Tension Stiffening Model

⎝𝜔+

√︃

𝜔²− 1 2

⎞

⎠𝜖_𝑜 (8.10)

where

𝜔 = 1 2

(︃1 2

𝐸𝑐𝜖𝑜

𝑓_𝑐𝑐 + 1

)︃

(8.11) Of convenience, the relation is slightly rewritten here compared to MC90. Also Kent and Park have proposed expressions for 𝜖ℎ, both unconfined as well as confined by rectangular hoops. The succeeding formulas are taken from [36], but modified here in order both to comply with the sign convention in this work and also to convert the strength-unit from psi to MPa. Thus, for the unconfined 𝜖ℎ

𝜖_ℎ𝑢 = 3 − 0.29𝑓_𝑐^′

145𝑓_𝑐^′ + 1000 (8.12)

and the confined 𝜖_ℎ

𝜖_ℎ𝑐 = 𝜖_ℎ𝑢 − 3 4𝜌_𝑠

√︃𝑏^′′

𝑠ℎ

(8.13) Here 𝑓_𝑐^′ is the concrete compressive cylinder strength (negative) in MPa, while 𝜌_𝑠 is the volume ratio of transverse reinforcement to concrete core measured to outside of hoops. Furthermore, 𝑏^′′ is the width of confined core measured to outside of hoops, and finally 𝑠_ℎ is the spacing of hoops.

With a 3D constitutive model and the Poisson effect included, confinement-dominated problems may have been treated more consistently. However, as already stated, that is considered to be outside the scope of this work.

8.1.3 Tension Stiffening Model

Consider a reinforced concrete bar subjected to a gradually increasing tensile force.

Initially, both concrete and reinforcement are in their linear elastic ranges. The first crack will open when the (randomly distributed) tensile strength of concrete is exceeded in the weakest section. Across the crack the tensile force is now carried by the reinforcement alone. However, due to bond between the two components, tensile stresses rebuild in the concrete and a corresponding stress reduction takes place in the reinforcement; until the original uniform strain state is recovered at a certain distance away from the crack. At slightly higher load levels additional cracks will form in the same manner. This crack formation process goes on until the spacing between cracks reaches a typical minimum value. Above this load level cracks are growing, but the crack pattern is almost stable. Finally, the load capacity of the bar is governed by yielding of the reinforcement. Since concrete after initial cracking still has an average tensile stress capability due to interaction with reinforcement,

the corresponding stiffness contribution may be interpreted as an additional stiffness of the tensile reinforcement. Thus, in the literature this effect is frequently termed tension stiffening.

𝐴_𝑐 𝐴_𝑠 =𝜌𝐴_𝑐

𝑙 𝑢

𝑁

𝐸_𝑑=𝜌𝐸_𝑠 𝑓_𝑑=𝜌𝑓_𝑦 𝜎 = _𝐴^𝑁

𝑐

𝑓_𝑑

𝑓_𝑐𝑡+ 𝐸_𝑑𝜖_𝑐𝑟

𝐸𝑐

𝐸_𝑑

𝜖_𝑐𝑟 𝜖_𝑠𝑐 𝜖_𝑠𝑟 𝜖_{𝑓 𝑦} 𝜖_𝑦 𝜖= ^𝑢_𝑙

Normal

Critical

Subcritical

Figure 8.2: Response of Reinforced Concrete Tension Member

Noakowski and Krawinkler [37] have presented a deformation model for reinforced concrete in tension based on the simplifying assumptions that the two consecutive phases of crack formation and crack growth are taking place under constant force and constant tension stiffening, respectively. These assumptions make the model purely ‘strain-driven’, which is attractive in conjunction with displacement-based finite elements, since average strains then are known when the constitutive model is entered. The behavior of the model in terms of equivalent concrete stress versus average strain is depicted by the ‘thick’ solid line in Fig.8.2. Here (𝜖_𝑐𝑟, 𝜖_𝑠𝑐, 𝜖_{𝑓 𝑦}) are the average strains at the incidents of first cracking, stabilized crack pattern and (first) yielding, respectively. The ‘thin’ solid line is for the response of reinforcement alone (or at cracks). Here (𝜖_𝑠𝑟, 𝜖_𝑦) are the steel strains corresponding to initial cracking and yielding, respectively. Thus, the former strain is expressed by

𝜖_𝑠𝑟 = 𝜖_𝑐𝑟 + 𝑓_𝑐𝑡

𝐸_𝑑 (8.14)

where 𝑓_𝑐𝑡 is the tensile strength of concrete and 𝐸_𝑑 is the ‘smeared’ modulus of elasticity of reinforcing steel. In this work the tension stiffening coefficient 𝑏_𝑡 will be defined through

𝜖_𝑠𝑐 = 𝜖_𝑠𝑟 − 𝑏_𝑡(𝜖_𝑠𝑟−𝜖_𝑐𝑟) (8.15)

This definition agrees with the adopted form in MC90 [35], but deviates slightly from the expression used in [37] where the𝜖_𝑐𝑟-term is omitted. The latter is believed to be a simplification, since the bond-slip arises from the strain difference (𝜖_𝑠𝑟−𝜖_𝑐𝑟) and not from𝜖𝑠𝑟 alone. In [37] a value of 0.42 for𝑏𝑡is suggested, while MC90 recommends 0.40 for short-term loading and 0.25 for long-term or repeated loading; all values refer to deformed bars. Combination of Eqs.(8.14,8.15) gives for the average strain at the end of crack formation

𝜖_𝑠𝑐 = 𝜖_𝑐𝑟 + (1 − 𝑏_𝑡)𝑓𝑐𝑡

𝐸_𝑑 (8.16)

Since the tension stiffening effect is assumed constant throughout the crack growth phase, the average strain at start of yielding can be expressed

𝜖_{𝑓 𝑦} = 𝜖_𝑦 − 𝑏_𝑡(𝜖_𝑠𝑟−𝜖_𝑐𝑟) (8.17) By using Eq.(8.14), the final form becomes

𝜖_{𝑓 𝑦} = 𝜖_𝑦 − 𝑏_𝑡𝑓_𝑐𝑡

𝐸_𝑑 (8.18)

The validity of the tension stiffening model limits to reinforcement ratios above the ‘critical’ level, as indicated in Fig. 8.2. In the ‘subcritical’ range the amount of reinforcement is insufficient to restrain the first crack that opens, and thus crack-ing converts to the localized phenomenon that characterizes the behavior of plain concrete. In a reinforced concrete beam section this situation may occur at locations (integration points) with light or no reinforcement in the vicinity. However, a properly designed beam will still exhibit an overall hardening response upon crack initiation.

Thus, the situation with subcritical reinforcement will here only be covered by a sim-ple softening model that approaches pure ‘tension cut-off’ for the bounding case of plain concrete. This behavior is depicted by the ‘thick’ dotted line in Fig. 8.2. The alternative to this vertical cut-off upon crack initiation would have been a softening formulation based on the fracture energy approach. However, that would involve the introduction of a maximum allowable element size in order to prevent ‘snap-back’

in the softening branch [38]. This requirement is related to the fracture energy of concrete, and it may limit the maximum element size to the order of 1 m. Such a restriction would be quite severe for a beam model, especially when the related phenomenon is believed to have minor influence on the overall behavior of reinforced concrete.

The presentation in this subsection has been focused on the combined response of concrete and reinforcement in tension. Of convenience, the two components will be modeled as separate materials in the sequel, but the important interaction effect of tension stiffening will be retained. This is simply done by modeling the reinforcement as ‘steel alone’ (indicated by ‘thin’ lines in Fig.8.2), and assigning the remaining part of the combined response to the concrete. The modeling of reinforcing steel will be covered in more details in Section 8.2. Then hardening of the yield plateau is also included, although that has not been accounted for in the tension stiffening model.

The modeling of concrete in principal tension follows in the next subsection.

Finally note that an inherent feature of the adopted tension stiffening model is that the sum of average forces in concrete and steel is always equal to the steel force at cracks. Thus, the need for a separate stress control at cracks is eliminated. This is in contrast to the tension stiffening formulation in MCFT.