Effects of Joint and Crack Geometry on Hydraulic Jacking in Lined and Unlined Spillways

Wahl, Tony L.; Heiner, Bryan J.

doi:10.1061/JHEND8.HYENG-13994

Open access

Technical Papers

Jul 2, 2024

Effects of Joint and Crack Geometry on Hydraulic Jacking in Lined and Unlined Spillways

Authors: Tony L. Wahl, P.E., M.ASCE https://orcid.org/0000-0002-2081-2263 [email protected], and Bryan J. Heiner, P.E., M.ASCE [email protected]Author Affiliations

Publication: Journal of Hydraulic Engineering

Volume 150, Issue 5

https://doi.org/10.1061/JHEND8.HYENG-13994

PDF

Abstract

Offsets into the flow at joints or cracks can cause hydraulic jacking of spillway chute linings or remove rock blocks from unlined chutes. Previous articles established experimentally validated relations for uplift pressure and flow into square-edged joints and cracks oriented perpendicularly to the flow and chute surface. This paper addresses chamfered and rounded edges, tilts into or away from the flow, skewed orientations, irregular midslab cracks, and bevels or relief slots used to remediate existing offsets. Chamfers and rounding reduce uplift by diminishing the effective offset height. Tilts and bevels away from the flow reduce uplift and joint flow in proportion to

(\cos ϕ)^{1 / 2}

where

ϕ

= tilt angle, whereas tilts toward the flow cause marginal increases. Very flat beveling is needed for major uplift reduction; as a result, relief slots that shift the offset downstream from the joint may be more effective remediation alternatives. For skewed joints, uplift is reduced in proportion to

(\cos ψ)^{2.5}

where

ψ

is the deviation from perpendicular orientation. Flow through various irregular joints was reduced compared with that through regular square-edged joints.

Practical Applications

Existing offsets into the flow at joints or cracks in spillway chutes create the potential for hydraulic jacking failures that can destroy a spillway chute and potentially lead to uncontrolled release of reservoir storage. This study used laboratory experiments to evaluate several treatments that can reduce the risk of hydraulic jacking failure, including chamfering, rounding, and beveling of offsets or milling slots to shift the offset further downstream from an open joint. If the goal is to reduce uplift to 30% or less of that caused by the untreated offset, very flat beveling is needed and relief slots are more effective than bevels that involve the same volume of material removal. The study also shows that skewed joints (i.e., twisted so that their axis is not perpendicular to the flow direction) can reduce uplift pressures by about 55%, although this may not be an economic or practical construction approach for most spillways. Data collected in the study also aid in the estimation of uplift pressures at rounded and chamfered joints and midslab cracks, and at natural joints in unlined rock chutes.

Introduction

The Oroville Dam spillway failure in 2017 highlighted the importance of hydraulic jacking as a potential failure mode of concrete spillway chutes (IFT 2018). Stagnation of flow at joints or cracks where offsets project up into the flow injects high-pressure water into the foundation of spillway chute slabs, which can cause erosion and potentially lift (jack) slabs further up into the flow. Wahl and Heiner (2024a, b) provided relations that can be used to predict uplift pressures and flow rates associated with square-edged joints or linear cracks oriented normal to the flow direction and chute surface. Offsets into the flow also have the potential in high-velocity conditions to trigger damaging cavitation. This research focused only on the generation of stagnation pressure that may lift slabs, jack rocks out of place, or drive flow into a joint. These effects with associated severe consequences are possible at flow velocities much lower than those needed to cause cavitation.

In field cases, many other joint orientations and configurations are possible, including skewness (rotation of the joint axis in the plane of the chute), tilt (rotation of the joint in the vertical plane that parallels the flow direction), and chamfered or rounded edges. In addition, natural cracks in the middle of slabs may be nonlinear, with skewed orientations that vary along the length of the crack. Since the Oroville spillway failure, there has been strong interest in rehabilitating joints that may be in poor condition (e.g., Smith et al. 2023; Gilbert et al. 2023). To remediate spillways with existing offset joints or cracks, offsets may be beveled down by grinding; another alternative remediation option is milling the downstream slab to effectively shift the offset downstream from the joint opening. This paper presents experimental data on the changes in uplift pressure and joint flow caused by these variations from idealized square-edged geometries.

Unlined spillway chutes that are cut through rock are prone to hydraulic jacking of individual rock blocks (e.g., Bollaert 2010; George 2015; Pells 2016; Koulibaly et al. 2021), which can lead to headcutting and an uncontrolled release of stored water. Concern for such a possibility prompted evacuation of downstream populations when headcut erosion occurred in the Oroville auxiliary spillway channel during its operation following the hydraulic jacking failure of the service spillway. Recent research has explored means for identifying other auxiliary spillways that could be at risk of similar erosion (Ghimire and Schulenberg 2021). The tests of skewed and tilted joints in the present paper provide insight into the hydraulic forces that are applied to rock blocks, which may facilitate various analytical methods for modeling their removal, which is a complex phenomenon involving rock strength, joint mechanics, and the forces exerted by the flow.

Literature Review

Wahl and Heiner (2024b) provided experiment-based equations for estimating the stagnation pressure generated at square-edged offsets into a spillway flow, which serves as the driving pressure head for flow through a joint or crack. Uplift pressure measured in sealed (nonvented) joints is shown to vary as a function of the velocity head of the boundary layer flow intercepted by the joint (

h_{v}^{*}

), the gap width to offset height ratio (

β = s / h

), and the ratio of flow depth to offset height (

y / h

), where

y

is the flow depth,

s

is the gap width, and

h

is the offset height (Fig. 1).

Fig. 1. Schematic diagram of flow at an offset joint in the test facility. Chamber below joint is not drawn to scale (actual chamber is deeper than shown). Uplift pressure head $Δ H$ is measured when the outlet valves are closed. Reduced uplift pressure head (backpressure head) $Δ H_{b}$ is measured with outlet valves partially or fully open.

For uniform or gradually varied flows of known depth and velocity, the velocity profile approaching an offset can be described by a power-curve equation

V \propto y^{1 / N}

, where

N

can be estimated as (Wahl and Heiner 2024b)

N = 0.82 κ \sqrt{8 / f}

(1)

where

f

is the Darcy–Weisbach friction factor; and

κ

is the von Kármán constant, which is assumed to have a value of 0.4. Eq. (1) is an adjustment of the relations

N = κ \sqrt{8 / f}

(Chen 1991) and

N = 1 / \sqrt{f}

(Nunner 1956). With

N

known, the velocity head of the boundary layer flow striking the offset can be computed from

h_{v}^{*} = α^{*} V^{2} / (2 g)

(2)

where

g

= acceleration due to gravity; and

α^{*} = α {(h / y)}^{3 / N}

, where

α = {(1 + 1 / N)}^{3} / (1 + 3 / N)

. The parameter

α

is the familiar energy coefficient used to compute the mean channel velocity head

h_{v} = α V^{2} / (2 g)

.

The uplift pressure head generated at an offset into the flow is given by a series of equations

b = 1.29 + (0.059) β^{(3.2 β^{- 0.175})}, with b = 1.29 when β = 0

(3)

c = 1.587 - 0.935 β for β < 0.6

(4a)

c = e^{(- 0.8 + 1.5 e^{- β})} for β \geq 0.6

(4b)

\frac{Δ H}{h_{v}^{*}} = \frac{{(y / h)}^{c}}{{(y / h)}^{c} + b}

(5)

where

e

is the base of natural logarithms. Uplift pressure head can be expressed in relation to the mean flow velocity head using

Δ H / h_{v} = (Δ H / h_{v}^{*}) (h_{v}^{*} / h_{v}) = (Δ H / h_{v}^{*}) (α^{*} / α) = (Δ H / h_{v}^{*}) {(h / y)}^{3 / N}

.

The flow through a joint can be computed by applying an orifice equation to the joint (Wahl and Heiner 2024a), taking

Δ H

from Eq. (5) as the upstream head and assuming either atmospheric pressure below the joint or a backpressure head,

Δ H_{b}

, related to the drainage characteristics of the site

q_{j} = C_{d} s \sqrt{2 g (Δ H - Δ H_{b})}

(6)

where

q_{j}

= discharge per unit width through the joint;

C_{d}

is a discharge coefficient;

s

= width of the gap (Fig. 1); and

g

= acceleration due to gravity. Both

Δ H

and

Δ H_{b}

are referenced to the piezometric head associated with the chute flow over the joint, and thus represent net head above the prevailing flow depth. The discharge coefficient for square-edged joints oriented normal to the flow direction and chute surface was shown to be

C_{d} = 0.05 + \frac{1}{1.03 + 0.07 {(V_{gap} / V_{chute})}^{- 1.1}}

(7)

where

V_{gap}

and

V_{chute}

= average velocities through the joint and in the chute, respectively. When determining an unknown flow rate, Eqs. (6) and (7) must be solved iteratively because the velocity through the gap depends on the discharge coefficient.

Two previous experimental studies of hydraulic jacking were conducted at the Bureau of Reclamation hydraulics laboratory in Denver, Colorado (Johnson 1976; Frizell 2007). The Johnson (1976) experiments were performed in a small open-channel facility with velocities as high as

4.6 m / s (15 ft / s)

and measured uplift pressures only in a sealed joint (i.e., there was no net flow through the joint). Johnson suggested future testing of joints with various tilted and skewed orientations. Frizell (2007) used a water tunnel facility with an adjustable joint tested at flow velocities as high as

14.6 m / s (48 ft / s)

, and this test program included joints with square-edged, rounded, and chamfered edges. Frizell (2007) observed that chamfered and rounded edges caused joints to perform as though they were effectively wider than their actual dimension, and provided test data and curves for rounded and chamfered edges of specific sizes but did not give general methods to account for effects of the different edge types. Sánchez (2022) performed computational fluid dynamics simulations of the Frizell (2007) experiments that confirmed Frizell’s general observations and also indicated that the chute slope

θ

has no special influence on the uplift pressure head generated at the offset. Both Johnson (1976) and Frizell (2007) discussed the potential importance of the boundary layer on uplift pressures and Frizell used laser-based particle image velocimetry (PIV) to map the complex velocity fields in the vicinity of one joint configuration. The relations given by Wahl and Heiner (2024b) incorporate the boundary layer via the exponent

N

.

Development of internal pressures in rock joints has been the subject of several experimental studies and modeling efforts. Bollaert and Schleiss (2005) measured the dynamic pressures in simulated joints of rock-lined plunge pools with water jets (aerated and nonaerated) directed roughly normal to the rock surface. Forces applied to rock blocks and joints at the surface of unlined chutes, in which flow generally is parallel to the surface, have been studied using flume-based experiments (George 2015; Pells 2016; Koulibaly et al. 2023), and computational models for spillway scour simulation have been proposed and demonstrated (Bollaert 2010; Gardner 2023; Hurst et al. 2023). Field studies of forces applied to instrumented rock blocks also were performed by George (2015).

Experimental Facility—General

An experimental facility was constructed in early 2021 in the Bureau of Reclamation’s hydraulics laboratory in Denver, Colorado to measure uplift pressures within sealed offset joints and flow rates at partially vented joints. Wahl and Heiner (2024b) provided a detailed description of the facility; the overview here focuses on aspects relevant to the tests of irregular joints.

The 0.61-m-wide (2 ft) smooth acrylic flume on a fixed 15° slope is equipped with an adjustable spillway joint located near the downstream end (Fig. 2). The approach distance to the joint is

8.76 m (28.75 ft)

, producing well-developed uniform or gradually varied flow profiles, depending on discharge. Flow enters through a jet box that can be used to adjust the incoming Froude number. The flume is supplied from the laboratory’s fixed pumps and venturi meter flow measurement systems which include meters from 76- to 356-mm (3- to 14-in.) diameter calibrated against a weight-based tank. Uncertainty of discharge measurements from this system generally is

\pm 0.25 %

or less.

Fig. 2. Hydraulic jacking research flume, showing (a) jet box (top), spillway joint (right foreground), and receiving channel and 90° V-notch weir (foreground); and (b) valves that regulate the backpressure and flow out of the chamber below the joint.

The model spillway joint consists of a

57 \times 7.62 - cm

(

22.5 \times 3 - in

.) chamber in the flume floor that spans most of the channel width. The chamber extends 44 cm (17.25 in.) below the flume floor and is equipped with three 50-mm-diameter (2-in.) outlet valves at the bottom and a 19-mm-diameter (¾-in.) air inlet valve near the top of the chamber. Plates installed on the flume floor downstream from the joint create offsets into the flow,

h

, to simulate a displaced spillway slab, and a portion of the chamber can be filled with solid materials to vary the width of the gap,

s

, and the thickness of the joint,

t

. Tested offsets were constructed from a variety of materials, including brass, aluminum, high-density foam, expanded PVC panels, and marine plywood. Care was taken to ensure that materials were water resistant and suitably rigid to avoid deformation during tests in which large uplift forces were generated. Gap widths and offset heights were measured for each configuration using digital calipers or feeler gauges and a machinist’s height gauge. Control of the gap width was less precise in the middle of the joints than at the chute surface where gap widths were measured. The floor of the flume was constructed from 19-mm-thick (

3 / 4 - in

.) acrylic, and there was a slight overhang of this floor piece over the upstream wall of the chamber below the flume [about 1.1 mm (0.043 in.)]. This caused the effective gap width to be slightly greater below the flume floor; this issue is more significant for very narrow gap widths.

Sealed joint testing to determine maximum uplift pressure head was performed with all valves on the chamber closed. For vented testing to evaluate flow rates through joints, valves were opened incrementally and joint discharge from the chamber was measured with a 90° V-notch weir installed in a 0.61-m-wide (2 ft), 1.52-m-long (5 ft) weir box. The weir operates in a partially contracted flow condition for most flow rates. The weir box is equipped with a stilling well and Lory Type-A hook gauge with vernier scale measurement precision of 0.0003 m (0.001 ft). Flow calibration of the weir is based on the Kindsvater–Carter weir equations as described in the Water Measurement Manual (Bureau of Reclamation 2001).

Tests were performed at chute discharges from 0.01 to

0.54 m^{3} / s

(

0.35 - 19 {ft}^{3} / s

), producing unit discharges as high as

0.88 m^{2} / s

(

9.5 {ft}^{2} / s

) and mean chute velocities at the tested joint as high as about

9.75 m / s

(

32 ft / s

). Froude numbers

F = V / {(g y)}^{1 / 2}

at the joint location ranged from about 6.7 to 13. Reynolds numbers

R = 4 R_{h} V / ν

ranged from

5.9 \times 10^{4}

to

3.2 \times 10^{6}

, where

R_{h}

is the hydraulic radius and

ν

is the kinematic viscosity. This put all tests at the edge of or within the zone of fully turbulent flow. All flows were predominantly nonaerated at the test location due to the moderate slope of the chute, which was insufficient to create and maintain highly aerated flow. For the largest tested flow rates [greater than about

0.5 m^{3} / s

(

19 {ft}^{3} / s

)], mildly aerated flow entered the flume from the jet box, but deaeration occurred quickly, and the flow near the channel bed was nonaerated when it reached the test section.

Regular square-edged joints were tested with offsets ranging from 0.75 mm (

1 / 32 in.

) to 12.6 mm (

1 / 2 in.

) and gap widths ranging from about 0.45 mm (0.018 in.) to 76.2 mm (3 in.). Tests spanned

β

ratios of 0.044 to 32—a ratio of

1 ∶ 725

, or almost 3 orders of magnitude. The thickness of joints,

t

(i.e., length of the flow path through the joint), was varied from about 5 to 28 cm (2–11 in.). Tests of regular joints were conducted mostly with a smooth acrylic approach flow surface leading up to the joint, although a few tests used roughened overlays applied to the upstream channel floor. Tests of irregular joints described in this paper were performed with the smooth floor, except one series of tests that used a cracked concrete specimen with a rough, broom finish on the specimen itself and a roughened 80-grit sandpaper surface approaching the specimen.

Velocity profiles were measured just upstream from the offsets using a 1.32-mm-diameter (0.052-in.) stainless steel Pitot tube installed with its inlet port 70 mm (2.75 in.) upstream from the edge of the chamber. Flow depths were determined by continuity from the integrated velocity profiles and validated against approximate measurements from an acoustic water level sensor mounted above the water surface. Pressures within the spillway joint were sensed via four manifolded piezometer taps spaced equally along the centerline of the bottom of the chamber, with the outlet valves centered between the taps. All Pitot tube measurements and uplift pressure measurements from the model spillway joint were obtained visually from a nearby manometer board. Pitot tube measurements were made at the centerline of the channel for all tests, and lateral surveys of velocity were made early in the study to develop correction factors to convert centerline velocities to width-averaged mean velocities.

Experimental Facilities—Irregular Joint Configurations

A variety of irregular joint configurations were tested, with unique devices designed to implement some of the geometries. Tests of regular square-edged joints detailed by Wahl and Heiner (2024a, b) were conducted over a broad range and size of joint geometries and flow conditions. Tests of irregular joint configurations generally were performed over narrower ranges of conditions to develop relations between the relative uplift pressure and joint flow rates of modified versus regular joints. However, testing still included a sufficient range of conditions to confirm that results were independent of size- or discharge-related scale effects. Configuration and flow test data are provided in Tables S1 and S2 in the Supplemental Materials for uplift pressure tests and joint flow rate tests, respectively.

Chamfered and Rounded Edges

A small set of chamfered and rounded edges was tested by machining the desired shape into the leading edge of the offset or gap filler piece installed at the model joint (Fig. 3); the piece representing the trailing edge of the upstream slab was square and sharp in all cases. Only 45° chamfers and ¼-round radius edges were tested; edges had a 6.35-mm (

1 / 4 - in.

) radius or

6.35 \times 6.35 - mm

chamfer. Total offset heights were similar to the dimension of the edge treatments, and gap widths were all of slightly larger dimension (i.e.,

β > 1

).

Fig. 3. Joint offsets with: (a) chamfers; and (b) rounded edges.

Tilted Joints

Tilted joints are those in which the axis of the joint is perpendicular to the flow direction, but the upstream face of the offset is rotated around the joint axis to produce a tilt into or away from the flow. A few tests were performed with joints constructed so that the tilt of the offset face also extended below the surface of the chute [Figs. 4(a and b)], but results quickly showed that the geometry of the opening below the chute surface had no effect on uplift pressure as long as the tilt extended at least down to the chute surface. As a result, most of these tests were performed with geometries like those shown in Figs. 4(c and d), with the tilted upstream face connected to a vertical slot passing through the floor of the flume. The gap width for these tests was defined as

s

(Fig. 4), in the plane of the chute surface, although in some cases it was more convenient to measure

s^{'}

physically and calculate

s = s^{'} / | \cos ϕ |

. In the field it probably would be convenient in most cases to measure

s

directly in the plane of the chute at the chute surface using a rule, scale, or inside calipers. Table S1 provides the test configurations and data, in which values of

ϕ < 0

indicate a tilt away from the flow, and

ϕ > 0

indicates a tilt into the flow.

Fig. 4. Alternate test configurations for tilted and beveled joint geometries: (a) tilt into flow extending through the slab at the tilt angle; (b) tilt away from the flow extending through the slab at the tilt angle; and (c and d) opening below the plane of the chute perpendicular to the chute surface. For equal gap widths, $s$ , and tilt angles, $ϕ$ , configurations (a and c) produce the same uplift pressure, as do configurations (b and d).

Tilts away from the flow are significant because they represent beveling that could be performed to reduce uplift pressure at existing offsets. A wide range of tilt angles away from the flow was tested to examine the value of minor beveling (similar to chamfers) versus major beveling similar to treatments that might commonly be applied for cavitation prevention [e.g., Fig. 4(d)]. Joints described here as tilted away from the flow correspond in geologic terms to rock strata dipping against the flow direction. Similarly, joints tilted into or toward the flow correspond to rock strata dipping in the flow direction. The testing considered cases in which downstream strata protruded into the flow, although the ease with which such layers can be stripped away may tend to eliminate such offsets into the flow naturally over time; these spillway surfaces may develop a profile more akin to that of a stepped spillway with skimming flow from tip to tip, unlike the type of flow tested here.

Relieved Joints

An alternative to beveling existing offsets to reduce uplift pressure is the milling of the offset to create a flush alignment of the slab surfaces at the opening and shift the offset to a location downstream from the open joint (Fig. 5). The distance from the downstream edge of the open joint to the upstream face of the shifted offset is the relief distance,

r

. With increasing

r

, the uplift pressure at the joint is reduced. Tests were performed to determine the reduction ratio compared with the nonrelieved condition.

Fig. 5. Schematic diagram showing a relieved offset located a distance $r$ downstream from an open joint.

Skewed Joints

Skewed joints have the axis of the joint rotated in the plane of the chute so that it is not perpendicular to the flow direction. Skewness may occur for exposed rock joints in unlined spillways, or in concrete-lined chutes with horizontal curvature and/or superelevation that causes the convenient layout of joints during construction to not align perfectly with the flow direction. For a joint that is skewed but not tilted, the face of the offset is normal to the plane of the chute surface (i.e., vertical if the chute is horizontal). Two different construction methods were used to create skewed joints for testing. Both performed similarly, and the data are combined in the results. The first construction had a skewed slot constructed at a fixed angle in a foam block that filled the entire chamber in the floor, with an offset installed at the downstream edge of the skewed joint opening [Fig. 6(a)]. This construction allowed measurement of uplift pressure and testing of flow rate through the joint, because the joint had a significant length. The second construction method used a receiving block mounted within the chamber in the chute floor, with the block designed to receive 25-mm (1-in.) square foam inserts with slots machined at various skew angles [Fig. 6(b)]. With the insert in place, an offset piece was installed above the insert and small ports through the insert (with diameter equal to the slot width) allowed uplift pressure to be transmitted to the sealed chamber below. The inserts could be installed and removed quickly to facilitate rapid uplift pressure testing, but the small ports produced flow rates that were too small to be measured accurately with the V-notch weir described previously. For all skewed joints, the gap width,

s

, was defined normal to the joint axis [Fig. 6(a)]. This allowed the relative uplift at a joint with a given gap width to be compared directly with the uplift that would be experienced for the same joint offset and gap in a nonskewed orientation.

Fig. 6. Test devices and arrangements for skewed joints: (a) slot constructed within a foam block fills the chamber in the flume floor; and (b) interchangeable inserts with different skew angles.

Cracked Concrete Specimen

The configurations described thus far address the relatively prismatic geometry of typical spillway joints. To assess the uplift associated with a nonprismatic, midslab crack, a specimen of concrete 238 mm long by 70 mm wide by 63.5 mm thick (

9 - 3 / 8 \times 2.75 \times 2.5 in.

) was cracked intentionally using a structural testing machine and then cut to size so that it could be mounted within a receiving block that fit into the chamber in the flume floor. The specimen was scored prior to loading to obtain a relatively straight break that would enable it to fit within the 76-mm-wide (3-in.) chamber; the break on one surface was relatively straight, whereas on the opposite surface the break was much more jagged and irregular (Fig. 7). The specimen used a typical concrete mix design with aggregate as large as 19 mm (

0.75 in.

). The straight break occurred on the smooth side of the specimen, which had a steel trowel surface finish, whereas the opposite side with the jagged break line had a rougher broom finish. The specimen was tested in three orientations. Two 180°-opposite installations with the smooth surface and straight crack exposed to the chute flow were used to test reversed flow directions, and the jagged crack and rough surface were tested in one orientation. For this latter installation, an overlay of 80-grit sandpaper was adhered to the flume floor for a distance of about 1.2 m (4 ft) upstream to produce a boundary layer condition that was more representative of what would occur in a spillway with a broom finish or other roughness, such as weathered concrete. The tests were performed at only one offset height [nominally,

h = 11 mm

(

7 / 16 in.

)] and gap width condition [

s = 17.5 mm

(

11 / 16 in.

)]; a relatively large gap was used to allow a significant offset to be created without interference between the jagged roughness elements within the crack. The offset heights were measured for each test after installation of the cracked specimen into the flume. The gap widths were determined based on dimensions of the specimen and the block into which it was mounted, since they could not be measured accurately after installation due to the jagged nature of the opening.

Fig. 7. Photos taken in the downstream direction of cracked concrete specimens installed in the flume: (a) steel trowel surface finish with straight crack alignment; and (b) broom finish with jagged crack alignment.

Test Procedures and Measurements

Detailed test procedures for determining sealed-joint uplift and flow through partially vented joints were described by Wahl and Heiner (2024a, b). The basic procedure was to measure uplift pressure and the chute velocity profile with the joint chamber sealed, and then to measure the joint flow rate and back-pressure below the joint as the joint chamber valves were opened incrementally. The difference between the sealed uplift and the vented backpressure was taken to be the driving head for orifice flow through the joint. Similar procedures were used for testing of the various irregular joint configurations, with one simplification; for those configurations that could be varied readily and rapidly from one orientation or configuration to the next, uplift pressures and joint flow rates were measured repeatedly for the same discharge in the chute without repetition of the velocity profile measurements. For each test series, a small number of tests were performed at other flow rates to verify that normalized results were not sensitive to the specific magnitude of the chute discharge.

Wahl and Heiner (2024b) discussed the detachment of the chute flow from the floor for large offsets, which causes increased uplift pressure. The majority of the irregular joint testing was conducted with flow attached to the chute floor; a few tests of the cracked concrete specimens also were conducted with detached conditions, and the results were consistent with the prior relationships developed for detached flow at regular joints (Wahl and Heiner 2024b).

Analysis and Results

Chamfers and Rounded Edges

Chamfering and rounding of the leading edge of the offset caused a small reduction of the uplift pressure measured in the sealed joint compared with that for sharp-edged offsets with similar

h

and

s

dimensions. Fig. 8 shows observed uplift pressures and predicted uplift pressures computed with Eqs. (3)–(5) using actual offset heights and adjusted or effective offset heights. For 45° chamfers (the only angle tested), reasonable agreement was obtained by taking the effective offset height to be

h - h_{ch} / 2

, where

h_{ch}

is the chamfer height, which in these tests was 6.35 mm (

1 / 4 in.

). This indicates that the effective tip of the offset is at the midpoint of the chamfered edge. For rounded edges with circular arcs, reasonable agreement was achieved when the effective offset height was

h - R [1 - \sin (π / 4)]

, which placed the effective tip of the offset at the height of a bisector of the 1/4-round edge. No tests of flow rate through joints were conducted with chamfered or rounded offsets, but the procedures described by Wahl and Heiner (2024a) should be effective using estimates of the driving stagnation pressure obtained for the modified offset heights described here.

Fig. 8. Predictions of uplift pressure head for rounded and chamfered edges, with $\pm 10 %$ error bands. Test conditions included offset height $h \approx 5.6$ and 10 mm, gap width $s \approx 7 mm$ , chamfer height and $radius = 6.35 mm$ , flow depth $y = 35 - 72 mm$ , $β = 0.74$ and 1.28, and $y / h = 3.6 - 7.3$ .

Tilted Away, or Beveled

Beveled grinding is a remediation method typically applied to existing offsets into the flow during spillway maintenance work. Tests were performed to determine the reduction of uplift pressure produced by different bevel angles. For each test, the measured uplift was compared with that which would be predicted using Eqs. (3)–(5) assuming a regular geometry with the face of the offset perpendicular to the chute floor. The fractional uplift is given by

(\cos ϕ)^{1 / 2}

(Fig. 9), which means that there still is significant uplift (more than 20% of the reference value) until the bevel angle reaches 88°, or 2° above the plane of the chute.

Fig. 9. Reduction of uplift pressure due to beveled offsets expressed as the fraction of normal uplift that occurs for a square-edged (nonbeveled) offset into the flow. Tests spanned offset heights $h = 3 - 12 mm$ , gap widths $s = 6 - 28 mm$ , $β = 0.5 - 8$ , and $y / h = 3.9 - 20$ .

Tilted into Flow

Offsets tilted into the flow overhang at least a portion of the open joint, and initially were observed to produce relatively small uplift pressure increases compared with the reductions found for offsets tilted at similar angles away from the flow. This is believed to be due to the fact that with the chamber below the joint sealed, the water within the joint and under the shelter of the overhang is effectively stagnant, so an offset tilted into the flow behaves much like an equivalent geometry in which the offset is vertical, but the face is located farther upstream [Fig. 10(b)].

Fig. 10. Offset tilted into the flow illustrating assumed stagnant zone under the overhang and within the joint: (a) conceptual pressure distribution; and (b) effective geometry.

A procedure for predicting the uplift pressure in the joint was developed by estimating the pressure distribution applied to the top of the open joint to be a combination of that due to the equivalent vertical-faced offset and a pressure applied beneath the overhanging tip of the offset. To determine this latter pressure, a series of tests was performed with offsets tilted far enough upstream to completely overhang the joint opening. A relation like those developed by Wahl and Heiner (2024b) for each

β

ratio was determined with the uplift pressure head given by

\frac{Δ H}{h_{v}^{*}} = \frac{{(y / h)}^{3}}{{(y / h)}^{3} + 6}

(8)

which is a special application of Eq. (5) with

b = 6

and

c = 3

determined by a visual fit (Fig. 11). This pressure was applied to the covered portion of the joint opening with width

h (\tan ϕ)

[Fig. 10(a)]. If the tilt completely overhangs the joint, this is the predicted uplift pressure for the entire joint. If the tilt does not completely cover the joint, the average pressure applied to the remaining portion of the joint with length

s - h (\tan ϕ)

is determined using Eqs. (3)–(5) with

β = [s - h (\tan ϕ)] / h

. A weighted average of the two pressures based on the length over which they are applied determines the pressure within the joint. The relative increase in uplift due to the tilt into the flow is limited when

β

is small and/or

y / h

is large, because these cases already have relatively large uplift with little opportunity for increase; when

β

is large and/or

y / h

is small, the situation is reversed, and tilt into the flow can produce larger relative increases in uplift. This method worked well when applied to limited test data obtained from two joints with

β \approx 0.55

and

\approx 3.55

(Fig. 12).

Fig. 11. Dimensionless uplift pressure generated at open joints fully covered by an overhanging offset tilted into the flow. These data were collected for offset heights $h \approx 11 - 12 mm$ , gap width $s \approx 6 mm$ , and $β = 0.55$ .

Relieved Joints

Reducing uplift pressure in a joint by use of a milled relief slot was illustrated in Fig. 5. Tests of a variety of joints modified in this way showed that the relative reduction of uplift pressure as a function of the normalized relief distance,

r / h

, is

1 / [1 + 0.2 {(r / h)}^{1.5}]

. For comparison, Fig. 13 also shows this relation and the relative reduction achieved by a straight bevel of the same length, which removes half the volume of surface material, as well as a bevel with length

2 r

, which removes the same volume as the milled slot. For

r / h > 2

the relief slot was more effective than a bevel of the same length, and for

r / h > 5

the milled slot also outperformed the double-length bevel. A slot with relief distance

r \geq 12 h

reduced uplift to 10% of the untreated value, whereas a bevel of 89.4° was needed to achieve a similar reduction. In many cases, especially when access for milling equipment is feasible in the spillway chute, the relief slot may be more economical and more effective than bevel grinding.

Fig. 13. Reduction of uplift pressure due to relief slots and comparable bevels, expressed as the fraction of normal uplift that occurs for a square-edged offset into the flow. Data spanned offset heights $h = 4 - 12 mm$ , gap widths $s = 3 - 16 mm$ , $β = 0.5 - 2.55$ , and $y / h = 3.2 - 14.4$ .

Skewed Joints

Skewed orientations were tested using the two experimental setups in Fig. 6. The tests were used to determine the fractional uplift relative to a joint with the same gap width oriented perpendicular to the flow. Two wide, integrated joints were tested, one with

ψ = 30 °

and the other with

ψ = 45 °

. The gap width for each setup was fixed, but each was tested with several offset heights and at several flow rates, with individual chute velocity profiles measured for each test. The removable inserts shown in Fig. 6(b) were tested at several preset flow rates which could be established quickly after each insert was installed; the chute velocity profile for each flow condition was measured only once, when the first insert was tested, and the same offset height was used for all tests. Inserts were not tested at

ψ = 0 °

(perpendicular to the flow direction) or at

ψ = 80 °

or 90° because the orientation almost parallel to the flow direction caused the end of the insert’s slot to create flow stagnation, so the results did not accurately represent the behavior of a very long slot. The fractional uplift for the skewed joints is well represented by

(\cos ψ)^{2.5}

(Fig. 14).

Combined Tilt and Skew

A limited series of tests was performed with combined tilt and skew, using two different

β

ratios. The skewness angle was

ψ = 30 °

and tilt angles were

ϕ = - 45 °

,

- 30 °

, 0°,

+ 30 °

, and

+ 45 °

, where negative

ϕ

values indicate tilt away from the flow and positive

ϕ

values indicate tilt into the flow. The relationships developed from the separate tests of tilted and skewed joints were applied to predict the uplift pressure heads, and the observed uplift pressures tended to be about 12% low for the combined cases (Fig. 15). Apparently, the incomplete stagnation that results from a skewed orientation tends to reduce the pressure increase caused by a tilt into the flow and amplifies the reduction due to a tilt away from the flow.

Fig. 15. Predicted versus observed uplift in tests of combined tilt and skew, with uplift pressure heads expressed as fractions of channel velocity head $h_{v}$ , with $\pm 10 %$ error bands. Tested flow conditions included offset heights $h = 2.9$ and 5.8 mm, gap width $s = 5.8 mm$ , $β = 1.0$ and 2.0, $y / h = 7.5$ and 15, $ϕ = - 45 °$ to $+ 45 °$ , and $ψ = 30 °$ .

Cracked Concrete Specimens

The cracked concrete specimens were tested at seven different flow rates for the first installation in which the smooth surface and straight crack were exposed to the flow, four flow rates for the same crack tested in the reversed flow direction, and three flow rates for the tests of the roughened side of the specimen with the more jagged crack shape. Fig. 16 presents the observed uplift pressures and those predicted using Eqs. (3)–(5). The results agreed within

\pm 10 %

, similar to the uncertainty observed in tests of regular square-edged joints by Wahl and Heiner (2024b). The average error in relation to the mean channel velocity head was

- 0.0085 h_{v}

, and the RMS variation of the differences was

0.0132 h_{v}

.

Fig. 16. Predicted versus observed uplift for cracked concrete specimens expressed as fractions of channel velocity head $h_{v}$ , with $\pm 10 %$ error bands. Test conditions were offset height $h \approx 11 mm$ , gap width $s \approx 17.5 mm$ , $β \approx 1.57$ , and $y / h = 2.44 - 8.27$ .

Flow into Joints

Testing of normally oriented square-edged joints (Wahl and Heiner 2024a) produced Eq. (7) for predicting the discharge coefficient

C_{d}

as a function of the velocity ratio,

V_{gap} / V_{chute}

. These discharge coefficients seem to primarily represent an entrance loss, with minimal influence of friction through the joint. Many of the modified joints discussed in this paper (skewed, beveled, relieved) generate less stagnation pressure than normally oriented square-edged joints, and thus also will cause less discharge through the joints even with no change in the discharge coefficient. However, limited testing of some of these modified joints showed that there was a general reduction of the discharge coefficient (with significant scatter) for nonrectangular joints (Fig. 17 and Table S2). For skewed orientations this probably is due to incomplete stagnation of the flow. For the joints with relief slots, the lack of an offset exactly at the joint also prevents true flow stagnation over the joint opening. Reduced discharge coefficients for the cracked concrete specimens probably were due to friction loss through the rough flow passage. No attempt was made to develop relations for predicting these reduced discharge coefficients; Eq. (7) will generally yield conservatively large estimates of the flow through a variety of modified joints. Flow testing was not performed for joints tilted into or away from the flow.

Fig. 17. Discharge coefficients of modified joints. Test conditions were offset heights $h = 0.7 - 12 mm$ , gap widths $s = 3 - 17 mm$ , $β = 0.26 - 4.2$ , and $y / h = 3.5 - 104$ .

Discussion

Prior tests of square-edged joints with regular perpendicular offsets into the flow have shown that large uplift pressures beneath concrete chute linings are possible, as well as large flow rates into joints when waterstops or other joint-filling materials are not present or effective (Wahl and Heiner 2024a, b). To mitigate against possible failure, existing poor joints and other defects should be treated to reduce or eliminate the effects of offsets into the flow. Elimination or reduction of offsets will reduce both uplift pressure and potential inflow through a joint. Several potential modifications were tested. Angled beveling of offsets was not highly effective until bevel angles are extremely flat, but the creation of relief slots holds significant promise for producing more pressure reduction with less volume of material removal for

r / h > 5

.

For unlined rock chutes, testing of skewed and tilted joints showed that both factors can change uplift pressure significantly; skewness always reduces uplift while tilt has the potential to either increase or decrease uplift. Some of the idealized configurations tested here may be relatively unlikely to occur in a rock-lined channel, except in localized cases, and may be transitory (Fig. 18). In addition, the existence of repeated, multiple offsets into the flow combined with tilts into or away from the flow could create complex flow conditions near the boundary that would differ significantly from the tests of isolated offsets reported here.

Fig. 18. Illustration of potential rock joint orientations: (a) offsets tilted toward and protruding into the flow; (b) offsets into the flow removed; (c) offsets tilted away but protruding into the flow; and (d) offsets removed.

Rock scour prediction methods have long recognized that the orientation of joints relative to flow direction is an important factor in erosion, even if solely due to additional degrees of rock block removal freedom provided by certain orientations. The rock rippability index of Kirsten (1982) and the headcut erodibility index of Annandale (1995) both contain a ground structure number that indicates that the most effective removal of rock occurs when the most closely spaced joint sets in a rock formation dip at about 30°–70° in the flow direction [Figs. 18(a and b)] or direction of ripping (i.e., with joint openings tilted into the flow). In this orientation the flow is effective at entering the joints, which also are oriented to enhance block removal. A second maximum occurs when primary joints dip against the flow or ripping direction by about 30°–60° [Figs. 18(c and d)]; in this orientation, primary joints may be less prone to erosion, but secondary joint sets still can provide freedom of movement that facilitates rock removal, even if pressures within primary joints are reduced by the joint orientation. In addition, the test results for joints tilted away from the flow showed that uplift forces remain relatively high until joints are tilted away from the flow at extreme angles.

Data from separate tests can be viewed as scale models of one another. For example, a

β

ratio of 1.0 can be produced with a 1-mm gap and 1-mm offset, or a 10-mm gap and 10-mm offset, and so forth. Many joints with similar

β

ratios were tested at differing scales. Similarly, tests with fixed joint size and geometry were performed at a wide range of depths, velocities, and discharges. Results from the full velocity and geometric scale range of these tests were represented accurately by the relations presented here, in which uplift pressure is normalized with respect to velocity head of the mean flow and velocity head within the boundary layer. As explained previously, these tests were all conducted in nonaerated flow conditions. Thus, as long as nonaerated flow is maintained, the relations presented here should be applicable to all velocities. Future research to test aerated flow conditions would be valuable, both for spillways that are naturally aerated and for those that are equipped with aerator structures to mitigate against cavitation damage.

For

β

ratios less than about 5, there also should be no limit to the application of these equations to joints of varying scale. For

β > 5

, there is a minor scale effect related to gap size and the nonlinear deterioration of the boundary layer that occurs as the approach flow crosses the open gap upstream from the offset face of the joint. This was discussed in detail by Wahl and Heiner (2024b).

Conclusions

Uplift pressures were measured at a variety of modified joints that may be encountered in spillway chutes, including joints that have chamfered or rounded edges, skewness relative to the flow direction, and tilts into or bevels away from the flow. The latter can be a typical remediation option for spillways with existing offsets into the flow, and this study also considered other geometries for that purpose, such as a novel relief gap that separates the joint opening from the offset itself. The experimental data were used to develop relations for predicting the increase or decrease of uplift pressure associated with modified joints. Testing of cracked concrete specimens showed that uplift pressures can be predicted accurately from the data already collected from normally oriented square-edged joints. Flow rates through most of the modified joints, including the cracked concrete specimens, should be lower than those that would be predicted for normally oriented square-edged joints.

Future spillway construction and repair practices can be informed by the results of this study. In particular, joints skewed 45° from a normal orientation experience less than half of the uplift of typical joints, although this may not be a practical change to incorporate into regular construction. The most effective means for eliminating existing spillway uplift issues is elimination of offsets into the flow. However, traditional beveled grinding of offsets is shown to have limited effectiveness unless bevel angles are extreme, with an 88° bevel needed to achieve 80% reduction of uplift. When the goal is more than 70% uplift reduction, relief slots are more effective than bevels that involve the same volume of material removal.

Notation

The following symbols are used in this paper:

$b$: curve-fitted parameter;
$c$: curve-fitted parameter;
$e$: base of natural logarithms, approximately 2.718;
$f$: Darcy–Weisbach friction factor;
$g$: acceleration due to gravity on Earth, $9.806 m / s^{2} (32.17 ft / s^{2})$ ;
$h$: offset height;
$h_{ch}$: chamfer height;
$h_{v}$: velocity head computed from mean channel velocity, $V^{2} / 2 g$ ;
$h_{v}^{*}$: velocity head of boundary layer flow between chute floor and the tip of offset;
$N$: exponent in velocity profile equation, $V \propto y^{1 / N}$ ;
$q_{j}$: discharge per unit width through joint or crack;
$R$: radius of rounded edges;
$r$: relief distance from downstream edge of open joint to face of offset;
$s$: gap width;
$V$: average velocity in spillway chute;
$w$: width of open joint across width of chute, perpendicular to flow direction;
$y$: flow depth;
$α$: energy coefficient relating true velocity head to $V^{2} / 2 g$ ;
$α^{*}$: energy coefficient relating $h_{v}^{*}$ to $V^{2} / (2 g)$ ;
$β$: joint gap width to offset height aspect ratio, $s / h$ ;
$Δ H$: uplift pressure head within sealed (nondrained) joint;
$Δ H_{b}$: uplift pressure head within simulated joint during joint flow;
$θ$: angle of chute slope below horizontal;
$κ$: von Kármán $constant \approx 0.4$ ;
$ϕ$: angle of joint rotation from perpendicular to chute surface (tilt); and
$ψ$: angle of joint rotation from perpendicular to flow direction (skewness).

Supplemental Materials

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Data Availability Statement

All data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

This work was jointly funded by the Bureau of Reclamation Dam Safety Office (Technology Development Program) and Research Office (Science and Technology Program). Model makers Jimmy Hastings, Jason Black, Jeffrey Falkenstine, and Dane Cheek constructed many of the unique test facility components. Trevor Stockton of the Concrete & Structural Laboratory performed the cracking of the concrete specimen. Joshua Mortensen in the Hydraulics Laboratory provided internal peer review.

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Information & Authors

Information

Published In

Journal of Hydraulic Engineering

Volume 150 • Issue 5 • September 2024

Copyright

This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Nov 12, 2023

Accepted: Mar 25, 2024

Published online: Jul 2, 2024

Published in print: Sep 1, 2024

Discussion open until: Dec 2, 2024

Authors

Affiliations

Tony L. Wahl, P.E., M.ASCE https://orcid.org/0000-0002-2081-2263 [email protected]

U.S. Dept. of the Interior, Bureau of Reclamation, Hydraulics Laboratory, 86-68560, P.O. Box 25007, Denver, CO 80225-0007 (corresponding author). ORCID: https://orcid.org/0000-0002-2081-2263. Email: [email protected]

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Bryan J. Heiner, P.E., M.ASCE [email protected]

U.S. Dept. of the Interior, Bureau of Reclamation, Hydraulics Laboratory, 86-68560, P.O. Box 25007, Denver, CO 80225-0007. Email: [email protected]

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Abstract

Practical Applications

Introduction

Literature Review

Experimental Facility—General

Experimental Facilities—Irregular Joint Configurations

Chamfered and Rounded Edges

Tilted Joints

Relieved Joints

Skewed Joints

Cracked Concrete Specimen

Test Procedures and Measurements

Analysis and Results

Chamfers and Rounded Edges

Tilted Away, or Beveled

Tilted into Flow

Relieved Joints

Skewed Joints

Combined Tilt and Skew

Cracked Concrete Specimens

Flow into Joints

Discussion

Conclusions

Notation

Supplemental Materials

Data Availability Statement

Acknowledgments

References

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