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THE USE OF YIELD LINE ANALYSIS AND PANEL TESTS FOR
THE DESIGN OF SHOTCRETE by
WC JOUGHIN* and GC HOWELL** SRK Consulting, Johannesburg
* Principal Mining Engineer * Principal Geotechnical Civil Engineer
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Presentation Summary
Information available from Test Work Observation of crack formation Requirements of Analysis Methods Relationship between CAPACITY and DEMAND Yield Line method and how it is used Integration into the Shotcrete Design Method Summary and Conclusions
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Shotcrete Design SchemaTEST WORK
• Cube Tests• Fibre Density• EFNARC Panels• RDP tests
STRUCTURAL ANALYSIS
• Loads• Moments• Shear force• Torsion
UNDERGROUND OBSERVATION
• Crack formation• Crack monitoring• Crack measurement
YIELD LINE METHOD
• Characteristic Strength• Allowable Moment• Moment CAPACITY
• Load/moment relationship• Load DEMAND• Moment DEMAND
• Veracity CHECK• Crack patterns• Rock Loading
SHOTCRETE DESIGN• Factor of Safety (Capacity/Demand)• Probability/Reliability (p(D – C) < 1.0)
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Contribution of TEST WORKTEST WORK
• Cube Tests• Fibre Density• EFNARC Panels• RDP tests
YIELD LINE METHOD
• Characteristic Strength• Allowable Moment• Moment CAPACITY
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Yield Line Pattern
EFNARC TEST RIG
LOAD Wpe
8pe
peW
m
8pe
peW
m 8pe
peW
m 8pe
peW
m
8pe
peW
m where:Wpe is the peak load (kN)
from Yield Linempe = Wpe/8
EFNARC TEST WORK
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EFNARC TEST WORK
0
20
40
60
80
100
120
140
0.0 5.0 10.0 15.0 20.0 25.0
Lo
ad
(kN
)
Deflection [mm]
A B C D
6ut
Figure 8: Example of EFNARC test results for steel fibre reinforced shotcrete (70 kg/m3)
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LOAD Wpe
Point support
Point supportPoint support
Yield Line PatternRDP TEST RIG
from Yield Linempe = Wpe/5.54
ASTM RDP TEST WORK
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Figure 9: Example of ASTM C1550 RDP test results for steel fibre reinforced shotcrete (70 kg/m3)
0
10
20
30
40
50
60
70
0 5 10 15 20 25 30 35 40 45
Lo
ad (k
N)
Deflection (mm)
A B C D
ASTM RDP TEST WORK
“Elastic” Energy Absorption“Plastic” Energy Absorption
Peak Load – Crack Formation
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TEST WORK Summary
Moment Capacity development using Yield Line for a standard test panel
Ratio of thickness of test panel to design thickness (on the wall) give the Design Moment Capacity
Method allows a Characteristic Moment Capacity to be specified (cf Cube Strength)
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Contribution of ObservationUNDERGROUND OBSERVATION
• Crack formation• Crack monitoring• Crack measurement
YIELD LINE METHOD
• Veracity CHECK• Crack patterns• Rock Loading
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Observation
Cracking in shotcrete is due to different mechanismsFlexure or Bending (moment)Punching shearAdhesion lossDirect shearAxial force (tension)
Sometimes difficult to categorize on the wall Long term monitoring required
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Observation 2 Look for patterns which resemble expected yield lines Take into account the in-plane axial (tensile) force
component Locate areas of shear dislocation Ultimately
Looking for yield line patterns
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14 March 2008
29 Mar ‘07
26 Apr ’07
11 May ‘07
08 Jun ‘07
15 Jun ‘07
23 Oct ‘07
20 Dec ’07
24 Jan ‘08
12 Feb ‘08
14 Mar ‘08
EXAMPLE
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Contribution of Structural AnalysisSTRUCTURAL
ANALYSIS• Loads• Moments• Shear force• Torsion
YIELD LINE METHOD
• Load/moment relationship• Load DEMAND• Moment DEMAND
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Structural Analysis Develop relationship between
DEMAND (load)CAPACITY (strength)
Moment CapacityPanel tests
Moment DemandRock Loading
Dead weight – simple prismQuasi Static – relationship with deformationRock Mass Assessment - QDynamic – Energy absorption method
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Contribution of Structural Analysis
YIELD LINE METHOD
Why YIELD LINEOne of the PLASTIC suite of methodsBased on Elastic Perfectly plastic behaviourAllows redistribution of stress
Relatively simple analysis methodDirectly integrated with designEconomical (less reinforcement/m2)VersatileClosed-form solution (cf FE, FD, BE numerical
methods)
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Yield Line Basics
External Work Done = Internal Work Done WD by Loads moving = WD by YL rotating
Simply supportedStatically determinate
vContinuous beamsStatically indeterminate
HINGE LOAD
Unit Displacement = δ
P (Load)
Lever Arm = L/2
Rotation = 2θ
θ Pδ = 4mθPδ = 2mθ
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Upper Bound Theorem
Any arbitrary crack pattern gives a design moment less than the maximum for a given load
Require MAXIMUM moment from all possible crack patterns
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Continuous Slab
Yield line moment for a given load w
From SANS 0100 (Concrete Design Code)
Figure 2: Yield Line Pattern for a rectangular panel
md = wab/48
Md average = wab/36.5
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Fan Mechanism Fan Mechanism
Figure 3: Yield line pattern for the fan mechanism
md = P/12.56
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Combined Mechanism
Figure 4: Yield Line Pattern for a combined mechanism panel
Figure 4: Yield Line Pattern for a combined mechanism panel
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Triangular Mechanism Triangular Mechanism
Compare with Rectangular Mechanism
1/144 : 1/48 = 66% economy
Figure 5: Yield Line Pattern for a triangular mechanism
md = wab/48
md = wc2/144
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Shotcrete Design SchemaTEST WORK
• Cube Tests• Fibre Density• EFNARC Panels• RDP tests
STRUCTURAL ANALYSIS
• Loads• Moments• Shear force• Torsion
UNDERGROUND OBSERVATION
• Crack formation• Crack monitoring• Crack measurement
YIELD LINE METHOD
• Characteristic Strength• Allowable Moment• Moment CAPACITY
• Load/moment relationship• Load DEMAND• Moment DEMAND
• Veracity CHECK• Crack patterns• Rock Loading
SHOTCRETE DESIGN• Factor of Safety (Capacity/Demand)• Probability/Reliability (p(D – C) < 1.0)
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Conclusions 1 Shotcrete Moment Capacity
Peak Moment Capacity reliably estimated for RDP PanelsSteel fibre in particular
Unreinforced panels give highly variable resultsMoment capacity reliably increases with fibre density/mesh areaResidual Moment capacities can be estimated using the same
method (see following paper)Actual underground capacities are variable
Dependant of local rock geometry and shotcrete application
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Conclusions 2 Shotcrete Moment Demand
Rock load influenced by the crack patternEspecially in irregular rock wall geometries
Select crack pattern to give lowest moment of resistanceConventional Yield Line Design = 15% ruleShotcrete Yield Line Design = 50% rule (suggested)
Yield Line methods used advantageouslyCalculation of Shotcrete capacity (strength)Calculation of Shotcrete demand (moment/load)
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Thank You
fromWilliam and Graham