- Open Access
Theoretical and Experimental Study of Effective Shear Stiffness of Reinforced ECC Columns
- Chang Wu^{1},
- Zuanfeng Pan^{2}Email authorView ORCID ID profile,
- Kang-Su Kim^{3} and
- Shaoping Meng^{4}
https://doi.org/10.1007/s40069-017-0219-2
© The Author(s) 2017
Received: 15 December 2016
Accepted: 23 October 2017
Published: 7 December 2017
Abstract
Engineered cementitious composites (ECC) possesses characteristics that make it suitable in the zones of high shear and ductility demand of structural elements; however, there is a lack of an adequate model to predict its shear stiffness. A theoretical model for the effective shear stiffness of reinforced ECC (RECC) columns is proposed on the basis of the truss-arch model, with the consideration of the unique property of ECC material. A total of six column specimens subjected to cyclic reverse loading are conducted, and the main test variables include the shear span-to-depth ratio, the transverse reinforcement ratio and the axial load ratio. Results show that the shear contribution to the total deflection in the diagonally cracked RECC beam is significant, and the proposed theoretical model can predict the shear deformation with reasonable accuracy.
Keywords
1 Introduction
Engineered cementitious composites (ECC) invented by Li et al. (Li 2012; Li and Leung 1992; Li et al. 1993) based on the basic principle of micromechanics and fracture mechanics is one of a family of high performance fiber reinforced cement composite (HPFRCC) (Naaman 1987), which exhibits pseudo strain hardening and multiple cracking properties under uniaxial tensile stress. This type of material consists of cement, mineral admixture, fine aggregates (maximum grain size usually 0.15 mm), water, admixtures to enhance strength and workability, and less than 2.0% volume of short fibers. Uniaxial tensile tests on ECC indicate that multiple fine cracks in ECC are formed uniformly over the length of the specimen, and the opening of each crack is usually less than 100 μm, subsequently, the ultimate tensile strain can exceed 2.0%, which is several hundred times that of normal concrete.
ECC has attracted the attentions of many researchers during the past two decades (Yoo and Yoon 2016), due to the advantages of unique macroscopic pseudo strain hardening, high energy dissipation capacity and good durability. The randomly distributed fibers in ECC help to transfer loads at the internal micro cracks, which leads to the fact that RECC member can have a relatively higher load carrying capacity and deformation capacity compared to normal RC member. Generally, both ultimate strength limit state and serviceability limit state requirements should be considered in the structural design. As there is no coarse aggregate in ECC, the elastic modulus of ECC is usually lower than that of concrete. Consequently, greater deformation of RECC member tends to be caused. The design of RECC members may be controlled by the serviceability limit rather than strength. In addition, for seismic design, the stiffness of RECC members of a structure strongly influences the calculated response under seismic action. Therefore, it is important to accurately predict the effective stiffness up to yielding of each structural component. Generally, the total deformation of a structural member can be regarded as the summation of flexural and shear deformations. However, even for RC members, the shear mechanisms is not as clearly elucidated compared to the sound understanding of flexural behavior, and the shear deformation is usually underestimated or just neglected. For the newly developed material, ECC, few researches so far have been reported on the estimation of shear deformation of ECC members. To address this issue, an approach to predict the effective shear stiffness of RECC columns is proposed in the present study, based on the truss-arch model.
Since the truss concept was first introduced a century ago, the truss models have wildly used for predicting the ultimate shear strength of RC members, such as the traditional 45° truss model, constant angle truss model (CATM), variable angle truss model (VATM), compression field theory (CFT), modified compression field theory (MCFT), rotating-angle softened truss model (RA-STM) and fixed-angle softened truss model (FA-STM), etc. (ASCE-ACI Committee 445 1998). In terms of the study on shear stiffness or deformation, Kim and Mander (1999) systematically researched the truss model and analyzed the shear stiffness of RC columns with the VATM which was derived using various numerical integration schemes. The Programs, VecTor2 (Won and Vecchio 2002) and Response 2000 (Bentz 2000), were developed based on MCFT, which can be used to analyze the load–displacement curves of RC members subjected to the combination of axial load, shear and flexure. Based on Softened Membrane Model (SMM) (Hsu and Mo 2010), Mo et al. (2008) developed the Simulation of Concrete Structures (SCS) program on the OpenSEES platform to simulate the load–displacement responses of shear-critical RC elements. Pan et al. (2014) derived the explicit expression of effective shear stiffness on the basis of CATM and VATM, which was verified by the experiment of RC T-section beams.
In the truss-arch model, the shear resistant mechanism is explicitly considered as the truss action superimposing an arch action. Arch action in RC members subjected to shear force has been recognized by many researchers (Ichinose 1992; Kim et al. 1998). Especially, for members with low shear span-to-depth ratio, if the arch action is not taken into account, the shear stiffness tends to be underestimated. Ichinose (1992) presented a truss-arch model and proposed a design equation to prevent shear failure after the inelastic flexural deformation, which has been adopted in the Architectural Institute of Japan Design Guidelines (AIJ 1994). On the basis of the experimentally measured steel stresses over the shear span in the RC beams, Kim et al. (1998) proposed an empirical coefficient, which represents arch action contribution to the total shear capacity. Pan et al. (Pan and Li 2013; Jin and Pan 2015) proposed a new type of truss-arch model with the consideration of the deformation compatibility for both truss model and arch action, and the proposed model was verified by the shear-critical RC column tests.
The unique tensile strain-hardening property allows cracked ECC members to carry tensile stresses, and the tensile stress of cracked ECC can not be directly neglected like brittle materials such as normal concrete. In the present study, a theoretical model is proposed on the basis of the truss-arch model, incorporating the unique properties of ECC, to predict the effective shear stiffness of RECC columns. Then, six RECC columns subjected to cyclic reverse loading with various shear span-to-depth ratios, transverse reinforcement ratios and axial load ratios were studied experimentally to verify the proposed model.
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
Consider a differential truss in Fig. 2(b) subjected to a differential shear force dV _{ t }, in which the in-plane width of the tie and strut are dx and dxsinθ _{0}, respectively. This differential truss can be expressed with the simplified diagram, as shown in Fig. 2(c). The member forces in the truss subjected to a unit shear force can be easily resolved by the static equilibrium, and the shear deformation of the differential truss subjected to dV _{ t } can be analyzed based on the principles of virtual work. When calculating the shear deformation, the flexural deformation can be eliminated by assuming that the longitudinal chords (AB and CD in Fig. 2(c)) are very rigid. Therefore, only the strut BC and the tie BD are considered when calculating the shear deformation.
Chord deformations of CATM by the principles of virtual work method.
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
Chord deformations of VATM by the principles of virtual work method.
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
Chord deformations of VATM by two-point Gauss quadrature.
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 _{1}cotα/2 | − x _{1}cotα/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
Summary of specimen information.
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 |
Properties of ECC.
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 |
The longitudinal steel bars employed in each specimen were four 20 mm-diameter deformed bars with a yield stress of 498 MPa. The stirrups used with the specimens were 8 mm-diameter deformed bars with a yield stress of 408 MPa.
3.2 Test Setup and Loading Configuration
The base mat of each specimen was fully fixed, while the top of the specimen was free to move horizontally. The transverse load was applied at the top of the column through a double-action actuator (with force and displacement control system) with a load capacity of 1000 kN fixed on a reaction wall. The axial load was applied on the centroid of the free end section of the specimen through a 1000 kN hydraulic jack keeping constant throughout the test. During the test, displacements were measured by means of linear variable differential transducers (LVDTs), while strains of reinforcement were measured by means of strain gauges. The location of strain gauges can be found in Fig. 8. The axial load was first applied to the target value and maintained constant by adjusting the readings of instrument panel in the hydraulic jack during the experiment. In the tests, a force-control loading program was applied before the longitudinal reinforcement yielded. The increment of load was initially 50 kN for each step until arriving at 80% of the predicted yielding load, and then changed the increment of 50 kN into 25 kN. The force-control loading procedure was not changed until the longitudinal reinforcement yielded. Then, the specimens were subjected to the cyclic shear force by the displacement-control loading method with the increment of each load step equals to the measured yield displacement up until the shear-resisting capacity drops by more than 20% of the maximum shear force. In the force-control loading procedure, all cycles were carried out once, while each displacement level was repeated three times in the displacement-control loading procedure. The detailed experiment setup and loading configuration can be found in the experimental study of ECC columns by Wu et al. (2017).
4 Discussion of Test and Predicted Results
4.1 Crack Patterns and Failure Modes
Summary of experimental and predicted results.
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 |
4.2 Deformation Comparison
4.3 Proportion of Flexural and Shear Deformation
5 Conclusions
- 1.
By assuming that the load–displacement response of RECC column from the first cracking diagonally to the yielding is linear, the effective shear stiffness between the elastic shear stiffness and fully diagonally cracking shear stiffness is derived using the linear interpolation method.
- 2.
Based on the truss-arch model, the fully diagonally cracked shear stiffness is considered as the combination of a truss component and an arch component. For the truss component, explicit formulas for calculating the fully diagonally cracked shear stiffness based on CATM and VATM were provided, respectively. The selection of truss model used is determined by the minimum crack angle which was obtained through the principle of minimum potential energy. In view of the tensile strain-hardening behavior of ECC material, the ties in the truss model for RECC columns were proposed to be consisted of the transverse reinforcement and the fiber bridging effect at cracks.
- 3.
Six RECC columns with various shear span-to-depth ratios, transverse reinforcement ratios and axial loads were studied experimentally to verify the proposed model. Comparison of the measured and calculated deformation of RECC columns indicate that the observed and calculated crack angles are comparable, and the theoretical results using the proposed model of effective shear stiffness are shown to be consistent with the shear behavior observed experimentally. By analyzing the proportion of shear and flexural deformation, it can be concluded that the shear contribution to the total deformation for the diagonally cracked RECC column is significant, especially for the RECC columns with low shear span-to-depth ratios.
Declarations
Acknowledgements
The authors acknowledge the funding supports of National Natural Science Foundation of China (Grant No. 51778462 and 5141101015), and national key research and development plan, China (Grant No. 2016YFC0701400).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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