1. 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

2. 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

3. 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)

4. Contribution of TEST WORK
Cube Tests
Fibre Density
EFNARC Panels
RDP tests
YIELD LINE METHOD
Characteristic Strength
Allowable Moment
Moment CAPACITY

5. EFNARC TEST WORK from Yield Line mpe = Wpe/8 LOAD Wpe
where:
Wpe is the peak load (kN)
from Yield Line
mpe = Wpe/8
LOAD Wpe
Yield Line Pattern
EFNARC TEST RIG

6. EFNARC TEST WORK
Figure 8: Example of EFNARC test results for steel fibre reinforced shotcrete (70 kg/m3)

7. ASTM RDP TEST WORK from Yield Line mpe = Wpe/5.54 LOAD Wpe
Point support
LOAD Wpe
Point support
Point support
Yield Line Pattern
RDP TEST RIG

8. ASTM RDP TEST WORK Peak Load – Crack Formation
“Elastic” Energy Absorption
“Plastic” Energy Absorption
Figure 9: Example of ASTM C1550 RDP test results for steel fibre reinforced shotcrete (70 kg/m3)

9. 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)

10. Contribution of Observation
UNDERGROUND OBSERVATION
Crack formation
Crack monitoring
Crack measurement
YIELD LINE METHOD
Veracity CHECK
Crack patterns
Rock Loading

11. 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

12. 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

13. EXAMPLE 14 March 2008 29 Mar ‘07 26 Apr ’07 11 May ‘07 08 Jun ‘07
23 Oct ‘07
20 Dec ’07
24 Jan ‘08
12 Feb ‘08
14 Mar ‘08

14. Contribution of Structural Analysis
Loads
Moments
Shear force
Torsion
YIELD LINE METHOD
Load/moment relationship
Load DEMAND
Moment DEMAND

15. Structural Analysis Develop relationship between Moment Capacity
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

16. Contribution of Structural Analysis
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/m2)
Versatile
Closed-form solution (cf FE, FD, BE numerical methods)
YIELD LINE METHOD

17. Yield Line Basics External Work Done = Internal Work Done
HINGE
LOAD v
Simply supported
Statically determinate
Continuous beams
Statically indeterminate
Lever Arm = L/2
P (Load)
Pδ = 2mθ
Pδ = 4mθ θ
Rotation = 2θ
Unit Displacement = δ
External Work Done = Internal Work Done
WD by Loads moving = WD by YL rotating

18. 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

19. Figure 2: Yield Line Pattern for a rectangular panel
Continuous Slab
Yield line moment for a given load w
From SANS 0100 (Concrete Design Code)
md = wab/48
Md average = wab/36.5
Figure 2: Yield Line Pattern for a rectangular panel

20. Figure 3: Yield line pattern for the fan mechanism
md = P/12.56
Figure 3: Yield line pattern for the fan mechanism

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

22. Figure 5: Yield Line Pattern for a triangular mechanism
Compare with Rectangular Mechanism
1/144 : 1/48 = 66% economy
md = wc2/144
md = wab/48
Figure 5: Yield Line Pattern for a triangular mechanism

23. 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)

24. 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

25. 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)

26. Thank You
from
William and Graham