Laboratory Measurements of Hydraulic Jacking Uplift Pressure at Offset Joints and Cracks

Either failure mode described above destroys the integrity of the chute lining, which can allow rapid erosion of underlying material. Headcut erosion following the initial failure of the chute lining at Oroville caused concern for possible uncontrolled release of the reservoir; use of the previously untested emergency spillway to minimize further damage to the main spillway also caused headcutting that prompted an evacuation order affecting about 180,000 people. Repairs and upgrades to the spillways eventually cost about $1.1 billion. Another recent failure attributed to hydraulic jacking was the fifth drop structure on the St. Mary Canal in Montana in May 2020, which was taken out of service for the entire 2020 irrigation season while two of the five drop structures on the project were rebuilt (Irrigation Leader 2020, 2021). Other recent uplift failures include spillways at Eping Dam in China (China Observer 2021), Bukan Dam in Iran (Bahramifar et al. 2022), Toddbrook Reservoir Dam in the United Kingdom (Hughes 2020; Mauney undated), and Rhymney Bridge 2 reservoir in Wales (Hughes and Williamson 2014). Other documented historic failures include Big Sandy Dam spillway (Wyoming) in 1983 and Dickinson Dam spillway (North Dakota) in 1954 (Hepler and Johnson 1988; Trojanowski 2004; Frizell 2007). A near-failure of the spillway at Hyrum Dam (Utah) (Trojanowski 2008) occurred when flow through open joints created large voids below the slab, and a similar near-failure incident at Fairbairn Dam in Australia took place in 2011 (Foster et al. 2016), although the Fairbairn case may have been related more closely to stilling basin pressure fluctuations than uplift at offsets within the chute. Masonry-lined spillways are also prone to uplift failures as demonstrated by cases in the U.K. at Ulley Dam in 2007 (Hinks et al. 2008), Boltby Dam in 2005 (Charles et al. 2011), and Butterly Dam in 2002 (Chesterton et al. 2018). Given these recent failures, rehabilitation projects and designs that can prevent uplift failure are being actively discussed in recent literature (Gilbert et al. 2023; Lux et al. 2023; Smith et al. 2023; Vensel et al. 2023).

The typical thickness of concrete spillway slabs has increased over time, ranging from about 0.2 to 0.3 m (0.75 to 1.0 ft) before the 1960s to about 0.3 to 0.61 m (1 to 2 ft) more recently (personal communications with practicing spillway engineers: Patrick Maier, Bureau of Reclamation; John Trojanowski, Trojanowski Dam Engineering Ltd.). The weight of the slab provides a first defense against uplift failure, but even a submerged 0.61-m (2-ft) thick slab on a flat slope can be displaced by a net uplift pressure head of only 0.86 m (2.8 ft) and a slab on an incline is even more easily displaced; such uplift pressures can easily be generated by flow stagnation at an offset. Thicker slabs provide space for installing two layers of reinforcement and allow adequate embedment of anchor bars. Other defensive measures include proactive design of joints with offsets slightly away from the flow so that small differential movements can be tolerated without immediately creating dangerous offsets, keyed or doweled joints to prevent differential movement, waterstops to prevent water intrusion, anchor bars to secure slabs to the foundation, and drainage systems to alleviate pressure buildup and safely convey flow out of the foundation. Design of the latter two features requires accurate estimates of uplift pressure and flow rates through affected joints. In spillways with existing offsets, mitigation may also include beveling and grinding down offsets or modifying joints to restore sealants or waterstops.

The Oroville failure was carefully investigated by an Independent Forensic Team (IFT 2018). The 1960s-era design lacked waterstops and keyed or doweled joints, had insufficient anchorage due to age-related corrosion and weak foundation materials (anchors embedded in soil-like materials rather than competent rock), and probably had insufficient drainage. Many older spillways have similar characteristics. The IFT evaluations of anchorage and drainage systems carried significant uncertainty due to limited experimental studies of offset joint hydraulics in either laboratory or field conditions. The IFT report presented calculations of estimated stagnation pressure at offsets of various heights. Those estimates were informed by previous experimental studies and used estimated boundary layer velocities at the tip or mid-height of offsets determined with a basic open-channel flow velocity profile equation (Rouse 1945). This approach had not been experimentally verified and did not include any influence of the gap width. The IFT report also provided estimates of seepage flow through open joints of various widths based on a simple orifice equation, considering the driving head to be only the hydrostatic pressure associated with the depth of flow in the chute and assuming a fully vented condition below the slab. This approach neglected the effect of offset height and the increased driving head created by stagnation of flow at an offset.

Only two previous experimental studies of hydraulic jacking are known, both performed at the Bureau of Reclamation (Reclamation) hydraulics laboratory in Denver, Colorado (Johnson 1976; Frizell 2007). The problem has been of great interest to Reclamation because of its many older spillways and chutes constructed on soil or rock foundations. These structures are more prone to hydraulic jacking than chutes integrated into mass concrete arch or gravity dams, since the concrete spillway lining is a relatively thin veneer over a weaker and variable foundation. Hydraulic jacking failure modes are commonly considered in Reclamation’s risk assessment studies.

The study by Johnson (1976) was aimed at supercritical chutes associated with small canal systems. It used a 2.44-m (8-ft) long, 0.15-m (0.5-ft) wide open-channel laboratory flume limited to velocities of $4.6\mathrm{m}/\mathrm{s}$ ($15\mathrm{ft}/\mathrm{s}$). Documentation is limited, but the flume was probably level. Measurements of static uplift pressure and dynamic pressure fluctuations were made at a model joint with variable offset height and gap width, sealed to prevent any net flow through the joint.

Frizell (2007) used a level water tunnel supplied by a high-head pump to generate velocities up to $14.6\mathrm{m}/\mathrm{s}$ ($48\mathrm{ft}/\mathrm{s}$) approaching an adjustable joint, with tests of square, chamfered, and radius joint edges. Measurements of static uplift pressure were made, but there were no measurements of dynamic pressures. Measurements of flow rate through joints were also attempted, but most of the flows were determined indirectly with unknown backpressure conditions below the joints and are thus unreliable with little predictive value. Both studies showed that uplift pressure was directly dependent on chute velocity, and that uplift increased with the offset height and decreased with increasing gap width. The Johnson study reported uplift pressures relative to the mean velocity head in the channel but presented these dimensionless values as functions of absolute (dimensional) velocities and joint dimensions. The Frizell study reported dimensional uplift pressures as functions of joint dimensions and flow velocities. Unfortunately, these presentations of the data do not facilitate application to other scales, joint configurations, or flow conditions. Neither study measured or estimated the boundary layer flow properties in the experimental facilities, although both discussed the potential influence of the boundary layer and suggested that such measurements be included in future studies. Frizell (2007) did some limited velocity vector mapping within and above the joint for one joint configuration ($3\mathrm{mm}=1/8\mathrm{in.}$ offset with $13\mathrm{mm}=1/2\mathrm{in.}$ gap) using particle image velocimetry (PIV) and commented on the potential influence of the observed boundary layer.

Investigation of past failures has been limited until the Oroville failure, with the best early documentation provided by Hepler and Johnson (1988) for the Dickinson and Big Sandy failures. The Oroville and Dickinson failures occurred well upstream from the toe of each chute, away from any possible tailwater influences. The Big Sandy failure was closer to its stilling basin, but probably still above the zone of influence of the basin’s hydraulic jump. Wahl et al. (2019) reviewed literature on stilling basin failures in zones affected by the turbulent flow of the hydraulic jump, but this problem is driven by much different flow mechanics than mid-chute failures. Smith (1995) describes an alternate uplift failure mode at the toe of a chute where supercritical flow enters a stilling basin. In this failure mode, hydrostatic pressures created by the tailwater depth in the basin cause uplift behind the chute lining and below the toe of the chute that cannot be counterbalanced by the depth of the incoming supercritical flow. The driving forces in this case do not develop from an offset into the flow, but only from seepage through joints, cracks, or drainage systems within the basin lining. Eductors that utilize offsets away from the flow to draw water out from beneath the lining through a venturi effect can help to protect against this mode of failure (Smith 1995).

In the first phase of the present study, Wahl et al. (2019) combined and reanalyzed data from the Johnson (1976) and Frizell (2007) laboratory studies. Adjustments of the Frizell (2007) data were made to account for effects of the water tunnel experimental setup, which increased the measured pressures due to factors that would not be present in an open-channel environment. This analysis showed that the normalized uplift pressure head $\mathrm{\Delta }H/[V^2/(2g)]$ illustrated in Fig. 1 was related to the dimensionless joint aspect ratio, $\beta =s/h$, aswhere $s$ = gap or slot width; $h$ = height of the offset into the flow; $\mathrm{\Delta }H$ = increased pressure head compared to the hydrostatic pressure associated with the flow depth; $V$ = mean velocity in the chute; and $g$ = acceleration due to gravity. The normalized uplift pressure head was shown to vary, with significant scatter, from about 0.1 to 0.7 within the range $\beta =6$ down to $\beta =0.083$, increasing with decreasing $\beta$. It was difficult to discern whether smaller values of $\beta$ might lead to still larger uplift pressures, but Eq. (1) represents that trend, indicating an upper bound of $\mathrm{\Delta }H/[V^2/(2g)]\approx 0.81$ for $\beta =0$. The reanalysis also attempted to develop relations between uplift pressure and boundary layer velocity similar to those used in the Oroville forensic analysis, although neither previous study directly measured boundary layer velocities. As a result, boundary layer velocities could only be estimated for the previous studies, and the resulting relations exhibited more data scatter than Eq. (1).

Sánchez (2022) used the ANSYS Fluent computational fluid dynamics (CFD) model to simulate the experimental work of Frizell (2007) and two open-channel flow situations with 30° and 45° slopes. For sealed conditions with no flow through joints, the results matched the Frizell experimental work closely, with maximum deviations of 6%. For vented conditions the deviations were up to 15%, which may be due to the unknown backpressure conditions and joint flow rates in the Frizell study. The open-channel simulations at the two slopes yielded almost identical uplift pressures, indicating that slope is an insignificant factor in the relation between flow velocity and the stagnation pressure occurring within a joint of fixed geometry.

Following the reanalysis of the Johnson (1976) and Frizell (2007) data sets, Reclamation constructed an experimental facility designed to address questions posed by IFT (2018) and Wahl et al. (2019). The objectives of the experimental work are as follows:

The first two objectives are addressed in this article. Test data related to flow rates through joints and cracks will be presented in a subsequent article.

All velocity measurements were fit well by power curve equations in which $V^{*}\propto {(y^{*})}^{1/N}$ or $y^{*}\propto {(V^{*})}^N$, where $V^{*}$ is dimensionless velocity normalized by the mean velocity and $y^{*}$ is dimensionless distance from the boundary normalized by the flow depth. The quality of the power curve fits indicates that the flow was well developed at the joint test station. The fitted profiles were used to determine the average velocity and flow depth. The exponent of the power curves varied with flow depth, corresponding to changes in the relative roughness and effective friction factor of the chute. The value of $N$ decreased with decreasing flow depth and decreased significantly for the handful of tests run with roughened floor conditions, consistent with expectations. It should be noted that the closest Pitot tube measurements to the boundary are located far above the likely upper bound of the viscous sublayer and buffer zone, and all offset heights tested were also many times taller than the thickness of the viscous sublayer. Thus, it is reasonable to represent the entire velocity profile with a single power law equation. Although the form of the velocity profile indicates well-developed flow, it should also be noted that the flow at the tested joint location was still accelerating for most tests; plots of lengthwise profiles of velocity, depth, and Froude number indicated that uniform, normal-depth flow was only reached at the joint for unit discharges less than about 0.107 m³/s/m (1.15 ft³/s/ft). This is consistent with potential applications for this research since many spillways operating at high discharges experience accelerating flow for much of their length.

Experimental measurements of uplift pressure have been normalized in this research with respect to the boundary layer velocity head, which depends on the velocity profile exponent $1/N$ in $V^{*}\propto {(y^{*})}^{1/N}$. In the laboratory studies, $N$ has been determined by fitting to the Pitot tube velocity measurements. For prototype application, $N$ has previously been related to the Darcy friction factor, $f$. Nunner (1956) used data from Nikuradse (1933) and Laufer (1954) along with his own experiments to develop the empirical relation $N=1/\sqrt{f}$, which can also be expressed as $N=0.354\sqrt{8/f}$. Subsequent investigations by many others, well summarized by Karim and Kennedy (1987), produced more complex expressions that have often been reduced to a relatively simple relation found in several newer references (e.g., Chen 1991; Chanson 1994):where $\kappa =0.4$ is the von Kármán constant. Such a relation will allow $N$ to be determined for prototype chutes without the need for direct velocity profile measurements.

A prominent feature of an offset into the flow is that flow will fully detach from the chute floor if the offset is large compared to the flow depth (Fig. 4). If the jet breaks up in the air before landing again in the chute, it will remain detached, since air can penetrate the disintegrated jet. For deep flows, full jet breakup does not occur, and the jet remains attached to the floor since the flow entrains any air that is present and additional air cannot get under the jet. This is the usual condition for most prototype spillway flows, but both conditions can be produced in the lab over a large range of flows. The attached jet condition can be established by starting at a large discharge that produces naturally attached flow, and then reducing the discharge. Alternately, at low flow rates, the detached jet can be physically forced down to the floor using a handheld deflector plate, and, once attached, the flow will remain attached down to a relatively small flow rate. If detached jet conditions are again desired, the flow can be easily detached by momentarily disrupting the flow near the offset face with an obstruction. Attached jet conditions were the primary focus and baseline condition for the study, but measurements were made for both conditions whenever possible; for a given flow condition approaching a joint, jet detachment leads to greater uplift (up to about 8% of the approaching velocity head). For each joint configuration, tests were performed from high discharges down to the lowest discharge that would allow for attached flow. A few tests were run at discharges so low that only detached flow could occur.

Testing has been conducted for a wide variety of joint configurations and flow conditions. This paper presents results from the following:

Several potential empirical equations were investigated to relate normalized uplift pressure head to other experimental parameters singly and in combination, including $\beta$, the velocity exponent $1/N$, the Froude number, several Reynolds numbers using different velocity and length references, and several variations of roughness Froude numbers (Wahl 2023) utilizing the offset height, $h$. Parameters used to normalize the uplift pressure head included the mean channel velocity head and the velocity head at various discrete positions within the boundary layer, such as the mid-height and tip of the offset. Relations were also investigated based on the concept of determining the distance from the bed to the hypothetical streamline in the velocity profile whose velocity head matched the measured uplift pressure head. However, none of these empirical methods were satisfactory over the full range of tests.

Ultimately, the most useful relations for modeling and predicting uplift pressure were based on the normalized uplift pressure head $\mathrm{\Delta }H/h_v*$, where $h_v^{*}$ is the velocity head of the flow in the boundary layer integrated between the chute floor and the tip of the offset. This can be determined from the velocity profiles measured with the Pitot tube.

Relating $h_v^{*}$ back to the mean channel velocity is convenient and provides insight into the relation between the velocity head of the boundary layer and that of the bulk flow. We follow the same development process given by Chow (1959) for the familiar energy coefficient or Coriolis coefficient, $\alpha$, which is used to calculate velocity head $h_v=\alpha V^2/(2g)$ using the mean velocity, $V$. The energy coefficient is given by $\alpha =(\int v^{3}dA)/(V^{3}A)$, where $v$ is the velocity through a differential area $dA$ and $A$ is the total flow area. Integrating from the bed to the water surface, this relates the total kinetic energy of the water passing through the area $A$ to that calculated for a uniform velocity $V$ through the same area. If the velocity profile is described by a power curve as shown in Fig. 3 with $V\propto y^{1/N}$, integration yields $\alpha ={(1+1/N)}^3/(1+3/N)$. Changing the limits of integration to cover only the zone from the channel bed to the offset height, $h$, the result isand the velocity head of the flow between the chute floor and the tip of the offset is $h_v^{*}={\alpha }^{*}{V}^2/(2g)$. The coefficient ${\alpha }^{*}$ is the ratio of the velocity head in the boundary layer, $h_v^{*}$, to the simple velocity head calculated from the average chute velocity, $V^2/(2g)$. Values of $N$ in the uplift research flume varied from about 5 for shallow flow in roughened channels up to 10 for deeper flow with the smooth acrylic floor. Thus, the range of $\alpha$ was about 1.025 to 1.08. Values of ${\alpha }^{*}$ range from zero to $\alpha$, depending on the relative offset height, $h/y$ (assuming $h$ never exceeds $y$). For the jet striking an offset to remain attached to the chute floor in these tests, the relative offset height had to be less than about 0.33 to 0.5, or alternately the relative submergence of the offset, $y/h$, had to exceed 2 to 3. The smallest attached-flow value of $y/h$ was about 2 with ${\alpha }^{*}=0.81$; for cases of large $y/h$, ${\alpha }^{*}$ was as small as 0.16.

Fig. 6 shows the normalized uplift pressure head $\mathrm{\Delta }H/h_v^{*}$ as a function of $y/h$ for selected $\beta$ values; approximately 40 different combinations of $h$ and $s$ were tested, but for clarity this figure shows data for only a few cases. For a given value of $\beta$ the normalized uplift pressure increases nonlinearly with $y/h$, and for any given $y/h$ the normalized uplift pressure increases nonlinearly with decreasing $\beta$, as shown in Fig. 7. The curves fitted to the data in Fig. 6 all use the equation form:which can be transformed into

Eq. (6) is linear and convenient for regression analysis to determine the fitting parameters $b$ and $c$. [A form was also considered in which the numeral 1 in the denominator of Eq. (5) was replaced with a third fitted parameter, $a$, but this offered too many degrees of freedom and did not produce useful results.] This equation indicates that for each value of $\beta$ the normalized uplift asymptotically approaches a value of 1.0 for large values of $y/h$. The asymptotic limit is visually apparent for the smaller $\beta$ values in Fig. 6 but less obvious for large $\beta$. However, the quality of the curve fits demonstrates that Eq. (5) provides a good representation of the behavior for all $\beta$ within the range of conditions that could be tested. Fig. 8 shows data from all tests of regular joints with the smooth acrylic floor.

The curve fitting parameters $b$ and $c$ in Eq. (5) both vary as functions of $\beta$, as shown in Figs. 9 and 10. Values of $b$ can be represented with a single curve:while values of $c$ are represented by two curveswhere $e$ = base of natural logarithms. Increased scatter of $b$ and $c$ is noticeable in Figs. 9 and 10 for approximately $5<\beta <9$. The source is evident in Fig. 8 where there is significant crossing of the fitted curves in the region of $y/h=3$ to 8 and $\mathrm{\Delta }H/h_v^{*}=0.25$ to 0.35. These crossing curves with differing values of $b$ and $c$ correspond to tests of joints with similar $\beta$ ratios, but different offset heights. The curves on the left side of this region are for large offsets (up to about $12\mathrm{mm}=0.47\mathrm{in.}$), which could only be tested at low $y/h$ ratios for even the largest possible discharges, while the curves on the right side of the region are obtained from offsets as small as 3 mm that could only be tested at larger $y/h$ ratios due to small flow depths and detachment at very low discharges. This behavior can be considered a scale effect caused by the interaction of the chute flow with the circulation that develops within the joint, below the plane of the chute surface. The shear between these flows creates a shear boundary layer that reduces the velocity of at least some of the flow impacting the offset. The thickness of this layer increases with the distance that the chute flow travels across the gap. When $\beta$ is small (offset height large relative to the gap width), this layer is thin in comparison to the offset height, so the effect is minimal. For larger $\beta$ the effect is increased as the layer becomes thick in comparison to the offset height. The effect begins gradually, becomes quite noticeable by about $\beta =5$, and continues to be visible up to about $\beta =9$. The maximum gap width in the test facility and limits on accurate construction and measurement of extremely small offset heights make it impractical to test a significant range of offset heights for larger values of $\beta$, but if such testing could be performed the effect should continue to be seen for all larger values of $\beta$.

To better understand the development of this shear layer, velocity profiles were measured above open gaps at incremental distances, $x$, downstream from the start of the gap, with no offset installed. Profiles were measured above a 76.2-mm (3-in.) wide gap for two discharges, designated Q3 ($0.084{\mathrm{m}}^3/\mathrm{s}=3{\mathrm{ft}}^3/\mathrm{s}$) and Q13 ($0.37{\mathrm{m}}^3/\mathrm{s}=13{\mathrm{ft}}^3/\mathrm{s}$), and a 38.1-mm (1.5-in.) wide gap at a third discharge, designated Q9 ($0.264{\mathrm{m}}^3/\mathrm{s}=9.33{\mathrm{ft}}^3/\mathrm{s}$). Normalizing velocity with respect to the mean velocity in the chute, $v^{*}=v/V$, and distances across the gap and above the chute floor with respect to the gap width, $x^{**}=x/s$ and $y^{**}=y/s$, the velocity profiles exhibit a consistent behavior shown in Fig. 11, with measurements over the narrower gap omitted from the figure for clarity. Velocity over the gap is gradually reduced from the initial chute velocity profile in a layer whose thickness grows linearly and is equal to about 11% of $x$. Over the first third of the gap, the entire layer is a transitional zone in which the top of the layer matches the initial chute velocity, and the bottom has a growing velocity reduction compared to the original profile. Over the last two thirds of the gap, the shear layer continues to grow, composed of the transitional layer and a thickening lower layer in which the velocity is reduced uniformly by about 12% from the original chute velocity. Tests with the narrower gap showed similar trends, and the velocity reduction in the lower layer remained about 12%, appearing to be independent of the gap width. The lower part of the shear layer exhibits a similar slope on the log–log plot to that of the original profile, while the transitional zone has a steeper gradient of velocity versus depth. As the scale of the offset and gap change, the shear layer has a nonlinear effect on uplift pressure due to the variation of the velocity reduction factors. Offsets of any scale with the same $\beta$ value have the same proportion of their height influenced by the shear layer, but for small-scale offsets the transitional portion makes up all or most of the layer while larger-scale offsets become dominated by the lower portion with its greater velocity reduction.

Eqs. (5), (7), and (8) provide a generalized method for predicting the normalized uplift pressure for a joint with a given value of $\beta$. To test the method, Eqs. (7) and (8) were used to determine values of $b$ and $c$ as functions of $\beta$ rather than obtaining $b$ and $c$ by curve-fitting to individual test data. Eq. (5) was then used to compute the normalized uplift, $\mathrm{\Delta }H/h_v^{*}$. To enable a more direct comparison to the previous evaluation of Eq. (3), which computed $\mathrm{\Delta }H/[V^2/(2g)]$, we define the velocity head of the full flow as $h_v=\alpha V^2/(2g)$ (since $\alpha$ can be determined when the velocity profile is known) and convert values of $\mathrm{\Delta }H/h_v^{*}$ to equivalent values of $\mathrm{\Delta }H/h_v$ using

This also provides a more realistic evaluation of relative errors than a comparison of $\mathrm{\Delta }H/h_v^{*}$ values, since large errors in $\mathrm{\Delta }H/h_v^{*}$ are less significant when the offset height and $h_v^{*}$ are small.

Fig. 12 shows predicted versus observed values of $\mathrm{\Delta }H/h_v$ for the full set of regular joint tests. The majority of predictions are inside of the $\pm 10\%$ error band, with most of the larger errors occurring in the zone influenced by the scale effect discussed previously. The root-mean-square (RMS) average of errors for the complete data set is 1.25% of $h_v$, reduced by a factor of 2.87 from the RMS errors of Eq. (3) applied to the same data.

Tests conducted with the smooth acrylic floor exhibited significant effects of changing roughness and variation of the boundary layer velocity profile, since relative roughness varied with discharge and flow depth over a range of about $15:1$. This caused $N$ to vary from about 6.8 to 9.9. To test an even wider range of conditions, surface roughness was added to the chute upstream from the joint using several different treatments: 80-grit sandpaper (approx. $0.177\mathrm{mm}=0.0006\mathrm{ft}$ sand particle diameter) adhered to the acrylic floor for a distance of about 1.2 m (4 ft) upstream from the joint; fabricated roughness panels with discrete triangular ridges perpendicular to the flow direction; and a 3.6 m (12 ft) length of expanded metal screen. Velocity profiles measured near the beginning, intermediate, and downstream points of the roughened sections showed that new well-developed velocity profiles existed at the joint, accurately represented by power curve equations with reduced values of $N$ between 5.0 and 5.8 and reduced velocity magnitudes near the floor consistent with the increased roughness. Although these profiles exhibited a power curve shape over the full depth that could be measured, it should be emphasized that uniform flow (i.e., normal depth) was not achieved for these tests; the flow was still decelerating at the tested joint, in contrast to the accelerating flow that existed for high-flow tests with the smooth acrylic floor.

Table 2 shows results of the rough-floor tests and compares observed uplift to that calculated by three methods: Eqs. (5)–(9) using the $N$ value fitting the measured velocity profile; Eqs. (5)–(9) using $N=8.5$ that represents a relatively smooth chute; and Eq. (3), which relates uplift only to $\beta$ without considering boundary layer velocity conditions. The generally lower errors for the first method compared to the second and third show that accounting for the effects of roughness on the velocity profile significantly improves the agreement between predictions and experimental results. Note that errors are expressed as percentages of $h_v$ to give an indication of their magnitude in relation to actual pressure differences. If errors were computed relative to observed values of $\mathrm{\Delta }H$ or $\mathrm{\Delta }H/h_v^{*}$ the percentage differences would be considerably larger in most cases. These results validate the use of the new uplift pressure equations for a wide range of channel roughness conditions, even though the relations were developed primarily from smooth-channel tests, and they demonstrate that through reduction of $N$, additional roughness clearly causes a significant reduction of uplift pressure.

Eqs. (5)–(9) can also be applied to a range of hypothetical conditions to investigate the maximum possible uplift pressure head. The experimental data suggest an upper limit of about 60% of $h_v$. To apply the equations, a value of $N$ must be assumed. Arbitrarily taking $N=8$ as a representative value (approximate median of observed values in the lab tests), Fig. 13 is obtained. Figs. 6 and 8 showed that uplift pressure normalized to $h_v^{*}$ approaches an asymptotic limit of 1.0 for large $y/h$, but at these conditions ${\alpha }^{*}$ is also quite small and the offset is exposed to boundary layer velocities that are much less than the mean velocity, so less uplift should be expected. Fig. 13 confirms this and shows that when normalized to the mean velocity head, peak uplift pressure head is about $0.545h_v$, occurring near $y/h=2.5$; uplift also drops significantly for larger values of y/h that are typical of design discharge conditions. Over the range of $N=5$ to 10 the peak varies almost linearly from $0.47h_v$ to $0.59h_v$ and the location of the peak varies almost linearly from $y/h=1.61$ to 2.94 (Fig. 14).

Eqs. (5)–(9) predict the uplift pressure head for jets that remain attached to the chute floor downstream from an offset into the flow. If the flow detaches from the floor due to a large offset that forces the flow into the air so that the jet breaks up before landing downstream, the uplift pressure will be increased due to the momentum thrust needed to deflect the jet. Some of the increased pressure on the chute floor is transmitted into the joint. Increases measured experimentally were as large as 20% of $h_v^{*}$, 8% of $h_v$, or 60% of the attached uplift. While the increases are significant for a given flow condition, detached uplift pressure will rarely be a design concern for most spillways, since even greater uplift will occur at large flow depths and velocities that produce attached flow. Procedures for determining whether flow will detach from the floor and empirical relationships for estimating the additional uplift are provided in Supplemental Materials.