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
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
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)
Contribution of TEST WORK
TEST WORK
•
Cube
Tests
•
Fibre
Density
•
EFNARC Panels
•
RDP tests
YIELD LINE METHOD
•
Characteristic Strength
•
Allowable Moment
•
Moment CAPACITY
Yield Line Pattern
EFNARC TEST RIG
LOAD
W
pe
8
pe
pe
W
m
8
pe
pe
W
m
8
pe
pe
W
m
8
pe
pe
W
m
8
pe
pe
W
m
where
:
W
pe
is
the
peak
load
(kN)
f
rom Yield Line
m
pe
=
W
pe
/8
EFNARC TEST WORK
EFNARC TEST WORK
0
20
40
60
80
100
120
140
0.0
5.0
10.0
15.0
20.0
25.0
Load (kN)
Deflection [mm]
A
B
C
D
6ut
Figure 8:
Example of EFNARC test results for steel fibre reinforced
shotcrete
(70 kg/m
3
)
LOAD
W
pe
Point support
Point support
Point support
Yield Line Pattern
RDP TEST RIG
from Yield Line
m
pe
=
W
pe
/5.54
ASTM RDP TEST WORK
Figure 9:
Example of ASTM C1550 RDP test results for steel fibre reinforced
shotcrete
(70 kg/m
3
)
0
10
20
30
40
50
60
70
0
5
10
15
20
25
30
35
40
45
Load (kN)
Deflection (mm)
A
B
C
D
ASTM RDP TEST WORK
“Elastic” Energy Absorption
“Plastic” Energy Absorption
Peak Load
–
Crack Formation
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)
Contribution of Observation
UNDERGROUND
OBSERVATION
•
Crack
formation
•
Crack monitoring
•
Crack measurement
YIELD LINE METHOD
•
Veracity
CHECK
•
Crack patterns
•
Rock Loading
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
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
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
Contribution of Structural Analysis
STRUCTURAL
ANALYSIS
•
Loads
•
Moments
•
Shear force
•
Torsion
YIELD LINE METHOD
•
Load/moment
relationship
•
Load DEMAND
•
Moment DEMAND
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
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)
Yield Line Basics
External Work Done = Internal Work Done
WD by Loads moving = WD by YL rotating
Simply supported
Statically determinate
Continuous beams
Statically indeterminate
HINGE
LOAD
Unit Displacement =
δ
P (Load)
Lever Arm = L/2
Rotation = 2
θ
θ
P
δ
= 4m
θ
P
δ
= 2m
θ
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
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
Fan Mechanism
Fan Mechanism
Figure 3: Yield line pattern for the fan mechanism
m
d
=
P/
12
.
56
Combined Mechanism
Figure 4: Yield Line Pattern for a combined mechanism panel
Figure 4: Yield Line Pattern for a combined mechanism panel
Triangular Mechanism
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
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)
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
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)
Thank You
from
William and Graham
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