## Design of apron slabs

It is common to place a concrete slab (i.e. an ‘apron’) below a dam structure to protect against scour or to form a basin to control flows. Design of this apron has to account for various hydraulic pressures, which I consider below in three categories: 1. static 2. kinetic and 3. kinetic-dynamic. Consider the case of Figure 1:

Using the water level line to define hydrostatic pressures along the ground surface, we can run a seepage analysis (Figure 2) and estimate the static pressures acting on the apron (Figure 3)

If there were an open joint at the upstream end of the apron (ie along line B), then pressures would also develop here. Considering Figure 3, static pressures would cause flow through this joint, and causing a new static distribution. Leaving this aside for now, lets turn to the injection of velocity into this joint such as shown in Figure 4 – ie kinetic and dynamic pressures.

When water with high velocity strikes a bluff object is it slowed (and/or turned) giving rise to a pressure in the same way that a moving tennis ball impacting a wall imparts force on the wall. This is known as form drag and arises where kinetic energy (½mv^{2} in physics, but in hydraulics we favour the symbol ‘u’ for velocity (its a 3D thing) and divide by mass and gravity to say u^{2}/2g – units of length which call velocity ‘head’) being transformed to potential energy (mgh in physics, but divided by mass and gravity we say pressure head ‘h’ in hydraulics). That is, kinetic pressure head ‘h_{k}‘ arises from conversion of velocity head u^{2}/2g. The amount of velocity head converted to kinetic pressure head depends on geometry so we say h_{k}=C_{P}u^{2}/2g, where C_{P} is a dimensionless ratio (ie C_{P} is a ‘coefficient of pressure’ – saying ‘how much was converted’).

The distinction I draw between ‘kinetic’ and ‘dynamic’ pressures relates to velocity. We know that fast flowing water is turbulent, and turbulence is characterised by the water having a concoction of internal velocities. To make life simple, we time-average these velocities to get the mean velocity u (the overline symbolises that it’s time-averaged). If we use this time-averaged velocity u we can estimate a time-averaged kinetic pressure head:

h_{k} = C_{P} u^{2}/2g (Equation 1)

We note that the C_{P} ratio here has an overbar, and we use it when we are using the time-averaged velocity to get the time-averaged kinetic pressure head. The value of C_{P }has been examined in many experimental studies and these give guidance based on various geometric conditions. If the upper edge of the apron in Figure 4 protrudes a small amount into the flow, C_{P} might be around 0.1 or 0.2, and if a large amount into the flow perhaps 0.5 or more. If there is no protrusion, C_{P} can be zero (examples of such testing can be found in Part II of my PhD).

Regarding non-time-averaged velocities, they are chaotic, but it is common to state that at any instant the velocity u is equal to the sum of a ‘mean’ component u and a fluctuating component u’ (ie u=u+u’). We don’t know what u’ is really, but we can observe it in experiments, and find that its fluctuations follow a standard deviation, and that the standard deviation increases with increased turbulence. The ratio of the standard deviation of fluctuations σ_{u’} to the mean velocity u is called the turbulence intensity T_{u}. ie

T_{u} = σ_{u’} / u = u_{RMS} / u

Turbulent intensity is normally a few percent, but may be around 5% or more for high energy flows such as in spillways (eg Ervine, D.A., H.T. Falvey, and W. Withers. “Pressure Fluctuations on Plunge Pool Floors.” *Journal of Hydraulic Research* 35, no. 2 (1997): 257–79. https://doi.org/10.1080/00221689709498430) – that is, the standard deviation of fluctuations is a relatively small fraction of the mean velocity. By inference, then, the fluctuating kinetic pressures are a fraction of the kinetic pressures.

In my PhD studies I measured pressure fluctuations in various experimental setups. I found they followed a standard deviation and I presented a ‘coefficient of fluctuation’ ‘C_{P,σ}‘, and therefore a way to calculate σ_{hk} following the form of Equation 1:

σ_{hk} = C_{P,σ} u^{2}/2g (Equation 2)

In these experiments, C_{P,σ} was generally a few percent, but up to 20% in the most adverse conditions. Again, the fluctuating kinetic pressures are a fraction of the kinetic pressures. For example, interpreted kinetic and dynamic pressures on the apron for the case in Figure 1 are presented in Figure 5.

It is my observation that, in industry at present, much attention is directed towards the fluctuating component when designing aprons. Fluctuations can cause vibrations and fatigue and so I agree that it is important to consider them. Some have argued can fluctuations can be subject to amplification and resonance in certain geometries, and I also understand that Figure 5 is a simplified representation. Nonetheless, in my view fluctuations are often assigned a mystical authority over the design, although it is illustrated in Figure 5 that such fluctuations may only be a small part of the design. My concern is that sole focus on fluctuations can cloud important issues, and that often the key matter is detailing, as per the two examples below.

A commonly used apron slab design equation is presented below and, in simple terms, balances kinetic pressures against self-weight of the slab. The shape factor Ω accounts for the various patterns of kinetic pressures above and below the slab for a given slab geometry, and is based on experimental tests such as shown in the figure above (taken from Bellin, Alberto, and Virgilio Fiorotto. “Direct Dynamic Force Measurement on Slabs in Spillway Stilling Basins.” *Journal of Hydraulic Engineering* 121, no. 10 (October 1, 1995): 686–93. https://doi.org/10.1061/(ASCE)0733-9429(1995)121:10(686) ).

The equation is very useful as both the mean and dynamic aspects of kinetic pressures are woven into the shape factor based on experiments, but it can be seen that the experimental setup cannot account for seepage pressures or other site-specific drainage details. In design examples, Barjastehmaleki et al 2016 use CP+ =CP- = 0.5, and for slab section dimensions of 5m by 5m gives Ω~0.1, giving s~0.5 for the case in Figure 1, inferring a hydraulic uplift of around 12 kPa. However, in this case, the uplift from static head alone (considering the difference between upper and lower profiles in Figure 3) is more than 20 kPa.

The failure of slabs at Karnafuli dam in Bangladesh in 1961 (the photo at the head of this article, taken from Bowers and Toso, 1988) is an example that, in my observation, is heralded as a reason to focus on fluctuating pressures. In this case, the hydraulic jump was located upon the lower slab, and research does indicate that fluctuations under a hydraulic jump can be large – measurements in Bowers, C. Edward, and Joel Toso. “Karnafuli Project, Model Studies of Spillway Damage.” *Journal of Hydraulic Engineering* 114, no. 5 (May 1, 1988): 469–83. https://doi.org/10.1061/(ASCE)0733-9429(1988)114:5(469) demonstrate this. However, another key matter in this case study is the detailing of under-slab drainage – its intent was to drain pressures from the under the chute slab – but its effect was to allow pressure from the tailwater to be directed back up under the chute slab. The fluctuations in the case of Karnafulli is understood to relate to reshuffling of the hydraulic jump (i.e lateral repositioning of the static pressure under the tailwater), rather than stagnation of u’ into kinetic pressures. These large pressure changes due to hydraulic jump positioning may be transferred up the drainage system, leading to a large differential between the under-slab and above-slab pressures. In fact, examination of this detail shows that large pressure differentials could also exist in the absence of any fluctuations (Figure 6)