stability evaluations of concrete dams - log into your...

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Stability Evaluations

of

Concrete Dams

Guy S. Lund, P.E.

Robert A. Kline, Jr., P.E.

Image Source: US Bureau of Reclamation

Concrete Dam Examples:• “Solid” Gravity Dam

Image Source: Wikipedia - Mark Quadling, Henley Quadling


• “Hollow” Buttress Dam

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Image Source: US Bureau of Reclamation



• Arch Dam



• Arch Dam

• Hydraulic Structures

Image Source: US Army Corps of Engineers

Sazilly - 1853

P

W

R

• From Retaining Wall Studies

• Unit Width – Vertical Cantilever

• Failure Modes

� Sliding

� Overturning

• Linear Internal Stress Distribution

• Equal Max. Vertical Compressive Stress

� Reservoir Full

� Reservoir Empty

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1) Resist Horizontal Sliding– (0.67 – 0.75) OR (33̊ - 36̊)

2) Limit Compressive Stresses– Max. Principal Stresses Parallel to Face

3) No Tension – Middle Third Rule

Design Load Cases– Reservoir Empty– Reservoir Full

Rational Design Method

PW

R

1853 Sazilly

1861 Delocre

1881 Rankine

1888 Wegmann

1) Increase Compressive Limit

2) Ignore “Equal Resistance” Rule

3) Increase Assumed Unit Weight

50 Meters

(164 feet)

Profile Comparison

Gravity DamTypical Analysis Sections:

• Non-Overflow

• Overflow

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Gravity Dam

Typical Construction Methods:• Monolithic Blocks (Conventional)

Typical Construction Methods:• Monolithic Blocks (Conventional)

• Horizontal Layers (RCC)

Design Layout / Analysis

– Layout establishes structural profile, section and details

• Vertical Monolith Joints

• Horizontal Lift Joints

• Drainage Galleries

• Water Passages

• Foundation Treatments

– Seepage Cutoff Wall

– Grout Curtain

– Drainage Curtain

– Analysis determines internal stresses and safety against overturning and sliding

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NWS

Crest of Dam, El. 280 ft

0.75

1

TWS

Sediment

El. 265 ft

El. 240 ft

Typical Design Features

Engineered

Materials

Natural

Materials

NWS

0.75

1

TWS

Sediment

El. 265 ft

El. 240 ft Monolith

Contraction Joint

Drainage

Gallery

Waterstops

Horizontal

Construction Joint

Internal Drain Line

Foundation Drain

Typical Design Features

Monolithic Construction

– Vertical Contraction Joint

• Double waterstop with

internal drain line

• Can be keyed and grouted to

transfer load to abutments

– Horizontal Construction Joint

• Length approx. 50 ft

• Height varies 5-10 ft

• Green cutting using air/water

jet prior to placement

Monolith

Contraction

Joint

Drainage

Gallery

Waterstops

Horizontal

Construction Joint

Internal Drain Line

Foundation Drain

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Drainage

Gallery

Waterstops Horizontal

Construction Joints

Internal Drain Line

Foundation Drain

Minimal Distance

Minimal

Distance

Loads

Static Loads

• Dead Load

• Reservoir

• Sedimentation

• Tailwater

• Backfill

• Ice

• Uplift

• Subatmospheric (Spillway)

• Wind

• Temperature

Dynamic Loads (Seismic)

• Inertia Forces due to

acceleration

• Hydrodynamic forces due to

reservoir/dam interaction

Loads

Dead Load

• Function of concrete mix design

– Aggregates

• Unit Weights

– Vary 145 lb/ft3 – 155 lb/ft3

– As low as 135 lb/ft3 in older

masonry (cyclopean) concrete

dams

– Recommended 150 lb/ft3 Wc

centroid

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Loads

Reservoir Load

• Hydrostatic pressure

– Based on normal reservoir

operation

• Minimum level

• Maximum Normal Level

• Inflow design flood level

– PR = H X 62.4 lb/ft3

NWS

PR

WR

H

Loads

Sedimentation Load

• Hydrostatic pressure due to

the effective weight of

sedimentation

– Effective horizontal unit weight

• 85 lb/ft3

– Effective vertical unit weight

• 120 lb/ft3

PS

WS

Sediment

Loads

Tailwater Load

• Hydrostatic pressure on

downstream face that

corresponds to the reservoir

level

– Minimum level

– Maximum Normal Level

– Inflow design flood level TWS

PT

WT

NWS

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Loads

Tailwater Load

• Overflow Section

– Retrogression (FERC, USBR, USACE)

• Account for air entrained flow

• 60 percent of total height

• Non-Overflow Section

– Overtopping Tailwater

Flood Level

TWS

PT

WT

Loads

Backfill Load

• Soil Backfill placed against the dam

– At-rest horizontal coefficient

Backfill

PB

WB

Loads

Passive Resistance Load

• Passive Rock Wedge

– Residual Strength (Interface/Foundation)

– Rock Mass Strength (Passive Wedge)

WP

PP

Weak Interface Plane

ΣH

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Loads

Ice Load

• Due to thermal expansion

and wind drag

• For United States, maximum

depth of 2 feet

– 5,000 lb/ft2

PiNWS

Loads

Wind Load

• Considered for construction

load case

• Uniform load applied to

downstream face

– 30 lb/ft2

PW

Loads

Temperature Load

• Most severe temperature

load is during construction

– Concrete cooling

– Typically not considered in

two-dimensional studies

• Transient effects due to

temperature fluctuations

– Reservoir

– Ambient Air

– Solar radiation

NWS

Month

Te

mp

era

ture

20

40

60

80Daily

Maximum

Daily

Minimum

Jan Mar May Jul Sep Nov

35 40 45 50 60 65

3550

3500

3450

3400

3350

3300

3250

3200

3150

3100

Range of

Mean

Reservoir

Water

Temperatures

Temperature

Ele

va

tio

n

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Loads

Subatmospheric Load

• Negative pressure created

when spillway discharge

exceeds design head

Reservoir Level

TWS

Loads

Uplift Load

• Undrained Condition

– Vary linearly from full

reservoir head (HR) to full

tailwater head (HT)

Reservoir Level

L

TWS

γHR

γHT

Loads

Uplift Load

• Undrained Cracked Condition

– Full Reservoir pressure within

crack length (T)

– Vary linearly from full reservoir

head (HR) at tip of crack to full

tailwater head (HT)

Reservoir Level

L

TWS

T

γHR

γHT

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Loads

Uplift Load

• Drained Condition

– Assume reduced head at the

foundation drains, H3

X

K = ( 1 – E )

X = Distance from heel to drains

Reservoir Level

L

TWS

E= Drain Effectiveness

H3 = K (HR – HT) + HT

L = Base length

H3

γHR

γHT

Loads

Uplift Load

• Drained Crack Condition


crack length (T)



H3 = K [ (HR – HT) ] + HT

X

T = Crack Length


( L – X )

( L – T )

K = ( 1 – E )

Reservoir Level

L

TWS

T

H3

γHR

γHT

Loads

Uplift Load

• Drained Crack Condition


crack length (T)



H3 = K (HR – HT) + HT

X

T = Crack Length


K = ( 1 – E )

Reservoir Level

L

TWS

T

H3

γHR

γHT

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Loads

Seismic Loads

• Inertia forces on dam NWS

Pe

WVi

Me

WHi

y

H

te = earthquake period, typically

taken as 1 second

= seismic acceleration

Loads

Seismic Loads

• Inertia forces on dam

Hydrodynamic Pressure Coefficient

θ

te = earthquake period, typically

taken as 1 second

= seismic acceleration

Load Combinations• Usual Load Combinations

– Normal reservoir elevation with appropriate dead loads, sedimentation, ice, tailwater, uplift, and temperature (if applicable)

• Unusual Load Combinations

– Maximum design reservoir elevation with appropriate dead loads, sedimentation, tailwater, uplift and appropriate temperature (if applicable)

• Extreme Load Combinations

– The usual loading plus effects of the maximum credible earthquake (MCE)

• Other Load Combinations

– Construction loading conditions

– Post-earthquake loading conditions

– Historic Record Event

– Interim Loading Condition (For Worst Case or Risk-Based Analyses)

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USACE Stability Criteria1) Sliding Resistance

2) Allowable Stress

3) Resultant Location Load

Case

Resultant

Location

Usual Middle 1/3

Unusual Middle 1/2

Extreme Within Base

Load

Case

Sliding Safety Factor

(Limit Equilibrium)

Well-

Defined

Ordinary

EQ No EQ

Usual 1.7 2.0 2.0

Unusual 1.3 1.5 1.7

Extreme 1.1 1.1 1.3

Load

Case

Internal

Stress

Usual σ = allowable w/

safety factor

Unusual σ = 115% of

usual allowable

Extreme σ = 150% of

usual allowable

Reference: EM 1110-2-2100 – Stability Analysis of Concrete Structures – 2005

Multiple Failure Planes

Limit Equilibrium Method

Single Horizontal Failure Plane

Shear-Friction Method

FERC Stability Criteria

Reference: FERC Engineering Guidelines – Chapter 3 – Gravity Dams – 2002

1) Sliding Resistance

2) Allowable Stress


Case

Resultant Location

Existing Dam Repair Design

Usual Within Base Middle 1/3A

Unusual Within Base Middle 1/3A

Extreme N/A N/A

Load

Case


C > 0 C = 0

Usual 3.0 1.5

Unusual1 2.0 1.3

Extreme 1.3 1.3

Load

Case

Internal Stress

Intact/Dam Cracked/Dam Interface/Foundation

Usual σc < f'c / 3.0

σt = 1.7 f'c2/3

σc < f'c / 3.0

τ < 1.4 σn

σt = 0

σc < σult / 3.0

σt = 0

Unusual σc < f'c / 2.0

σt = 1.7 f'c2/3

σc < f'c / 2.0

τ < 1.4 σn

σt = 0

σc < σult / 2.0

σt = 0

Extreme

N/A

Post-Seismic

N/A

Post-Seismic

N/A

Post-Seismic

1 SF = 1.3 if SDF=PMF, C=0

A To Extent Possible


R = ΣVtanΦ + CA

USBR Stability Criteria

Reference: USBR Design of Small Dams 1987

1) Sliding Resistance

2) Allowable Stress


Case

Resultant Location

New Dam Existing Dam

Usual Middle 1/3A Middle 1/3A

Unusual Middle 1/3A Within BaseA,B

Extreme Within BaseA Within BaseA

Load

Case


Dam/

Interface

Foundation

C > 0 C = 0

Usual 3.0 4.0 2.0

Unusual 2.0 2.7 1.5

Extreme 1.0 1.3 1.0

Load

Case

Internal Stress

Dam Interface/Foundation

Usual

σc < f'c / 3.0

σc < 1,500 psi

σt = 0

σc < σult / 4.0

σt = 0

Unusual

σc < f'c / 2.0

σc < 2,250 psi

σt = 0

σc < σult / 2.7

σt = 0

Extreme

σc < f'c /1.0

σt allowed

σc < σult / 1.3

σt = 0

A Inferred by Stress Criteria

B Drains Inoperative


R = ΣVtanΦ + CA

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BREAK # 1

Questions

Image Source: Development of Dam Engineering in the United States, 1988

Material Properties

• Concrete– Compressive Strength

– Tensile Strength

– Density

– Poisson’s Ratio

– Modulus of Elasticity• Sustained

• Instantaneous

– Diffusivity

– Coefficient of Thermal Expansion

– Shear Strength• Intact Concrete

• Bonded and Unbonded lift lines

– Damping

• Foundation– Modulus of Deformation

– Poisson’s Ratio

– Density

– Shear Strength

Concrete Strength

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Stages of Stress vs. Strain

• Stage I– Small deformations and strains

– linear elastic behavior

• Stage II– Some inelastic behavior

– Minor permanent deformations (hysteresis)

• Stage III– Large inelastic strains

– Observed deformation

– Stable crack growth

• Stage IV– Fracture stage

– Unstable crack growth

– Failure

ST

RE

SS

STRAINReference: USACE, EM 1110-2-6051

Static tensile strength

ft= 1.7 f’

C

2/3

E1

A

D

B

C

Stage I

Stage II

Stage III

Stage IV


• Stage I– Small deformations and strains

– linear elastic behavior

• Stage II– Some inelastic behavior

– Minor permanent deformations (hysteresis)

• Stage III– Large inelastic strains

– Observed deformation

– Stable crack growth

• Stage IV– Fracture stage

– Unstable crack growth

– Failure

ST

RE

SS

STRAINReference: USACE, EM 1110-2-6051


ft= 1.7 f’

C

2/3

B

E1

A

D

B

C

Stage I

Stage II

Stage III

Stage IV

Apparent seismic tensile strength

2ft= 3.4 f’

C

2/3

Seismic tensile strength

ft= 2.6 f’

C

2/3

Concrete Compressive Strength

• USACE

– Usual

• 35 % f’C (3.0)

– Unusual

• 50 % f’C (2.0)

– Extreme

• 90 % f’C (1.1)

Reference: USACE EM 1110-2-2200

ST

RE

SS

STRAIN

35 % f’c

50 % f’c

90 % f’c

E1

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Concrete Tensile Strength

USACE• Usual

– Zero (0)

• Unusual– 0.6 f’C

2/3 (2.8)

• Extreme– 1.5 f’C

2/3 (1.1)

FERC• Usual

– 0.57 f’C2/3 (3.0)

• Unusual– 0.85 f’C

2/3 (2.0)

• Post-Earthquake– 1.3 f’C

2/3 (1.3)

ST

RE

SS

STRAIN

Reference: USACE EM 1110-2-2200

FERC Chapter 3

0.6 f’C

2/3

0.85 f’C

2/3

1.5 f’C

2/3

E

1

B

A

D

C

A

D

B

C


USACE

• Extreme

– 3.4 f’C2/3 (2 x ft)

ST

RE

SS

STRAIN

B

Reference: USACE, EM 1110-2-6051 E1


ft= 1.7 f’

C

2/3

A

D

B

C

Apparent seismic tensile strength

2ft= 3.4 f’

C

2/3

Seismic tensile strength

ft= 2.6 f’

C

2/3

Allowable Stress of Concrete

• Usual Load (Normal)

– Linear elastic behavior 33 %

• Unusual Load (Flood)

– Primarily linear elastic behavior,

some hysteresis 50 %

• Extreme Load (Seismic)

– Non-linear behavior with permanent damage,

below fracture stage

– Increase strength due to rapid load 90 %

• Post-Earthquake Load

– Non-linear behavior with permanent damage,

below fracture stage 75 %

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Foundation Shear Strength

• Traditionally shear

strength parameters

based on Mohr’s Circle

– Fiction Angle

– Cohesive Strength

• Current practice

– Neglects effect of

cohesion

E. Hoek & E.T.Brown, Underground Excavations in Rock


• Estimating shear

strength parameters

– Direct shear tests

• Small samples

– In Situ tests

– Empirical methods

• Account for first and

second order undulations

along failure surface

E. Hoek & J.W.Bray, Rock Slope Engineering


• Estimating Effective

Friction Angle

– 1st order roughness

• Major undulation

– 2nd order roughness

• Bumps on the surface

– Effective Friction Angle

•

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Start Simple

• Simplified studies typically conservative

• Verification

• Stop if conservative study indicates adequate

safety

• Sharpen the pencil, when necessary

Simplified Study

• Gravity Method of analysis

– Two-dimensional

– Rigid body mechanics (SF = 0, SM = 0)

– Stress distribution based on beam theory

– Static analysis

Two-Dimensional Study

• Gravity Method

• Pseudo Dynamic

– Simple dynamic study

– Capture basic dynamic behavior of structure

• Finite Element Method

– Complex geometric and material behaviors

– Dynamic Behavior

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Three-dimensional Studies

• Capture cross-canyon behavior– Monolithic interaction

• Trial-Load Method of Analysis– Divides dam into horizontal (arch) and vertical (cantilever)

elements

– Solves separate structural elements, and adjusts until the deformation at coincident points equal

• Finite Element Analysis– Linear and non-linear geometric and material behavior

– Implicit and Explicit

• Finite Difference Analysis (FLAC)– Large strain/plastic systems (soils)

Analysis driven by Answer

• Analysis type based on desired answer

• Failure Modes

– Structural Capacity (overstressing)

• Cracking or crushing of concrete

• Potential load redistribution within structure

– Structural Stability (SF = 0, SM = 0)

• Overturning

• Traditional sliding stability

• Rock block stability

• Rock Scour due to overtopping

Gravity Analysis

Simple Gravity Dam

• Crest, El. 280.0

• Base, El. 160.0

• Slope,

– U/S 0.10: 1.0

– D/S 0.75: 1.0

• gc = 145 pcf

• NWS, El. 275.0

• TWS, El. 183.0

• Sediment, El. 200.0

NWS

Crest of Dam, El. 280 ft

0.75

1

TWS

Sediment

El. 265 ft

El. 240 fte

e

e

e

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Gravity Analysis

Simple Gravity Dam

• Loads

– Gravity, Reservoir, Sediment,

Ice, Tailwater, Uplift, etc

• Foundation Considerations

– Strength Parameters

• Stress Evaluation

– Cracking or crushing

• Stability

– Overturning

– Stability

W

Gravity Analysis

Calculated Loads

Load FX FY MToe

(kip) (kip) (k-ft)

Gravity -- (828.1) (51,047)

Reservoir 413.3 (37.5) 12,319

Sediment 28.1 (7.2) (218)

Tailwater (16.5) (12.4) (198)

Uplift -- 419.5 24,940

Sum 424.9 (465.6) (14,205)

W

Gravity Analysis

Stress Evaluation

• Beam theory

• Area, A

• Moment @ center, Mc

• Moment of inertia, I

• Distance from center to edge

of base, c

P

As = ±

Mc c

I

+4 psi -72.2 psi

-70 psi

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Gravity Analysis

Stress Evaluation

• Beam theory

• Area, A

• Moment @ center, Mc

• Moment of inertia, I

• Distance from center to edge

of base, c

P

As = ±

Mc c

I

SM = 0

+4 psi -72.2 psi

-70 psi

Gravity Analysis

• Cracked Base Analysis

– Dam/Foundation

Contact

• No tension zone

W

W

Gravity Analysis


– Dam/Foundation

Contact

• No tension zone

– Crack develops

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Gravity Analysis


– Dam/Foundation

Contact

• No tension zone

– Crack develops

– Uplift increases

• Increases load

• Increases moment

• Re-calculate stress

distribution

W

Gravity Analysis

Revised loads

Load FX FY MToe

(kip) (kip) (k-ft)

Gravity -- (828.1) (51,047)

Reservoir 413.3 (37.5) 12,319

Sediment 28.1 (7.2) (218)

Tailwater (16.5) (12.4) (198)

Uplift -- 419.5 24,940

Sum 424.9 (465.6) (14,205)

Uplift -- 455.6 27,128

Total 424.9 (429.5) (12,017)

W

Gravity Analysis

• Structural Stability

– Moment Equilibrium

• No crack propagation

– Sliding Stability

• Area of intact base

• Normal Force

• Driving Force

• Effective angle of friction

• Apparent cohesion

FX

F.O.S.= A c + FY Tan(F)

Lu = 84.0 ft

Lc = 13.3 ft

L = 97.3 ft

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Gravity Analysis

• Structural Capacity


– Sliding Stability

– Stress Evaluation

• Comparison of computed

stress with concrete

strength

+0 psi -71.1 psi

-33psi

-25 psi

-48psi

-20 psi

Finite Element Analysis

• Use same dam

– Generate finite element

model

– Verify loads & moments

Comparison of Results

Gravity Analysis

Load FY FX MToe

(kip) (kip) (k-ft)

Gravity (828.1) -- (51,047)

Reservoir (37.5) 413.3 12,319

Sediment (7.2) 28.1 (218)

Tailwater (12.4) (16.5) (198)

Uplift 419.5 -- 24,940

Sum (465.6) 424.9 (14,205)

Finite Element Analysis

Load FY FX MToe

(kip) (kip) (k-ft)

Gravity (828.0) -- (51,047)

Reservoir (37.5) 412.3 12,248

Sediment (7.8) 78.3 549

Tailwater (12.4) (16.2) (198)

Uplift 419.5 -- 24,940

Sum (466.2) 474.1 13,507

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Gravity Method Finite Element Method

+4 psi -70 psi+24 psi -62 psi



+4 psi -70 psi

-40 psi -63 psi -40 psi -56 psi

+24 psi -62 psi



+4 psi -70 psi

-40 psi -63 psi

-34 psi -49 psi

-40 psi -56 psi

-36 psi -48 psi

+24 psi -62 psi

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+4 psi -70 psi

-40 psi -63 psi

-34 psi -49 psi

-28 psi -35 psi

-40 psi -56 psi

-36 psi -48 psi

-27 psi -33 psi

+24 psi -62 psi



+4 psi -70 psi

-40 psi -63 psi

-34 psi -49 psi

-28 psi -35 psi

-25 psi -20 psi-26 psi -20 psi

-40 psi -56 psi

-36 psi -48 psi

-27 psi -33 psi

+24 psi -62 psi



+4 psi -70 psi

-40 psi -63 psi

-34 psi -49 psi

-28 psi -35 psi

-25 psi -20 psi

-13 psi -16psi-15 psi -13 psi

-26 psi -20 psi

-40 psi -56 psi

-36 psi -48 psi

-27 psi -33 psi

+24 psi -62 psi

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+0 psi -73.2 psi

-33psi

-25 psi

-48psi

-20 psi

84.7-71 psi

-66 psi

12.6

97.3 ft

82.215.0

Structural Analysis

Static Results Evaluation

• Overstressing Criteria

– Compare computed stress to the allowable strength of concrete

• Overturning Criteria

– Compare stress distribution along base with required criteria


• Sliding Criteria

– Compare computed Factor of Safety with Minimum

-71 psi

-40 psi -63 psi

-34 psi -49 psi

-28 psi -35 psi

-25 psi -20 psi

-13 psi -16psi

FOS = c A + FY TAN ( fe )

FX

= 1.1

BREAK # 2


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Seismic Analysis of Concrete Dam

• Traditionally designed based on the seismic

coefficient method

– Simple method for preliminary studies

– Considered inadequate because it fails to

recognize dynamic behavior of the dam

Dynamic Behavior

Dynamic Behavior

• Key properties to model

dynamic behavior

– Elasticity (stiffness)

– MassM

K

Dynamic Behavior

Dynamic Behavior


dynamic behavior


– Mass

• Structures exhibit specific

shapes under vibration

loads K

M

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Dynamic Behavior

Dynamic Behavior


dynamic behavior


– Mass

• Structures exhibit specific

shapes under vibration

loads

– Mode shape

K

MM

System displacement w/o damping

K

M M

System displacement with damping

K

M M

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Dynamic Behavior

Seismic Loads

• Reservoir Interaction

– Additional mass on the

structure

NWS

WVi

WHi

y

H

Dynamic Behavior

Seismic Loads



structure

• Thought of as layers of ice

attached to face of dam

NWS

Dynamic Behavior

Seismic Loads



structure



• Ice follows the deformation

of structure

NWS

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Dynamic Behavior

Seismic Loads



structure



• Ice follows the deformation

of structure

NWS

Dynamic Behavior

Fundamental Period

τ = 0.103

Dynamic Behavior

NWS

Fundamental Period

τ = 0.19

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Dynamic Behavior

Seismic Loads

• Reservoir Interaction has

significant effect on

dynamic behavior

• Method to evaluate

reservoir interaction

– Westergaard’s Added Mass

– Incompressible Fluid Element

– Compressible Fluid Element

NWS

WVi

WHi

y

H

Dynamic Behavior

Seismic Loads

• Westergaard’s Added Mass NWS

b

λ= Peak ground acceleration WVi

WHi

y

H

ϒ = unit weight of water

g = acceleration due to gravity

H = Depth of Reservoir

g = Depth below reservoir

surface

Dynamic Behavior

Seismic Loads

• Westergaard’s Added Mass

– Not considered as accurate as

compressible or

incompressible fluid method

– Typically produces overly

conservative results

NWS

b

WVi

WHi

y

H

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Dynamic Analysis

• Most concrete gravity dams were designed

using pseudo static methods

– Applies horizontal acceleration to simulate ground

motion

– No longer considered acceptable

• Does not account for dynamic characteristics

Dynamic Analysis

• Acceptable Methods of Analysis– Simplified Pseudo dynamic

• Developed by University of California, Berkeley– Chopra Method

– Takes into account fundamental modes of dam

– Response Spectrum Method

– Response Spectrum Method• Combines the absolute contribution from modes

– Square Root Sum of the Squares (SSRS)

– Time History Method• Combines contribution from modes

• Allows for contrasting modes to cancel

Dynamic Analysis

• Includes dynamic

characteristics of dam

• Reservoir interaction

• Damping effects

• Earthquake motion

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Dynamic Analysis

Response Spectra Time-History

Dynamic Analysis

• Concrete Dam– Dynamic properties

• Instantaneous modulus of elasticity

• Dynamic tensile and compressive strength

– Density (mass)

• Foundation– Dynamic stiffness

– Massless Foundation• Removes inertia from foundation

• Can results in unrealistically high stresses in the dam

– Including Foundation Mass • Use infinite non-reflecting boundaries at the outer edge of the

foundation

Dynamic Analysis

• Hydrodynamic Loads

– Lumped Mass

• Westergaard’s theory of added mass

• Zangar’s Added Mass

Shear force at a plane y-depth below reservoir level

Moment at a plane y-depth below reservoir level

Coefficient based on angle of incline

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Dynamic Analysis

• Hydrodynamic Loads

– Lumped Mass

• Westergaard theory of added mass

– Compressible Fluid Elements in the Frequency

Domain

– Three-dimensional Fluid Elements

Dynamic Analysis

Static Analysis Pseudo Static Analysis

Dynamic Analysis

• Simplified Pseudo

Dynamic

– Computes stress

distribution in dam using

gravity method

– Computes addition

stress based on response

spectra and fundamental

modes

Sa = 0.27

SPGA = 0.15

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Dynamic Analysis

Simplified Pseudo Dynamic Pseudo Static Analysis

Dynamic Analysis

Simplified Pseudo Dynamic

• Characteristics of the

Pseudo Dynamic Analysis

– Significant increase in

potential separation at the

base

– Increase in tensile stress on

the upstream face

Dynamic Analysis

• FEM Response Spectra

Analysis

– Model complex

geometries

– Includes dynamic

characteristics

– Reservoir Interaction

– Conservative

• Sums absolute effects of

dynamic modes

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Dynamic Analysis


Analysis

– Include effects of

damping

• 3% - 7%

– Inputs ground motion

using response spectrum

Dynamic Analysis

Simplified Pseudo Dynamic FEM Response Spectra

+64 psi

+36 psi

+50 psi

+60 psi

+120 psi -165 psi

+46 psi

-82 psi

-134 psi

-118 psi

-98 psi

-64 psi

Dynamic Analysis


– Conservative results

• Less conservative than

the Pseudo dynamic

analysis+64 psi

+36 psi

+50 psi

+60 psi

+120 psi -165 psi

+46 psi

-82 psi

-134 psi

-118 psi

-98 psi

-64 psi

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Dynamic Analysis

Time History Analysis

• Includes dynamic

characteristics of dam

• Reservoir interaction

• Based on structures mode

shapes

• Includes potential cancelling

effects of contrasting modes

+50 psi

+22 psi

+37 psi

+47 psi

+90 psi -186 psi

-73 psi

-125 psi

-107 psi

-89 psi

+19 psi -48 psi

Seismic Time

t = 1.41 sec

Time-History Analysis

Response Spectrum Analysis Time History Analysis

+64 psi

+36 psi

+50 psi

+60 psi

+120 psi -165 psi

+46 psi

-82 psi

-134 psi

-118 psi

-98 psi

-64 psi

+50 psi

+22 psi

+37 psi

+47 psi

+90 psi -186 psi

-73 psi

-125 psi

-107 psi

-89 psi

+19 psi -48 psi

Seismic Time

t = 1.41 sec


• Time History Analysis

– Results less conservative

– Must evaluate the

behavior at several times

10/8/2013

38






• 1.41 seconds






• 1.41 seconds

• 1.51 seconds






• 1.41 seconds

• 1.51 seconds

• 3.21 seconds

10/8/2013

39

Structural Evaluation

Dynamic Results Evaluation


– Compare computed stress to

the allowable strength of

concrete

f’C = 4000 lb/in2

ft = 400 lb/in2




– Compare computed stress to

the allowable strength of

concrete


– Compare stress distribution

along base with required

criteria





– Compare computed stress to the allowable strength of concrete


– Compare stress distribution along base with required criteria


• Sliding Criteria

– Compare computed Factor of Safety with Minimum

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40



• Results show tensile stress at dam/foundation interface

• Requires Post-Earthquake Analysis

– Crack extends through tensile zone

– Residual friction coefficient

• Damage at interface due to oscillatory deformation

– Full uplift within cracked zone

– Normal static loads86.710.6

97.3 ft

Understanding the Analysis

• Sensitivity Studies

– Material Parameters

• Concrete Strengths

• Foundation Stiffness and Strengths

– Dynamic Parameters

• Natural frequencies

• Peak dynamic response


– Methods of Analysis

BREAK # 3


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41

Deterministic Criteria

• Overstressing

– Potential for cracking or crushing of concrete

• Overturning

– Failure through rotational instability

• Sliding

– Failure along slip plane

– Rock block instability

Concrete Dam Failure Modes

• Overstressing

– Stress develops in

concrete that is greater

than the strength of the

material


• Overstressing

– Stress develops in

concrete that is greater

than the strength of the

material

• Overturning

– Toppling failure

10/8/2013

42


• Overstressing

– Stress develops in concrete

that is greater than the

strength of the material

• Overturning

– Toppling failure

• Sliding

– Driving forces are greater

than resisting forces.

• Rock Block Stability

Arch Dam Analysis

• Double Curvature Arch

– 209 feet high

Arch Dam Analysis

• Double Curvature Arch

– 209 feet high

– Concrete properties

• f’c = 5,000 lb/in2

• ft = 500 lb/in2

• Ecs = 2.4x106 lb/in2

• Eci = 3.5x106 lb/in2

– Foundation properties

• Ef = 3.3x106 lb/in2

• Effective Friction, 54 deg

• Neglect Cohesion

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43

Arch Dam Analysis

• USLC-1– Gravity, stress-free temperatures (spring/fall),

NWS reservoir El. 7811 feet, tailwater El. 7618 feet, and sediment El. 7692 feet.

• USLC-2– Gravity, summer temperatures, reservoir NWS

El. 7811 feet, tailwater El. 7618 feet, and sediment El. 7692 feet.

• USLC-3– Gravity, winter temperatures, reservoir NWS

El. 7811 feet, ice, tailwater El. 7618 feet, and sediment El. 7692 feet.

• UNLC-1– Gravity, stress-free temperatures (spring/fall),

reservoir PMF El. 7814.8 feet, tailwater El. 7618 feet, and sediment El. 7692 feet.

• EXLC-1– Gravity, stress-free temperatures (spring/fall),

reservoir NWS El. 7811 feet, tailwater El. 7618 feet, sediment El. 7692 feet, hydrodynamic added mass, and ground acceleration due to the MDE.

Arch Dam Analysis

Evaluation of Results

• Start with Deformation

– Behavior can hide behind

stresses,

but not deformations or loads

– Stress Free Temperature

• Spring/Fall Condition

Crest Deformations

Arch Dam Analysis


• Review Stress Contours

– Horizontal arch stress

• Tangent to the curvature of

the arch

Arch Stress Contours

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44

Arch Dam Analysis





the arch

– Vertical cantilever stress

• Parallel to slope

Cantilever Stress Contours

Usual Load 1Cantilever Stress

Arch Dam Analysis





the arch



• Evaluate sections through

dam

Crown Section, Cantilever Stresses

Arch Dam Analysis




• Tangent to the curvature of the arch



• Evaluate sections through dam

• Evaluate stress conditions along dam/foundation interface

Dam/Foundation Interface Stress Results

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Arch Dam Analysis

Usual Load 1Sliding Planes

Sliding Factors of Safety

Arch Dam Analysis

Usual Load 2, Summer temperature load

• Reduction in crest

deflection

– Increased temperature

expands concrete deforms

upstream

– Effectively reduces radius of

structure

Crest Deflections

Arch Dam Analysis


• Arch stress results show

dam entirely in compression


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Arch Dam Analysis


• Arch stress results show

dam entirely in compression

• Vertical cantilever stress

results show mostly

compressive load


Usual Load 2Sliding Planes

Sliding Factors of Safety

Arch Dam Analysis

Arch Dam Analysis

Usual Load 3, Winter temperature load

• Increased crest deflection

– Reduced temperature

contracts concrete so crest

deforms downstream

– Effectively increases radius of

structure

Crest Deflections

10/8/2013

47

Arch Dam Analysis


• Usual Load 2

– Arch stress results show

significant tension

– Greater than allowable

tensile strength of concrete


Arch Dam Analysis


• Tensile stress is limited to

area near surface of dam

– Temperature related

– Limited cracking


Arch Dam Analysis

Usual Load 3a, Winter temperature load

• Concern is load on foundation, not strength of concrete

• Modify model to simulate surface cracking

• Allows for redistribution of load onto foundation

• Deformation

– Before: 1.0 inch

– After: 1.0 inch

Crest Deflections

Usual Load 3aDam Crest Deflection

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48

Arch Dam Analysis


• Surface stress affected by

modifications


Arch Dam Analysis


• Arch stress through section

– Mostly compressive


Arch Dam Analysis

Extreme Load 1, Seismic Load Combination

• Transient Analysis

– Modal Superposition

– Time-history

Mode Shapes

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49

Arch Dam Analysis


• Seismic Analysis

– Transient Analysis

• Modal Superposition

• Time-history

– Earthquake Records

• Use 3 – 5 different records

• Envelope behavior

– Reservoir Interaction

• Generalized Westergaard’s

added mass

• Damping 5%

Time History Records

Arch Dam Analysis


• Extreme Load

– Behavior

• Primarily influenced by

mode shape No. 1

– Maximum deformation

downstream

• 1.6 inches

– Time into earthquake

• 5.18 seconds

Crest Deflections

Mode Shape 1

Arch Dam Analysis


• Extreme Load

– Arch Stress Evaluation

• Stress results indicate

adequate concrete strength


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50

Arch Dam Analysis


• Extreme Load

– Arch Stress Evaluation

• Stress results indicate

adequate concrete strength

– Cantilever Stress Evaluation

• Maximum stress less than

allowable strength of

concrete


“We believe that while a cookie-cutter approach

is great for making cookies, it doesn’t

necessarily translate to other pursuits.”

A thought to Consider

Stability Analysis

• Understand the basic failure modes

– Overstressing

– Overturning

– Abutment Stability

• Understand the loads and the impact to the structural behavior– Seasonal temperature variations on three-dimensional systems

• Gain significant understanding on behavior of structure

– Hydrodynamic interaction between dam and reservoir

• Understand the strength of the material properties

– Concrete

– Foundation

• Understand the impact uncertainties can have on the behavior

– Sensitivity studies

10/8/2013

51

BREAK # 4


Stability Evaluations

of

Concrete Dams

Guy S. Lund, P.E. – [email protected]

Robert A. Kline, Jr., P.E. – [email protected]

Post Event Evaluation & Quiz

Please click the following link to take the Seminar Evaluation and Quiz:

http://e02.commpartners.com/users/asdso/posttest.php?id=10945

You must complete the Seminar Evaluation and Quiz to receive PDH credit hours

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