1 Introduction
2 Shear Stiffness for RECC Columns
2.1 Pre-cracking Shear Stiffness
2.2 Truss-arch Model for Fully Diagonally Cracked Shear Stiffness
2.2.1 Shear Stiffness of CATM
Member | Force | Unit load | Length | Rigidity | Deformation |
---|---|---|---|---|---|
F
|
f
|
l
|
EA
|
Ffl/EA
| |
BC |
\( - \frac{{dV_{t} }}{{\sin \theta_{0} }} \)
|
\( - \frac{1}{{\sin \theta_{0} }} \)
|
\( \frac{{d_{v} }}{{\sin \theta_{0} }} \)
|
E
c
bdx sinθ
0
|
\( \frac{{d_{v} dV_{t} }}{{E_{c} b\sin^{4} \theta_{0} dx}} \)
|
BD | + dV
t
| + 1 |
d
v
|
\( E_{s} \left( {\frac{{A_{sh} }}{s} + \frac{b}{n}} \right)dx \)
|
\( \frac{{d_{v} dV_{t} }}{{E_{s} \left( {\frac{{A_{sh} }}{s} + \frac{b}{n}} \right)dx}} \)
|
2.2.2 Shear Stiffness of VATM
Member | Force | Unit load | Length | Rigidity | Deformation |
---|---|---|---|---|---|
F
|
f
|
l
|
EA
|
Ffl/EA
| |
BD |
\( - \frac{{dV_{t} }}{{\sin \theta_{1} }} \)
|
\( - \frac{1}{{\sin \theta_{1} }} \)
|
\( \frac{{d_{v} }}{{\sin \theta_{1} }} \)
|
\( \frac{{E_{c} bL\sin \theta_{1} dx}}{2} \)
|
\( \frac{{2d_{v} dV_{t} }}{{E_{c} bL\sin^{4} \theta_{1} dx}} \)
|
CE |
\( - \frac{{dV_{t} }}{{\sin \theta_{2} }} \)
|
\( - \frac{1}{{\sin \theta_{2} }} \)
|
\( \frac{{d_{v} }}{{\sin \theta_{2} }} \)
|
\( \frac{{E_{c} bL\sin \theta_{2} dx}}{2} \)
|
\( \frac{{2d_{v} dV_{t} }}{{E_{c} bL\sin^{4} \theta_{2} dx}} \)
|
BE | + dV
t
| + 1 |
d
v
|
\( E_{s} \left( {\frac{{A_{sh} }}{s} + \frac{b}{n}} \right)Ldx \)
|
\( \frac{{d_{v} dV_{t} }}{{E_{s} \left( {\frac{{A_{sh} }}{s} + \frac{b}{n}} \right)Ldx}} \)
|
2.2.3 Shear Stiffness of Arch Component
2.2.4 Determination of the Minimum Inclined Crack Angle
Member | Force | Unit load | Length | Rigidity | Deformation |
---|---|---|---|---|---|
F
|
f
|
l
|
EA
|
Ffl/EA
| |
AB | cotα
| cotα
|
x
1
L
|
\( E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right] \)
|
\( \frac{{x_{1} L\cot^{2} \alpha }}{{E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right]}} \)
|
BC | (2 − x
1)cotα/2 | (2 − x
1)cotα/2 | (1 − 2 x
1) L
|
\( E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right] \)
|
\( \frac{{L\cot^{2} \alpha \left( {1 - x_{1} } \right)^{2} \left( {1 - 2x_{1} } \right)}}{{4E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right]}} \)
|
CD | cotα/2 | cotα/2 |
x
1
L
|
\( E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right] \)
|
\( \frac{{x_{1} L\cot^{2} \alpha }}{{4E_{s} \left[ {\frac{{A_{st} }}{2} + \frac{{\gamma \left( {1 - \xi } \right)bh}}{n}} \right]}} \)
|
EF | − cotα/2 | − cotα/2 |
x
1
L
|
\( E_{s} \left( {\frac{{A_{st} }}{2} + \frac{\xi bh}{n}} \right) \)
|
\( \frac{{x_{1} L\cot^{2} \alpha }}{{4E_{s} \left( {\frac{{A_{st} }}{2} + \frac{\xi bh}{n}} \right)}} \)
|
FG | − x
1cotα/2 | − x
1cotα/2 | (1 − 2 x
1) L
|
\( E_{s} \left( {\frac{{A_{st} }}{2} + \frac{\xi bh}{n}} \right) \)
|
\( \frac{{x_{1}^{2} \left( {1 - 2x_{1} } \right)L\cot^{2} \alpha }}{{4E_{s} \left( {\frac{{A_{st} }}{2} + \frac{\xi bh}{n}} \right)}} \)
|
GH | 0 | 0 |
x
1
L
|
\( E_{s} \left( {\frac{{A_{st} }}{2} + \frac{\xi bh}{n}} \right) \)
| 0 |
2.2.5 Determination of Constant Crack Angle
2.3 Post-cracking Effective Shear Stiffness
3 Experimental Program
3.1 Specimens and Material Properties
Specimen |
λ
|
P (kN) |
ρ
sv (%) |
---|---|---|---|
E1 | 1.42 | 350 | 0.45 |
E2 | 1.75 | 350 | 0.45 |
E3 | 2.75 | 350 | 0.45 |
E4 | 1.75 | 700 | 0.45 |
E5 | 1.75 | 350 | 0.22 |
E6 | 1.75 | 350 | 0 |
Material |
f
cu (MPa) |
f
c (MPa) |
f
t (MPa) |
ε
t (%) |
E (GPa) |
---|---|---|---|---|---|
ECC | 49.7 | 45.0 | 4.39 | 4.46 | 22.2 |
3.2 Test Setup and Loading Configuration
4 Discussion of Test and Predicted Results
4.1 Crack Patterns and Failure Modes
Specimen |
V
cr
(kN) |
V
y
(kN) |
δ
y
(mm) |
θ
avg
(°) |
θ
pre (°) |
δ
pre (mm) |
---|---|---|---|---|---|---|
E1 | 200 | − 330.05 | − 1.84 | 40 | 42.5 | 1.91 |
330.12 | 1.92 | |||||
E2 | 150 | − 280.00 | − 3.02 | 42 | 42.5 | 2.90 |
279.91 | 3.02 | |||||
E3 | 75 | − 165.24 | − 3.68 | 45 | 42.5 | 4.11 |
165.92 | 3.60 | |||||
E4 | 200 | − 382.09 | − 3.09 | 42 | 42.8 | 2.89 |
380.80 | 2.84 | |||||
E5 | 100 | − 202.47 | − 2.56 | 41 | 41.2 | 2.54 |
240.16 | 2.35 | |||||
E6 | 100 | − 196.58 | − 2.50 | 40 | 39.4 | 2.26 |
216.24 | 2.60 |