SANCOLD Conference 2017: “Management of Dams and Reservoirs in Southern Africa”

Centurion, Tshwane, South Africa, 15 to 17 November 2017 © SANCOLD, ISBN 978-0-620-76981-5

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STABILITY ANALYSIS OF A CONCRETE GRAVITY DAM TONGUE WALL

RO Cassells1, GD Roberts2

1. ARQ Consulting Engineers, Pretoria, South Africa 2. ARQ Consulting Engineers, Pretoria, South Africa

PRESENTER: RYAN CASSELLS

ABSTRACT

A concrete gravity dam tongue wall comprises a concrete structure that transitions from the concrete gravity spillway section of a dam, to the non-overflow earthen embankment section.

The stability analysis of a concrete gravity dam tongue wall is affected by lateral earth loads imposed on it due to the embankment fill, and a specific analysis methodology is not generally defined. The magnitude of the active and passive lateral earth load acting on the tongue wall is directly related to the deflection of the structure under such loading.

This paper discusses how active and passive lateral earth loads were applied by progressive mobilisation of wedge forces in the fill, and proposes a defined methodology for the stability evaluation of tongue walls.

1. BACKGROUND

The Nwamitwa Dam is being developed by the DWS to increase the assurance of yield and enhance the water resources of the Groot Letaba River in the Limpopo Province. ARQ was subcontracted by LTE Consulting to prepare the tender design for the dam.

The dam is designed as a concrete gravity dam with zoned earthfill embankments as indicated in Figure 1 . The central concrete section is 350 m in length with a maximum height above foundation of 43.5 m. The concrete section comprises a 54 m long tongue wall on the right flank, a 190 m long RCC gravity spillway and a 96 m long tongue wall on the left flank. The zoned earthfill embankments are 1900 m in length on the right flank and 1200 m in length on the left flank.

Figure 1. Nwamitwa Dam Layout

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2. INTRODUCTION

The Nwamitwa Dam incorporates tongue walls into the design as an interface between the gravity concrete section and the embankment section. The tongue wall is a concrete gravity structure that transitions from the concrete gravity spillway section to the earthen embankment section as can be seen in Figure 2.

Figure 2. Plan view of the Nwamitwa Dam right bank tongue wall

The cross-section profile of the tongue wall gradually changes from a wide section like that of the NOC section, to a narrow section flanked on either side by the embankment dam fill as indicated in Figure 3. The tongue wall allows for the gradual lowering of the embankment dam fill level by way of wrap-around along its axis.

Figure 3. Side profile section of right bank tongue wall wide and narrow sections

The tongue wall needs to be designed for stability. There is currently no well-developed method used for determining the stability of a dam tongue wall supported laterally on either side by earth fill.

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3. METHODOLOGY

3.1 Gravity Dam Stability Analysis

The stability analysis of the tongue wall was undertaken based on principles as required for stability analysis of a concrete gravity dam. The relevant loading conditions were defined as per the Engineering Manual Gravity Dam Design (USACE, 1995). The tongue wall was analysed under each loading condition to determine various results that were compared to stability design criteria.

Most of the loadings on the tongue wall are similar to that of the spillway section and are determined simply from first principle mechanics. The typical loads imposed on a gravity dam are indicated in Figure 4. The seismic loadings are applied as pseudo-static loads.

Figure 4. Typical loads imposed on a gravity dam

3.2 Lateral Earth Loads of a Retaining Structure

An earth retaining wall is subject to lateral earth loads imposed by the earth it supports. Lateral earth pressures imposed onto a retaining structure are directly related to the magnitude of movement due to rotation of the structure. An active pressure is associated with lateral dilation of the soil and a passive pressure is associated with lateral compression. A fill that causes a soil retaining structure to deflect exists in an active state and a fill resisting deflection exists in a passive state as shown in Figure 5. If an earth fill causes no deflection on a retaining structure, ie no expansion or compression is experienced, the fill is said to be in an at-rest state.

The active, passive and at-rest lateral earth pressure coefficients are represented by Ka and Kp and Ko respectively. These coefficients are used to calculate the lateral pressure load of the fill on the retaining wall.

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Figure 5. Active and passive soil states as a result of deflection of wall

The peak Ka, Kp and Ko values for a given sloped earth fill can be calculated using equations 3-45, 3-47 and 3-4 as per the Engineering Manual Retaining and Flood Walls (USACE, 1989).

𝐾𝑎 =cos2(𝜙 − ψ − θ)

cos ψ cos2 𝜃 cos(ψ + θ + δ) [1 + √sin(𝜙 + 𝛿) sin(𝜙 − ψ − β)cos(𝛽 − 𝜃) cos(ψ + θ + δ)

]

2

𝐾𝑝 =cos2(𝜙 − ψ + θ)

cos ψ cos2 𝜃 cos(ψ − θ + δ) [1 − √sin(𝜙 + 𝛿) sin(𝜙 − ψ + β)cos(𝛽 − 𝜃) cos(ψ − θ + δ)

]

2

𝐾𝑜 = 1 − sin 𝜙

Where:

Φ = Internal friction angle Ψ = Seismic inertia angle θ = Inclination of wall with respect to vertical δ = Wall friction angle

β = Inclination of soil surface

The graph in Figure 6 indicates a general relationship between the movement of the earth retaining structure and the extent of mobilisation of the active and passive pressures in the fill. It is clear from the graph that a considerable amount of rotational deflection is needed for a peak passive pressure to be achieved. It cannot be assumed that peak active and passive pressures will be applicable. It is necessary to develop a more detailed assessment of the earth pressures based on the extent of mobilisation.

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Figure 6. Relationship between wall movement and mobilisation of earth pressures (Rhodes and Moses, 2015)

The ranges of wall movement (Δx) for cohesive soil types, as a ratio of the wall height (H) required to develop peak active and passive earth pressures for cohesive soils are indicated in Table 1.

It is evident that passive earth pressures require significantly more wall movement to reach their peak values than that needed for the active earth pressures to reach their peak values.

Table 1. Wall movement required to develop peak active and passive earth pressures (SAICE, 1989)

Soil Type Movement of Wall (Δx/H)

Active Passive

Stiff cohesive soils 0.002 – 0.01 0.01 – 0.02

Soft cohesive soils 0.005 – 0.02 0.02 – 0.04

Using the values given Table 1 and applying a curve fit to Figure 6 the extent of mobilisation for active and passive earth pressures were normalised and are indicated in Figure 7. This graph is useful in determining the degree of active and passive mobilisation for a given displacement,

Figure 7. Normalised active and passive wedge mobilisation

0

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edge M

obilis

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Normalised Wedge Mobilisation vs Displacement

Active

Passive

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3.3 Application to Tongue Wall Stability

The tongue wall has fill on either that imposes loads on it by way of lateral earth pressures. The lateral earth pressures applied on the tongue wall comprise active, passive and at-rest pressures which are a function of mobilisation due to deflection.

For the tongue wall to overturn or slide, a failure mechanism would need to develop within the earth fill. The tongue wall behaves like an earth retaining wall with fill on both sides.

The following is noted:

• When the dam reservoir is empty, the lateral earth pressures applied by the earth fill on either side of the dam structure will be similar, and the dam structure will experience negligible deflection. The earth fill will be in a near at-rest state.

• When the dam structure is loaded with the hydrostatic loading of the impounded reservoir, the load on the upstream face of the dam will cause the structure to deflect in the downstream direction. This deflection will lead to the development of active and passive earth pressures in the fill on the upstream and downstream of the dam respectively.

• The amount of movement the dam experiences will determine the degree with which such pressures change from at-rest Ko to peak active Ka and peak passive Kp.

3.4 Analysis Approach

To determine the extent of active and passive wedge mobilisation under loading, the tongue wall is modelled as a cantilever with a varying section profile using elemental beam theory. The various loadings, including the lateral earth loads, are applied to the tongue wall as indicated in Figure 8 and it is analysed for rotational deflection. The deflection experienced by the tongue wall is then used to determine the lateral earth loads acting on it.

Figure 8. Modelling of the dam structure for displacements

This analysis is undertaken using an iterative approach. The initial iteration is undertaken applying the lateral earthfill load under at-rest conditions. The second iteration is run applying the lateral earthfill load (as well as the hydrostatic load) under active and passive conditions, based on the displacement of the dam in the initial analysis.

Further iterations are undertaken applying revised lateral earth pressures based on dam displacements from the prior result. This iterative process is followed until the dam displacements converge. When the dam displacement converges, the active and passive lateral earth loads corresponding to such displacement are used for the stability analysis of the tongue wall. The dam stability analysis is then carried out.

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The iterative process is streamlined by an automated programme written in excel or similar software package with programming capability. The programme is set up to determine the amount of deflection and mobilisation required to achieve given design criteria under different loading conditions.

4. IMPLEMENTATAION OF ANALYSIS

The defined loading conditions for the tongue wall are provided in Table 2. All load cases include the lateral earth loadings. The tongue wall was analysed with undrained uplift conditions.

Table 2. Loading Conditions for Analysis

Load Case

Loading Condition

Upstream Water Level

Tail water Level

Self-weight

Embankment Fill Load

Uplift Seismic

1 Normal nil nil Yes Yes N/A -

2 Normal FSL Min Yes Yes Undrained -

3 Unusual RDF RDF Yes Yes Undrained -

6 Unusual FSL Min Yes Yes Undrained OBE

7 Extreme SEF SEF Yes Yes Undrained -

8 Extreme FSL Min Yes Yes Undrained MCE

The stability analysis design criteria are provided in Table 3 and Table 4. The allowable stress refers to the allowable stress under the base of the dam and the tensile joint strength.

Table 3. Stability Analysis Design Criteria

Stability Condition Loading Condition Required Factor of Safety

Sliding Normal 2.0

Unusual 1.7

Extreme 1.3

Overturning Loading Condition Location of Reaction

Normal Middle Third

Unusual Middle Half

Extreme Inside Base

Table 4. Allowable Stresses for Gravity Dam Structures

Stress Condition Loading Condition Allowable Stress

Compression Normal 0.30 * fc

Unusual 0.50 * fc

Extreme 0.90 * fc

Tension Normal 0

Unusual 0.15 * fc(2/3)

Extreme 0.05 * fc(2/3)

Fc = 180 day concrete cube strength in MPa.

The material properties and input parameters for the dam and foundation are provided in Table 5.

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Table 5. Material properties and input parameters for the dam and foundation

Material Property Value

Concrete Elastic Modulus (E) 16.5 GPa

Concrete Density 2450 kg/m3

Residual Base Friction (ϕ) 37 °

Residual Base Cohesion (c) 70 kPa

OBE PGA (Kijko, 2016) 0.025g

MCE PGA (Kijko, 2016) 0.200g

Pseudo-static coefficient factor (Haynes, et al, 1984) 0.5

Westergaard Ce (Kroon, 1984) 0.81 kN.sec.m

The material properties for the earth fill of the embankment section of the dam (Figure 3) are provided in Table 6. These properties were obtained from geotechnical testing done on fill materials to be used.

Table 6. Material properties of embankment earthfill

Fill layer Φ (deg) C (kPa)

γ-sat (kN/m3)

γ-eff (kN/m3)

δ

(degrees)

Fine Rockfill 40 0 21.58 11.77 13.33

Impervious 24.7 8.7 15.79 5.98 8.23

Semi-pervious 28.3 8 18.30 8.49 9.43

The Nwamitwa Dam tongue wall was analysed for stability using the various loadings, material parameters and input parameters provided.

For purposes of this analysis, the allowable tensile stress under the heel of the dam was set as the design criteria. The automated analysis programme was configured to determine crest deflection required to achieve this given bearing stress.

5. REVIEW OF RESULTS

The stability analysis results for the various loading conditions are provided in Table 7. It is evident from the results that the dam tongue wall is stable under all the defined loading conditions. It can be seen in the results that the maximum tensile bearing stress was used as the design criteria for the various loading conditions.

Table 7. Stability analysis results for tongue wall

Loading Condition

FoS Sliding

FoS Overturning

Heel Bearing

Stress (kPa)

Toe Bearing Stress (kPa)

Resultant Position

(%)

Section of Base of

Resultant

1 21.95 2.39 1095 735 46.73 Middle Third

2 5.20 1.36 0 1384 66.67 Middle Third

3 3.98 1.21 -300 1557 74.62 Middle Half

5 4.04 1.26 -300 1661 74.02 Middle Half

7 2.51 1.06 -900 2111 91.44 Within Base

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Loading Condition

FoS Sliding

FoS Overturning

Heel Bearing

Stress (kPa)

Toe Bearing Stress (kPa)

Resultant Position

(%)

Section of Base of

Resultant

8 2.88 1.08 -900 2192 89.88 Within Base

Note: - = tension += compression

The amount of deflection of the tongue wall required to mobilise sufficient passive earth resistance for it to be stable under the various loading conditions is indicated in Table 8. The extent of active and passive wedge mobilisation required for it to be stable is also indicated.

Table 8. Deflection required and associated wedge mobilisation

Loading Condition

Defection at Crest (mm)

Active Wedge Mobilisation at Crest

(%)

Passive Wedge Mobilisation at Crest (%)

1 Negligible Negligible Negligible

2 53.75 93 25

3 67.35 97 30

5 90.70 98 43

7 18.41 37 7

8 376.03 100 87

Note: - deflection = upstream direction + deflection = downstream direction

At first, it seems illogical that some loading conditions with greater destabilising loads require less passive wedge mobilisation to attain stability than those with lesser de-stabilising loads. This is due to the higher allowable tensile stress required for the tongue wall to be stable under the load condition specified.

The active and passive wedge mobilisation, lateral earth pressure and deflection (displacement) under load case 2 are indicated in Figure 9.

6. CONCLUSION

There is currently no well-developed method used for determining the stability of a dam tongue wall supported laterally on either side by earth fill. Various simplistic methods for determining lateral earth loads from and embankment fill have been used. The method provided in this paper improves upon simplistic methods using geotechnical theory of lateral earth pressures.

The lateral earth pressures imposed on a tongue wall by an embankment fill should not simply be calculated assuming peak active and passive earth pressures, as is generally done in retaining wall design. The actual active and passive earth pressures imposed on a soil retaining structure are directly related the rotation movement of the structure. To accurately define the loads imposed on a soil retaining structure, the movement of the structure under such loads needs to be determined, resulting in the need for an iterative approach to be undertaken.

Upon correctly defining the lateral earth loads imposed on the tongue wall the stability analysis can be correctly carried out and results compared to design criteria.

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Figure 9. Summary of dam and earthfill interaction for loading condition 2

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7. ACKNOWLEDGEMENTS

The authors would like to thank LTE Consulting and the DWS for allowing the work undertaken on the Nwamitwa Dam to be used for this paper.

8. REFERENCES

Hynes-Griffin, ME and Franklin, AG. 1984. Rationalizing the Seismic Coefficient Method. Department of the Army Corps of Engineers

Kijko, A. 2016 Probabilistic Seismic Hazard Assessment for the site of Nwamitwa Dam, Limpopo Province, South Africa.

KROON, J. 1984. Swartekrag Damstrukture. University of Pretoria.

Nwamitwa Dam Design Criteria Memorandum, Report Number 7602/15357, November 2016, ARQ Consulting Engineers and LTE Consulting.

Rhodes, S. and Moses, J. 2016. Integral Bridges and the Modelling of Soil-Structure Interaction, LUSAS.

SAICE. 1989. Lateral Support in Surface Excavations (Code of Pratice).

United States Army Corps of Engineers. June 1995. Gravity Dam Design. Engineering Manual, EM 1110-2-2200.

United States Army Corps of Engineer. September 1989. Retaining and Flood Walls. Engineering Manual, EM 1110-2-2502.

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