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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 Schema TEST 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 WORK TEST 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 W pe where: W pe is the peak load (kN) from Yield Line m pe = W pe /8 EFNARC TEST WORK

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Figure 8:Example of EFNARC test results for steel fibre reinforced shotcrete (70 kg/m 3 )

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LOAD W pe Point support Yield Line Pattern RDP TEST RIG from Yield Line m pe = W pe /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/m 3 ) 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 Observation UNDERGROUND 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 mechanisms Flexure or Bending (moment) Punching shear Adhesion loss Direct shear Axial 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 Analysis STRUCTURAL 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 Capacity Panel tests Moment Demand Rock Loading Dead weight – simple prism Quasi Static – relationship with deformation Rock Mass Assessment - Q Dynamic – Energy absorption method

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Contribution of Structural Analysis YIELD LINE METHOD Why YIELD LINE One of the PLASTIC suite of methods Based on Elastic Perfectly plastic behaviour Allows redistribution of stress Relatively simple analysis method Directly integrated with design Economical (less reinforcement/m 2 ) Versatile Closed-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 supported Statically determinate v Continuous beams Statically indeterminate HINGELOAD 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 m d = wab/48 M d average = wab/36.5

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Fan Mechanism Figure 3: Yield line pattern for the fan mechanism m d = P/12.56

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Combined Mechanism Figure 4: Yield Line Pattern for a combined mechanism panel

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Triangular Mechanism Compare with Rectangular Mechanism 1/144 : 1/48 = 66% economy Figure 5: Yield Line Pattern for a triangular mechanism m d = wab/48 m d = wc 2 /144

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Shotcrete Design Schema TEST 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 Panels Steel fibre in particular Unreinforced panels give highly variable results Moment capacity reliably increases with fibre density/mesh area Residual 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 pattern Especially in irregular rock wall geometries Select crack pattern to give lowest moment of resistance Conventional Yield Line Design = 15% rule Shotcrete Yield Line Design = 50% rule (suggested) Yield Line methods used advantageously Calculation of Shotcrete capacity (strength) Calculation of Shotcrete demand (moment/load)

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Thank You from William and Graham

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