• With the U-wrap strengthening scheme, both failure modes are possible but when each failure mode might occur is a question that has not yet been answered. Special attention should be paid to this question.
• The effective bond length of the FRP plates influences their axial effective strain.
The published results regarding this variable are contradictory. In order to determine the suitable sheet height, more investigations are required.
• In this study, the shear-strengthened beams were analysed only under static loading.
It appears that no study has been carried out when the loads are applied in a cyclic manner in order to investigate the fatigue in bridge design. To understand such behaviour, further studies are recommended.
• The concrete material modelling available in the current finite element packages are not appropriate for complicated applications. The last proposition on future research concerns the needs of advanced concrete material modelling for use in the finite element analysis.
Bibliography
M. Abdel-Jaber, P. Walker, and A. Hutchinson. Shear strengthening of reinforced concrete beams using different configurations of externally bonded carbon fibre reinforced plates.
Materials and Structures, 36(259):291-301, 2003.
Committee 440 ACL Guide for the Design and Construction of Externally Bonded FRP Systems for Strengthening Concrete Structures. American Concrete Institute, Michigan, 2002.
B. Adhikary and H. Mutsuyoshi. Behaviour of concrete beams strengthened in shear with carbon-fibre sheets. Journal of Composites for Construction, ASCE, 8(3):258-264, 2004.
B. Adhikary and H. Mutsuyoshi. Shear strengthening of reinforced concrete beams using various techniques. Construction and Building Materials, 20(6):366-373, 2006.
B. Adhikary, H. Mutsuyoshi, and M. Ashraf. Shear strengthening of reinforced concrete beams using fiber-reinforced polymer sheets with bonded anchorage. ACI Structural
Journal, 101(5):660-668, 2004.
B. Adhikary, H. Mutsuyoshi, and M. Sano. Shear strengthening of reinforced concrete beams using steel plates bonded on beam web:experimental and analysis. Construction and Building Materials, 14(5):237-244, 2000.
ADINA. Automatic Dynamic Incremental Nonlinear Analysis, Finite Element Software, Version 8.4. ADINA R&D Inc., Watertown, MA, USA, 2004a.
ADINA. Theory and Modeling Guide: Volume I, Version 8.4- ADINA R&D Inc., Water- town, MA, USA, 2004b.
BIBLIOGRAPHY B. Aedy, J. San Roman, and E. Bruhwiller. Carbon fibre shear strengthening of rectangular concrete beams. Final Report 97.02, Ecole Polytechnique de Lausanne, Switzerland, 1999.
B. Agarwal and L. Broutman. Analysis and Performance of Fibre Composites. John Wiley and Sons, USA, 1st edition, 1990.
R. Al-Mahaidi, K. Lee, and G. Taplin. Behavior and analysis of RC T-beams partially damaged in shear and repaired with CFRP laminates. In P.C. Chang, editor, 2001 Structural Congress and Exposition, Washington, D C , May 21-23 2001. ASCE.
G. Al-Sulaimani, A. Istem, A. Basunbul, M. Bluch, and B. Ghaleb. Shear repair for reinforced concrete by fibre glass plate bonding. ACI Structural Journal, 91(4):458- 464, 1994.
J. Alexander and R. Cheng. Shear design model of concrete girders strengthened with advanced composite materials, pages 531-542, Calgary, Canada, 1996. The International Conference on Short and Medium Span Bridges.
W. An, H. Saadatmanesh, and M. Ehsani. RC beams strengthened with FRP plates:
Analytical and parametric study. Journal of Structural Engineering, ASCE, 117(11):
3434-3455, 1991.
N. Araki, Y. Matsuzaki, K. Nakano, T. Katoaka, and H. Fukuyama. Shear capac- ity of retrofitted re members with continuous fibre sheets. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforcement for Concrete Structures, volume 1, pages 515-522, Tokyo, Japan, 1997. Japan Concrete Institute.
M. Arduini, A. Nanni, A. Di Tommaso, and F. Focacci. Shear response of continuous RC beams strengthened with carbon FRP sheets. In Non-Metallic (FRP) Reinforcement for Concrete Structures, volume 1, pages 459-466, Tokyo, Japan, 1997. Japan Concrete Institute.
Committee ASCE ASCE. Recent approaches to shear design of structural concrete. Jour- nal of Structural Engineering, ASCE, 124(12):1375-1417, 1998.
R. Barnes, P. Baglin, G. Mays, and N. Subedi. External steel plate systems for the shear strengthening of reinforced concrete beams. Engineering Structures, 23(9): 1162-1176, 2001.
A. Bousselham and O. Chaallal. Shear strengthening reinforced concrete beams with fibre-reinforced polymer: Assessment of influencing parameters and required research.
ACI Structural Journal, 101(2):219-227, 2004a.
A. Bousselham and O. Chaallal. Behavior of reinforced concrete T-beams strengthened in shear with carbon fiber-reinforced polymer—an experimental study. A CI Structural
Journal, 103(3) :339-347, 2006a.
A. Bousselham and O. Chaallal. Effect of transverse steel and shear span on the perfor- mance of RC beams strengthened in shear with CFRP. Composites Part B:Engineering, 37(l):37-46, 2006b.
G. Box and K. Wilson. On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society Series B, 13(1):1—45, 1951.
Concrete Society Committee BS. Design for Strengthening Concrete Structures using Fibre Composite Materials. British Concrete Society, UK, 2000.
S. Cao, J. Chen, J. Teng, Z. Hao, and J. Chen. Debonding in RC beams shear strengthened with complete FRP wraps. Journal of Composites for Construction, ASCE, 9(5):417- 428, 2005.
A. Carolin. Carbon Fibre Reinforced Polymers for Strengthening of Structural Elements.
PhD thesis, Lulea University of Technology, Sweden, 2003.
A. Carolin and B. Taljsten. Theoretical study of strengthening for increased shear bearing capacity. Journal of Composites for Construction, ASCE, 9(6):497-506, 2005.
O. Chaallal, M. Nollet, and D. Perraton. Strengthening of reinforced concrete beams with externally bonded fibre-reinforced plastic plates: Design guidelines for shear and flexure. Canadian Journal of Civil Engineering, 25(4):692-704, 1998a.
O. Chaallal, M. Nollet, and D. Perraton. Shear strengthening of reinforced concrete beams by externally bonded side CFRP strips. Journal of Composites for Construction, ASCE, 2(2):111-113, 1998b.
O. Chaallal, M. Shahawy, and M. Hassan. Performance of reinforced concrete T-girders strengthened in shear with carbon fibre-reinforced polymer fabric. ACI Structural Jour- nal, 99(3):335-343, 2002.
BIBLIOGRAPHY M. Chajes, T. Januszka, D. Mertz, T. Thomson, and W. Finch. Shear strengthening of reinforced concrete beams using externally applied composite fabrics. ACI Structural Journal, 92(3):295-303, 1995.
J. Chen and J. Teng. Shear capacity of fiber-reinforced polymer-strengthened reinforced concrete beams: Fiber reinforced polymer rupture. Journal of Structural Engineering, ASCE, 129(5) :615-625, 2003a.
J. Chen and J. Teng. Shear capacity of FRP-strengthened RC beams: FRP debonding.
Construction and Building Materials, 17(1):27-41, 2003b.
M. Collins and P. Mitchell. Shear and torsion design of prestressed and non-prestressed concrete beams. PCI Journal, 25(5):32-100, 1980.
CSA-A23.3-04. Design of Concrete Structures for Buildings. Canadian Standards Asso- ciation, Rexdale, Ontario, Canada, 240 p, 2004.
Oakdale Engineering DataFit. DataFit curve fitting (nonlinear regression) ana data plot- ting software, version 8.1.69. Oakdale Engineering Inc., Oakdale, PA, USA, 2005.
C. Deniaud. Behaviour of Reinforced Concrete Beams Strengthened in Shear with FRP Sheets. Doctoral Thesis, University of Alberta, Edmonton, 2000.
C. Deniaud and J. Cheng. Shear rehabilitation of G-girders bridges in Alberta using FRP sheets. Canadian Journal of Civil Engineering, 27(5):960-971, 2000.
C. Deniaud and J. Cheng. Shear behaviour of reinforced concrete t-beams with externally bonded fibre-reinforced polymer sheets. ACI Structural Journal, 98(3):386-394, 2001a.
C. Deniaud and J. Cheng. Shear behaviour of reinforced concrete T-beams with externally bonded fiber-reinforced polymer sheets. ACI Structural Journal, 98(3).'386-394, 2001b.
C. Deniaud and J. Cheng. Reinforced concrete T-bams strengthened in shear with fiber reinforced polymer sheets. Journal of Composites for Construction, ASCE, 7(4):302- 310, 2003.
Stat-Ease Design-Expert. Design of experiments (DOE) software, version 6.0.1. Stat-Ease Inc., Minneapolis, MN, USA, 2000.
C. Diagana, A. Li, B. Gedalia, and Y. Delmas. Shear strengthening effectiveness with CCF strips. Engineering Structures, 25(4):507-516, 2003.
DIANA. DIANA Finite Element Analysis User's Manual Manual Nonlinear Analysis Release 7.2. DIANA R&D Inc., Delft, Netherland, 2000.
I. Elyasian, N. Abdoli, and H. Ronaph. Evaluation of parameters effective in FRP shear strengthening of RC beams using FE method. Asian Journal of Civil Engineering
(Building and Housing), 7(3):249-257, 2006.
P.H. Emmons. Concrete Repair and Maintenance Illustrated. R.S. Means Company. Inc., USA, 1993.
P. Fanning and O. Kelly. Shear strengthening of reinforced concrete beams: an experi- mental study using CFRP plates. UK, 13-15 July 1999. 8th International concrete on Structural Faults and Repair.
Task Group 9.3 FRP Reinforcement for Concrete Structures FIB. Design and Use of Externally Bonded Fibre Reinforced Polymer Reinforcement (FRP EBR) for Reinforced Concrete Structures. International Federation for Structural Concrete, Switzerland, 2001.
H. Floegl and H. Mang. On tension stiffening in cracked reinforced concrete slabs and shells considering geometric and physical nonlinearity. Journal of Applied Mechanics, 51(3):215-242, 1981.
G. Gendron, A. Picard, and M. Guerin. A theoratical study on shear strengthening of reinforced concrete beams using composite plates. Composite Structures, 45(4):303-309, 1999.
S. Giuseppe. Finite Element Analysis of RC Beams retrofitted with Fibre Reinforced Polymers. PhD thesis, Universita Delgi Studi di Napoli Federico II, Italy, 2005.
A. Godat, K.W. Neale, and P. Labossiere. Numerical modelling of FRP shear-strengthened reinforced concrete beams. Jornal of Composites for Construction, ll(6):640-649, 2007a.
BIBLIOGRAPHY A. Godat, K.W. Neale, and P. Labossiere. Towards modelling FRP shear-strengthened
reinforced concrete beams. In Triantafillou, editor, 8th Int. Symposium, FRP Reinforce- ment for Concrete Structures (FRPRCS-8), Greece, 6-18 July 2007b.
E. Grande, M. Imbimbo, and A. Rasulo. Experimental behaviour of RC beams strength- ened in shear by FRP sheets. In T. Triantafillou, editor, 8th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-8), Greece, 16-18 July 2007.
S. Hassan, C. Morley, and J. Lees. Effect of effective depth and longitudinal steel ratio on the behaviour of precracked reinforced concrete T-beams strengthened in shear with CFRP fabrics. In Triantafillou, editor, 8th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-8), Greece, 16-18 July 2007.
R. Hutchinson, A. Adelrahman, S. Rizkalla, and G. Smith. Shear strengthening using FRP sheets for a highway bridge in manitoba, Canada. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforcement for Concrete Structures, volume 1, pages 531-538, Tokyo, Japan, 1997. Japan Concrete Institute.
Canada ISIS. Strengthening Reinforced Concrete Structures with Externally-Bonded Fibre Reinforced Polymers, Design Manual No.4- Intelligent Sensing for Innovative Structures (ISIS) Canada, University of Manitoba, Winnipeg, Manitoba, CANADA, 2001.
D. Kachlakev and D. McCurry. Behaviour of full-scale reinforced concrete beams retrofitted for shear and flexure with FRP laminates. Composites Part B: Engineering, 31(6-7):445-452, 2000.
V. Kaliakin, M. Chajes, and T. Januszka. Analysis of concrete beams reinforced with externally bonded woven composite fabrics. Composites Part BEngineering, 27(3-4):
235-244, 1996.
A. Khalifa, W. Gold, A. Nanni, and M. Aziz. Contribution of externally bonded FRP to shear capacity of RC flexural members. Journal of Composites for Construction,ASCE, 2(4): 195-202, 1998.
A. Khalifa and A. Nanni. Improving shear capacity of existing RC T-section beams using CFRP composites. Cement and Concrete Composites, 22(3): 165-174, 2000.
A. Khalifa and A. Nanni. Rehabilitation of rectangular simply supported RC beams with shear deficiencies using CFRP composites. Construction and Building Materials, 16(3):
135-146, 2002.
F. Kong and R. Evans. Reinforced and Prestressed Concrete. Spon Press, UK, 3rd edition, 1990.
H. Kwak and F. Filippou. Nonlinear finite element analysis of reinforced concrete struc- tures under monotonic loads. Computers and Structures, 65(1):1-16, 1997.
K. Lee, R. Al-Mahaidi, and G. Taplin. Non-linear finite element modeling of shear dam- aged concrete T-beams repaired with CFRP laminates. In Composites in the Trans- portation Industry, Sydney. Australia, 2001. ACUN-2 International Composites.
T. Lee. Shear Strengthening of Reinforced Concrete T-Beams Strengthened Using Carbon Fibre Reinforced Polymer (CFRP) Laminates. Doctoral Thesis, Monash University, Victoria, Australia, 2003.
T. Lee and R. Al-Mahaidi. An envestigation on shear behaviour of RC T-beams strength- ened with CFRP using photogrametry. Composite Structures, 82(1):185-193, 2008.
C. Leung, Z. Chen, S. Lee, M. Ng, M. Xu, and J. Tang. Effect of size on the failure of geometrically similar concrete beams strengthened in shear with FRP strips. Journal
of Composites for Construction, ASCE, ll(5):487-496, 2007.
A. Li, C. Diagana, F. Buyle-Bodin, and Y. Delmas. Shear strengthening of reinforced concrete beams. Concrete Science and Engineering, 3:250-256, 2001.
A. Li, C. Diagana, and Y. Delmas. Shear performance with externally bonded carbon fibre fabrics. In K. Tan, editor, 6th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-6), pages 497-506, Singapore, 8-10 July 2003.
R. Loov. Review of A23.3-94 simplified method of shear design and comparison with results using shear friction. Canadian Journal of Civil Engineering, 25(3):437-450, 1998.
L. Lorenzis and A. Nanni. Shear strengthening of reinforced concrete beams with near- surface mounted fibre-reinforced polymer rods. ACI Structural Journal, 98(l):60-68, 2001.
BIBLIOGRAPHY X.Z. Lu, L.P. Ye, J.G. Teng, and J.J. Jiang. Mesoscale finite element model for FRP sheets/plates bonded to concrete. Journal of Engineering Structures, 27:264-575, 2005.
A. Malek and H. Saadatmanesh. Analytical study of reinforced concrete beams strength- ened with web-bonded fiber reinforced plastic plates or fabrics. ACI Structural Journal, 95(3):343-352, 1998a.
A. Malek and H. Saadatmanesh. Ultimate shear sapacity of reinforced concrete beams strengthened with web-bonded fiber-reinforced plastic plates. ACI Structural Journal, 95(4):391-399, 1998b.
F. Mecilli, R. Annaiah, and A. Nanni. Strengthening of short shear span reinforced concrete T joists with fibre-reinforced plastic composites. Journal of Composites for
Construction, ASCE, 6(4):264-271, 2002.
G. Melo, A. Araujo, and Y. Nagato. Strengthening of RC beams in shear with carbon sheet laminates (CFRP). In K. Tan, editor, 6th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-6), pages 477-486, Singapore, 2003.
Y. Mitsui, K. Murakami, K. Takeda, and H. Sakai. A study on shear reinforcement of reinforced concrete beams externally bonded with carbon fibre sheets. Composite Interfaces, 5(4):285-295, 1998.
G. Monti and M. Liotta. Tests and design equations for FRP-strengthening in shear.
Construction and Building materials, 21(4):799-809, 2007.
A. Mosallam and S. Banerjee. Shear enhancement of reinforced concrete beams strength- ened with FRP composite laminates. Composites Part:B Engineering, 38(5-6):781-793, 2007.
K.W. Neale. FRPs for structural rehabilitation: a survey of recent progress. Progress in Structural Engineering and Materials, 2(2):133-138, 2000.
M. Neto, M. Melo, and Y. Nagato. T-beams strengthened in shear with carbon sheet laminates (CFRP). In C. Burgoyne, editor, 5th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-5), pages 239-248, UK, 2001.
T. Norris, H. Saadatmanesh, and M. Ehsani. Shear and flexural strengthening of R/C beams with carbon fiber sheets. Journal of Structural Engineering, ASCE, 123(7):
903-911, 1997.
C. Pellegrino and C. Modena. Fibre reinforced polymer shear strengthening of reinforced concrete beams with transverse steel reinforcement. Journal of Composites for Con- struction, ASCE, 6(2):104-111, 2002.
C. Pellegrino and C. Modena. Fiber-reinforced polymer shear strengthening of reinforced concrete beams: Experimental study and analytical modeling. ACI Structural Journal, 103(5):720-728, 2006.
Z. Qu, X. Lu, and L. Ye. Size effect of shear contribution of externally bonded FRP U- jackets for RC beams. In Chen J. and J. Teng, editors, Int. Symposium, Bond Behaviour
of FRP in Structures (BBFS 2005), volume 1, pages 363-371, Hong Kong, China, 2005.
International Institute for FRP in Construction.
Australian Standard AS3600-2001 SAL Concrete Structures. International Federation for Structural Concrete, Sydney, Australia, 2001.
R. Santhakumar, E. Chandrasekaran, and R. Dhanarraj. Analysis of retrofitted reinforced concrete shear beams using carbon fiber composites. Electronic Journal of Structural Engineering, 4:66-74, 2004.
Y. Sato, T. Ueda, and Y. Kakuta. Shear reinforcement effect of carbon fiber sheet attached to sides of reinforced concrete beams. In M. El-Badry, editor, Advanced Composites Materials in Bridges and Structures, pages 621-628, Montreal, Quebec, 1996.
Y. Sato, T. Ueda, Y. Kakuta, and Y. Kobatake. Shear strengthening of existing reinforced concrete beams by CFRP sheet. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforce- ment for Concrete Structures, volume 1, pages 507-514, Sapporo, Japan, 1997a.
Y. Sato, T. Ueda, Y. Kakuta, and S. . Ono. Ultimate shear capacity of reinforced concrete beams with carbon fibre sheet. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforce- ment for Concrete Structures, volume 1, pages 499-506, Tokyo, Japan, 1997b. Japan Concrete Institute.
BIBLIOGRAPHY Y. Sato and F. Vecchio. Tension stiffening and crack formation in reinforced concrete mem-
bers with fiber-reinforced polymer sheets. Journal of Structural Engineering, ASCE, 129 (6):717-724, 2003.
M. Shamsaia, H. Sezena, and A. Khaloo. Behavior of reinforced concrete beams post- tensioned in critical shear region. Engineering Structures, 29(7): 1465-1474, 2007.
T. Suntharavadivel and T. Aravinthan. Shear strengthening of cracked RC beam using external post-tensioning. Christchurch, NZ., 29 Nov - 1 Dec 2006. 19th Australasian Conference on the Mechanics of Structures and Materials.
R. Swamy, R. Jones, and J. Bloxham. Structural behaviour of reinforced concrete beams strengthened by epoxy-bonded steel plates. The Structural Engineer, 65(2):59-68, 1987.
R. Swamy, P. Mukhopadhyaya, and C. Lynsdale. Strengthening for shear of RC beams by external plate bonding. The Structural Engineer, 77(12):19-30, 1999.
L. Taerwe, H. Khalil, and S. Matthys. Behaviour of RC beams strengthened in shear by external CFRP sheets. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforcement for
Concrete Structures, volume 1, pages 483-490, Sapporo, Japan, 1997. Japan Concrete Institute.
B. Taljsten. Strengthening of Existing Concrete Structures with Epoxy Bonded Plates of Steel or Fibre Reinforced Plastics. Doctoral Thesis, Lulea University of Technology, Lulea, Sweden, 1994.
B. Taljsten. FRP Strengthening of Existing Concrete Structures. Lulea University, Sweden, 1st edition, 2002.
B. Taljsten. Strengthening concrete beams for shear with CFRP sheets. Construction and Building Materials, 17(1), 2003.
B. Taljsten and A. Carolin. Shear strengthening of concrete beams in shear-theory and test. In J. Teng, editor, FRP Composites in Civil Engineering, pages 657-667, Hong Kong, 2001.
B. Taljsten and L. Elfgren. Strengthening concrete beams for shear using CFRP-materials:
Evaluation of different application methods. Composites Part B: Engineering, 31(2):
87-96, 2000.
J. Teng, J. Chen, S. Smith, and L. Lam. FRF'-Strengthened RC Structures. John Wiley and Sons, UK, 1st edition, 2002.
J.G. Teng, X.Z. Lu, L.P. Ye, and J.J. Jiang. Recent research on intermediate crack- induced debonding in FRP strengthening RC beams. In Advanced Composite Materials
in Bridges and Structures, ACMBS-IV, Canadian Society for Civil Engineering, M.
El-Badry and L. Dunaszegi, Eds., Calgary, CANADA, 12 p, 2004.
T. Triantafillou. Shear strengthening of concrete members using composites. In 3rd Int. Symposium, Non-Metallic (FRP) Reinforcement for Concrete Structures, volume 1, Tokyo, Japan, 1997. Japan Concrete Institute.
T. Triantafillou. Shear strengthening of reinforced concrete beams using epoxy-bonded FRP composites. ACI Structural Journal, 95(2):107-115, 1998.
T. Triantafillou and C. Antonopoulos. Design of concrete flexural members strengthened in shear with FRP. Journal of Composites for Construction, ASCE, 4(4):198-205, 2000.
K. Uji. Improving shear capacity of existing reinforced concrete members by applying carbon fibre sheets. Transaction of the Japan Concrete Institute, 14:254-266, 1992.
F. Vecchio and M. Collins. The modified compression field theory for reinforced concrete elements subjected to shear. ACI Strucrural Journal, 83(2):219-231, 1986.
R. Wong. Towards modeling of reinforced concrete members with externally bonded fibre reinforced polymer (FRP) composites. M.A.Sc. Thesis, University of Toronto, Toronto, Ontario, 2001.
R.S.Y. Wong and F.J. Vecchio. Toward modeling of reinforced concrete members with externally bonded fiber-reinforced polymer composites. ACI Structural Journal, 100(1):
47-55, 2003.
X. Yang, A. Nanni, and G. Chen. Effect of corner radius on the performance of ex- ternally bonded frp reinforcement. In C. Burgoyne, editor, 5th Int. Symposium, FRP Reinforcement for Concrete Structures (FRPRCS-5), pages 197-204, UK, 2001.
Z. Zhang and C. Hsu. Shear strengthening of reinforced concrete beams using carbon- fiber-reinforced polymer laminates. Journal of Composites for Construction, ASCE, 9 (2):158-169, 2005.
Appendix A
A D I N A Concrete Constitutive Model
It is well accepted that concrete is a very complex material. The concrete constitutive model provided in ADINA may not be capable to simulate all structural applications.
However, it works well for simple applications, such as FRP shear-strengthening of rein- forced concrete beams. The concrete model is a hypoelastic concrete model. It is based on three features: concrete in compression; post-cracking model; and failure envelope.
The nonlinear stress-strain relation allows for weakening of the material under increasing compressive stresses. Failure envelopes that define failure in tension and crushing of con- crete in compression. A fixed smeared crack model is used to describe the post-cracking behaviour of concrete. The objective of this appendix is to provide a brief description of the characteristics of ADINA, version 8.4, concrete material model.
A . l Concrete in Compression
The general multiaxial stress-strain relations are derived from a uniaxial (incremental) stress-strain relation. When the concrete in tension the stress-strain relation is linear until tensile failure. In this case, an incremental relation between stress (a) related to the strain (i) through a modulus of elasticity ([C]). Such relation can be expressed as:
H = [cm (A.I)
e* strain, Epi
Figure A . l : Equivalent uniaxial stress-strain relationship under multiaxial state of stress
When the concrete is under compression and before the cracking initiated, the following relation is used to calculate t h e matrix of modulus of elasticity:
R •
•ism. pi
Ec[l - B{^)2 - 2C(^)3]
[l + A(£-£ + B(£-£)2 + C(£f)3}2 (A.2)
where, A = -^ f ^ + ( P3- 2 P2| £ - ( 2 P3- 3 P2+ l ) ]
T^-*P+m • B = [(2|-3)-2A], C = [(2-f+A]
and t h e values of Es = V-, P = A"- and t h e parameters ( eu, em, a ' anda' ) are computed in relation t o their corresponding uniaxial values (eu, em, auandcrm), as shown in Figure A.l as:
eu = (Ci7i2 + C 2 7 1 K (A.3)
4 = (Ci7i + C27 i ) ^ (A.4)
l\Vu (A.5)
CTm = 71*11 (A.6)