Aggregate Interlock vs. Crack Interlock
by
Thomas T. C. Hsu
Moores Professor of Civil Engineering
Dept. of Civil & Environmental Engineering
University of Houston
Houston, Texas, USA
ACI Committee 123 – ACI Forum
April 15, 2013, Minneapolis, MN
Walraven’s Aggregate Interlock,1981
=
τ
w
ft ( w, ∆) and =
σ
f n ( w, ∆)
d τ   Bnn Bnt  dw 
 
 = 
d σ   Btn Btt  d ∆ 
∆
∂f n
∂f n
∂ft
∂ft
Bnn
=
=
, Bnt =
, Btn =
, Btt
∂w
∂∆
∂w
∂∆
How to formulate f t ( w, ∆ ) , f n ( w, ∆ ) ?
Vecchio and Collins, 1986
vci= 0.18vci max + 1.64 f ci − 0.82
f ci2
vci max
− f c'
vci max =
0.31 + 24 w / (a +16)
w = ε1⋅ sθ
sθ =
1
sin θ cos θ
+
smx
smy
Vecchio and Collins, 1986
Mattock’s Modified Shear Friction Theory, 1976
Push-off tests
f c' = 2000 to 6000 psi
Shear
Plane
=
vu 4.5 f
'0.545
c
+ 0.8(ρv f y + σn ) psi
Walraven et al. Shear Friction Capacity, 1987
88 push-off tests
f c' = 2000 to 8550 psi
Using statistical best fit method:
=
vu C3 (0.007ρv f y )C4 ( psi )
where C3 = 15.686 f cc' 0.406
C4 = 0.0353 f cc' 0.30
f cc' = cube strength,
f c' = 0.85 f cc'
Mattock Vs. Walraven et al.
Mau and Hsu’s Shear Transfer Strength, 1987
τu
='
fc
0.66 ω ≤ 0.3
where
ρv f y
ω = '
fc
(b)
Walraven: “Prof. Mau and Hsu are to
be congratulated for finding such an
equation, which is really simple and
is almost as good as the more
complicated statistical function
proposed by the authors (Walraven et
al).”
(a)
Compare with Walraven et al’s test results
τu
f c'
vu
τ
u = 0.66 ω ≤ 0.3

'
'
ffcc
ω
Hsu’s Softened Membrane Model, 2002
Based on Fixed-Angle Shear Theory
Equilibrium Equations
c
σ  = σ 1c cos 2 α1 + σ 2c sin 2 α1 − τ12
2sin α1 cos α1 + ρ f 
σ t = σ 1c sin 2 α1 + σ 2c cos 2 α1 + τ12c 2sin α1 cos α1 + ρt ft
τ t =
(σ 1c − σ 2c )sin α1 cos α1 + τ12c (cos 2 α1 − sin 2 α1 )
Derived from the three equilibrium equations, and
assume yielding of steel
=
τty
(τ )
+ ρ f y ρt f ty
2 ρ f y ρt f ty
Vu =
c 2
12
Vc
+
Vs
Fixed-Angle Shear Theory
The crack is crooked, not a plane. So shear stress
along cracks is caused by crack interlock.
c
Shear stress τ12 is a function of f c' .
Softened Membrane Model
Stress-Strain Relationship of Concrete
c
c
σ 1   E1
 c  
c
σ 2  = ν 21E2
 c 
τ12   0
c
ν 12 E1
c
E2
0
0   ε1 


0   ε2 
c  γ / 2 
G12   12 
Constitutive Law of Concrete in Shear, 2001
Zhu, Hsu & Lee, ACI Struct. Jour., July-Aug., 2001
Using Smeared Crack Concept:
c
G12 =
c
c
σ1 − σ 2
2(ε1 − ε 2 )
σ
ε = Tensile stress-strain curve is a function of
σ
ε
c
1 vs . 1
f c'
c
2 vs . 2 = Compressive stress-strain curve is a function of
f c'
Universal Panel Tester at University of Houston
North View
South View
Softened Membrane Model
Concrete in Tension, σ
c
1 vs.
σ 1c
f cr
 0.00008 

σ = f cr 
 ε1 
0.4
c
1
σ 1c = Ecε1
ε cr = 0.00008
ε1
ε1
Softened Membrane Model
Concrete in Compression, σ
f c′
σ
c
2
c
2 vs.
ε2
NONSOFTENED
  ε ζε − 1 2 
=
σ 2c ζ f c′ 1 −  2 0  
2 ζ −1  
 

ζ f c′
  ε   ε 2 
=
σ 2c ζ f c′  2  2  −  2  
  ζε 0   ζε 0  
ζε 0
ε0
2ε 0
 5.8

β 

1
ζ=
< 0.9 
1−


 (1 + 400ε )  24° 
′
f
1
 c

ε2
Conclusion
Aggregate Interlock:
P
P
Lw
165.1mm
Top Beam (152.4mmx152.4mm)
V/2
V/2
Column (152.4mmx152.4mm)
Wall Panel
Thickness
76.2mm
Column (152.4mmx152.4mm)
1066.8mm
165.1mm
1397mm
V/2
P-V
1066.8mm
1397mm
Crack Interlock:
V/2
Bottom Beam (152.4mmx152.4mm)
P+V
165.1mm
hw
165.1mm
At the edge of the wall when a large
local plane crack is observed, shear
stress along such a crack is a function
of aggregate size a and crack width w.
Then shear stress is caused by
aggregate interlock.
In the main body of wall when
smeared cracks can be assumed,
shear stress along a crooked cracks is
simply a function of f c' , not a and
w. Shear stress is caused by crack
interlock.