# Hydrodynamic pressures

20 September 2005Dr Martin Wieland investigates hydrodynamic pressures in penstocks, bottom outlets, plugs of diversion tunnels and gates of intake structures and spillways caused by earthquakes

Penstocks, plugs of diversion tunnels, gates in bottom outlets, intake structures and spillways, valves and other hydromechanical components of the pressurised water system in hydro power plants have rarely been designed for the hydrodynamic pressures, which may be caused during a strong earthquake. In general the hydrodynamic pressure according to Westergaard is assumed. This assumption is acceptable for gates of surface spillways, but not for gates located in tunnels or valves in large diameter penstocks.

Although no case is known where a penstock has failed or been damaged during a strong earthquake, the question of earthquake safety still has to be addressed. As earthquakes affect all components of a hydro power plant, hydrodynamic actions also have to be checked for all hydromechanical components. It may be argued that in the pressurised water system the water hammer is already investigated and the emergency shut-down of a penstock may cause the maximum hydrodynamic pressures. As discussed in the subsequent sections, the hydrodynamic pressures due to valve regulation are larger than those due to earthquakes in long penstocks or pressure tunnels with a natural period of vibration of the water of several seconds. However, in relatively short penstocks with fundamental eigenfrequency of the oscillating water mass in the range of the dominant frequencies of the earthquake ground motion, the situation may be the opposite.

Damage and failure of buried water and gas mains has been caused by quite a number of earthquakes. The vulnerability of these lifelines is not a new phenomenon. However, the implications of these observations may not yet have been fully realised elsewhere. Penstocks and gates of hydro power plants are not intrinsically safe against earthquakes. Earthquakes can cause damage to penstocks by different mechanisms as discussed below:

• Hydrodynamic pressures due to ground shaking: The magnitude of the pressure depends on the response spectrum of the ground shaking and the natural period of the oscillating mass in the pressurized water system.

• Relative displacements along active faults.

• Support movements due to earthquake-triggered slides.

• Rock falls on penstocks on the surface in steep terrain.

• Dynamic forces in penstock due to support movements and deformations of rigid anchor points.

• Dynamic soil pressures in buried penstocks; imposed (quasi-static) deformations due to soil liquefaction, differential settlements and creep movements of slopes, etc.

It is evident from the above list that the seismic actions, which can cause damage to a penstock, cover a wide spectrum. To ignore the earthquake action because it may not have been considered in the past would not be an acceptable argument. Also, to simply declare the earthquake action on a penstock as an acceptable risk, does not comply with today’s state of knowledge practice, particularly in view of the fact that rather sophisticated dynamic analyses are already carried out for the seismic design of dams.

In mountainous regions the main hazard is from rock falls and mass movements. For example, during the 1999 Chi-Chi earthquake in Taiwan (magnitude 7.5) and the 1990 Manjil earthquake in Iran (magnitude 7.6) several thousand rock falls have occurred (Figs. 1 and 2). Therefore, rock falls and mass movements have to be expected and appropriate action shall be taken in the design of penstocks (soil cover, concrete protection, pressure shaft instead of surface penstock etc.).

Fault crossings may require special solutions, as shown in the examples in Figure 3, where the Trans Alaska pipeline is crossing the Denali fault. During the magnitude 7.9 Denali earthquake of 3 November 2002, significant horizontal fault movements occurred at this location. Due to the sliding bearings, the fault movements could be absorbed without major damage and with a very short interruption of the pipeline operation (Sorenson and Meyer, 2003).

Besides the penstocks, the gates of spillways, bottom outlets, intakes etc. can also be affected by high hydrodynamic forces. This is the case when the gate is located in a pressure tunnel or shaft. For example, during the 1990 Manjil earthquake in Iran, one of the two radial gates of the intermediate level spillway of the Sefid Rud buttress dam was damaged due to hydrodynamic forces resulting from a mass oscillation of the relatively short intake tunnel. One of the arms of the radial gates buckled and one of the trunnion plates cracked (Jalalzadeh, 2000). Due to the deformations, the damaged gate could not be operated after the earthquake and was leaking.

So what should be done in order to improve the seismic safety of a penstock? First, it is necessary to identify the type of seismic hazard (see list above) and the weak elements in a penstock.Such checks are not very costly and it is worthwhile to perform them in the context of an overall safety assessment involving other hazards from the natural and man-made environment.

In order to minimise the potential damage in the case of failure of a penstock, the proper functioning of valves during and after an earthquake must be guaranteed. Thus, the seismic safety of these devices shall be carefully checked and assessed.

The rock fall and landslide hazard is best assessed together with an experienced geologist, who is familiar with the local conditions. The best protection of a penstock against rock falls is by burying the penstock.

Anchor points in potentially unstable slopes need special solutions. Of course, the best solution would be to avoid such locations. However, for high head penstocks in the mountains, the slopes are generally very steep with slope angles up to 45°, which exceeds the friction angle of the slope wash or the weathered rock, therefore, the slope stability of supports is a general problem. Earthquake damage can be accepted when, due to the release of the water in the penstock, people are not affected and damage to the environment and property is limited and the economical losses can be covered by insurance. Earthquake alarm systems, which allow the automatic shut-down of a penstock and the installation of additional valves in long penstocks may be more effective than designing a penstock for the worst seismic actions (Griesser et al., 2004).

In the subsequent sections the earthquake-induced hydrodynamic pressures in penstocks are discussed, which for low head schemes can greatly exceed the hydrostatic pressures. These pressures may not only jeopardise the safety of the penstock but also the gates and valves located in relatively short intake tunnels and penstocks, respectively. Moreover, the plugs in closed diversion tunnels may experience very high hydrodynamic forces during an earthquake.

**Hydrodynamic pressures in penstocks due to earthquake action**

The hydrodynamic pressures in penstocks can be characterized by the one-dimensional wave equation, which can be expressed as follows:

(1)

where c is the wave propagation velocity of a pressure wave in an elastic pipe (depending on the flexibility of the pipe and the embedment, c varies between ca. 1000 to1300m/s), subscripts ’t’ und ’x’ denote partial derivation with respect to time and space, x is the coordinate along the pipe axis and u(x,t) is the time-dependent displacement of the elastic fluid in direction of the pipe axis.

In case of an earthquake ground acceleration üg(t) in direction of the pipe axis, the relative acceleration utt has to be replaced by the total acceleration, which is the sum of the ground acceleration and the relative acceleration, i.e. the wave equation takes the following form:

(2)

The hydrodynamic pressure, p(x,t), can be obtained from the following relationship, assuming the fluid in the pipe being an elastic material:

(3)

where E = ? c^{2} represents the equivalent modulus of elasticity of the water and the elastic pipe and r is the mass density of water (1000 kg/m^{3}).

In a penstock with the length L, which is closed at one end and terminates at the surface of a reservoir or surge chamber, the hydrodynamic pressure due to a ground excitation can be determined with the mode superposition method (Clough and Penzien, 1975). Accordingly, the modes of vibration X_{n}(x) (subscript n denotes the n-th mode) can be expressed as follows (note: x = 0 at the free surface end of the pipe):

(4)

The modes of vibration satisfy the following orthogonality relationship:

(5)

Using u(x,t) = ? X_{n}(x) u_{n}(t) we obtain from equation 4

(6)

where u_{n}(t) represents the n-th modal coordinate.

By substituting equation 6 into equation 2, multiplication with Xn(x), introduction of a viscous damping term and integration along ther whole pipeline, the following equations of motion result:

(7)

where ?n: n-th damping ratio

?n: n-th circular frequency of oscillating fluid in pipe in rad/s

(8)

d_{n}: n-th modal partizipation factor,

(9)

üg(t): ground acceleration in direction of pipe axis.

The eigenfrequencies of the pressure system are as follows:

(10)

The fundamental frequency (n=1) can be taken as f_{1} = c/(4L).

The maximum of u_{n}(t) can be obtained directly from the acceleration response spectrum, S_{a}(f_{n}, ?_{n}), i.e.

(11)

Using equation 3 the hydrodynamic pressure at the fixed end (x = L) of the n-th mode of vibration can be obtained in an analogous way.

(12)

By substituting equation. 11 and the expressions for f_{n} and d_{n} in equation 12, we obtain the following maximum n-th pressure at the fixed end of the pipe:

(13)

The total hydrodynamic pressure at the closed end of the pipe can be obtained by using the the following standard superposition method (SRSS-method) for the modal maxima:

(14)

If only the contribution of the first mode is taken into account then we obtain the following maximum hydrodynamic pressure at the fixed end:

(15)

In case of a pipe (or tunnel) filled with an incompressible fluid where the flexibility of the pipe is not accounted for, we obtain

(16)

where ao is the peak ground acceleration. This means the total mass in the pipe (or tunnel) behaves like a rigid body and the maximum inertia force is equal the total mass of water in the pipe of length L multiplied with the peak ground acceleration.

The ratio between the maximum hydrodynamic pressures of the compressible and incompressible fluid, characterized by K, can be determined as follows:

(17)

If the design response spectra shown in Fig. 5 are used, then the maximum spectrum amplification factor for the absolute acceleration s = Sa,max/a0 varies between 2.5 (soil type A: rock) and 3.5 (soil type E) for a damping ratio of 5%. However, the damping ratio of the pressure oscillations in a penstock is usually less than 5%. For seismic design purposes a damping ratio of about 2% may be assumed. The effect of the damping ratio on the elastic acceleration response spectra can be approximated by the following correction factor (Eurocode 8, 2004):

(18)

The response spectra for 5% damping have to be multiplied by the above factor for periods exceeding 0.15 s (soil type A). In the case of an undamped pressure oscillation, we obtain h = 1.414. Therefore, for a rock foundation (s = 2.5) we obtain a maximum K-value of 2.9 (equation 17) for the undamped penstock. This means that the maximum hydrodynamic pressure in a compressible liquid in a penstock can be more than twice as large as in an incompressible one.

In the case of soil type E (s = 3.5) we obtain a maximum value of K of 8x3.5x1.41/3.142 = 4.0.

If a pressure tunnel is located in the body of a massive concrete dam (e.g. orifice spillway in arch dam) then the hydrodynamic pressure must be calculated using the so-called floor response spectrum at the location of the pressure tunnel rather than the acceleration response spectrum of the ground motion.

**Example 1: 1000 m long penstock**

Determine the maximum hydrodynamic pressure in a 1000 m long penstock with the following properties: rock site with a peak ground acceleration of 0.3 g (3m/s2); wave propagation velocity in the elastic pipe of c=1100 m/s, damping ratio of 0%, mass density of water of 1000 kg/m3. The fundamental period of vibration according to equation 10 is T1 = 1/f1 = 4L/c = 3.64 s. The maximum hydrodynamic pressure of the first mode of vibration is obtained from equation 15 using the response spectrum shown in Figure 5 for soil type A and a vibration period of 3.64 s (spectrum value: 0.264 for 5% damping). For an undamped pressure oscillation the 5% damped response spectrum has to be multiplied by h = 1.414 according to equation 18. Therefore, we obtain

p_{1,max} = 8 x 1000 kg/m^{3} x 1000 m x 1.414 x 0.264 x 3 m/s^{2} / 3.14^{2} = 0.91 MPa or 9.1 bar

This maximum hydrodynamic pressure is independent of the head or the average slope of the penstock. Therefore, for high head penstocks with a hydrostatic pressure of several MPa’s, a maximum dynamic pressure of about 1 MPa is not a serious concern. However, for low head penstocks with static pressures in the order of the dynamic pressure – this applies mainly to pipes in relatively flat terrain – the hydrodynamic pressure has to be checked carefully. For penstocks with a length of say 3000m the natural period of vibration is three times that of the 1000 m long one and increases to 10.9 s. For this period the spectrum acceleration drops to roughly 1/9 of that of the 1000m long pipe as the tail of the response spectrum is proportional to one over the square of the period. This results in a hydrodynamic pressure of ca. 0.1 MPa or 1 bar.

The maximum hydrodynamic pressure of the second mode of vibration (n=2) with a period of vibration of T2 = 4L/(3c) = 1.212 s can be determined from equation 13 (spectrum value for 1.212 s for soil type A in Fig. 5: 0.825):

p_{2,max} = 8 x 1000 kg/m^{3} x 1000 m x 1.414 x 0.825 x 3 m/s^{2} /(9 x 3.142) = 0.32 MPa (3.2 bar)

Similarly, for the third mode of vibration we obtain (period of vibration: 0.727 s, spectrum value: 1.375):

p_{3,max} = 8 x 1000 kg/m^{3} x 1000 m x 1.414 x 1.375 x 3 m/s^{2} /(25 x 3.142) = 0.19 MPa (1.9 bar)

From these numerical values it can be seen that the maximum hydrodynamic pressures decrease with increasing mode number despite the fact that the response spectrum is still increasing for periods down to 0.4 s. This is due to the fact that the hydrodynamic pressures are inversely proportional to (2n-1)2, where n is the mode number.

Finally the total hydrodynamic pressure has to be calculated using the SRSS superposition of the modal maxima according to equation 14. Thus for the first three modes we obtain:

p_{tot} = (0.91^{2} + 0.32^{2} + 0.19^{2})^{1/2} = 0.98 MPa (9.8 bar)

By comparing the total hydrodynamic pressure of the first three modes with that of the first mode we can note that the total pressure is only 8% larger than that of the first mode contribution. In view of the different assumptions made, this is a negligible difference. Therefore, for most applications it is sufficient to include only the contribution of the first mode.

**Example 2: 130m long pressure tunnel**

Determine the maximum hydrodynamic pressure in a 130m long pressure tunnel with the following properties: rock site with a peak ground acceleration of 0.3g (3m/s2); wave propagation velocity in the elastic pipe of c=1300m/s, damping ratio of 5%, mass density of water of 1000kg/m3. The fundamental period of vibration according to equation 10 is T1 = 1/f1 = 4L/c = 0.4 s. The maximum hydrodynamic pressure of the first mode of vibration is obtained from equation 15 using the response spectrum shown in Figure 5 for soil type A and a vibration period of 0.4 s (spectrum value: 2.5 for 5% damping). Therefore, we obtain

p_{max} = 8 x 1000 kg/m^{3} x 130 m x 2.5 x 3 m/s^{2} / 3.14^{2} = 0.79 MPa (7.9 bar)

The maximum hydrodynamic pressure is of the same order as that of the 1000m long penstock in Example 1. If zero damping would be assumed then the maximum hydrodynamic pressure would reach 11.2 bar, which is more than that in Example 1. For short penstocks with small head or gates and plugs in relatively short tunnels, this dynamic pressure is quite important. The example is selected in such a way that for the given response spectrum the hydrodynamic pressure would become a maximum, i.e. if the pressure tunnel is longer or shorter than 130m then the hydrodynamic pressure will decrease.

The contributions of the higher modes of vibration play a smaller role in this example than in Example 1 as the spectrum values of the rock response spectrum do not increase further if the periods of vibration decrease to below 0.4 s (in Example 2 the spectrum values of the 2nd and 3rd modes were larger than those of the 1st mode)..

In an incompressible pipe or tunnel the maximum hydrodynamic pressure acting on a gate or valve is given by equation 16. Accordingly in Example 1 we get a maximum dynamic pressure of

p_{max} = 1000 kg/m^{3} x 1000 m x 3 m/s^{2} = 3 MPa (30 bar)

and in Example 2 we obtain

p_{max} = 1000 kg/m^{3} x 130 m x 3 m/s^{2} = 0.39 MPa (3.9 bar).

We have to keep in mind that the slope of a penstock or pressure tunnel does not have any effect on the maximum hydrodynamic pressure caused by an earthquake.

Furthermore, during pressure oscillation no negative pressures are possible. If such pressures are calculated based on the linear analysis presented above then nonlinear processes would change the nature of the oscillations. These phenomena are not discussed in this paper.

**Conclusions drawn from numerical examples**

Based on the above illustrative numerical examples the following conclusions may be drawn regarding the hydrodynamic pressures:

• The hydrodynamic pressures in short penstocks (typical situation: dam with power house at the base of the dam, see Figure 6) and short pressure tunnels can be quite high during strong earthquake shaking. In this case resonance-like pressure oscillations can develop under earthquake shaking, i.e. the lowest eigenfrequency of the pressurised water system is in the range of the dominant frequencies of ground shaking. In this case the assumption of an incompressible fluid leads to an underestimate of the hydrodynamic pressures.

• In penstocks with a length of several hundred meters, the lowest eigenfrequency of the pressurized water system is far below the dominant frequencies of ground shaking. In this case the maximum hydrodynamic pressures in a compressible fluid are below those of an incompressible fluid.

• The contributions of higher modes of vibration of the pressurized water system are relatively small and may be neglected.

• The maximum hydrodynamic pressures depend on the shape of the acceleration response spectrum. In the case of long penstocks the fundamental period of vibration is of the order of several seconds, which is beyond that of most buildings and bridges. Thus spectrum values result, which are rather low. Moreover, the design response spectra given in building codes may be rather inaccurate in that range of periods. Therefore, the design response spectra used for such analyses have to be reviewed carefully.

**Design of penstocks and gates in pressure tunnels **

In the seismic design and safety check of penstocks and gates and valves in pressure tunnels, the following items have to be taken into consideration:

• The design earthquake (DE) or safety evaluation earthquake (SEE) occurs very seldom, i.e. every 475 years when the DE is used. However, for safety-relevant elements like the bottom outlet, which must function after an earthquake, the SEE with a return period of say 10,000 years has to be taken into account.

• The high hydrodynamic pressures act during a relatively short period of time, i.e. for a sinusoidal oscillation with a period of 5 s the duration of high dynamic pressures is roughly 2 s in each cycle.

• In the case of extreme events with very low probability of occurrence average strength values may be taken into account together with load and safety factors of 1.0 if no specific regulations and guidelines exist for such design situations.

• Local plastification in ductile steel elements is acceptable as long as the safety-relevant gates are still operable after the earthquake. This applies mainly to structural elements subjected to high flexural stresses where, e.g., a small plastified zone in a flange does not cause any significant deformations, which would impair the serviceability of a gate.

Using the above concepts – if applicable - then penstocks or gates, which have been designed for a safety factor of 1.8 should also be able to cope with the increased stresses due to earthquake-induced hydrodynamic pressures. In any case a check of the seismic stresses and deformations is necessary.

**Conclusions**

Penstocks and other elements of the pressurised water system in hydro power plants have seldom been designed for earthquake action.

Earthquakes can cause substantial hydrodynamic forces in penstocks and pressure tunnels. Depending on foundation conditions (soil type) and length of a penstock or pressure tunnel the maximum hydrodynamic pressures are up to 1 MPa for a peak ground acceleration of 0.3g. These dynamic pressures are not a problem for high head penstocks in steep terrain as they amount to less than 20% of the hydrostatic pressures in a penstock with a head of 500m. However, in pipes in rather flat terrain with low head, the maximum hydrodynamic pressures can be of the same magnitude as the hydrostatic ones.

High dynamic pressures are also expected in short pressure tunnels located within the body of a dam, where at the location of the tunnel a higher acceleration response is expected than on the ground.

However, the main seismic hazard for long surface penstocks in mountainous regions results from mass movements and rock falls. In steep terrain large numbers of rock falls can be triggered by strong earthquakes. This was, for example, the case during strong earthquakes in Taiwan (1999) and Iran (1990).

The analysis procedure discussed in this paper can be used to determine the maximum hydrodynamic pressures in penstocks at the location of closed valves, the dynamic pressures acting on valves, gates and plugs installed in penstocks, pressure tunnels and diversion tunnels respectively.

Author Info:

Dr. Martin Wieland, Chairman, ICOLD Committee on Seismic Aspects of Dam Design, Electrowatt-Ekono Ltd. (Jaakko Pöyry Group), Hardturmstrasse 161, CH-8037 Zurich, Switzerland. Tel. 076 356 28 62; Fax 044 355 55 61; E-mail: [email protected]