18.2.7 Compressive Behavior

Concrete subjected to compressive stresses shows a pressure-dependent behavior, i.e., the strength and ductility increase with increasing isotropic stress. Due to the lateral confinement, the compressive stress-strain relationship is modified to incorporate the effects of the increased isotropic stress. Furthermore, it is assumed that the compressive behavior is influenced by lateral cracking. To model the lateral confinement effect, the parameters of the compressive stress-strain function, f_cf and $\varepsilon_{{\mathrm{p}}}^{}$ , are determined with a failure function which gives the compressive stress which causes failure as a function of the confining stresses in the lateral directions.

If the material is cracked in the lateral direction, the parameters are reduced with the factor $\beta_{{\varepsilon _{\mathrm{cr}}}}^{}$ for the peak strain, and with the factor $\beta_{{\sigma _{\mathrm{cr}}}}^{}$ for the peak stress. A possible relationship is given in §18.2.7.2. It is tacitly assumed that the base curve in compression is determined by the peak stress value f_p = $\beta_{{\sigma _{\mathrm{cr}}}}^{}$f_cf , and the corresponding peak strain value $\alpha_{{\mathrm{p}}}^{}$ = $\beta_{{\varepsilon _{\mathrm{cr}}}}^{}$$\varepsilon_{{\mathrm{p}}}^{}$ . In summary,

$$\beta_{{\sigma _\mathrm{cr}}}^{}$$ f_p (18.93)

$$\alpha_{{\mathrm{p}}}^{} \beta_{{\varepsilon _{\mathrm{cr}}}}^{} \varepsilon_{{\mathrm{p}}}^{}$$ (18.94)

The base function in compression, with the parameters f_p and $\alpha_{{\mathrm{p}}}^{}$ , is modeled with a number of different predefined and user-defined curves. The predefined curves are the constant curve and the brittle curve. Also available are the linear hardening curve and the saturation hardening curve. Figure 6.6 shows the available hardening-softening curves in compression. For the curve according to Thorenfeldt and the parabolic curve we give some background theory.

Thorenfeldt et al. [102]

Figure 18.10: Thorenfeldt compression curve

This curve [Fig.18.10] is described by

$${\frac{{ \alpha }}{{ \alpha_{\mathrm{p}} }}} \left(\vphantom{ \frac{ n }{ n - \left( 1 - \left( \dfrac{ \alpha } { \alpha_{\mathrm{p}} } \right)^{\!nk}\right) } }\right. {\frac{{ n }}{{ n - \left( 1 - \left( \dfrac{ \alpha } { \alpha_{\mathrm{p}} } \right)^{\!nk}\right) }}} \left.\vphantom{ \frac{ n }{ n - \left( 1 - \left( \dfrac{ \alpha } { \alpha_{\mathrm{p}} } \right)^{\!nk}\right) } }\right)$$ (18.95)

with

$${\frac{{ f_{\mathrm{cc}} }}{{ 17 }}} \begin{cases} 1 & \text{if $\alpha_{\mathrm{p}} < \alpha < 0... ...\quad & \text{if $\alpha \leq \alpha_{\mathrm{p}}$} \end{cases}$$ (18.96)

Parabolic.

The parabolic curve in DIANA is a formulation based on fracture energy, according to Feenstra [27]. The parabolic curve is described by three characteristic values [Fig.18.11].

Figure 18.11: Parabolic compression curve

The strain $\alpha_{{\mathrm{c}/\scriptscriptstyle{3}}}^{}$ , at which one-third of the maximum compressive strength f_c is reached, is

$$\alpha_{{\mathrm{c}/\scriptscriptstyle{3}}}^{} {\frac{{1}}{{3}}} {\frac{{ f_{\mathrm{c}} }}{{ E }}}$$ (18.97)

The strain $\alpha_{{\mathrm{c}}}^{}$ , at which the maximum compressive strength is reached, is

$$\alpha_{{\mathrm{c}}}^{} {\frac{{5}}{{3}}} {\frac{{ f_{\mathrm{c}} }}{{ E }}} \alpha_{{\mathrm{c}/\scriptscriptstyle{3}}}^{}$$ (18.98)

Note that $\alpha_{{\mathrm{c}/\scriptscriptstyle{3}}}^{}$ and $\alpha_{{\mathrm{c}}}^{}$ are determined irrespective of of the element size or compressive fracture energy. Finally, the ultimate strain $\alpha_{{\mathrm{u}}}^{}$ , at which the material is completely softened in compression, is

$$\alpha_{{\mathrm{u}}}^{} \alpha_{{\mathrm{c}}}^{} {\frac{{ 3 }}{{ 2 }}} {\frac{{ G_{\mathrm{c}} }}{{ h \, f_{\mathrm{c}} }}}$$ (18.99)

The parabolic compression curve in DIANA is now described by

$$\begin{cases} - f_{\mathrm{c}} \, \dfrac{ 1 }{ 3 } \, \dfrac... ... 0 & \text{if $\alpha_{j} \le \alpha_{\mathrm{u}}$} \end{cases}$$ (18.100)

It could now easily be verified that the fracture energy G_c and the characteristic element length h govern the softening part of the curve only:

$$\int\limits_{{\alpha_{\mathrm{c}}}}^{{\alpha_{\mathrm{u}}}} \alpha_{{j}}^{} \left.\vphantom{ \left( \alpha_{j} - \frac{ 1 }{ 3 } \left( \frac... ...{ \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} } \right)^{\!3} \: \right) }\right. \left(\vphantom{ \alpha_{j} - \frac{ 1 }{ 3 } \left( \frac{ \alph... ...m{c}} } { \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} } \right)^{\!3} \: }\right. \alpha_{{j}}^{} {\frac{{ 1 }}{{ 3 }}} \left(\vphantom{ \frac{ \alpha_{j} - \alpha_{\mathrm{c}} } { \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} } }\right. {\frac{{ \alpha_{j} - \alpha_{\mathrm{c}} }}{{ \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} }}} \left.\vphantom{ \frac{ \alpha_{j} - \alpha_{\mathrm{c}} } { \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} } }\right)^{{\!3}}_{} \left.\vphantom{ \alpha_{j} - \frac{ 1 }{ 3 } \left( \frac{ \alph... ...m{c}} } { \alpha_{\mathrm{u}} - \alpha_{\mathrm{c}} } \right)^{\!3} \: }\right) \left.\vphantom{ \left( \alpha_{j} - \frac{ 1 }{ 3 } \left( \frac... ...^{\!3} \: \right) }\right\vert _{{\alpha_{\mathrm{c}}}}^{{\alpha_{\mathrm{u}}}} {\frac{{ G_{\mathrm{c}} }}{{ h }}}$$ (18.101)

The total compressive fracture energy which has been found in experiments ranges from 10 to 25 [Nmm/mm²] which is about 50 to 100 times the tensile fracture energy [27].

18.2.7.1 Compressive Behavior with Lateral Confinement

The increase of the strength with increasing isotropic stress is modeled with the four-parameter Hsieh-Ting-Chen failure surface which is defined as

$${\dfrac{{J_{2}}}{{f_{\mathrm{cc}}^{2}}}} {\dfrac{{\sqrt{J_{2}}}}{{f_{\mathrm{cc}}}}} {\dfrac{{f_{\mathrm{c1}}}}{{f_{\mathrm{cc}}}}} {\dfrac{{I_{1}}}{{f_{\mathrm{cc}}}}}$$ (18.102)

with the invariants J₂ and I₁ defined in terms of the stress in the concrete $\sigma_{{ci}}^{}$ according to

$${\tfrac{{1}}{{6}}} \left(\vphantom{ (\sigma _{\mathrm{c1}} - \sigma _{\mathrm{c2}})^... ...mathrm{c3}})^{2} + (\sigma _{\mathrm{c3}} - \sigma _{\mathrm{c1}})^{2} }\right. \sigma_{{\mathrm{c1}}}^{} \sigma_{{\mathrm{c2}}}^{} \sigma_{{\mathrm{c2}}}^{} \sigma_{{\mathrm{c3}}}^{} \sigma_{{\mathrm{c3}}}^{} \sigma_{{\mathrm{c1}}}^{} \left.\vphantom{ (\sigma _{\mathrm{c1}} - \sigma _{\mathrm{c2}})^... ...mathrm{c3}})^{2} + (\sigma _{\mathrm{c3}} - \sigma _{\mathrm{c1}})^{2} }\right)$$ J₂ (18.103)

$$\sigma_{{\mathrm{c1}}}^{} \sigma_{{\mathrm{c2}}}^{} \sigma_{{\mathrm{c3}}}^{}$$ I₁ (18.104)

and f_c1 the maximum concrete stress

$$\sigma_{{\mathrm{c1}}}^{} \sigma_{{\mathrm{c2}}}^{} \sigma_{{\mathrm{c3}}}^{}$$ (18.105)

which is not the maximum tensile stress but the maximum principal stress [12]. The parameters in (18.102) are determined by fitting of the uniaxial tensile and compressive strength, the biaxial compressive strength, and experimental data of triaxial tests on concrete specimen.

The stress f_c3 is assumed to result in failure and is determined by scaling the linear elastic stress vector $\boldsymbol\sigma$_c = s E $\boldsymbol\varepsilon$_nst such that the equation (18.102) holds. The compressive failure stress in multi-axial stress situation is then given by

$$\sigma_{{\mathrm{c1}}}^{} \sigma_{{\mathrm{c2}}}^{} \sigma_{{\mathrm{c3}}}^{}$$ (18.106)

If the scaling factor s is negative, thus resulting in a positive failure stress f_c3 , the stress vector is scaled to the tensile side of the failure surface and the failure strength is set equal to a large negative value ( -30  f_cc ). The failure strength f_cf is given by

f_cf = - f_c3 (18.107)

The peak stress factor K_$\sigma$ is given by Selby [95, Eq.(2.7)]

$$\sigma {\dfrac{{f_{\mathrm{cf}}}}{{f_{\mathrm{cc}}}}} \geq$$ (18.108)

and the peak strain factor is assumed to be given by

$$\varepsilon \sigma$$ (18.109)

In unconfined compression, the values at the peak are given by the uniaxial values compressive strength, and the peak stress factor is equal to one. The parameters of the compressive stress-strain function now become

$$\sigma$$ f_cf (18.110)

$$\varepsilon_{{\mathrm{p}}}^{} \sigma \varepsilon_{{0}}^{}$$ (18.111)

with the value of the initial strain $\varepsilon_{{0}}^{}$ is given by the relationship

$$\varepsilon_{{0}}^{} {\dfrac{{n}}{{n-1}}} {\dfrac{{f_{\mathrm{cc}}}}{{E}}}$$ (18.112)

The equations given above result in a gradual increase of the maximum strength in confined compression, with an initial slope of the stress-strain diagram given by the Young's modulus. In a full triaxial stress situation the failure surface cannot be reached and a linear stress-strain relation is obtained [Fig.18.12].

Figure 18.12: Influence of lateral confinement on compressive stress-strain curve

The increased ductility of confined concrete is modeled by a linear adaption of the descending branch of the Thorenfeldt curve according to

$$\left(\vphantom{ 1 - ( 1 - r ) \dfrac{\alpha_{j} - \alpha_{\mathrm{p}}}{\alpha_{\mathrm{u}} - \alpha_{\mathrm{p}}} }\right. {\dfrac{{\alpha_{j} - \alpha_{\mathrm{p}}}}{{\alpha_{\mathrm{u}} - \alpha_{\mathrm{p}}}}} \left.\vphantom{ 1 - ( 1 - r ) \dfrac{\alpha_{j} - \alpha_{\mathrm{p}}}{\alpha_{\mathrm{u}} - \alpha_{\mathrm{p}}} }\right) \leq$$ (18.113)

with r the factor which models the residual strength of the material [Fig.18.13].

Figure 18.13: Compressive behavior under lateral confinement

The ultimate strain in compression is assumed to be determined by the ratio between the peak strength and the compression strength and the strain at peak according to

$$\alpha_{{\mathrm{u}}}^{} \left(\vphantom{ \dfrac{f_{\mathrm{p}}}{f_{\mathrm{cc}}} }\right. {\dfrac{{f_{\mathrm{p}}}}{{f_{\mathrm{cc}}}}} \left.\vphantom{ \dfrac{f_{\mathrm{p}}}{f_{\mathrm{cc}}} }\right)^{{\gamma}}_{} \alpha_{{\mathrm{p}}}^{}$$ (18.114)

with the scalar $\gamma$ to be determined, currently $\gamma$ = 3 is assumed. The residual strength r f_p also depends on the ratio between the peak strength and the compressive strength according to

$$\left(\vphantom{ \dfrac{f_{\mathrm{p}}}{f_{\mathrm{cc}}} }\right. {\dfrac{{f_{\mathrm{p}}}}{{f_{\mathrm{cc}}}}} \left.\vphantom{ \dfrac{f_{\mathrm{p}}}{f_{\mathrm{cc}}} }\right)^{{\gamma}}_{}$$ (18.115)

with r₀ an initial value, assume r₀ = 0.1 .

The linear compression-softening relationship is only applied for the Thorenfeldt curve if the peak value f_p is considerably larger than the compressive strength f_cc , assume f_p/f_cc > 1.05 . In case of lateral compression and lateral cracking resulting in f_p/f_cc < 1.05 , will not increase the ductility of the material.

18.2.7.2 Compressive Behavior with Lateral Cracking

In cracked concrete, large tensile strains perpendicular to the principal compressive direction reduce the concrete compressive strength. The compressive strength f_p is consequently not only a function of the internal variable $\alpha_{{j}}^{}$ , but also a function of the internal variables governing the tensile damage in the lateral directions, $\alpha_{{l,1}}^{}$ and $\alpha_{{l,2}}^{}$ . The reduction factors due to lateral cracking are denoted as $\beta_{{\varepsilon _{\mathrm{cr}}}}^{}$ = $\beta_{{\varepsilon _{\mathrm{cr}}}}^{}$($\alpha_{{\mathrm{lat}}}^{}$) and $\beta_{{\sigma _{\mathrm{cr}}}}^{}$ = $\beta_{{\sigma _{\mathrm{cr}}}}^{}$($\alpha_{{\mathrm{lat}}}^{}$) , which are functions of the average lateral damage variable given by $\alpha_{{\mathrm{lat}}}^{}$ = $\sqrt{{\alpha_{l,1}^{2}+\alpha_{l,2}^{2}}}$ .

The relationship for reduction due to lateral cracking is the model according to Vecchio & Collins [108, model B] [Fig.18.14]

$$\beta_{{\sigma _{\mathrm{cr}}}}^{} {\dfrac{{1}}{{1 + K_{c} }}} \leq$$ (18.116)

in which

$$\left(\vphantom{ -\dfrac{\alpha_{\mathrm{lat}}}{\varepsilon _{0}} - 0.37 }\right. {\dfrac{{\alpha_{\mathrm{lat}}}}{{\varepsilon _{0}}}} \left.\vphantom{ -\dfrac{\alpha_{\mathrm{lat}}}{\varepsilon _{0}} - 0.37 }\right)$$ (18.117)

The factor $\beta_{{\varepsilon _{\mathrm{cr}}}}^{}$ is equal to one.

Figure 18.14: Reduction factor due to lateral cracking