COMPUTATIONAL MECHANICS New Trends and Applications S.Idelsohn, E. Oñate and E. Dvorkin (Eds.) ©CIMNE, Barcelona, Spain 1998
A PROGRAMMING SYSTEM FOR NONLINEAR DYNAMIC AND STATIC ANALYSIS OF TALL BUILDINGS Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti Department of Structural and Foundation Engineering Escola Politécnica da Universidade de São Paulo Caixa Postal 61548 CEP 05424-970 São Paulo Brazil e-mail:
[email protected],
[email protected]
Key words: Finite elements, Fortran 90, Nonlinear analysis, Tall buildings
Abstract. The paper presents a recently developed finite element code for the nonlinear static and dynamic analysis of structures and its application to the analysis of tall buildings. The code is intended for academic purposes (teaching and research). Nevertheless it was used in the nonlinear stability analysis of an inclined tall building in Santos, Brazil, which is near to collapse and will be brought to the right position through the application of hydraulic jacks. This very interesting case is also reported in the paper. The most important features of the system are discussed. The language Fortran 90/95 was chosen in order to allow the reutilization of existing FORTRAN77 routines, to retain numerical efficiency and to take profit of new capabilities of the language, that permitted the development of a very modular and compact code, which is readable, easy to maintain and virtually self-explained. The paper also briefly displays the implemented element and algorithm modules and some special features which are appropriate to Civil Engineering structures and required by the stability analysis of tall buildings.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
1.
INTRODUCTION
The paper presents the PEFSYS programming system, a recently developed finite element code for the nonlinear static and dynamic analysis of structures and its application to the analysis of tall buildings. The code is intended for the academic purposes of teaching and research. In the following section an unified theory of rods and shells is presented, which is employed in the system. Three kinematics are possible within this theory, namely the geometrically exact, the consistent second-order and the linear one. It is shown that such an approach permits an intensive use of tensor algebra and parameterized programming. The most important features of the system are discussed in section 3. The language Fortran 90/95 was chosen in order to allow the reutilization of existing FORTRAN77 routines, to retain numerical efficiency and to take profit of new capabilities: data encapsulation, information hiding, function overloading, array processing, and derived types. The new capabilities of the language permitted the development of a very modular and compact code, which is readable, easy to maintain and virtually self-explained. The section briefly displays the implemented element and algorithm modules. The paper also briefly displays the implemented element and algorithm modules. These include truss, cable, membrane, frame and shell elements with geometrically exact, second-order and linear kinematics, the usual static and dynamic algorithms and some special features which are appropriate to Civil Engineering structures, like birth and dead of elements and/or element parts, reinforced and prestressed concrete properties and verifications according to the Brazilian national building code. Although the code has academic purpose, it was used in the nonlinear safety analysis of an inclined tall building in Santos, Brazil, which is near to collapse and will be brought to the right position through the application of hydraulic jacks. This very interesting case is reported in section 4.
2.
AN UNIFIED NONLINEAR THEORY OF RODS AND SHELLS
This section shortly presents an unified theory of rods and shells, which is employed in the PEFSYS programming system. The theory can be consistently derived for three kinematics, namely, linear, consistent second order and fully nonlinear. It is based on several previous works1,2,3,4,5,6,7 and is very useful for numerical procedures employing Galerkin projection like the Finite Element Method. A very compact program was obtained through the parameterization here developed.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
In this section, vectors, tensors and matrices are denoted by bold letters. The summation convention over repeated Greek indices is adopted from 1 to n, where n=1 for rods and n=2 for shells. Latin indices ranges from 1 to 3, as usual. The mathematical model accommodates arbitrarily large displacements as well as rotations and accounts for shear distortion. For rods the theory assumes that cross sections orthogonal to the rod axis in the reference configuration remain plane and undistorted during the motion, as displayed in figure 1. For shells it assumes that straight fibers orthogonal to the shell middle surface in the reference configuration remain rigid during the motion, as shown in figure 2. a
r
a
Figure 1: Rod kinematics
Figure 2: Shell kinematics
As shown in figures 1 and 2 an orthogonal vector basis {er1, er2, er3} is defined in the reference configuration, which is assumed to be straight for rods or plane for shells. Initially curved configurations can be mapped from this configuration. In the rod, er1 is placed along the axis while in the shell er1 and er2 are placed on the middle surface. Points in the reference configuration are described by xr = zr + ar , (1) r where z describes the position of points on the rod axis or on the shell middle surface and ar describes the position of points on the rod cross section relative to the axis or points on the shell transversal fibers relative to the shell middle surface. During the motion the position of a material point is described by x = z+a (2) The displacement vector of points on the rod axis or on the shell middle surface is given by u = z − zr (3) while a by a = Q ar , (4) where Q is a rotation tensor. Q can be expressed by the Euler Rodrigues formula, viz.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
2
sin θ 1 sin θ 2 2 Q=I+ (5) Θ + Θ . 2 θ 2 θ To second order approximation, Q is expressed by 1 Q = I + Θ + Θ2 . (6) 2 In (5) and (6) Θ is an anti-symmetric tensor, whose axial vector is θ , and θ=||θ θ ||. The components of u and θ are the translational and rotational degrees of freedom of the model. The deformation gradient can be expressed by
[ (
]
(7)
κ α = Γ T θ ,α
(8)
)
F = Q I + η α + κ α × a r ⊗ e αr , where, with (•),α = ∂(•)/∂xα, η α = Q T z , α − eαr are generalized strains of the model.
and
In (8) the following tensor has been introduced
sin θ 2 1− 1 sin θ 2 θ Θ2 . Θ+ Γ = I + 2 θ 2 θ2 which has the following second order expansion 1 1 Γ = I + Θ + Θ2 2 6 Let the non-symmetric Piola stress tensor P be given by P = t i ⊗ e ir and write τ α = Q T tα . Hence, the following cross section resultants can be defined nαr = ∫ τ α dS and m αr = ∫ a αr × τ α dS . S
S
(9)
(10) (11) (12) (13)
In (13) nrα is a backward rotated normal force while mrα is a backward rotated moment, both in the α direction. S represents the rod cross section on the shell transversal fiber. Now the following strain, stress and displacement vectors can be defined: nr η u σ α = αr , (14) εα = α , d = . κ α θ m α By consistent linearization one can obtain the relation between the virtual strains and displacements. This is written below as δ ε α = Bα δd , (15) where
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
1 2 1 1 Z ,α + Z , α Θ − Θ Z ,α 0 I − 2 Θ + 6 Θ 2 Bα = (16) 1 1 1 1 1 2 0 Θ + Θ ,α Θ − Θ Θ ,α I − Θ + Θ 2 6 3 2 6 to second order. A geometrically exact expression for Bα has been derived elsewhere7. In (16), Z is the anti-symmetric tensor, whose axial vector is z. The virtual work expression for a domain Ω is then given by δ W = ∫ σ α ⋅ B α δ d − q ⋅ δ d dΩ Ω
(
)
(17)
where q is the vector of the external load.
The Fréchet derivative of (17) is very important for the solution of non-linear equations by the Newton Method as well as for stability and linear vibration analyses. The resulting expression is (18) δ 2 W = ∫ (Dα β Bα δd )⋅ (Bβ δd ) + (Gα ∆ α δd )⋅ (∆ α δd ) − (L δd ) ⋅ δd ]dΩ , Ω
[
where Dα β =
0 G αu′ θ 0 u′ θ G α = G α G αθ θ G αθ θ′ 0 Gαθ θ′ 0
∂ σα , ∂ εα
and
0 0 L= θθ 0 L
(19)
To second order 1 r N α Θ − Θ N αr , 2 1 1 1 Gαθ θ′ = M αr + ΘM αr − M αr Θ , (20) 2 6 3 1 1 Gαθ θ = Z ,α N αr + N αr Z ,α + Θ ,α M αr + M αr Θ ,α . 2 6 θθ geometrically exact expressions for (20), and for L can be found elsewhere7. In (20) Nrα and Mrα are anti-symmetric tensor whose axial vectors are nrα and mrα, respectively. Note that Gα is always symmetric, even far from an equilibrium state. Gαu′ θ = − N αr +
(
) (
)
The constitutive relation between σα and ε α can be taken from the geometrically linear theory. For the drilling degrees of freedom an arbitrarily chosen linear law has been used in the applications. More sophisticated constitutive laws for thin-walled sections can be found elsewhere8.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
3.
PEFSYS A STRUCTURES
SYSTEM
FOR
THE
NONLINEAR
ANALYSIS
OF
PEFSYS is a programming system for the nonlinear static and dynamic analysis of structures. The code is intended for the academic purposes of teaching and research and was written in Fortran 90. It is only a solver without any pre and post-processing capabilities, which are left to available commercial packages. Current applications employs the ANSYS system, for which appropriate interface files are written. The choice of Fortran 90 allowed to reuse existing FORTRAN77 routines, to retain numerical efficiency and to take profit of new capabilities of the language: data encapsulation, information hiding, function overloading, array processing and derived types. They permitted the development of a very modular and compact code, which is readable, easy to maintain and virtually self-explained. Using the above capabilities, PEFSYS approaches, in many senses, the object orientation of finite element codes written in languages as C++. Object orientation indeed allowed a higher level of abstraction compared to our legacy codes. However, it was employed in PEFSYS with some caution, in order to keep the efficiency of the finite element algorithms. PEFSYS is organized in modules. There are data modules, organized in a tree structure allowing some inheritance (which is not an intrinsic feature of Fortran 90 language). These modules are intended as a database. We have a high level module called structure which embodies all the data that are present in static and dynamic structural analyses. This module collects the statics and dynamics submodules, where the data specific to each kind of analysis are stored. By their turn, these submodules collect the data related to incremental, linear stability and vibration analyses, encompassing several algorithms. For instance, the module static_incremental_analysis currently groups together the data related to Newton/BFGS (with optional line search and arch-length methods) and the Wriggers stability algorithm. There are procedure modules which perform typical finite element operations on the database, basically mirroring the data structure above. The procedures related to finite element computations are collected in a modular tree called finite_elements. This module contains generic procedures to assemble the structure matrices, alongside with more specific submodules (structural_elements and solid_elements) where the element matrices are evaluated. For example, the module structural_elements module uses two submodules called one_dimensional_elements and two_dimensional_elements. Again, these modules encompass submodules related to specific finite element and structural theories. To date they include truss, cable, membrane, frame and shell elements with geometrically exact, consistent second-order and linear kinematics. Some special features which are appropriated to Civil Engineering structures, like birth and dead of elements and/or
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
element parts, reinforced and prestressed concrete properties and verifications according to the Brazilian national code are also present. There is a linear_algebra module where we have encapsulated our legacy FORTRAN77 code for the solution of linear systems and eigenproblems. This module use routines from the BLAS package (in order to achieve high computational efficiency) and can as well encapsulate routines from LAPACK or similar packages. Since PEFSYS is intended for educational purposes, we have included some procedures totally written in Fortran 90, providing a more elegant and readable source code, at the cost of some loss of efficiency. Another module called tensor_algebra, available everywhere in the code, has revealed as an useful and convenient abstract programming tool to calculate the finite element quantities. As an example of the internal coding of PEFSYS, let us consider the generalized strains η and κ of the rod element discussed in the previous section, which expressions are recollected bellow 1 Q = I + Θ + Θ2 2 1 1 Γ = I + Θ + Θ2 2 6 T η α = Q z , α − eαr κ α = Γ T θ ,α In these expressions, η , κ, z,α and eα are vectors, Q is the cross-section rotation tensor, Θ is an anti-symmetric tensor, Θ 2 is a symmetric tensor, Γ is a non-symmetric tensor and I is the identity tensor. In Fortran 90, these quantities are declared as objects of derived types (vector, rotation_tensor, etc. defined in the module tensor_algebra) as below . use tensor_algebra . . type (vector), dimension (n) type (rotation_tensor) type (antisymmetric_tensor) type (symmetric_tensor) type (identity_tensor) type (unsymmetric_tensor) . .
:: :: :: :: :: ::
eta, kappa, der_z, der_theta, e Q Theta Theta2 I Gamma
The use of polimorphic operators, acting on derived types, allows to code the above displayed math in a very natural fashion, without loss of numerical efficiency, as shown in the following source lines:
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
. . Theta2 = Theta .x. Theta Q = I + Theta + (half * Theta2) Gamma = I + (half * Theta ) + (one_sixth * Theta2) do alpha=1,n eta(alpha) kappa(alpha) end do . .
= ( Q .tx. der_z(alpha) ) - e(alpha) = Gamma .tx. der_theta(alpha)
Note that +, - and * are overloaded operators which generalize the sum of tensor and vectors and the product of tensors and vectors by scalars, while .x. and .tx. stand for polimorphic operators that perform the internal product (the latter simultaneously transposes the first operand, retaining the algebraic efficiency of our old FORTRAN 77 code). Many others operators are already available. For instance, .vector. and .tensor. perform the vector and tensor products, respectively. Fourth order tensor algebra is also at hand. 4.
APPLICATION TO A REAL WORLD PROBLEM
A challenging application of the new system was the stability analysis and the rehabilitation project of a tall building in Santos. This city, 60 km far from São Paulo, houses the largest Brazilian harbor and is the home of the famous soccer player Pelé. During the sixties several tall buildings were constructed along the beach shore. They were built on direct foundation on the superficial sand layer, which is supported by deep soft clay layers. Due to differential consolidation of the clay subsoil most of these buildings are leaning on a progressively manner, threatening their structural stability. Figure 3 shows the most critical case in Santos: the tower A of the “ Malzoni”. This 17-storied building was designed in 1962 and built in 1963-1964 on a shallow foundation at a 1.50 m depth in the superficial sand layer. The skyscraper is 62.0 m high, showing an inclination of 4.2%, that corresponds to a 1.93 m difference from vertical reference∗.
∗
This case resembles the most famous inclined construction in the world: the leaning tower of Pisa10, which is 58.0 m high and presents an inclination of 9.5%.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
The structure is a conventional reinforced concrete spatial frame with 33 rectangular columns and the floors were designed as smooth concrete slabs. The characteristic concrete strength adopted in the design phase was fck=15 MPa. Recent extraction by coring of concrete samples has shown that today the concrete presents a characteristic strength fck=23 MPa. This result has been confirmed by sclerometric instruments.
Figure 3:: Tower A of the “Condomínio Núncio Malzoni” in Santos The soil profile is shown in figure 4 along with the results of the Standard Penetration Test. Short after construction of Tower A, a similar building was erected on its left side, as one can see at figure 4. The additional pressure on the deep soft clay layers caused both buildings to lean, one against the other. The inclination of both buildings is being monitored by settlement measures of several columns with respect to a fixed reference. The measured settlements of columns A1 and A33, which are the extreme cases, are displayed in figure 5. Column A1 has settled 250 mm since the construction while column A33 presents the value 520 mm. This difference is responsible for the inclination of more than 2o observed at the frontal frame. Figure 6 shows the daily settlement rate for the columns A1 and A33. With exception of a tentative foundation strengthening done in 1978, both columns present an almost constant settlement velocity in the past 30 years! This corroborates the conclusion of a previous thesis9 that the classical Therzaghi’s Consolidation Theory is no longer applicable to the soft clay layers of Santos. Several modern consolidation theories were examined elsewhere9 with the purpose of better fitting the experimental data. It was necessary to include viscous effects in the soil constitutive model in order to reproduce this long term consolidation.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
Figure 4: The soil profile Figure 6 shows the daily settlement rate for the columns A1 and A33. With exception of a tentative foundation strengthening done in 1978, both columns present an almost constant settlement velocity in the past 30 years! This corroborates the conclusion of a previous thesis9 that the classical Consolidation Theory of Therzaghi is no longer applicable to the soft clay layers of Santos. Several modern consolidation theories were there examined with the purpose of better fitting the experimental data. It was necessary to include viscous effects in the soil constitutive model in order to reproduce this long term consolidation.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
Figure 5: Measured settlements of columns A1 and A33
Figure 6: Settlement increase per day for the columns A1 and A33 In the last two years the city authorities became concerned with the stability of 94 inclined buildings. Tower A of the “Condomínio Núncio Malzoni”, as the most extreme case, was the first building to have its safety verified. Prof. Maffei’s office was then contracted by the condominium to undertake such an analysis, whose computational part was conducted by Prof.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
Pimenta at the University of São Paulo. For this analysis a preliminary version of the computational system PEFSYS was used.
Figure 7: The mesh for the spatial frame. The complete structure of the building was discretised in over 30000 frame and plate elements. The analysis was performed in hierarchical form. In phase I only the contribution of the frame elements were considered. In phase II an estimated contribution of the floor slabs as shear panels was taken into account. Finally, in phase III, the slabs were considered as plates and an estimated contribution of the masonry walls was incorporated. Figure 7 shows the mesh for the spatial frame. The reinforced concrete material requires the description of every steel bar in the rod and plate elements. The element behavior in tension, compression and bending is computed through integration on the cross sections. As constitutive model, the elastic-plastic uniaxial models for the concrete and steel from CEB-FIP model code 1990 was used, as qualitatively displayed in figure 8. Shear and torsion effects were assumed to follow a simplified elastic model.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
σ
−σ
ε
−ε Figure 8: Elastic-plastic uniaxial models for the concrete and steel
The real dead loading on the structure was evaluated after a detailed visit to the building. An error lower than 5% is expected in this evaluation. The footings are connected by very stiff beams, what was a correct design decision therefore. It was assumed a mean settlement plane, which was applied as prescribed displacements after the application of the dead loading. The resulting inclination is shown in figure 9. The normal forces and bending moments in the members of the frontal frame at the current situation are displayed in figures 10 and 11. Due to the overturning moment the normal forces on the left column have increased by 20% over the values in the vertical position. Additional bending moments have strongly changed the stress distribution in columns and beams, as well.
Figure 9: Inclination of the frontal frame.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
Figure 10: Normal forces in the frontal frame
Figure 11: Bending moments in the frontal frame
After this analysis, a vast measurement program was conducted on the structure in order to confirm the mathematical model. Through stress relief tests on several reinforcement steel bars it was possible to estimate the current stresses on them. The measured values were compared with the computed ones. The constitutive parameters of concrete were then reformulated in order to keep the stress data fitting within 10% error. After that, a new analysis was performed, applying the settlement displacements with the measured average velocity. Figure 10 show the collapse of the structure through a combined rupture of columns and beams. According to our mathematical model, the collapse will occur
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
eight years from now, if the average inclination rate is maintained and if the contribution of slabs and walls are neglected.
Figure 10: The collapse of the structure through a combined rupture of columns and beams The condominium has contracted the authors to propose a rehabilitation project for the structure. The proposal is a new deep foundation at both sides of the building, as shown in the figure 14. Fourteen excavated piles 63.0 m long and 1.20 m diameter are prescribed. After that seven reinforcements, similar to the one whose stress analysis is shown in figure 15, will be built under the second floor. They will transfer the loading to the new foundation. Then fourteen hydraulic jack will be placed between the structure and the foundation in order to bring the building back to the vertical position. During this process the old footing foundation will be disconnected and abandoned. The authors expect that the PEFSYS system will be a useful predictive tool during the whole rehabilitation process.
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
Figure 14: Subfoundation proposal
Figure 15: Stress analysis of proposed reinforcement
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Paulo M. Pimenta, Carlos E. M. Maffei, Heloisa H. S. Gonçalves, Ruy M. O. Pauletti
ACKNOWLEDGEMENTS The authors acknowledge the “Fundação para o Amparo à Pesquisa do Estado de São Paulo” (FAPESP) for the financial support to the development of PEFSYS under the grant Proc.0553-7/94. The authors acknowledge also the graduate student Luis Alberto Tello Arévalo for his generous help in the preparation of this paper. REFERENCES [1] Pimenta, P.M. and Yojo, T.; “Geometrically exact analysis of spatial frames”, Applied Mechanics Reviews, 46(11), Part 2, 1993. [2] Pimenta, P.M.; “On a geometrically exact finite-strain model”, In: proceedings of PACAM III, Third Pan American Congress of Applied Mechanics, São Paulo, 1993. [3] Simo, J.C. and Vu-Quoc, L.; “A finite strain beam formulation. The three-dimensional dynamic problem. Part I”, Computer Methods in Applied Mechanics and Engineering, 58, 79116, 1986. [4] Wriggers, P. and Gruttmann, F.; “Thin shells with finite rotations formulated in Biot stresses: theory and finite element formulation”, International Journal for Numerical Methods in Engineering, 36, 2049-2071, 1993. [5] Pimenta, P.M. and Yojo, T.; “Geometrically exact analysis of spatial frames with consideration of torsion warping”, In: proceedings of XIV CILAMCE, XIV Congresso Iberolatino-americano de Métodos Computacionais em Engenharia, IPT, São Paulo, 1993. [6] Pimenta, P.M. and Fruchtengarten, J.; “Carga crítica de barras com consideração de empenamento XV Congresso por torção – uma formulação consistente”, In: proceedings of XV CILAMCE, Ibero-latino-americano de Métodos Computacionais em Engenharia, 14861495, Belo Horizonte, 1994. [7] Sansour, C. and Bufler, H.; “An exact finite rotation shell theory, its mixed variational formulation and its finite element implementation”, International Journal for Numerical Methods in Engineering, 34, 73-115, 1992. [8] Fruchtengarten, J.; Sobre a instabilidade por distorção de perfis de seção aberta e parede delgada, PhD Thesis, Escola Politécnica da Universidade de São Paulo, 1995. [9] Gonçalves, H.H.S.; Análise crítica, através de modelos visco-elásticos, dos resultados de ensaios de laboratório, para previsão de recalques dos solos da Baixada Santista, PhD Thesis, Escola Politécnica da Universidade de São Paulo, 1992. [10] Leonhardt, F; “The committee to save the tower of Pisa: a personal report”, Structural Engineering International, 3, 201-212, 1997.
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