Confined Concrete
4.4 Analytical Modeling of Confined Concrete
4.4.3 Stress–Strain Relation
Several algebraic forms have been proposed to represent the stress–strain curve for confined concrete. The models generally fall into one of three categories. Figure 4.19a is representative of a class of models that have an ascending branch followed by a linear descending branch with or without a residual plateau (e.g., Park et al., 1982; Razvi and Saatcioglu, 1999; Bing et al., 2001). Figure 4.19b is similar but with a plastic plateau at the peak stress (e.g., Sheikh and Uzumeri, 1982). Figure 4.19c shows an example of a single equation that gives a continuous stress–strain curve (e.g., Ahmad and Shah, 1982; Mander et al., 1988a).
FIGURE 4.19 Different forms of confined concrete stress–strain relations.
Each of the models cited has been shown to produce good correlation with measured laboratory test results. Here we adopt the algebraic form proposed by Mander et al. (1988a), as shown in Figure 4.20. The longitudinal compressive stress fc is given by
FIGURE 4.20 Confined concrete stress–strain relation. (After Mander et al., 1988a.)
In Eq. (4.21), the modulus of concrete, Ec, can be estimated as described in Chapter 3. The variables and εcc are determined as described previously in this chapter. Note that Eq. (4.19) with appropriate input quantities can be used to describe the behavior of plain concrete as well.
Example 4.3. Construct the unconfined and confined concrete stress–strain relations for the cross section (Figure 4.21) of Example 4.3.
FIGURE 4.21 Stress–strain relations.
Solution
From Example 4.2, ke = 0.72, femin = 378 psi (2.6 MPa), Cfc′ = 4250 psi
(29 MPa), and = 6700 psi (46 MPa). Taking ε0 = 0.002, Eq. (4.16) gives εcc
= 0.008 and Eq. (4.17) gives εcmax = 0.023. From Chapter 3, we can
estimate Young’s modulus as Equation
(4.19) is then used to plot the stress–strain relations. Note that for unconfined concrete, the material properties in Eqs. (4.19) through (4.21) are set equal to those of unconfined concrete, that is, and εcc = ε0. The results are plotted in the accompanying figure.
The proposed model has been used to compute stress–strain relations for several columns or walls tested in axial compression in the laboratory.
Figure 4.22 compares calculated and measured stress–strain relations for a sample of the tests, showing generally good correlation between calculated and measured results.
FIGURE 4.22 Measured and calculated stress–strain relations for a sample of columns tested in axial compression in different laboratories, for different confinement configurations, and at different strain rates. The arrows at 0.85 fc′ show strengths expected for unconfined concrete. x marks the strain (measured or calculated) at first hoop fracture. (Data after Sheikh and Uzumeri, 1980;
Scott et al., 1982; Moehle and Cavanagh, 1985; Mander et al., 1988b)
References
Ahmad, S.H., and S.P. Shah (1982). “Stress-Strain Curves of Concrete Confined by Spiral Reinforcement,” ACI Journal, Vol. 79, No. 6, pp.
484–490.
Balmer, G.G. (1949). Shearing Strength of Concrete Under High Triaxial Stress—Computation of Mohr’s Envelop as a Curve, Structural
Research Laboratory Report No. SP-23, U.S. Bureau of Reclamation, Denver, CO.
Bing, L., R. Park, and H. Tanaka (2000). “Constitutive Behavior of High- Strength Concrete under Dynamic Loads,” ACI Structural Journal, Vol.
97, No. 4, pp. 619–629.
Bing, L., R. Park, and H. Tanaka (2001). “Stress-Strain Behavior of High- Strength Concrete Confined by Ultra-High- and Normal-Strength Transverse Reinforcements,” ACI Structural Journal, Vol. 98, No. 3, pp. 395–406.
Blume, J.A., N.M. Newmark, and L.H. Corning (1961). Design of Multi- Story Reinforced Concrete Buildings for Earthquake Motions, Portland Cement Association, Chicago, IL, 318 pp.
Bresler, B., and V.V. Bertero (1975). “Influence of Strain Rate and Cyclic Loading on Behavior of Unconfined and Confined Concrete in
Compression,” XVII Jornadas Sudamericanas de Ingenieria
Estructural, V Simposio Panamericana de Estructuras, Caracas, 8 al 12 de Diciembre de 1975.
Burdette, E.G., and H.K. Hilsdorf (1971). “Behavior of Laterally
Reinforced Concrete Columns,” Journal of the Structural Division, Vol. 97, No. ST2, pp. 587–602.
CSA (2004). Design of Concrete Structures, CSA A23.3-04, Canadian Standards Association, Mississauga, Canada.
Dodd, L.L., and N. Cooke (1992). “Dynamic Response of Circular Bridge Piers,” Proceedings, 10th World Conference on Earthquake
Engineering, A.A. Balkema, Rotterdam, pp. 3035–3039.
Ghali, A., and S.A. Youakim (2005). “Headed Studs in Concrete: State of the Art,” ACI Structural Journal, Vol. 102, No. 5, pp. 657–667.
Iyengar, K.T., P. Desayi, and K.N. Reddy (1970). “Stress-Strain
Characteristics of Concrete Confined in Steel Binders,” Concrete Research, Vol. 22, No. 72, pp. 173–184.
Kaar, P.H., A.E. Fiorato, J.E. Carpenter, and W.G. Corley (1978). Limiting Strains of Concrete Confined by Rectangular Hoops, Research and Development Bulletin RD053.01D, Portland Cement Association, Skokie, IL, 12 pp.
Khaloo, A.R., K.M. El Dash, and S.H. Ahmad (1999). “Model for
Lightweight Concrete Columns Confined by Either Single Hoops or
Interlocking Spirals,” ACI Structural Journal, Vol. 96, No. 6, pp. 883–
890.
Mander, J.B., M.J.N. Priestley, and R. Park (1988a). “Theoretical Stress- Strain Model for Confined Concrete,” Journal of Structural
Engineering, Vol. 114, No. 8, pp. 1804–1826.
Mander, J.B., M.J.N. Priestley, and R. Park (1988b). “Observed Stress- Strain Behavior of Confined Concrete,” Journal of Structural Engineering, Vol. 114, No. 8, pp. 1827–1849.
Manrique, M.A., V.V. Bertero, and E.P. Popov (1979). Mechanical Behavior of Lightweight Concrete Confined by Different Types of Lateral Reinforcement, Report No. UCB/EERC-79/05, Earthquake Engineering Research Center, University of California, Berkeley, CA, 103 pp.
Martin, C.W. (1968). “Spirally Prestressed Concrete Cylinders,” ACI Journal, Vol. 65, No. 10, pp. 837–845.
Martinez, S., A.H. Nilson, and F.O. Slate (1984). “Spirally Reinforced High-Strength Concrete Columns,” ACI Journal, Vol. 81, No. 5, pp.
431–441.
Moehle, J.P., and T. Cavanagh (1985). “Confinement Effectiveness of
Crossties in RC,” Journal of Structural Engineering, Vol. 111, No. 10, pp. 2105–2120.
Park, R., M.J.N. Priestley, and W.D. Gill (1982). “Ductility of Square- Confined Concrete Columns,” Journal of the Structural Division, Vol.
108, No. ST4, pp. 929–950.
Paultre, P., and F. Légeron (2008). “Confinement Reinforcement Design for Reinforced Concrete Columns,” Journal of Structural Engineering, Vol. 134, No. 5, pp. 738–749.
Razvi, S., and M. Saatcioglu (1999). “Confinement Model for High-Strength Concrete,” Journal of Structural Engineering, Vol. 125, No. 3, pp.
281–289.
Richart, F.E., A. Brandtzaeg, and R.L. Brown (1928). A Study of the
Failure of Concrete under Combined Compressive Stresses, Bulletin No. 185, Engineering Experiment Station, University of Illinois,
Urbana, IL, 104 pp.
Richart, F.E., A. Brandtzaeg, and R.L. Brown (1929). The Failure of Plain and Spirally Reinforced Concrete in Compression, Bulletin No. 190, Engineering Experiment Station, University of Illinois, Urbana, IL, 74 pp.
Richart, F.E., and R.L. Brown (1934). An Investigation of Reinforced
Concrete Columns, Bulletin No. 267, Engineering Experiment Station, University of Illinois, Urbana, IL, 94 pp.
Rood, M., and J.P. Moehle (2006). “Investigation of Welded Reinforcement Grids,” http://nees.berkeley.edu/Projects/, 32 pp.
Saatcioglu, M., and M. Grira (1999). “Confinement of Reinforced Concrete Columns with Welded Reinforcement Grids,” ACI Structural Journal, Vol. 96, No. 1, pp. 29–39.
Saatcioglu, M., and S.R. Razvi (1992). “Strength and Ductility of Confined Concrete,” Journal of Structural Engineering, Vol. 118, No. 6, pp.
1590–1607.
Scott, B.D., R. Park, and M.J.N. Priestley (1982). “Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates,” ACI Journal Proceedings, Vol. 79, No. 1, pp. 13–27.
Shah, S.P., A. Fafitis, and R. Arnold (1983). “Cyclic Loading of Spirally Reinforced Concrete,” Journal of the Structural Division, Vol. 109, No. 7, pp. 1695–1710.
Sheikh, S.A., and S.M. Uzumeri (1980). “Strength and Ductility of Tied Concrete Columns,” Journal of the Structural Division, Vol. 106, No.
ST5, pp. 1079–1102.
Sheikh, S.A., and S.M. Uzumeri (1982). “Analytical Model for Concrete Confinement in Tied Columns,” Journal of the Structural Division, Vol. 108, No. ST12, pp. 2703–2722.
Tanaka, H., and R. Park (1987). “Effectiveness of Transverse
Reinforcement with Alternative Anchorage Details in Reinforced Concrete Columns,” Proceedings, Pacific Conference on Earthquake Engineering, Wairakei, New Zealand. Vol. 1, pp. 225–235.
____________
1In the reinforced concrete literature, D is sometimes defined as the core diameter measured to the centerline or the inside of the spiral or circular hoop reinforcement. Such definitions are consistent with cover concrete spalling geometry, which commonly leaves a concrete core with diameter somewhat smaller than the outside diameter of the spiral or hoops. Here we adopt the alternative definition of D, also commonly used in the literature, because it is simpler from a design perspective and produces acceptably accurate results.
2The design and behavior of spiral reinforcement and circular hoop reinforcement for flexural, axial, and shear loadings are essentially identical. Therefore, to simplify the presentation in the text that follows, spiral reinforcement will be emphasized, but the discussion applies equally to circular hoop reinforcement. Distinctions between the two types of reinforcement will be necessary for
reinforcement detailing and for torsional loading (the spiral pitch results in torsion directionality that does not arise for circular hoops).
3Confinement of cylindrical cores and buckling restraint for compressed longitudinal bars derives from the curvature of the circular hoop or spiral. For very large diameter columns, the hoop curvature becomes small, raising concerns about the ability of a large-diameter hoop to resist localized bursting forces from the core or from buckling longitudinal bars.