STRESS FIELD EVALUATION IN CONCRETE GRAVITY DAMS USING THE PSEUDO-STATIC AND PSEUDO-DYNA MIC APPROACH ES Paulo Marcelo VIEIRA RIBEIRO Doctorate Student, Universidade de Brasília – UnB Lineu José PEDROSO Professor, Universidade de Brasília – UnB Silvio CALDAS Civil Engineer, Centrais Elétricas do Norte do Brasil - ELETRONORTE Brazil Traditionally, concrete gravity dams have been designed and analyzed by a simplified procedure, which does not consider the structural elasticity and the fluid compressibility(Pseudo-Static Method). Some authors have already found out that the consideration of these aspects can produce significant stresses in the dam, indicating that the simple procedure might not be appropriate for the design of this type of structure [1,2]. A study conducted by [1] indicates an alternative procedure for evaluation of the seismic actions, incorporating the effects of the elasticity of the structure and the compressibility of the water. It is a simplified analytical procedure, based on the spectral response of the structure. Its application consists in the calculation of the seismi c loading, which should be applied to the dam in a equivalent sta ti c analysis. In design conditions where the dam can be treated as a rigid body (period ofvibration is approximately equal to zero), the proposed model by the Pseudo-Static method is appropriate, and a constant distribution of seismic accelerations along the dam’s height can be adopted, neglecting relative motion contribution. This method considers that the dam is provided with the same acceleration of an infinitely rigid foundation, and the peak ground acceleration produces the largest seismic effects on the structure.
STRESS FIELD EVALUATION IN CONCRETE GRAVITY DAMS USING THE
PSEUDO-STATIC AND PSEUDO-DYNAMIC APPROACHES
Paulo Marcelo VIEIRA RIBEIRODoctorate Student, Universidade de Brasília – UnB
Lineu José PEDROSOProfessor, Universidade de Brasília – UnB
Silvio CALDASCivil Engineer, Centrais Elétricas do Norte do Brasil - ELETRONORTE
Brazil
Traditionally, concrete gravity dams have been designed and analyzed by a
simplified procedure, which does not consider the structural elasticity and the fluidcompressibility(Pseudo-Static Method). Some authors have already found out thatthe consideration of these aspects can produce significant stresses in the dam,indicating that the simple procedure might not be appropriate for the design of thistype of structure [1,2].
A study conducted by [1] indicates an alternative procedure for evaluation of theseismic actions, incorporating the effects of the elasticity of the structure and thecompressibility of the water. It is a simplified analytical procedure, based on thespectral response of the structure. Its application consists in the calculation of theseismic loading, which should be applied to the dam in a equivalent staticanalysis.
In design conditions where the dam can be treated as a rigid body (period of vibration is approximately equal to zero), the proposed model by the Pseudo-Staticmethod is appropriate, and a constant distribution of seismic accelerations alongthe dam’s height can be adopted, neglecting relative motion contribution. Thismethod considers that the dam is provided with the same acceleration of aninfinitely rigid foundation, and the peak ground acceleration produces the largestseismic effects on the structure.
However, there are conditions where the effects of the flexibility of the damshould be considered, and analysis of the dynamic response of the structure isneeded. There are reports of accidents with dams that were designed by thetraditional procedure, which were submitted to seismic actions much higher than
those laid down by the usual rigid body procedure [1]. The Koyna dam, for example, was designed with a seismic coefficient of 0.05g. However, the 1967earthquake, occurred in India, was able to produce actions that exceeded thedesign considerations. As a result, many cracks were formed along the dam.
The structural flexibility is of fundamental importance for understanding theactions produced by the earthquake. A large amplification of the groundacceleration can occur with the consideration of this effect. Fig. 1 and 2 illustratethe total acceleration responses produced in a system of 1 degree of freedom(obtained by numerical integration, using a fourth order Runge-Kutta procedure),
when subjected to a short range of the north-south component of the El Centroearthquake - 1940 [3]. The first system has a period of vibration equal to 0.02s,while the second system presents a period of vibration equal to 1s. The dampingfor the two situations is equal to 2%. It can be observed that in the first system(Fig. 1) the response approaches the ground acceleration. Thus, this indicates arigid body motion. The second system presents an amplification of theacceleration response, with respect to the earthquake produced component. At acertain instant the total acceleration achieved by the system is twice the value of the ground acceleration.
The behavior shown in Fig. 1 reveals an important characteristic of the
seismic design spectra. On this diagram the spectral acceleration tends to movecloser to the peak ground acceleration as the period of vibration of the structuredecreases. This indicates that the system acquires properties similar to those of arigid body motion, with the total acceleration response approximately equal to theground acceleration. In these cases the use of Pseudo-Static method isacceptable. According to [4] the use of this procedure is valid for dams withfundamental periods of vibration less than 0.03s. Thus, the dam can be treated asa rigid body, with a constant coefficient equal to the ground produced acceleration,distributed throughout its height (sometimes called seismic coefficient method).Structural relative motion is neglected on this type of analysis. The behavior shown in Fig. 2 illustrates a dynamic amplification of the ground motion. In somecases the gain reaches 75% of the ground acceleration. A dam with the specificcharacteristics of this dynamic system, designed by the traditional procedure,would have a peak ground acceleration of approximately 0.04g, while the flexiblestructure reaches a maximum acceleration of 0.07g.
The Pseudo-Static method seismic actions are derived from the hypothesis
of a rigid body moving towards an incompressible fluid. Thus, the structural massis treated with a uniform distribution of acceleration, equal to the infinitely rigidfoundation acceleration. The seismic loading (distributed per unit of height) to beapplied in an equivalent static analysis is composed of two parts: the inertia forceand the hydrodynamic pressures. The first one is given by the product of thecorresponding mass distribution in the analyzed section with the designestablished rigid body acceleration. The hydrodynamic pressure distribution isbased on studies developed by [5], which were recently reviewed by [6], andrepresents the inertial effects of the reservoir along the fluid-structure interface.
0.03
0.04
0.05
total acc
ground a
Fig. 1Single degree of freedom total acceleration response (0.02s period)
0.04
0.06
0.08
total acc
ground a
Fig. 2Single degree of freedom total acceleration response (1s period)
This procedure is also known as the Seismic Coefficient Method and waswidely used for the seismic resistant design of dams. The simplified character andthe routine procedure is one of the major advantages of this method, despite itsdrawbacks, which include not considering: the structural elasticity, damping forces,
variation of the foundation acceleration over time, and the alternation and shortduration characteristics of the seismic loading [7].
The Pseudo-Static Method seismic actions are given by Eq.[1]:
( ) ( ) ( ). .g
S
vC S y w y g p y
g = +
[1]
where g v stands for the horizontal ground acceleration, g indicates the gravity
acceleration, S w represents the distributed dam weight per unit of height and g p
is related to the hydrodynamic pressure distribution along the fluid-structureinterface (relative to a rigid body motion towards an incompressible fluid [2]). Thus,the resulting seismic forces and moments in an analyzed section are given byapplication of this seismic loading in an equivalent static analysis.
2.1. EQUIVALENT STATIC ANALYSIS
Stress analysis procedures can be applied to the structure once the seismicloading (distributed per unit of height) has been established. The procedure
involved in the Pseudo-Static method is simple, and consists in applying thisloading to an equivalent static analysis. The stress analysis calculation may bedriven by analytical or numerical methods. One the highlights of the analyticaltools is the Gravity Method [8] with a formulation that provides the internal stressdistribution in the dam geometry. These are divided into three in planecomponents: two normal stresses (to the vertical and horizontal planes,respectively) and a shear stress. These are given, respectively, by PolynomialEquations [2], [3] and [4]. A great attraction of this method is the ease of use, inaddition to the excellent results obtained when compared to more refinednumerical solutions [9]. Fig. 3 and 4 illustrate, respectively, results from the stressfield distribution obtained with this methodology and with the application of the
Finite Element Method. A clear similarity between these two stress fields can beobserved. Generally, this comparison tends to lose quality as the selected sectionapproaches the foundation. In sections where the hypothesis of trapezoidaldistribution of vertical normal stresses is not verified, the method collapses, sincethis a basic assumption of this procedure. However this method remains as one of the most widely used approaches for preliminary dam design.
where a, b, a1, b1, c1, a2, b2, c2 and d2 are constants related to the analyzed section,which depend on parameters such as geometry, forces and moments on thedesign elevation. A more detailed explanation can be found on [2].
Fig. 3Gravity Method results
(units: kPa)
Fig. 4Finite Element Method results
(units: kPa)
Another alternative for the analytical stress evaluation is the application of the procedure developed by [10]. This author proposed the stress calculation onlyat the dam’s upstream and downstream slopes, which usually provide the criticaldesign regions of this type of structure when facing seismic actions. This reducesthe design equations to a procedure simpler than the Gravity Method, based onthe infinitesimal prisms equilibrium at upstream and downstream slopes. On theseformulations the involved terms are reduced to: normal stresses on horizontalplanes, hydrostatic pressures, hydrodynamic pressures and slope angles. Allthese components are immediately provided, except for the normal stresses,which can be obtained by application of the classical beam theory. The latter is afunction of the moment of inertia, and the resultant of forces and moments in theanalyzed section. These actions can be obtained in practice with the interpretationof seismic loading by means of line segments [10], forming trapezoidal loadingareas. Thus, the calculation of the seismic action is reduced to the solution of aloading formed by a combination of straight segments, largely simplifying theprocedure for calculating the seismic resultant forces and moments and providing
excellent results if an appropriate number of design divisions is chosen.
This analytical procedure was developed by [1] as an alternative to moregeneral procedures, which required the use of a computer in order to evaluate thestructural seismic response. It consists on a simplified analysis of the spectralresponse, which determines the structure’s response in the fundamental vibration
mode due to a horizontal ground motion [11]. This author observed that theresponse of structures with short periods of vibration, such as concrete gravitydams, when subjected to seismic actions, was largely influenced by thefundamental vibration mode. It was also concluded that the vertical components of the ground acceleration exerted little influence on the structural response. Basedon these conclusions this author suggested a simplified methodology for preliminary analysis of concrete gravity dams.
The dam, which in the Pseudo-Static Method was supposed rigid, is nowtreated as flexible (see Fig. 5), and the water in the reservoir considered as acompressible fluid. The seismic loading now depends on the fundamental mode
seismic response of the structure, associated with the corresponding horizontalground motion.
Fig. 5Pseudo-Dynamic Method seismic response [1]
The seismic coefficient of the Pseudo-Dynamic Method takes into accountthe particular characteristics of each ground motion. The spectral acceleration isobtained by a acceleration response spectrum corresponding to the design
earthquake, and depends on the fundamental vibration period and the structuraldamping. The seismic forces calculated using the spectral acceleration are used inan equivalent static analysis. The main disadvantages of this procedure involvethe alternation and short duration characteristics of the seismic loading, which are
not considered [7]. It is, in fact, an estimate of the maximum response produced bythe fundamental mode. A more refined analysis would consist on a study of thedynamic response of the structure, where the entire history of displacements andother response values would be studied over time. The seismic loading of thislevel of analysis, including the effects of the reservoir, is given by Eq.[5]. A moredetailed explanation of the origin and application of this loading can be obtained in[1,11].
( ) ( ) ( ) ( )1. . 4 aS
S C S y w y y g p y
g ψ = ⋅ ⋅ ⋅ + [5]
where aS indicates the spectral acceleration corresponding to the structure’sfundamental vibration period (considering the reservoir effects), g is the gravity
acceleration, S w represents the distributed dam weight per unit of height, ψ
stands for the fundamental mode shape function and 1 g p is related to the
hydrodynamic pressure distribution along the fluid-structure interface (relative to aflexible boundary motion towards a compressible fluid [1]).
3.1. PROPOSED MODIFICATIONS TO THE EQUIVALENT STATIC ANALYSIS
PROCEDURE
The Gravity Method [8] is an excellent analytical tool for evaluation of thestress field distribution. However, its original formulation was conceived for thePseudo-Static actions, and does include those provided by the Pseudo-DynamicMethod. Through a more detailed interpretation of this procedure, [2] introduced ina simplified manner, the seismic actions produced by the procedure developed by[1].
The modifications proposed by [2] include the adoption of a second degreepolynomial fundamental mode shape function (simplifying largely the analyticalprocedure for evaluation of the equivalent forces and moments produced by theinertia force) and the use of hydrodynamic pressure functions similar to thoseproposed by [5], with the inclusion of correction coefficients. Fig. 6 and 7 illustrategraphics with comparisons of these simplifications with the considerationsoriginally proposed by [1].
It can be observed that the proposed fundamental mode shape functionleads to conservative results in evaluation of the seismic loading, producing valuesalways higher than those obtained with Chopra’s proposed fundamental modeshape. It should be also noticed that the proposed solution to the hydrodynamicpressure is non-conservative, producing results lower than those achieved by [1].
In general, it is expected a compensation between the overestimated inertia forceand the underestimated hydrodynamic pressures. Results of tests conducted by[2] confirm the latter.
The Pine Flat dam non-overflow section (see Fig. 8) was analyzed by boththe Pseudo-Static and the Pseudo-Dynamic Methods. This study will present the
main differences obtained on the stress field distribution for this profile. In theequivalent static analysis the Gravity Method was applied, with the modificationsproposed by [2], for the introduction of the pseudo-dynamic seismic actions in thisprocedure.
In the pseudo-static analysis the dam will be treated as rigid, acceleratedwith the peak ground acceleration, which will be adopted as equal to 0.2g. Thepseudo-dynamic seismic loadings are evaluated using the simplifications proposedby [2]. Fig. 9 illustrates the seismic response spectrum applied on these analyses.
Due to the difficulties of obtaining a typical seismic response spectrum for the Brazilian territory, a curve was adopted for a specific earthquake in the NorthAmerican region. This spectrum is suitable for seismic design - in regions of firmground in California – to ground motions with a similar intensity of the earthquakesrecorded in Taft, during the Kern Country earthquake in July 1952 [1]. This is acharacteristic response spectrum of an earthquake with PGA (peak groundacceleration ground) equal to 0.2g.
Fig. 10 and 11 illustrate, respectively, the analyses of maximum andminimum principal stresses (were positive sign indicates compression), obtainedfrom the normal operation actions, including the concrete self-weight and thehydrostatic pressures. Uplift pressures were neglected on both analyses.
In this type of analysis it is assumed that the structure is rigid moving towardthe incompressible fluid reservoir. For the seismic coefficient, values rangingbetween 0.05g and 0.10g are usually adopted (or a fraction of the peak groundacceleration). The response spectrum illustrated on Fig. 9, for a vibration periodequal to zero, results in a seismic coefficient equal to 0.2g, which corresponds to a
value much higher than usually adopted. Still, this is the value that should be used,because it corresponds to the peak ground acceleration (PGA) in this example.
In this analysis the following loadings are considered: concrete self-weight,hydrostatic pressures, inertia forces and hydrodynamic pressures. The dam will beexamined with horizontal seismic acceleration towards the upstream direction.This means that seismic forces will act in the downstream direction.
Fig. 12 and 13 illustrate, respectively, the analyses of maximum andminimum principal stresses, obtained with the application this procedure.
In this analysis the applied seismic loading will be similar to the oneproposed by [1], with the simplifications made by [2]. This is not a pseudo-dynamicanalysis itself, because the seismic loading is not implemented exactly as defined
by [1]. However, the results of previous studies performed by [2] demonstrate anexcellent agreement of the simplified procedure (defined as Modified Pseudo-Dynamic Method) with the results obtained by [1].
Fig. 14 and 15 illustrate the maximum and minimum principal stressdistribution, obtained with the application of the Modified Pseudo-DynamicMethod, for a horizontal seismic acceleration oriented towards the upstreamdirection. It can be observed an increase in the magnitude of the seismic actionswhen compared to the previous item (resulting in smaller stresses at the upstreamslope and in an increase of these values at the downstream slope).
Two application examples of procedures for seismic analysis of typicalprofiles of concrete gravity dams, subjected to a design spectrum, were presented.The pseudo-static procedure treats the dam as a rigid body, ignoring theamplification effects of the dynamic response. However, there are some case werethis effect cannot be neglected. Only structures with very small periods of vibrationpresent dynamic responses that approach the ground motion (see Fig. 1),featuring a rigid body movement. In other situations, the dynamic characteristics of the dam should be taken into account, in order to include the structural response
amplification effects (see Fig. 2).
The pseudo-static procedure, which was traditionally used in the design of many dams (also known as the Seismic Coefficient Method), produces results thatunderestimate the seismic actions when the structure cannot be treated as a rigidbody. Application of this procedure is recommended only to structures with afundamental vibration period less than 0.03s [4]. The achieved results provide thatthe actions produced by the Pseudo-Static Method are far lower than thoseproduced by the Pseudo-Dynamic procedure.
The stress distribution in the pseudo-static analysis exerts little influence in
the design of a dam, because the tensions are almost nonexistent (see Fig. 13,with maximum tension equal to 50 kPa), and the added compression will hardlyexceed the concrete strength usually employed in this type of structure. However,in the modified pseudo-dynamic procedure, tensions can be significant (see Fig.
15, with values up to 1500 kPa), directly influencing the choice of the concretedesign resistance, due to the low tensile strength resisted by this material. In somecases, special mixtures are needed to ensure the necessary strength to regionswere very high tensions are expected.
ACKNOWLEDGEMENTS
The authors are grateful for the financial support provided by CAPES andCNPq agencies.
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