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The effect of the asynchronous ground motion on hydrodynamic pressures
PERGAMON
Computers and Structures 68 (1998) 271±282
The eect of the asynchronous ground motion on hydrodynamic pressures A. Bayraktar, A. A. DumanogÆlu Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Received 17 October 1995; received in revised form 1 January 1997
Abstract In this paper, the eects of the asynchronous ground motion on frequency contents and amplitudes of hydrodynamic pressures are investigated considering dam±reservoir±foundation interaction. The behaviour of the dam, the reservoir and the foundation is expressed in terms of the displacements (Lagrangian approach) in the formulations obtained using ®nite element method. Hydrodynamic pressures are calculated for various earthquake wave velocities. It is observed that asynchronous ground motion aects the frequency contents of hydrodynamic pressures slightly. However, the amplitudes of the hydrodynamic pressures decrease considerably when asynchronous ground motion is taken into account. # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved.
1. Introduction Hydrodynamic pressures (excess of hydrostatic pressure) occurring on the upstream face of the dam due to earthquakes play an important role in the design of dams. There are a lot of factors such as the compressibility of water in the reservoir, the shape of the reservoir, the shape of the dam, the ¯exibility of the dam and foundation, the elastic properties of the dam and foundation and the silt deposits at the upstream face of the dam that in¯uence the hydrodynamic pressures. In addition, the eect of asynchronous ground motion to be investigated in this paper also in¯uences hydrodynamic pressures considerably. The ®rst solution of the hydrodynamic pressures on dam with a vertical upstream face was found by Westergaard [1] in 1933. Since 1933, many researches on this subject have been performed, and the dierent aspects of the problem have been studied [2±10]. In those papers, it was assumed that the earthquake waves propagated with in®nite velocity. That is to say, the earthquake waves reach all support points along the structure±foundation interface at the same time. However, in reality the earthquake waves propagate with dierent velocities depending on the type of soil.
Seismic wave velocities near the earth's surface vary in the range from about 100 m/s to 3000 m/s [11]. Consequently, the seismic waves which travel from its source with a ®nite speed arrive at dierent points at dierent times. This situation causes support points to have an asynchronous motion. Asynchronous ground motion creates quasi-static displacements in the structure in addition to the dynamic displacements. Although the latter is caused by inertia, the former is caused by dierent movement of the support points at the structure-foundation interface. In this paper, the hydrodynamic pressures occurring on the upstream face of the concrete gravity dams due to asynchronous ground motion were calculated for various cases. A sample concrete gravity dam was chosen and the dam±reservoir±foundation interaction was considered in the analyses. Two dimensional ®nite solid and ¯uid elements with various nodes based on the Lagrangian approach were programmed to take into account dam±reservoir±foundation interaction together as sub programs in Fortran 77 language by the authors. These sub programs were incorporated into MULSAP [12] and SAP IV [13] programs. The modi®ed version of the MULSAP [12] program for ¯uid-structure systems was used in the asynchronous
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 2 3 - 6
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A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
dynamic analysis of dam-reservoir-foundation system. The SAP IV [13] program was utilized for the comparison of the results corresponding to in®nite velocity case. 2. Theoretical formulation In the Lagrangian approach, the behaviour of the ¯uid and structure is expressed in terms of displacements. For this reason, compatibility and equilibrium are automatically satis®ed at the nodes along the interfaces between the ¯uid and structure. Therefore, a ¯uid element based on the Lagrangian approach can be easily incorporated into a general-purpose computer program for structural analysis. 2.1. Fluid systems Fluid is assumed to be linear-elastic, inviscid and irrotational [10, 14]. For this ¯uid, the relation between pressure and volumetric strain is given by 1
P ben
where P is pressure, b is the bulk modulus of the ¯uid, and en is the volumetric strain which can be expressed in terms of the displacements. For two dimensional problems, en can be expressed as follows; en
@nfy @nfz @y @z
2
where nfy and nfz are the components of the displacement in the y and z directions, respectively. Irrotationality of the ¯uid is taken into account like penalty methods [15] in the formulations. Therefore, rotations and constraint parameters related to these rotations are included in the stress±strain equations of the ¯uid. For the two-dimensional case, there is only one rotational relation which is de®ned by 1 @nfy @nfz 3 ÿ w 2 @z @y where w is the rotation about the axis normal to the plane. The relation between the stress and stiness associated with this rotation is given by Pw aw
4
where Pw and a are the rotational stress and stiness (constraint parameter), respectively. The total strain energy of the ¯uid system can be expressed as follows; 1 T e Cf edV 5 pe 2 T
where e is a vector of strains given by e =[en w] and Cf is a diagonal matrix whose elements are
given by the elasticity parameters in Eqs. (1) and (4). Using the ®nite element method, Eq. (5) may be expressed as 1 pe n Tf Kf n f 2
6
where Kf and nf are the stiness matrix and the nodal displacement vector of the ¯uid system, respectively. An important behaviour of ¯uid systems is the ability to displace without changing in volume. For reservoir and storage tanks, this movement is as sloshing waves in which the displacement is in the vertical direction. The increase in the potential energy of the system due to the free surface motion can be written as; 1 7 rgn2fs dA ps 2 where r and g are the mass density of the ¯uid and the gravitational acceleration, respectively, and nfs is the free surface vertical displacement of the ¯uid. Using the ®nite element method, the free surface potential energy, Eq. (7), is expressed in terms of the vertical node displacements at the surface as 1 ps n Tfs Sf n fs 2
8
where Sf and nfs are the free surface stiness matrix and the free surface vertical displacement vector of the ¯uid system, respectively. Finally, the kinetic energy of the ¯uid is given by 1 9 P r n_ 2fy n_ 2fz dV 2 where nÇ fy and nÇ fz are the components of the velocity in the y and z directions, respectively. Using the ®nite element method, Eq. (9) can be written as; 1 T n_ Tf Mf n_ f 2
10
where Mf and nÇ f are the mass matrix and the nodal velocity vector of the ¯uid system, respectively. When the energy equations which are formed using ®nite element method are substituted into the Lagrange's equation [16], the equation of motion of the ¯uid system can be obtained as follows; Mf n f Kf n f Ff
11
where K*f and Ff are the system stiness matrix including the free surface stiness and time-varying nodal forces vector for the ¯uid system, respectively.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
2.2. Fluid±structure systems Equation of motion for ¯uid system, Eq. (11), has similar form with that of the structure when Lagrangian approach is used. Hence, the equations of motion of ¯uid±structure system can be obtained by combining the equation of two systems. However, it requires a dierent sensitivity to determine interface condition of the system. Because of the fact that the ¯uid is assumed to be inviscid, only the displacements in normal direction to the interface are continuous at the interface of the coupled system. This condition is imposed by the constraint equations [15]. Using the interface condition, the equation of motion (including damping eects) of the ¯uid±structure system is given by M n Cn_ Knn F
12
where M, C, and K are mass, damping and stiness matrices, respectively, and n, nÇ and n are total displacement, velocity and acceleration vectors, respectively, and F is a vector of input forces. In the absence of external forces applied directly to the structure, F becomes zero-vector. It is possible to separate the degrees of freedom given Eq. (12) into two groups as known and unknown. The known degrees of freedom are associated with those of the structure±foundation interface. In fact, the acceleration of these interface degrees of freedom is given as a function of time, and corresponding velocity and displacement can be obtained by numerical integration of the ground acceleration. The unknowns are related to degrees of freedom of the structure. The former degrees of freedom will be denoted henceforth as the vector ng, and the latter as nr. Here, sux g denotes `ground degrees of freedom' (GDOF) and sux r denotes `response degrees of freedom' (RDOF). Eq. (12), partitioning into RDOF and GDOF, can be expressed as follows;
Mrr Mgr
Mrg Mgg
nr Crr ng Cgr
Crg Cgg
Krr Kgr
Krg Kgg
n_ r n_ g
nr ng
n sr n dr n sg n dg
Mrr n dr Crr n_ dr Krr n dr Feff
nr ng
0 0
13
14
In fact, ndg is zero, n>sg is equal to ng. Substitution of
15
where Fe is used to represent a vector of eective forces acting on the RDOF. It is equal to n sr n_ sr n sr ÿ Crr Crg ÿ Krr Krg ÿMrr Mrg n sg n_ sg n sg 16 If all time-dependent terms are omitted form Eqs. (15) and (16), only the last term of Eq. (16) remains and this must be equal to zero. This means that n sr ÿKÿ1 rr Krg n sg ÿRrg n sg
17
If the eective force expression, Eq. (16), is arranged by using Eq. (17), and Eq. (15) is rewritten again, the equation of motion of the dynamic component of the RDOF can be written as; Mrr n dr Crr ndr Krr n dr ÿMrr Rrg nsg
18
where Rrg contains r vectors (ground displacement shape vectors) which describe the displaced shape of the structure when a unit displacement is given to the single GDOF while all other GDOF are held ®xed. This matrix becomes a single vector consisting zeros and ones in the classical dynamic analysis [16]. The r vectors are obtained using penalty methods [15]. The dynamic components of the total displacements of the RDOF are calculated from X fi Yi t 19 n dr i
where i is mode number, fi is ith mode vector and Yi(t) is the time-dependent modal amplitude of the response, which is calculated mode by mode from the uncoupled replacement of Eq. (18), that is i t2xoi Y_ i t o2 Yi t Y i ÿ
When considered asynchronous ground motion, it is possible to separate the total displacement vectors as quasi-static, ns, and dynamic, nd, into two groups which are shown as follows;
Eq. (14) into Eq. (13), the equation of motion of the dynamic component of the RDOF can be written as
"
273
# fTi Mrr n1g t1 ; t r2 n2g t2 ; t r1 fTi Mrr fi 20
where oi, x and ti are the natural circular frequency of the ith mode, a damping ratio applied to all modes of vibration, and the arrival time of the ground motion, ng, to the ith region from the reference point, respectively. On the right-hand side of Eq. (20), there will be as many terms as the number of r vectors. The total displacement requires the additional determination of nsr (quasi-static displacement), and this is obtained from Eq. (17) or more explicitly as follows; n sr r1 n1g t1 ; t r2 n2g t2 ; t
21
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Fig. 1. The dimensions of the Sariyar dam and ®nite element model of the dam±reservoir±foundation system subjected to asynchronous ground motion. (a) Dimensions of the Sariyar dam [17]. (b) Finite element model.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
275
Fig. 2. Corrected ground acceleration record and displacement of east±west (E-W) component of the 13 March 1992 Erzincan, Turkey, earthquake.
with as may terms on the right-hand side as there are r vectors. The total displacements, nr, are obtained by summing up quasi-static and dynamic displacements at each interval of time as follows; n r n sr n dr
22
3. The response of dam±reservoir±foundation system subjected to asynchronous ground motion The Sariyar concrete gravity dam constructed on Sakarya river which is 120 km in the northwest of Ankara, Turkey, was selected to determine the response of the dam±reservoir±foundation system subjected to asynchronous ground motion. The
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dimensions of the tallest section of the dam and the selected ®nite element model of dam±reservoir±foundation system are shown in Fig. 1. The reservoir length (3H = 255 m) was taken three times of the reservoir height (H) [18]. It was assumed that the reservoir has constant depth. The depth of the foundation and the length, which is on the part of downstream of foundation, were taken as many as reservoir height. The dam itself with foundation block and the reservoir were represented by 51 eight-noded and 24 nine-noded isoparametric quadrilateral ®nite elements, respectively. The nodes representing the bottom and edges of the
foundation were ®xed while the nodes representing the extreme edge of the reservoir were free to move in the vertical direction only. Because of the fact that ¯uid was assumed to be inviscid in this study, the normal components of the displacements at the reservoir±dam and reservoir±foundation interfaces are to be continuous. This condition is accomplished by using short and axially almost rigid truss elements in the normal direction of the interfaces. In the analysis, the plane strain assumption was employed. The dam and foundation material were assumed to be linear-elastic, homogeneous and isotro-
Fig. 3. Corrected ground acceleration record and displacement of north±south (N±S) component of the 13 March 1992 Erzincan, Turkey, earthquake.
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277
Fig. 4. Hydrodynamic pressure envelopes.
pic. The elasticity modulus, mass density and Poisson's ratio of the dam are taken as 35 109 N/m2, 2447 kg/m3 and 0.15, respectively. Elastic properties of the foundation were chosen the same as those of the dam. As for the reservoir, the bulk modulus (b0), mass density and unit weight are taken as 207 107 N/m2, 1000 kg/m3 and 9810 N/m3, respectively. Numerical problems as zero-energy modes occur in the modal analysis of ¯uid systems when the Lagrangian approach is used. The use of reduced of integration techniques and the introduction of the rotational constraints in the formulation of the ¯uid element stiness eliminates all unnecessary zero-energy modes. The rotation constraint parameter is chosen as 100 times of the bulk modulus of ¯uid [14]. The ¯uid element matrices were computed using 2 2 reduced integration orders [14]. For the solid elements, 3 3 normal integration orders were utilized. The selection of the mode number in the modal analysis of coupled system based on Lagrangian approach is very important. Because, the sloshing, volume change and rotational modes occur when the Lagrangian approach is used in the analysis. The number of surface sloshing modes which depends on ®nite element model of the reservoir becomes very much. The eect of these modes which form the ®rst ranges in the frequency table is a negligible level in the response of the dam. Because of that, the ®rst 30 modes were considered in this study [10]. The base of the coupled system was divided into the four regions for the asynchronous dynamic analysis as indicated in Fig. 1b. The lengths of the 1st, 2nd, 3rd and 4th regions vary between 0±105 m, 105±223.125 m, 223.125±335.75 m and 335.75±408.25 m, respectively. It is accepted that the amplitudes and frequency contents of the ground motion do not change along these regions, and the propagation speed of the seismic wave
is constant in all regions. Arrival time to each region was calculated using the relation of lr/Vs, where lr is the distance which varies from the reference point to the initial of each region; Vs is the propagation velocity of the seismic wave. For instance, for the wave velocity of 250 m/s, the arrival times to each region are 0.0, 0.42, 0.8925 and 1.343 second, respectively. The arrival times are taken into account as 0.0 second for all regions when the wave velocity is assumed in®nite. This situation corresponds to classical dynamic analysis. The ground displacement shape vectors (r vectors) of the coupled system are obtained for each region [17]. The penalty method [15] is used in the calculation of the r vectors as also mentioned in the section of formulation. When the r vectors are added to each other, unit horizontal displacements occur at every nodal point of the system. This event corresponds to rigid body motion in horizontal direction. If the summation of r vectors is not equal to unit value or close to unit value, incorrect displacements and stresses will be obtained. The ®ltering as well as the base-line correction routine has to be applied on the ground acceleration record. Otherwise, incorrect and residual ground displacements can occur after double integration [17, 19]. This procedure is not important in the classical dynamic analysis. But, it is important in the asynchronous dynamic analysis. Because, the ground displacements obtained form the double integration of the earthquake ground acceleration record are used in the calculation of the quasi-static displacements. Corrected ground acceleration records and displacements, which were obtained from the double integration of the accelerations, of the east±west (E±W) and north±south (N±S) components of the 13 March 1992 Erzincan, Turkey, earthquake to be used in the calculation of
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hydrodynamic pressures are given in Figs. 2 and 3, respectively. 3.1. Hydrodynamic pressures Hydrodynamic pressures were calculated using the time interval of 0.01 second. A damping value of 5% was taken into account for all modes. The east±west (E±W) component of the 13 March 1992 Erzincan, Turkey, earthquake is taken as ground motion (Fig. 2).
Earthquake wave velocities are chosen as 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressure envelopes were plotted using the absolute maximum values at Gauss points which are in the ¯uid elements near the face of the reservoir±dam interaction. The results of the MULSAP [12] and SAP IV [13] programs are compared to each other. The results obtained from the program MULSAP [12] using in®nite wave velocity should be equal or close to those of the program SAP IV [13], since using in®nite wave vel-
Fig. 5. Time histories of the hydrodynamic pressures at Gauss point P.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
ocity in the asynchronous dynamic analysis corresponds to classical dynamic analysis. The hydrodynamic pressures obtained from the program MULSAP [12] using in®nite wave velocity, and those of the program SAP IV [13] are depicted in Fig. 4. Hydrodynamic pressures calculated from the two programs are identical as seen in Fig. 4. Time histories for the wave velocities of 2000 m/s and in®nite were depicted at Gauss point P (Fig. 1b) to investigate the eect of the asynchronous ground motion on the frequency contents of the hydrodynamic pressures (Fig. 5). As seen from Fig. 5, the asynchronous ground motion aects the frequency contents of the hydrodynamic pressures slightly. But, the frequency contents of the stresses in the dam and foundation change considerably, and the amplitudes increase when considered asynchronous ground motion [17]. Quasi-static displacements cause to change the frequency contents and to increase amplitudes of the stresses [17]. However, the eects of the quasi-static displacements on the frequency contents of hydrodynamic pressures are very small as seen in Fig. 5. Asynchronous horizontal ground motion decreases the dynamic displacements as mentioned in reference [17, 20]. This situation is observed on the hydrodynamic pressures envelopes clearly (Fig. 6). Fig. 6 was plotted using the absolute maximum hydrodynamic pressures obtained for the velocities of 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressures decrease considerably when the wave velocity decreases. 3.2. The in¯uence of ground motion on hydrodynamic pressures The north±south (N±S) component of the 13 March 1992 Erzincan, Turkey, earthquake (Fig. 3) is selected to see the in¯uence of dierent ground motion on the hydrodynamic pressures. The north±south (N±S) component was applied to dam±reservoir±foundation system in horizontal direction. At the Gauss point P of the coupled system, the time histories of the hydrodynamic pressures were plotted using the propagation wave velocities of 2000 m/s and in®nite as shown in Fig. 7. Frequency content of the hydrodynamic pressure changes very little as in the case of east±west (E± W) component. The absolute maximum hydrodynamic pressures on the upstream face of the dam are depicted for the wave velocities of 250 m/s, 2000 m/s and in®nite in Fig. 8. Although the dierent earthquake record was used, the hydrodynamic pressures decrease when the wave velocities decrease. The hydrodynamic pressures obtained using north±south (N±S) component is smaller than those of the east±west (E±W) component. This situation arises from the values of the acceleration of the north±south (N±S) component. Because, the
279
Fig. 6. Hydrodynamic pressure envelopes.
values of the north±south (N±S) component are smaller than those of the east±west (E±W) component. 3.3. The in¯uence of bulk modulus on hydrodynamic pressures The in¯uence of the bulk modulus on hydrodynamic pressures is investigated by increasing the value of the bulk modulus (b0). The values of the bulk modulus of the ¯uid were chosen as 10b0, 50b0 and 1000b0. The east±west (E±W) component of the 13 March 1992 Erzincan, Turkey, earthquake was used in the analysis. The earthquake wave velocities were selected as 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressure envelopes were plotted in Fig. 9 taking the above mentioned values of the bulk modulus. Hydrodynamic pressures decrease for all velocities when the bulk modulus of the ¯uid increases. Hydrodynamic pressures obtained for 50b0 and 1000b0 are similar to each other which are shown in Fig. 9 b and c. This situation has shown that water approximates to the case of incompressibility after a certain value of the bulk modulus. 4. Conclusions In this paper, hydrodynamic pressures occurring on the upstream face of the dam due to asynchronous ground motion were calculated for various cases. Dam±reservoir±foundation interaction was taken into account in the analysis. Formulations based on the Lagrangian approach were obtained using ®nite element method. Asynchronous ground motion in¯uences the frequency contents of the hydrodynamic pressures insig-
280
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Fig. 7. Time histories of the hydrodynamic pressures at Gauss point P.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
Fig. 8. Hydrodynamic pressure envelopes.
ni®cantly. However, the amplitudes of the pressures decrease considerably when earthquake wave velocity decreases. When the dierent earthquake record is considered, hydrodynamic pressures decrease as wave velocity decreases. Increasing the value of the bulk modulus decreases hydrodynamic pressures. However, with the increase of the bulk modulus beyond certain limit, hydrodynamic pressures do not change appreciably.
References [1] Westergaard HM. Water Pressures on Dams During Earthquakes. Trans. ASCE 1933;98:418±33. [2] Zangar CN. Hydrodynamic Pressures on Dams Due to Horizontal Earthquake Eects, United States, Bureau of Reclamation, Engineering Monograph, No. 11, 15 pages., 1952. [3] Zienkiewicz OC. Hydrodynamic Pressures Due to Earthquakes. Water Power 1964;6:382±8. [4] Chopra AK. Hydrodynamic Pressures on Dams During Earthquakes. J. Eng. Mech. Div. ASCE 1967;93:205±23. [5] Chakrabarti P, Chopra AK. Hydrodynamic Pressures and Response of Gravity Dams to Vertical Earthquake Component. Earthquake Eng. Struct. Dyn 1973;1:325± 35. [6] Paul DK, Zienkiewicz OC, Hinton E. Hinton E, Bettess P, Lewis RW, editors. Transient Dynamic Analysis of Reservoir±Dam Interaction Using Staggered Solution Schemes, Numerical Methods for Coupled Problems. Pineridge Press, Swansea, U.K., 1981:321±334. [7] Humar J, Roufaiel M. Finite Element Analysis of Reservoir Vibration. J. Eng. Mech. ASCE 1983;109:215± 30. [8] Greeves EJ, Dumanoglu AA. The Implementation of an Ecient Computer Analysis for Fluid±Structure Systems
Fig. 9. Hydrodynamic pressure envelopes.
281
282
[9] [10] [11] [12] [13]
[14]
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282 Using the Eulerian Approach within SAP IV. Report No. UBCE-EE-89-10, Department of Civil Engineering, University of Bristol, Bristol, 1989. Sharan SK. Ecient Finite Element Analysis of Hydrodynamic Pressure on Dams. Comput. Struct. 1992;42:713±23. Calayir Y, Dumanoglu AA. Static and Dynamic Analysis of Fluid and Fluid±Structure Systems by the Lagrangian Method. Comput. Struct. 1993;49:625±32. Sanchez-Sesma FJ. Site Eects on Strong Ground Motion. Soil Dynamics Earthquake Engng 1987;6:124± 32. Dumanoglu AA, Severn RT. Dynamic Responses of Dams and Other Structures to Dierential Ground Motions. Proc. Instn Civ. Engrs 1984;77:333±52. Bathe K-J, Wilson EL, Peterson FE. SAP IV: A Structural Analysis Program for Static and Dynamic Response of Linear Systems, ch. Report No. EERC 7311. Earthquake Engineering Research Center, University of California, Berkeley, California, 1973. Wilson EL, Khalvati M. Finite Elements for the Dynamic Analysis of Fluid±Solid Systems. Int. J. Numer. Meth. Engng 1983;19:1657±68.
[15] Zienkiewicz OC, Taylor RL. The Finite Element Method. Vol. 1, Fourth Edition, McGraw-Hill Book Company, London, 1989. [16] Clough RW, Penzien J. Dynamics of Structures. McGraw-Hill Book Company, Singapore, 1975. [17] Bayraktar A. Dynamic Response of Dam±Reservoir± Foundation Systems Subjected to Asynchronous Ground Motion. Ph.D. Thesis, Department of Civil Engineering, Karadeniz Technical University, Trabzon, Turkey (in Turkish), 1995. [18] Newmark NM, Rosenblueth E. Fundamentals of Earthquake Engineering. Prentice-Hall Inc., Englewood Clis, New Jersey, 1971. [19] Erdik M. Processing of Strong Motion Accelerograms in Earthquake Engineering: Corrected Accelerogram, Velocity and Displacement Traces. METU Journal of Pure and Applied Sciences 1979;12:277±304. [20] Priscu R, Popovici A, Stematiu D, Stere C. Earthquake Engineering for Large Dams. Second Edition, Editura Academiei, Bucuresti, 1985.
Computers and Structures 68 (1998) 271±282
The eect of the asynchronous ground motion on hydrodynamic pressures A. Bayraktar, A. A. DumanogÆlu Department of Civil Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey Received 17 October 1995; received in revised form 1 January 1997
Abstract In this paper, the eects of the asynchronous ground motion on frequency contents and amplitudes of hydrodynamic pressures are investigated considering dam±reservoir±foundation interaction. The behaviour of the dam, the reservoir and the foundation is expressed in terms of the displacements (Lagrangian approach) in the formulations obtained using ®nite element method. Hydrodynamic pressures are calculated for various earthquake wave velocities. It is observed that asynchronous ground motion aects the frequency contents of hydrodynamic pressures slightly. However, the amplitudes of the hydrodynamic pressures decrease considerably when asynchronous ground motion is taken into account. # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved.
1. Introduction Hydrodynamic pressures (excess of hydrostatic pressure) occurring on the upstream face of the dam due to earthquakes play an important role in the design of dams. There are a lot of factors such as the compressibility of water in the reservoir, the shape of the reservoir, the shape of the dam, the ¯exibility of the dam and foundation, the elastic properties of the dam and foundation and the silt deposits at the upstream face of the dam that in¯uence the hydrodynamic pressures. In addition, the eect of asynchronous ground motion to be investigated in this paper also in¯uences hydrodynamic pressures considerably. The ®rst solution of the hydrodynamic pressures on dam with a vertical upstream face was found by Westergaard [1] in 1933. Since 1933, many researches on this subject have been performed, and the dierent aspects of the problem have been studied [2±10]. In those papers, it was assumed that the earthquake waves propagated with in®nite velocity. That is to say, the earthquake waves reach all support points along the structure±foundation interface at the same time. However, in reality the earthquake waves propagate with dierent velocities depending on the type of soil.
Seismic wave velocities near the earth's surface vary in the range from about 100 m/s to 3000 m/s [11]. Consequently, the seismic waves which travel from its source with a ®nite speed arrive at dierent points at dierent times. This situation causes support points to have an asynchronous motion. Asynchronous ground motion creates quasi-static displacements in the structure in addition to the dynamic displacements. Although the latter is caused by inertia, the former is caused by dierent movement of the support points at the structure-foundation interface. In this paper, the hydrodynamic pressures occurring on the upstream face of the concrete gravity dams due to asynchronous ground motion were calculated for various cases. A sample concrete gravity dam was chosen and the dam±reservoir±foundation interaction was considered in the analyses. Two dimensional ®nite solid and ¯uid elements with various nodes based on the Lagrangian approach were programmed to take into account dam±reservoir±foundation interaction together as sub programs in Fortran 77 language by the authors. These sub programs were incorporated into MULSAP [12] and SAP IV [13] programs. The modi®ed version of the MULSAP [12] program for ¯uid-structure systems was used in the asynchronous
0045-7949/98/$19.00 # 1998 Elsevier Science Ltd and Civil-Comp Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 0 2 3 - 6
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A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
dynamic analysis of dam-reservoir-foundation system. The SAP IV [13] program was utilized for the comparison of the results corresponding to in®nite velocity case. 2. Theoretical formulation In the Lagrangian approach, the behaviour of the ¯uid and structure is expressed in terms of displacements. For this reason, compatibility and equilibrium are automatically satis®ed at the nodes along the interfaces between the ¯uid and structure. Therefore, a ¯uid element based on the Lagrangian approach can be easily incorporated into a general-purpose computer program for structural analysis. 2.1. Fluid systems Fluid is assumed to be linear-elastic, inviscid and irrotational [10, 14]. For this ¯uid, the relation between pressure and volumetric strain is given by 1
P ben
where P is pressure, b is the bulk modulus of the ¯uid, and en is the volumetric strain which can be expressed in terms of the displacements. For two dimensional problems, en can be expressed as follows; en
@nfy @nfz @y @z
2
where nfy and nfz are the components of the displacement in the y and z directions, respectively. Irrotationality of the ¯uid is taken into account like penalty methods [15] in the formulations. Therefore, rotations and constraint parameters related to these rotations are included in the stress±strain equations of the ¯uid. For the two-dimensional case, there is only one rotational relation which is de®ned by 1 @nfy @nfz 3 ÿ w 2 @z @y where w is the rotation about the axis normal to the plane. The relation between the stress and stiness associated with this rotation is given by Pw aw
4
where Pw and a are the rotational stress and stiness (constraint parameter), respectively. The total strain energy of the ¯uid system can be expressed as follows; 1 T e Cf edV 5 pe 2 T
where e is a vector of strains given by e =[en w] and Cf is a diagonal matrix whose elements are
given by the elasticity parameters in Eqs. (1) and (4). Using the ®nite element method, Eq. (5) may be expressed as 1 pe n Tf Kf n f 2
6
where Kf and nf are the stiness matrix and the nodal displacement vector of the ¯uid system, respectively. An important behaviour of ¯uid systems is the ability to displace without changing in volume. For reservoir and storage tanks, this movement is as sloshing waves in which the displacement is in the vertical direction. The increase in the potential energy of the system due to the free surface motion can be written as; 1 7 rgn2fs dA ps 2 where r and g are the mass density of the ¯uid and the gravitational acceleration, respectively, and nfs is the free surface vertical displacement of the ¯uid. Using the ®nite element method, the free surface potential energy, Eq. (7), is expressed in terms of the vertical node displacements at the surface as 1 ps n Tfs Sf n fs 2
8
where Sf and nfs are the free surface stiness matrix and the free surface vertical displacement vector of the ¯uid system, respectively. Finally, the kinetic energy of the ¯uid is given by 1 9 P r n_ 2fy n_ 2fz dV 2 where nÇ fy and nÇ fz are the components of the velocity in the y and z directions, respectively. Using the ®nite element method, Eq. (9) can be written as; 1 T n_ Tf Mf n_ f 2
10
where Mf and nÇ f are the mass matrix and the nodal velocity vector of the ¯uid system, respectively. When the energy equations which are formed using ®nite element method are substituted into the Lagrange's equation [16], the equation of motion of the ¯uid system can be obtained as follows; Mf n f Kf n f Ff
11
where K*f and Ff are the system stiness matrix including the free surface stiness and time-varying nodal forces vector for the ¯uid system, respectively.
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2.2. Fluid±structure systems Equation of motion for ¯uid system, Eq. (11), has similar form with that of the structure when Lagrangian approach is used. Hence, the equations of motion of ¯uid±structure system can be obtained by combining the equation of two systems. However, it requires a dierent sensitivity to determine interface condition of the system. Because of the fact that the ¯uid is assumed to be inviscid, only the displacements in normal direction to the interface are continuous at the interface of the coupled system. This condition is imposed by the constraint equations [15]. Using the interface condition, the equation of motion (including damping eects) of the ¯uid±structure system is given by M n Cn_ Knn F
12
where M, C, and K are mass, damping and stiness matrices, respectively, and n, nÇ and n are total displacement, velocity and acceleration vectors, respectively, and F is a vector of input forces. In the absence of external forces applied directly to the structure, F becomes zero-vector. It is possible to separate the degrees of freedom given Eq. (12) into two groups as known and unknown. The known degrees of freedom are associated with those of the structure±foundation interface. In fact, the acceleration of these interface degrees of freedom is given as a function of time, and corresponding velocity and displacement can be obtained by numerical integration of the ground acceleration. The unknowns are related to degrees of freedom of the structure. The former degrees of freedom will be denoted henceforth as the vector ng, and the latter as nr. Here, sux g denotes `ground degrees of freedom' (GDOF) and sux r denotes `response degrees of freedom' (RDOF). Eq. (12), partitioning into RDOF and GDOF, can be expressed as follows;
Mrr Mgr
Mrg Mgg
nr Crr ng Cgr
Crg Cgg
Krr Kgr
Krg Kgg
n_ r n_ g
nr ng
n sr n dr n sg n dg
Mrr n dr Crr n_ dr Krr n dr Feff
nr ng
0 0
13
14
In fact, ndg is zero, n>sg is equal to ng. Substitution of
15
where Fe is used to represent a vector of eective forces acting on the RDOF. It is equal to n sr n_ sr n sr ÿ Crr Crg ÿ Krr Krg ÿMrr Mrg n sg n_ sg n sg 16 If all time-dependent terms are omitted form Eqs. (15) and (16), only the last term of Eq. (16) remains and this must be equal to zero. This means that n sr ÿKÿ1 rr Krg n sg ÿRrg n sg
17
If the eective force expression, Eq. (16), is arranged by using Eq. (17), and Eq. (15) is rewritten again, the equation of motion of the dynamic component of the RDOF can be written as; Mrr n dr Crr ndr Krr n dr ÿMrr Rrg nsg
18
where Rrg contains r vectors (ground displacement shape vectors) which describe the displaced shape of the structure when a unit displacement is given to the single GDOF while all other GDOF are held ®xed. This matrix becomes a single vector consisting zeros and ones in the classical dynamic analysis [16]. The r vectors are obtained using penalty methods [15]. The dynamic components of the total displacements of the RDOF are calculated from X fi Yi t 19 n dr i
where i is mode number, fi is ith mode vector and Yi(t) is the time-dependent modal amplitude of the response, which is calculated mode by mode from the uncoupled replacement of Eq. (18), that is i t2xoi Y_ i t o2 Yi t Y i ÿ
When considered asynchronous ground motion, it is possible to separate the total displacement vectors as quasi-static, ns, and dynamic, nd, into two groups which are shown as follows;
Eq. (14) into Eq. (13), the equation of motion of the dynamic component of the RDOF can be written as
"
273
# fTi Mrr n1g t1 ; t r2 n2g t2 ; t r1 fTi Mrr fi 20
where oi, x and ti are the natural circular frequency of the ith mode, a damping ratio applied to all modes of vibration, and the arrival time of the ground motion, ng, to the ith region from the reference point, respectively. On the right-hand side of Eq. (20), there will be as many terms as the number of r vectors. The total displacement requires the additional determination of nsr (quasi-static displacement), and this is obtained from Eq. (17) or more explicitly as follows; n sr r1 n1g t1 ; t r2 n2g t2 ; t
21
274
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Fig. 1. The dimensions of the Sariyar dam and ®nite element model of the dam±reservoir±foundation system subjected to asynchronous ground motion. (a) Dimensions of the Sariyar dam [17]. (b) Finite element model.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
275
Fig. 2. Corrected ground acceleration record and displacement of east±west (E-W) component of the 13 March 1992 Erzincan, Turkey, earthquake.
with as may terms on the right-hand side as there are r vectors. The total displacements, nr, are obtained by summing up quasi-static and dynamic displacements at each interval of time as follows; n r n sr n dr
22
3. The response of dam±reservoir±foundation system subjected to asynchronous ground motion The Sariyar concrete gravity dam constructed on Sakarya river which is 120 km in the northwest of Ankara, Turkey, was selected to determine the response of the dam±reservoir±foundation system subjected to asynchronous ground motion. The
276
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dimensions of the tallest section of the dam and the selected ®nite element model of dam±reservoir±foundation system are shown in Fig. 1. The reservoir length (3H = 255 m) was taken three times of the reservoir height (H) [18]. It was assumed that the reservoir has constant depth. The depth of the foundation and the length, which is on the part of downstream of foundation, were taken as many as reservoir height. The dam itself with foundation block and the reservoir were represented by 51 eight-noded and 24 nine-noded isoparametric quadrilateral ®nite elements, respectively. The nodes representing the bottom and edges of the
foundation were ®xed while the nodes representing the extreme edge of the reservoir were free to move in the vertical direction only. Because of the fact that ¯uid was assumed to be inviscid in this study, the normal components of the displacements at the reservoir±dam and reservoir±foundation interfaces are to be continuous. This condition is accomplished by using short and axially almost rigid truss elements in the normal direction of the interfaces. In the analysis, the plane strain assumption was employed. The dam and foundation material were assumed to be linear-elastic, homogeneous and isotro-
Fig. 3. Corrected ground acceleration record and displacement of north±south (N±S) component of the 13 March 1992 Erzincan, Turkey, earthquake.
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277
Fig. 4. Hydrodynamic pressure envelopes.
pic. The elasticity modulus, mass density and Poisson's ratio of the dam are taken as 35 109 N/m2, 2447 kg/m3 and 0.15, respectively. Elastic properties of the foundation were chosen the same as those of the dam. As for the reservoir, the bulk modulus (b0), mass density and unit weight are taken as 207 107 N/m2, 1000 kg/m3 and 9810 N/m3, respectively. Numerical problems as zero-energy modes occur in the modal analysis of ¯uid systems when the Lagrangian approach is used. The use of reduced of integration techniques and the introduction of the rotational constraints in the formulation of the ¯uid element stiness eliminates all unnecessary zero-energy modes. The rotation constraint parameter is chosen as 100 times of the bulk modulus of ¯uid [14]. The ¯uid element matrices were computed using 2 2 reduced integration orders [14]. For the solid elements, 3 3 normal integration orders were utilized. The selection of the mode number in the modal analysis of coupled system based on Lagrangian approach is very important. Because, the sloshing, volume change and rotational modes occur when the Lagrangian approach is used in the analysis. The number of surface sloshing modes which depends on ®nite element model of the reservoir becomes very much. The eect of these modes which form the ®rst ranges in the frequency table is a negligible level in the response of the dam. Because of that, the ®rst 30 modes were considered in this study [10]. The base of the coupled system was divided into the four regions for the asynchronous dynamic analysis as indicated in Fig. 1b. The lengths of the 1st, 2nd, 3rd and 4th regions vary between 0±105 m, 105±223.125 m, 223.125±335.75 m and 335.75±408.25 m, respectively. It is accepted that the amplitudes and frequency contents of the ground motion do not change along these regions, and the propagation speed of the seismic wave
is constant in all regions. Arrival time to each region was calculated using the relation of lr/Vs, where lr is the distance which varies from the reference point to the initial of each region; Vs is the propagation velocity of the seismic wave. For instance, for the wave velocity of 250 m/s, the arrival times to each region are 0.0, 0.42, 0.8925 and 1.343 second, respectively. The arrival times are taken into account as 0.0 second for all regions when the wave velocity is assumed in®nite. This situation corresponds to classical dynamic analysis. The ground displacement shape vectors (r vectors) of the coupled system are obtained for each region [17]. The penalty method [15] is used in the calculation of the r vectors as also mentioned in the section of formulation. When the r vectors are added to each other, unit horizontal displacements occur at every nodal point of the system. This event corresponds to rigid body motion in horizontal direction. If the summation of r vectors is not equal to unit value or close to unit value, incorrect displacements and stresses will be obtained. The ®ltering as well as the base-line correction routine has to be applied on the ground acceleration record. Otherwise, incorrect and residual ground displacements can occur after double integration [17, 19]. This procedure is not important in the classical dynamic analysis. But, it is important in the asynchronous dynamic analysis. Because, the ground displacements obtained form the double integration of the earthquake ground acceleration record are used in the calculation of the quasi-static displacements. Corrected ground acceleration records and displacements, which were obtained from the double integration of the accelerations, of the east±west (E±W) and north±south (N±S) components of the 13 March 1992 Erzincan, Turkey, earthquake to be used in the calculation of
278
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hydrodynamic pressures are given in Figs. 2 and 3, respectively. 3.1. Hydrodynamic pressures Hydrodynamic pressures were calculated using the time interval of 0.01 second. A damping value of 5% was taken into account for all modes. The east±west (E±W) component of the 13 March 1992 Erzincan, Turkey, earthquake is taken as ground motion (Fig. 2).
Earthquake wave velocities are chosen as 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressure envelopes were plotted using the absolute maximum values at Gauss points which are in the ¯uid elements near the face of the reservoir±dam interaction. The results of the MULSAP [12] and SAP IV [13] programs are compared to each other. The results obtained from the program MULSAP [12] using in®nite wave velocity should be equal or close to those of the program SAP IV [13], since using in®nite wave vel-
Fig. 5. Time histories of the hydrodynamic pressures at Gauss point P.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
ocity in the asynchronous dynamic analysis corresponds to classical dynamic analysis. The hydrodynamic pressures obtained from the program MULSAP [12] using in®nite wave velocity, and those of the program SAP IV [13] are depicted in Fig. 4. Hydrodynamic pressures calculated from the two programs are identical as seen in Fig. 4. Time histories for the wave velocities of 2000 m/s and in®nite were depicted at Gauss point P (Fig. 1b) to investigate the eect of the asynchronous ground motion on the frequency contents of the hydrodynamic pressures (Fig. 5). As seen from Fig. 5, the asynchronous ground motion aects the frequency contents of the hydrodynamic pressures slightly. But, the frequency contents of the stresses in the dam and foundation change considerably, and the amplitudes increase when considered asynchronous ground motion [17]. Quasi-static displacements cause to change the frequency contents and to increase amplitudes of the stresses [17]. However, the eects of the quasi-static displacements on the frequency contents of hydrodynamic pressures are very small as seen in Fig. 5. Asynchronous horizontal ground motion decreases the dynamic displacements as mentioned in reference [17, 20]. This situation is observed on the hydrodynamic pressures envelopes clearly (Fig. 6). Fig. 6 was plotted using the absolute maximum hydrodynamic pressures obtained for the velocities of 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressures decrease considerably when the wave velocity decreases. 3.2. The in¯uence of ground motion on hydrodynamic pressures The north±south (N±S) component of the 13 March 1992 Erzincan, Turkey, earthquake (Fig. 3) is selected to see the in¯uence of dierent ground motion on the hydrodynamic pressures. The north±south (N±S) component was applied to dam±reservoir±foundation system in horizontal direction. At the Gauss point P of the coupled system, the time histories of the hydrodynamic pressures were plotted using the propagation wave velocities of 2000 m/s and in®nite as shown in Fig. 7. Frequency content of the hydrodynamic pressure changes very little as in the case of east±west (E± W) component. The absolute maximum hydrodynamic pressures on the upstream face of the dam are depicted for the wave velocities of 250 m/s, 2000 m/s and in®nite in Fig. 8. Although the dierent earthquake record was used, the hydrodynamic pressures decrease when the wave velocities decrease. The hydrodynamic pressures obtained using north±south (N±S) component is smaller than those of the east±west (E±W) component. This situation arises from the values of the acceleration of the north±south (N±S) component. Because, the
279
Fig. 6. Hydrodynamic pressure envelopes.
values of the north±south (N±S) component are smaller than those of the east±west (E±W) component. 3.3. The in¯uence of bulk modulus on hydrodynamic pressures The in¯uence of the bulk modulus on hydrodynamic pressures is investigated by increasing the value of the bulk modulus (b0). The values of the bulk modulus of the ¯uid were chosen as 10b0, 50b0 and 1000b0. The east±west (E±W) component of the 13 March 1992 Erzincan, Turkey, earthquake was used in the analysis. The earthquake wave velocities were selected as 250 m/s, 2000 m/s and in®nite. Hydrodynamic pressure envelopes were plotted in Fig. 9 taking the above mentioned values of the bulk modulus. Hydrodynamic pressures decrease for all velocities when the bulk modulus of the ¯uid increases. Hydrodynamic pressures obtained for 50b0 and 1000b0 are similar to each other which are shown in Fig. 9 b and c. This situation has shown that water approximates to the case of incompressibility after a certain value of the bulk modulus. 4. Conclusions In this paper, hydrodynamic pressures occurring on the upstream face of the dam due to asynchronous ground motion were calculated for various cases. Dam±reservoir±foundation interaction was taken into account in the analysis. Formulations based on the Lagrangian approach were obtained using ®nite element method. Asynchronous ground motion in¯uences the frequency contents of the hydrodynamic pressures insig-
280
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Fig. 7. Time histories of the hydrodynamic pressures at Gauss point P.
A. Bayraktar, A.A. DumanogÆlu / Computers and Structures 68 (1998) 271±282
Fig. 8. Hydrodynamic pressure envelopes.
ni®cantly. However, the amplitudes of the pressures decrease considerably when earthquake wave velocity decreases. When the dierent earthquake record is considered, hydrodynamic pressures decrease as wave velocity decreases. Increasing the value of the bulk modulus decreases hydrodynamic pressures. However, with the increase of the bulk modulus beyond certain limit, hydrodynamic pressures do not change appreciably.
References [1] Westergaard HM. Water Pressures on Dams During Earthquakes. Trans. ASCE 1933;98:418±33. [2] Zangar CN. Hydrodynamic Pressures on Dams Due to Horizontal Earthquake Eects, United States, Bureau of Reclamation, Engineering Monograph, No. 11, 15 pages., 1952. [3] Zienkiewicz OC. Hydrodynamic Pressures Due to Earthquakes. Water Power 1964;6:382±8. [4] Chopra AK. Hydrodynamic Pressures on Dams During Earthquakes. J. Eng. Mech. Div. ASCE 1967;93:205±23. [5] Chakrabarti P, Chopra AK. Hydrodynamic Pressures and Response of Gravity Dams to Vertical Earthquake Component. Earthquake Eng. Struct. Dyn 1973;1:325± 35. [6] Paul DK, Zienkiewicz OC, Hinton E. Hinton E, Bettess P, Lewis RW, editors. Transient Dynamic Analysis of Reservoir±Dam Interaction Using Staggered Solution Schemes, Numerical Methods for Coupled Problems. Pineridge Press, Swansea, U.K., 1981:321±334. [7] Humar J, Roufaiel M. Finite Element Analysis of Reservoir Vibration. J. Eng. Mech. ASCE 1983;109:215± 30. [8] Greeves EJ, Dumanoglu AA. The Implementation of an Ecient Computer Analysis for Fluid±Structure Systems
Fig. 9. Hydrodynamic pressure envelopes.
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[15] Zienkiewicz OC, Taylor RL. The Finite Element Method. Vol. 1, Fourth Edition, McGraw-Hill Book Company, London, 1989. [16] Clough RW, Penzien J. Dynamics of Structures. McGraw-Hill Book Company, Singapore, 1975. [17] Bayraktar A. Dynamic Response of Dam±Reservoir± Foundation Systems Subjected to Asynchronous Ground Motion. Ph.D. Thesis, Department of Civil Engineering, Karadeniz Technical University, Trabzon, Turkey (in Turkish), 1995. [18] Newmark NM, Rosenblueth E. Fundamentals of Earthquake Engineering. Prentice-Hall Inc., Englewood Clis, New Jersey, 1971. [19] Erdik M. Processing of Strong Motion Accelerograms in Earthquake Engineering: Corrected Accelerogram, Velocity and Displacement Traces. METU Journal of Pure and Applied Sciences 1979;12:277±304. [20] Priscu R, Popovici A, Stematiu D, Stere C. Earthquake Engineering for Large Dams. Second Edition, Editura Academiei, Bucuresti, 1985.