g > > ½A41 ½A42 ½A43 ½A44 : fV g ; n 8 9 2 3 fa3 g > ½0 > > > < = 6 ½0 7 fb3 g 7 qp ð19Þ þ6 ¼ 4 5 0 ½ C f g > > 3 > > : ; ½C4 f0g in which [A11 ], [A14 ], [A22 ], [A23 ] are diagonal N N matrices including the nodal values of the functions a1 , a2 , b1 , b2 of Eq. (2) and [Aij ], ði ¼ 3; 4; j ¼ 1; 2; 3; 4Þ are square N N known coefficient matrices originating from the integration of the kernels on the boundary elements; {a3 }, {b3 } are N 1 known vectors including the N nodal values of the functions a3 , b3 in Eq. (2) and ½Ci ði ¼ 3; 4Þ are N M rectangular known matrices originating from the integration of the kernels on the domain cells. Finally, fwp g, fowp =ong, fMn g, fVn g are vectors including the N unknown nodal values of the respective boundary quantities and {qp } is a vector including the M unknown nodal values of the fictitious load.
463
The discretized counterpart of Eq. (15) when applied to all nodal points in X yields owp wp ¼ ½C qp ½A1 wp þ ½A2 on ð20Þ þ ½A3 fMn g þ ½A4 fVn g where [C] is an M M known matrix and [Ai ], ði ¼ 1; 2; 3; 4Þ are M N also known matrices. Elimination of the boundary quantities from Eq. (20) using Eq. (19) yields wp ¼ ½ P qp ð21Þ where [P] is an M M matrix. Further, differentiating Eq. (15) for x 2 X, using the same boundary and domain discretization and eliminating the boundary quantities by means of Eq. (20) we obtain wp;x ¼ ½Px qp ð22aÞ wp;xx ¼ ½Pxx qp
ð22bÞ
wp;yy ¼ Pyy qp
ð22cÞ
wp;xy ¼ Pxy qp
ð22dÞ
where [Px ], [Pxx ], [Pyy ], [Pxy ] are known M M coefficient matrices. The final step of AEM is to apply Eq. (1) to the M nodal points inside X. This yields D qp ½Nx ½Pxx þ 2 Nxy Pxy þ Ny Pyy qp ¼ fgg ½Z fqz g ½ X fqx g
ð23Þ
where [Nx ], [Nxy ] and [Ny ] are unknown diagonal M M matrices including the values of the inplane forces; {qz } and {qx } are vectors with L elements; L is the total number of the nodal points at the interfaces; [Z] is a position vector which converts the vector {qz } into a vector with length M. The matrix [X] results after approximating the derivative of Mp with finite differences. Its dimensions are also M L. 3.2. The analog equation method for the plane stress problem Following the procedure presented in Katsikadelis and Kandilas [17] and using the same boundary and domain discretization the membrane forces for homogeneous boundary conditions (7) (c3 ¼ d3 ¼ 0) are expressed as follows
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fNx g ¼ ½Fx fqx g Nxy ¼ Fxy fqx g Ny ¼ Fy fqx g
ð24Þ
where [Fx ], [Fxy ] and [Fy ] are known matrices with dimensions M L. 3.3. The analog equation method for the deflection of the beams Let wb be the sought solution of the boundary value problem described by Eqs. (8) and (9). Differentiating this function four times yields 4
d wb ¼ qb dx4
ð25Þ
Eq. (25) indicates that the solution of the original problem can be obtained as the deflection of a beam with unit flexural rigidity subjected only to a transverse fictitious load qb under the same boundary conditions. The fictitious load is unknown. However, it can be established using BEM as follows. The solution of Eq. (25) is given in integral form as l Z l ow
owb
M wb ðxÞ ¼ w b qb ds þ w b V b M þ wb V ox ox 0 0 ð26Þ
where w b is the fundamental solution which is given as w b ¼
1 3 l ð2 þ jqj3 3jqj2 Þ 12
ð27Þ
with q ¼ r=l, r ¼ n x, x 2 Lk , k ¼ 1; 2; . . . ; K, n at the beam ends x ¼ 0; l and the quantities M and V are given as M ¼ 12lð1 jqjÞ
ð28aÞ
V ¼ 12sgn q
ð28bÞ
Eq. (26) and its derivative for x ¼ 0 and x ¼ l yield wb ¼
Z
l 0
l ow
owb
M w b qb ds þ w b V b M þ wb V ox ox 0 ð29Þ
owb ¼ oxb
Z
l
ow b ow b o2 w b M qb ds þ V ox ox ox b b b ox 0 l o owb o þ wb V M
ox oxb oxb 0
ð30Þ
Eqs. (29) and (30) together with Eq. (9) can be employed to express the unknown boundary quantities wb , owb =on, M, V in terms of qb . This is accomplished numerically. If
k is the number of parallel beams and L that of the nodal points at the interfaces this procedure yields the following set of linear equations 9 2 38 ½E11 ½0 ½0 ½E14 > fwb g > > > 6 ½0 ½E22 ½E23 ½0 7< owb = on 6 7 4 ½E31 ½E32 ½E33 ½E34 5> f M g > > > : ; ½E41 ½E42 ½E43 ½E44 fV g 8 9 2 3 ½0 fa3 g > > > > < = 6 fb3 g ½0 7 7fq g 6 ð31Þ þ4 ¼ ½ f 0 g F3 5 b > > > > : ; ½F4 f0g in which [E11 ], [E14 ], [E22 ], [E23 ] are diagonal 2k 2k matrices including the nodal values of the functions a1 , a2 , b1 , b2 of Eq. (9) and [Eij ], ði ¼ 3; 4; j ¼ 1; 2; 3; 4Þ are square 2k 2k known coefficient matrices; {a3 }, {b3 } are 2k 1 known column matrices including the boundary values of the functions a3 , b3 in Eq. (9) and [Fi ] ði ¼ 3; 4Þ are 2k L rectangular known matrices originating from the integration of the kernels on the axis of the beams. Finally, fwb g, fowb =ong, fMg, fV g are vectors including the 2k unknown nodal values of the respective boundary quantities and {qb } is a vector including the L unknown nodal values of the fictitious load. The discretized counterpart of Eq. (26) when applied to all nodal points at the interfaces yields owb fwb g ¼ ½ F fqb g ½E1 fwb g þ ½E2 on ð32Þ þ ½E3 f M g þ ½E4 fV g where [F] is an L L known matrix and [Ei ], ði ¼ 1; 2; 3; 4Þ are L 2k also known matrices. Elimination of the boundary quantities from Eq. (32) using Eq. (31) for homogeneous boundary conditions (9) (a3 ¼ b3 ¼ 0) yields fwb g ¼ ½ B fqb g
ð33Þ
where [B] is an L L matrix. Further, differentiating Eq. (26) for x 2 Lk and eliminating the boundary quantities by means of Eq. (31) we obtain wb;x ¼ ½Bx fqb g ð34aÞ
wb;xx ¼ ½Bxx fqb g
ð34bÞ
where [Bx ], [Bxx ] are known L L coefficient matrices. The final step of AEM is to apply Eq. (8) to the L nodal points at the interfaces. This yields Eb Ib fqb g ½Nb ½Bxx fqb g ¼ fqz g ½Q fqx g
ð35Þ
where [Nb ] is unknown diagonal L L matrix including the values of the inplane forces; {qz } and {qx } are vectors with L elements. The matrix [Q] results after approxi-
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
mating the derivative of Mb with finite differences. Its dimensions are also L L. 3.4. The analog equation method for the axial deformation of the beams The solution of Eq. (10) is given in integral form as l Z l dub du b ub u b qx ds þ u b ð36Þ ub ðxÞ ¼ dx dx 0 0 where u b ¼
1 ðl jrjÞ 2Eb Ab
ð37Þ
Following the same procedure as for the plane stress problem the axial force at the neutral axis of the beam for homogeneous boundary conditions (11) (c3 ¼ 0) can be expressed as follows fNb g ¼ ½Fb fqx g
ð38Þ
where [Fb ] is known matrix with dimensions L L. Eqs. (23) and (35) after elimination of the quantities Nx , Ny , Nxy , Nb using Eqs. (24) and (38) together with continuity conditions (12) and (13) which after discretization at the L nodal points at the interfaces are written as wp ¼ fwb g ð39Þ hp hb wp;x ¼ fub g þ wb;x up 2 2 constitute a non-linear system of equations with respect to qz , qx , qp , qb . This system is solved using iterative numerical methods. Subsequently, the deflection at any interior point x 2 X of the stiffened plate is established using the discretized counterpart of Eq. (15) when applied to this point. For the solution of the non-linear system of equations the two-term acceleration method [18] has been employed. According to this method, an initial vector, say qð0Þ x ¼ 0 is assumed. Using this vector and Eqs. (35) ð0Þ ð0Þ and (39) the values of the vectors qð0Þ are z , qp , qb computed. Introducing these values into Eq. (23) a 0Þ vector qð1 is obtained. To accelerate convergence the x new initial vector is given as qð1Þ x
¼
aqð0Þ x
þ
0Þ bqð1 x
465
where [Fp ], [Fb ] are known L L flexibility coefficient matrices the elements of which are evaluated numerically following the steps of the AEM. 4. Presentation of additional models In Fig. 4 the structural model of ‘‘model I’’ is presented. According to this the stiffened plate is approximated by a simply supported beam with a cross-section consisting of a part of the plate corresponding to the effective breadth and the beam cross-section. The analysis of the model is achieved by solving the beam bending governing differential equation EI
d4 w ¼ q in the interior of the beam dx4
ð42Þ
satisfying the boundary conditions a1 w þ a2 V ¼ a3 ow þ b2 M ¼ b3 b1 ox
at the beam ends
ð43Þ
where w ¼ wðxÞ is the transverse deflection of the stiffened plate along the axis of the stiffening beam; EI is the flexural rigidity of the approximating beam; V, M are the reaction and the bending moment at the beam ends, respectively and ai , bi (i ¼ 1; 2; 3) are coefficients specified at the boundary of the beam. In Figs. 5 and 6 a typically discretized stiffened plate according to ‘‘model II’’ and ‘‘model III’’, respectively, are presented. According to these the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, while only qz tractions at the
Fig. 4. Cross-section and structural system of model I.
ð40Þ
where a, b appropriately chosen weight factors with a þ b ¼ 1. It is worth here noting that in Eq. (39) the column matrices {up }, {ub } are evaluated from the relations ð41aÞ up ¼ Fp fqx g fub g ¼ ½Fb fqx g
ð41bÞ
Fig. 5. Discretized stiffened plate according to model II.
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Fig. 8. Discretized stiffened plate according to model V. Fig. 6. Discretized stiffened plate according to model III.
fictitious interfaces are taken into account (Fig. 2). These tractions result in the loading of the beam as well as the additional loading of the plate. The difference between the two models is due to the eccentricity of the structural model of the beam with respect to that of the plate, equal with the distance between the center of gravity axis of the beam and the middle surface of the plate, which is taken into account in model III. The analysis of the plate is accomplished solving the plate bending governing differential equation Dr4 wp ¼ g
ð44Þ
in the interior of the plate, satisfying the boundary conditions (2) at its boundary, where wp is the transverse deflection of the plate. Moreover, the beam bending problem is governed from Eq. (42) and has to satisfy the boundary conditions (43). Both the plate and beam problems are solved using the FEM [19], while the distribution of the qz interface forces is established by imposing the displacement continuity condition (12) at the interfaces. In Fig. 7 a typically discretized stiffened plate according to ‘‘model IV’’ is presented. The difference between this model and model III is restricted in the analysis of the beam, which is accomplished by solving the plane stress problem in terms of displacements, given as 2 o u o2 w lr2 u þ ðk þ lÞ þ þ bx ¼ 0 ox2 ox oz ð45Þ 2 o u o2 w lr2 w þ ðk þ lÞ þ 2 þ bz ¼ 0 ox oz oz
and applying the continuity conditions (12) and (13) at the interfaces. In Eq. (45) u is the displacement along the beam axis; w is the deflection of the beam; k, l are Lame constants and bx , bz are body forces. Both the plate and beam problems are solved using the FEM [19]. Till now, in all the presented models of this section the inplane forces and deformations due to the shear forces at the interfaces have been neglected. Finally in Fig. 8 a typically discretized stiffened plate according to ‘‘model V’’ is presented, which takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. The analysis of this model is accomplished by solving for both the plate and the beams the 3-D elasticity problem in terms of displacements, given as 2 ou o2 v o2 w lr2 u þ ðk þ lÞ þ ð46aÞ þ þ bx ¼ 0 ox2 ox oy ox oz lr2 v þ ðk þ lÞ
lr2 w þ ðk þ lÞ
o2 u o2 v o2 w þ 2þ ox oy oy oy oz
o2 u o2 v o2 w þ þ 2 ox oz oy oz oz
þ by ¼ 0
ð46bÞ
þ bz ¼ 0
ð46cÞ
and satisfying prescribed displacement uðx; y; zÞ ¼ u0
ð47aÞ
vðx; y; zÞ ¼ v0
ð47bÞ
wðx; y; zÞ ¼ w0
ð47cÞ
or traction
Fig. 7. Discretized stiffened plate according to model IV.
tx ðx; y; zÞ ¼ tx0
ð48aÞ
ty ðx; y; zÞ ¼ ty0
ð48bÞ
tz ðx; y; zÞ ¼ tz0
ð48cÞ
boundary conditions of the stiffened structure. The analysis of this model and the solution of the aforementioned problem is achieved using the FEM [19]. It is worth here noting that the continuity conditions at
E.J. Sapountzakis, J.T. Katsikadelis / Computers and Structures 80 (2002) 459–470
467
the fictitious interfaces are automatically satisfied for this model, as the stiffened structure is examined as a whole. It is obvious that model V approximates better the behaviour of the stiffened plate. The disadvantage of this model compared with the proposed one is focalized in its inability to estimate the interface forces, the knowledge of which is very important in the design of composite or prefabricated ribbed plates.
5. Numerical examples On the basis of the analytical and numerical procedures presented in the previous sections, representative examples have been studied to verify the accuracy, the efficiency and the range of applications of the proposed model. Example 1: A rectangular plate with dimensions ap bp ¼ 18:0 m 9:0 m and thickness hp ¼ 0:20 m subjected to a uniform load g ¼ 10 kN/m2 and stiffened by a beam of width 1.0 m through its centerline has been studied (Ep ¼ Eb ¼ 3:0 107 kN/m2 , m ¼ 0:154). The plate is simply supported along its small edges, while the other two edges are free according to the transverse boundary conditions and Nx ¼ Nxy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nyx ¼ 0 along the edges y ¼ bp =2 and y ¼ bp =2 according to the inplane boundary conditions. In Table 1 the deflections of the plate at its center for different beam heights hb are shown as compared with those obtained from the five additional presented models. Moreover, in Fig. 9 the distributions of the deflection w along the axis of symmetry of the interface, according to the proposed model and model V, are presented for various values of the beam height hb . The discrepancy of the results of the proposed model (AEM results) from the corresponding ones of models I, II, III and IV is obvious. The results of model V, especially for high beams, are somewhat similar with those of the proposed model. The observed discrepancy between the proposed model and model V is attributed to the different implementation of the boundary conditions between the two models. Example 2: The same plate as in Example 1, with all edges simply supported according to the transverse
Fig. 9. Deflections along the axis of the beam of the stiffened plate of Example 1 for various values of the beam height hb .
boundary conditions and ux ¼ 0, Nxy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nyx ¼ 0 along the edges y ¼ bp =2, y ¼ bp =2 according to the inplane boundary conditions has been studied. In Table 2 the deflections of the plate at its center for different beam heights hb are shown as compared with those obtained from the five additional presented models. Moreover, in Fig. 10 the distributions of the deflection w along the axis of symmetry of the interface, according to the proposed model and model V, are presented for various values of the beam height hb . The same conclusions as in Example 1 can easily be drawn. Example 3: A rectangular concrete plate (Ep ¼ 3:0 107 kN/m2 , m ¼ 0:154) with dimensions a b ¼ 18:0 m 9:0 m subjected to a uniform load g ¼ 10 kN/ m2 and stiffened by an I-section steel beam (Eb ¼ 3:0 108 kN/m2 ) through its centerline has been studied (Fig. 11). The plate is simply supported along its small edges, while the other two edges are free according to the transverse boundary conditions and ux ¼ uy ¼ 0 along the edges x ¼ ap =2 and x ¼ ap =2, Ny ¼ Nxy ¼ 0 along the edges y ¼ bp =2 and y ¼ bp =2 according to the
Table 1 Deflections w (mm) at the center of the stiffened plate of Example 1 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.60 1.00 1.25 2.00
33.44 9.55 5.30 1.49
42.72 21.91 7.37 2.25
174.61 46.07 24.31 6.08
38.30 10.79 6.02 1.70
44.79 12.80 7.12 1.56
24.85 7.76 4.57 1.48
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Table 2 Deflections w (mm) at the center of the stiffened plate of Example 2 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.60 1.00 1.25 2.00
7.86 2.18 1.22 0.35
42.72 21.91 7.37 2.25
27.38 16.10 10.71 3.42
14.40 5.63 3.37 1.01
7.90 4.11 2.73 0.97
4.58 1.61 1.02 0.39
6. Concluding remarks In this paper an optimized model for the analysis of plates reinforced with beams is presented as compared with other models previously used. The adopted model contrary to the already used models takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beams, due to combined response of the system. The main conclusions that can be drawn from this investigation are
Fig. 10. Deflections along the axis of the beam of the stiffened plate of Example 2 for various values of the beam height hb .
inplane boundary conditions. In Tables 3 and 4 the deflections of the plate at its center and at the middle of the free edge, respectively, for different beam heights hb are shown as compared with those obtained from the five additional presented models. The conclusions of Example 1 are once again verified. Finally, in Fig. 12 the distributions of the interface forces qx are presented for various values of the height hb of the beam.
(a) The discrepancy of the results of the proposed model from the corresponding ones of models I, II, III and IV, which ignore the inplane forces and deformations due to the shear forces at the interfaces, is remarkable. This discrepancy necessitates the consideration of the inplane forces and deformations. (b) The results of model V, which analyzes the stiffened plate solving a 3-D elasticity problem, are somewhat similar with those of the proposed model. The observed discrepancy between the proposed model and model V is attributed to the different implementation of the boundary conditions between the 3-D elasticity model and the plate-beam model, which is more intensive in flexible structures. (c) The adopted model permits the evaluation of the inplane shear forces at the interface between the plate and the beams, the knowledge of which is very important in the design of composite or prefabricated plate-beams structures (estimation of bondage, shear connectors or welding).
Fig. 11. Plan (a) and cross-section (b) of the stiffened plate of Example 3.
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469
Table 3 Deflections w (mm) at the center of the stiffened plate of Example 3 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.50 1.00 1.50 2.00
9.00 2.873 1.342 0.756
51.819 18.091 9.244 5.711
117.16 35.532 16.848 9.881
46.271 16.168 8.310 5.165
18.240 7.181 4.154 2.851
13.898 5.871 3.490 2.434
Table 4 Deflections w (mm) at the middle of the free edge of the stiffened plate of Example 3 (AEM) compared with those of the five additional models for various beam heights hb
AEM
Model I
Model II
Model III
Model IV
Model V
0.50 1.00 1.50 2.00
28.56 22.58 21.09 20.51
1197.2 679.2 472.7 363.3
138.673 56.983 38.291 31.324
67.616 37.568 29.730 26.594
34.070 23.155 20.148 18.846
29.180 21.662 19.427 18.433
Fig. 12. Distribution of the qx forces along the beam axis of the stiffened plate of Example 1.
Acknowledgements The author wishes to thank his postgraduate student, Diplom Civil Engineer Mr. A. Kontizas for his assistance in obtaining the numerical results.
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