Ap1_02.vp 1 Introduction A number of numerical methods may be applied when estimating the bearing capacity of existing as well as planned buildings with random properties of structural elements, especially of vertical and horizontal joints. At present, probabilistic methods can be broadly classified into two major categories – methods using a statistical ap- proach and methods using a nonstatistical approach [1]. Sta- tistical methods are based on simulation. The direct Monte Carlo method and the Latin Hypercube Sampling (LHS) technique are fairly known, as well as improved simulation methods known as “Importance Sampling” and “Adaptive Sampling”. Nonstatistical methods include numerical inte- gration, the method of second order moments and the proba- bilistic finite element method. The horizontal and vertical joints of precast buildings and their properties are structural elements of the utmost importance. Calculations based on statistical methods and taking into account the random material properties of joints and panels, as well as the random properties of loading, especially due to temperature impact, are rather complicated and time consuming. That is why a different approach using reliability index � is preferred to the direct determination of failure probability. It is well known that very low values of � are attained (� � 2) when deterioration of the joint due to an extreme inelastic deformation and/or due to a certain type of cyclic loading is developed to such an extent that the consecu- tive static stiffness approaches its residual value. A typical loading path of a reinforced vertical joint published in [5] is displayed in Fig. 1. Based on this observation, the proposed procedure is as follows. Index � is determined using the second order reli- ability method. In the parts of joints where values of � are rather low, the initial stiffnesses of the joints are reduced to 20 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 41 No. 2/2001 Reliability – Based Design of Precast Buildings M. Kalousková, E. Novotná, J. Šejnoha We present a numerical analysis of a precast structure regarding random properties of the material characteristics of joints, as well as the random character of loading, especially due to temperature impact. Using FEM we compare some of our results with a deterministic nonlinear solution. Keywords: reliability, precast buildings, joints of panels, two-parameter subgrade, reliability index. Fig. 1: Loading paths for a vertical joint their residual values. The aim of this paper is to demonstrate that • this simple algorithm, which does not require an exam- ination of the whole loading path from Fig. 1, makes it possible to describe the propagation of the deteriorated regions of joints, • the image of these regions is similar to that obtained by the well-tried finite element deterministic solutions. 2 Reliability analysis of joints Probabilistic analysis is carried out using the NASREL (Numerical Analysis of Structures for Reliability) code. NASREL is the high performance finite element NASCOM (Numerical Analysis of Structures and Combined Objects) code integrated with COMREL (Componental Reliability Analysis). The second order reliability method SORM is used to determine reliability index b at selected points of the joints. Under the assumption that a failure domain � �g u � 0, u being the normalized basic uncertainty variables, is twice differentiable, the failure surface � �g u � 0 in the vicinity of the critical point u* with the distance � � u* to the origin is approximated by its supporting hyperparaboloid. Expanding the function g into the Taylor series up to the second order terms and introducing certain orthogonal transformations, the failure surface can be written as: � �� �g U Un i i i n U � � � � � � � � � � �0 12 2 1 1 � � . (1) Parameters, �i, i = 1, 2, … , n, stand for the second order derivatives in the principal directions of the failure surface. An expression for the failure probability can be found in [6] in this form � �� � � � � �P g i i n U � � � �0 1 1 2 1 1 � � � � , (2) where � is the Laplace function. The Rackwitz/Fiessler optimization procedure is used to find the design point. 3 Model of a precast building The FEM code NASCOM is called when analyzing the state of stress in walls and joints of a precast structure. In this paper, 3D elements have been used for panels and joints, beam elements to model a continuous footing, and beam and truss elements for the equivalent subgrade structures. A 3D Coulomb condition describes the failure envelope in joints [3] as � � � � �i j g i j i j � � � � � � � � � �cot , , , , , , , 2 4 2 1 2 3 1 2 3red (3) � � � red � 2 1 c cos sin , (4) where �i, �j principal stresses, c cohesion coefficient � friction angle. If � � �1 2 3� � , condition (3) reduces to � � � � �1 3 2 4 2 � � � � � � �cot g red . (5) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 21 Acta Polytechnica Vol. 41 No. 2/2001 Fig. 2: Interaction diagram for a joint Regarding the fact that the material of joints is plastically anisotropic, condition (5) can be written as [4] � � 1 3 2 � c , (6) where � R R c i ratio of the strength in compression to the strength in tension. The following three loadings are combined – the dead load, the loading transferred from the ceiling panels, and the temperature. All of these are supposed to be randomly distributed. In the deterministic solution the 2D finite elements are preferred to the 3D formulation discussed above. Fig. 2 shows a material model describing the interaction between the shear and normal stresses. Diagram � � has been derived from experimental results for a layer of lower strength [2]. The model is characterized by the following parameters – the characteristic strength fck, the ultimate strains �ck and �c. The proportional limit �e is equal to 0.4 fck. The deter- mination of reduced stiffnesses Ered, Gred at the n-th iteration step is based on the assumption that strains �(n�1) and (n�1) from the preceding iteration step are known. The algorithm starts by determining the reduced stiffness Ered. Next, the corresponding ultimate shear strength of the joint max is determined. Finally, Gred is assigned to the known (n�1) . This algorithm is implemented in the finite element code FEAT, where contact elements are used to model the joints. For more details, see [2]. 4 Model of a subgrade The proposed analysis of a precast structure takes into ac- count the structure-subgrade interaction. A straightforward way to solve this problem by the NASCOM code is to use 3D elements for both the structure and subsoil. An alternative and more effective approach is based on the Winkler – Pasternak model with two parameters, which is not imple- mented in the NASCOM code. This model is described in a concise manner in what follows. The stiffness of the subgrade is replaced by the stiffness of an equivalent construction com- posed of truss and beam elements, as shown in Fig. 3. As for the model of the subgrade, noninteracting founda- tion structures are considered [1]. Three basic types of elements are used (Fig. 4) and the deformation of each of them is given by the vertical displacements of end-points 1 and 2. a) Inner element I The stiffness matrix of the subgrade element is expressed [1] as � �K b C b C b C b C b C b C b C WP I � 2 3 2 3 2 3 2 2 3 1 2 1 2 1 2 � � � � � � � * * * * * * 1 2 2* * � ! ! ! " # $ $ $ b C � , (7) where 2b width of foundation � length of element C1, C2 stiffness parameters of the Winkler-Pasternak model C C1 2 * *, modified parameters defined as C C b C C1 1 1 2 1* � , C C b C C 2 2 2 3 1 1 2 * � . The corresponding stiffness matrix [KBT] of the equiva- lent beam and truss element (see Fig. 5) is given by � � � � � �K E A h E I E I E IBT I T T B B B B B B � 1 2 12 1 2 12 1 2 12 1 2 3 3 3 � � � � � � � � �� � 1 2 12 1 23 E A h E IT T B B � ! ! ! ! " # $ $ $ $� , (8) where A AT B, cross-section areas of truss and beam, respectively E ET B, Young moduli of truss and beam, respectively � length of beam element h length of truss IB moment of inertia of the beam cross section � � 6 B B B E I kGA �2 coefficient expressing the influence of shear. Comparing the equivalent stiffness matrices (7) and (8) gives � � 1 2 12 1 2 2 3 2 3 1 2 E A h E I b C b CT T B B � � � �� * *, (9) � � 12 1 2 3 2 3 1 2 E I b C b CN N � � � � � * *. (10) The determination of the beam and truss characteristics is evident. 22 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 41 No. 2/2001 Fig. 3: Beam and truss construction “flange” “web” 2221 1 1 Fig. 4: Basic types of subgrade elements b) End – point element II The stiffness matrix of the subgrade element is obtained from (7) by adding a complementary matrix � �% K b C C CWP II � � ! ! " # $ $ 0 0 0 2 3 2 1 2 2 . (11) c) Inner corner element III The interaction of the crossing beams cannot be ne- glected. Substituting the following expression for the displacement of the subgrade in the vicinity of the inner corner, � �w x y W e e e e C C x C C y C C x C C y , � & � � � � � � � � � � 0 1 2 1 2 1 2 1 2 , into the principle of virtual displacements yields formulas for the shear forces qx, qy acting along the crossing beams (unit corner displacement W0 is considered): � � � � q C C y q C C x x y � & � & 1 2 1 2 � � , (12) where � �� y e e C C y C C y � 1 3 4 1 2 1 2 2 . (13) For y = 0 or x = 0 we have q q C Cx y� � 3 4 1 2. (14) For y x� �� �0 0or (� 0 being length of the shear depres- sion) q q C Cx y� � 1 2 . (15) Applying equations (12) through (15) to the elements in the vicinity of the inner corner (Fig. 6) yields � �K k k k kWP III � � ! " # $ 11 12 21 22 , where for the “flange” shown in Fig. 6 � �� �k k b C C C x b C11 22 1 1 2 2 2 3 2 3 1 2F F s� � � � � � *, (16) � �� �k k b C C C x b C12 21 1 1 2 2 3 3 1 2F F s� � � � � � * , (17) and for the “web” � �k k b C C C y b C11 22 1 1 2 2 2 3 2 3 2W W s� � � � � � * , (18) � �k k b C C C y b C12 21 1 1 2 2 3 3 2W W s� � � � � � * . (19) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 23 Acta Polytechnica Vol. 41 No. 2/2001 I II Fig. 5: Equivalent beam and truss construction – elements of types I, II �2 2 0 0 � �w �1 1 0 0 � �w �2 2 0 0 � �w �1 1 0 0 � �w �� Fig. 6: Distribution of shear forces in the vicinity of the inner corner � 0 5 Numerical example Part of a seven-story precast building of type G57 was analysed (Fig. 7). The construction with the continuous footing lies on a sandy loam subgrade (C1 = 15 MNm �3, C2 = 5 MNm �1, E0 = 35 MPa). The statistical properties of the basic variables applied to the reliability analysis are listed in Table 1. The aim of this paper is to demonstrate the propaga- tion of deteriorated regions rather than to describe truthfully the random properties of the building. For simplicity, all variables except for the friction angle, which is a constant, are supposed to be normally distributed with the coefficient of variation 0.1. The temperature loading is caused by exposing one side of the building to thermal radiation from the sun. 24 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 41 No. 2/2001 . . . . Fig. 7: Ground plan and analyzed part of building G57 a b c Fig. 8: Distribution of the failed joints The maximum value of the temperature change 10 K is con- sidered on the outside surface and conducted to the inner wall (thermal conductivity coefficient � being 1.43 Wm�1K�1). Table 1: Basic variables Basic variables Dimension Mean cohesion c MPa 2.5 friction angle � rad 0.52 dead load – material density � kgm�3 2300 loading transferred from ceilings q kNm�2 6.67 � t1 K 4 Temperature increments � t2 K 4 � t3 K 2 Three temperature levels were used together with the dead load and the loading transferred from the ceilings (Ta- ble 1). At the first temperature level (� t1 = 4 K) the values of � attained in the whole structure were greater than 5. At the se- cond level (the total temperature increment � t = 8 K) the stiffnesses in the regions of joints with � � 1.5 were reduced to their residual values and the procedure was repeated. In this example, 10 % of the initial stiffness Kin has been chosen for the residual stiffness Kres, even though this value some- what overestimates the values obtained experimentally [5]. The vertical joints in precast buildings of this type are not equipped with reinforcing bars. Their stiffness is assured by the ceiling panels, which overlap the vertical fissures between the wall panels. The resulting distribution of the failed joints is drawn in Fig. 8a (solid lines). The third temperature level (total increment � t = 10 K) caused the failure distribution demonstrated in Fig. 8b. The detailed distribution of the deteriorated regions at the top of the building is displayed in Fig. 8c. When comparing the results obtained in this way with a deterministic non-linear solution by the 2D FEM mentioned in Section 3, nearly the same images of deteriorated regions of joints were reached. It should be pointed out that the two images become different when the coefficient of variation increases. 6 Conclusion This paper discusses a model describing the failure of a precast construction with random properties of joints and loading by means of index �. It is evident that introducing the residual stiffnesses in joints with � < 1.5 leads to results that are comparable with the deterministic solution, provid- ing the failure condition is of an adequate type. It appears that the results are almost the same when the level of index � used to reduce the individual stiffnesses varies from 1 to 2. Nevertheless, a fully probabilistic approach, using for example the Monte Carlo method, especially in conjunction with the response surface method, will provide more com- plex information about the construction behaviour and its reliability. Acknowledgments The financial support was provided by GAČR 103/99/0944 and by research project J04/98:210000001. References [1] Bittnar, Z., Šejnoha, J.: Numerical Methods in Structural Mechanics. ASCE Press, New York, Thomas Telford, London, 1996 [2] Blažek, V., Fajman, P., Šejnoha, J.: Východiska nelineární analýzy konstrukcí panelových budov (Theoretical background of nonlinear analysis of precast structures). Beton a zdivo (v tisku) [3] Ducháček, J.: Nauka o pružnosti a pevnosti II (Theory of elas- ticity II). SNTL Praha, 1964 [4] Chen, W., F.: Plasticity in Reinforced Concrete. McGraw- -Hill, New York, 1982 [5] Pume, D.: Structural Models of Joints between Concrete Wall Elements. CTU Report, No. 2/1997 [6] STRUREL, Theoretical Manual. RCP Consult, München, 1996 Ing. Marie Kalousková, CSc. phone: +420 2 2435 4489 e-mail: kalousko@fsv.cvut.cz Ing. Eva Novotná phone: +420 2 2435 4483 e-mail: novotnae@fsv.cvut.cz Prof. Ing. Jiří Šejnoha, DrSc. phone: +420 2 2435 4492 e-mail: sejnoha@fsv.cvut.cz Dept. of Structural Mechanics Czech Technical University in Prague Faculty of Civil Engineering Thákurova 7, 166 29 Praha 6, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25 Acta Polytechnica Vol. 41 No. 2/2001 << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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