# 3d stress strain relationship of confined and unconfined concrete

Stress–strain relationship of concrete; a unconfined concrete, b CFRP-confined concrete Failure modes; a unconfined columns, b CFRP-confined columns . Three-Dimensional Finite-Element Analysis of FRP-Confined Circular Concrete. Therefore, the stress-strain relationship of a steel-confined concrete than the strength of the unconfined concrete (very lightly confined concrete); a in Proceedings of the 3d International Conference on Concrete Repair. ing behavior (heavily confined concrete), and the stress-strain curves feature an strength lower than the strength of the unconfined concrete mortars,” in Proceedings of the 3d International Conference on. Concrete.

The value of fcc can be obtained from Eq. Strain corresponding to fcc, Ecc can be determine by Eq. According to the equations proposed by Nematzadeh for STCC columns, the parameterkis considered as a constant value equal to 2.

However, a larger value is obtained by Richart et al. Due to the existence of the longitudinal compressive stress of steel tube in STCC columns, the confinement effect and thus the confinement effectiveness coefficient is reduced compared with the case without the longitudinal stress.

The parameter fl can be calculated by the trial and error method and matching the FE results with the experimental data, as follows 4 The above equation indicates that the parameter fl is independent of the concrete compressive strength. The second stage of the stress-strain curve of confined concrete is nonlinear and is determined from Eq.

Also, RE and R are modular ratio and ratio relation, respectively, and are obtained by the following equations. They are taken as 4. Since these two parameters influence the ascending branch of the concrete stress-strain curve, they are can be used to define the second stage of the confined concrete stress-strain curve.

The third stage of the confined concrete stress-strain curve is bilinear. The final stress is equal toK'f cc where, K'is the product of the two parameters K ' c and K ' s which are related to the compressive strength of unconfined concrete and the tube outer diameter-to-wall thickness ratio, respectively.

It can be found from these equations that the value of K ' c and consequently K' decreases with increasing the concrete compressive strength because of the reduction in concrete ductility. Stress and strain at the intersection between two lines of the third stage is considered 1.

Therefore, the stress and strain in the middle of the third stage are obtained equal to 0. As mentioned for the steel stress-strain curve, it should be noted that the end point of the third stage in the confined concrete stress-strain curve does not imply the failure point, but the ABAQUS program considers a constant stress condition after endpoint.

Drucker-Prager model Chen and Saleeb can be used as one of the yield criteria of concrete, in which the concrete shear strength is expressed based on the hydrostatic pressure.

Also, by considering a linear relationship between the shear strength and the confining pressure of concrete, the linear Drucker-Prager can be used for the concrete yield surface as Eq. In this model, an additional parameter known as the flow stress ratio is adopted.

This parameter controls the dependence of the yield surface on the value of the intermediate principal stress and is physically defined as the ratio of the yield stress in triaxial tension to that in triaxial compression, equal to the shear strength ratio of concrete under equal biaxial compression to that under triaxial compression. According to this figure, the yield surface function F of the extended Drucker-Prager model is expressed as Eq.

This figure indicates that two elements with the same material and the same hydrostatic pressure p can exhibit different yield strengths q. Hence, the initial and final failure surfaces correspond to the beginning of the concrete nonlinear stage and the concrete failure, respectively.

In order to make sure that the yield surface remains convex, it is necessary to apply the condition of 0. In circular STCC columns, the lateral stresses applied on the concrete core in different directions are the same, as shown in Fig. It should be noted that in this study Eqs.

### Stress-Strain Law for Confined Concrete with Hardening or Softening Behavior

The results obtained from the FE model of STCC columns indicate that the compressive behavior of these columns exhibits little sensitivity to the K parameter. This fact can be clearly observed in Fig. The reason for this is that according to Eqs.

This parameter can be calculated based on the triaxial and uniaxial compressive tests. It should be noted that the improved behavior of concrete material is used for the modeling in ABAQUS program so that its compressive strength and stress-strain curve are not in accordance with the experimental results of unconfined concrete, but are a function of the lateral confining pressure.

For this reason, the friction angle achieved by the calculations leads to a poor agreement between the analytical and the experimental results. The cohesion parameter d is geometrically the intercept of the linear yield surface, and is related to the yield stress of the uniaxial compression as: Physically, the dilation angle is defined as the ratio of plastic volume change to plastic shear strain and geometrically, it is the slope of the potential function in the p-t plane, as shown in Fig.

In this research, a non-associated flow rule is used to define the direction of the plastic flow. Increment of deformation capacity of the structure in critical regions, where large plastic deformations are expected, is one of the most efficient strategies in this field. Jacketing over the entire length of the element by FRP or FRCM wraps is often preferred, due to the favourable properties of this retrofitting methodology: In order to evaluate the effects of such a retrofitting strategy on the seismic behavior of the structure, stress-strain relationship of confined concrete is needed to evaluate the moment-curvature response of the elements [ 1 ].

When steel transverse reinforcement as a confining system is utilized, as soon as the transversal stress on the concrete reaches the cracking limit, the lateral strain suddenly grows and the confining steel hoops yield.

From this point, a nearly constant confining pressure is applied to the column concrete core, and the concrete behaves as an active confined material [ 2 ]. Therefore, the stress-strain relationship of a steel-confined concrete member is characterized by a steep increasing branch up to the yielding of the transversal reinforcements, followed by a softening branch, with a slope related to the effectiveness of the confinement.

FRP jackets, as opposed to steel hoops, have an elastic behavior up to failure and exert a monotonic increasing confining action. Usually, the stiffness and strength of the confining jacket is sufficient to ensure the absence of softening behavior heavily confined concreteand the stress-strain curves feature an ascending bilinear shape.

Such behavior has been detected in most of the experimental tests, performed on specimens of small scale. In these cases, the thickness of a single or few layers of FRP is able to confer enough stiffness to the wrapping to avoid a softening behavior. Lam and Teng [ 3 ] have stressed that for large structural elements, such as piers of bridges, or when fibers with low strength and low elasticity modulus are utilized such as glass or aramid fibersor when elements with rectangular cross section are considered, the lateral pressure cannot always provide enough confinement to ensure a hardening behavior.

Elements confined with FRCM often exhibit a flat or a descending postpeak branch, due to the progressive cracking of the binding mortar in the range of large deformation [ 4 — 6 ].

Thus, depending on the confinement level three different behaviors in the large strain range have been observed in experimental tests: In Figure 1the three shapes of the stress-strain curve obtained in experimental tests [ 78 ] are shown. Experimental stress-strain relationships with different confinement levels: In practical application, a reliable design of FRP jackets requires that the constitutive behavior of the FRP-confined concrete is accurately modeled.

Since the last decade, many constitutive models have been proposed for the behavior of RC member wrapped with FRP jacket, and many reviews are given in the literature [ 9 — 11 ]. The models are classified in several ways: Most of the analysis-oriented constitutive laws are well based on mechanical models and are effective in reproducing stress-strain curves of any shape, but the use of incremental or iterative procedures make the models suitable for use in computer analysis only, such as nonlinear finite element analysis NLFEA.

By contrast, most of the design-oriented models available in the literature are able to reproduce only the hardening behavior of the heavily confined concrete [ 10 ] or are obtained by two different analytical relations in the low and high strain field.

This paper provides a new general design-oriented stress-strain law, obtained by a suitable modification of the well-known Sargin curve [ 13 ] for steel confined concrete.

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It will be shown that if the values of the parameters of the model are deduced from experimental tests, the model is able to accurately reproduce the experimental curve. If they are evaluated on the basis of an analysis-oriented model, the proposed law provides a handy equivalent design model.

Most of the models have been formulated by generalization of an existing stress-strain relationship derived for steel-confined concrete.

Three expressions have been used most frequently: The tests conducted by the cited authors showed a descending branch of the stress-strain curve in the postpeak region, due to the small stiffness and strength of the confining filament.

Therefore, this model conserves the main limitation of the Sargin model that is not able to reproduce the characteristic hardening behavior of the FRP heavily confined concrete. Toutanji [ 18 ] and Saafi et al. The major drawbacks of these models are the use of two different analytical expressions for modeling the entire range of the constitutive curve and the ineffectiveness for lightly confined element, in modeling the possible stiffness recovering for high strain values.

In order to obtain a single relation for modeling stress-strain curve with an almost linear behavior in the high strain range, Richard and Abbot [ 14 ] proposed to describe the elastic-plastic constitutive law by the following four parameter curve: