SMM-PSFC (Hoffman 2010) was developed to simulate the entire behavior of PSFC elements under monotonic loading. SMM-PSFC consisted of the stress equilibrium and strain compatibility equations along with the constitutive models of materials. However, SMM-PSFC contains the uniaxial constitutive models of materials under monotonic loading only.

### 2.1 Stress Equilibrium and Strain Compatibility Equations

Figure 2 shows an in-plane element model. The element considered is reinforced with two orthogonal grids of prestressing tendons and mild steel bars. The first Cartesian \( \ell - \, t \) coordinate system is along the steel bar directions. The second Cartesian 1–2 coordinates is along the principal stress. For the analytical purposes, it is assumed that the membrane element thickness is uniform with the steel bars are uniformly distributed in two orthogonal directions. The four applied stresses exerting on the element edges are assumed to be uniformly distributed.

In a membrane element, the external applied stresses (\( \sigma_{\ell } \),\( \sigma_{t} \) and \( \tau_{\ell t} \)) can be indicated by the prestressing steel stresses (\( f_{\ell p} \) and \( f_{tp} \)), the mild steel stresses (\( f_{\ell } \) and \( f_{t} \)), and the internal stresses of concrete (\( \sigma_{2}^{c} \),\( \sigma_{1}^{c} \) and \( \tau_{21}^{c} \)), the three stress equilibrium equations are given in Eqs. (1) to (3).

$$ \sigma_{\ell } = \sigma_{2}^{c} \cos^{2} \alpha_{2} + \sigma_{1}^{c} \sin^{2} \alpha_{2} + \tau_{21}^{c} 2\sin \alpha_{2} \cos \alpha_{2} + \rho_{\ell } f_{\ell } + \rho_{\ell p} f_{\ell p} $$

(1)

$$ \sigma_{t} = \sigma_{2}^{c} \sin^{2} \alpha_{2} + \sigma_{1}^{c} \cos^{2} \alpha_{2} - \tau_{21}^{c} 2\sin \alpha_{2} \cos \alpha_{2} + \rho_{t} f_{t} + \rho_{tp} f_{tp} , $$

(2)

$$ \tau_{\ell t} = ( - \sigma_{2}^{c} + \sigma_{1}^{c} )\sin \alpha_{2} \cos \alpha_{2} + \tau_{21}^{c} (\cos^{2} \alpha_{2} - \sin^{2} \alpha_{2} ) $$

(3)

The strains (\( \varepsilon_{1} \), \( \varepsilon_{2} \) and \( \gamma_{21} \)) in 1–2 coordinates can be converted to the strains (\( \varepsilon_{\ell } \), \( \varepsilon_{t} \) and \( \gamma_{\ell t} \)), as shown in Eqs. (4) to (6) (Pang and Hsu 1996).

$$ \varepsilon_{\ell } = \varepsilon_{2} \cos^{2} \alpha_{2} + \varepsilon_{1} \sin^{2} \alpha_{2} + \frac{{\gamma_{21} }}{2}2\sin \alpha_{2} \cos \alpha_{2} $$

(4)

$$ \varepsilon_{t} = \varepsilon_{2} \sin^{2} \alpha_{2} + \varepsilon_{1} \cos^{2} \alpha_{2} - \frac{{\gamma_{21} }}{2}2\sin \alpha_{2} \cos \alpha_{2} $$

(5)

$$ \frac{{\gamma_{\ell t} }}{2} = ( - \varepsilon_{2} + \varepsilon_{1} )\sin \alpha_{2} \cos \alpha_{2} + \frac{{\gamma_{21} }}{2}(\cos^{2} \alpha_{2} - \sin^{2} \alpha_{2} ) $$

(6)

### 2.2 Biaxial Strains Convert to Uniaxial Strains

Since general lab experiments and reference literatures can give only the uniaxial constitutive laws of steel and concrete, only the uniaxial constitutive laws can be utilized by a general analytical software, the biaxial strains in above equations should be transformed to uniaxial strains. Thus, using the Hsu/Zhu ratios (\( \nu_{12} \),\( \nu_{21} \)), four equations has been derived **(**Zhu and Hsu 2002) to represent the relationship between the uniaxial strains (\( \overline{{\varepsilon_{1} }} ,\overline{{\varepsilon_{2} }} ,\overline{{\varepsilon_{l} }} \) and \( \overline{{\varepsilon_{t} }} \)) and the biaxial strains (\( \varepsilon_{1} \),\( \varepsilon_{2} \),\( \varepsilon_{\ell } \) and \( \varepsilon_{t} \)), as shown in Eqs. (7) to (10).

$$ \bar{\varepsilon }_{1} = \frac{1}{{1 - \nu_{12} \nu_{21} }}\varepsilon_{1} + \frac{{\nu_{12} }}{{1 - \nu_{12} \nu_{21} }}\varepsilon_{2} $$

(7)

$$ \bar{\varepsilon }_{2} = \frac{{\nu_{21} }}{{1 - \nu_{12} \nu_{21} }}\varepsilon_{1} + \frac{1}{{1 - \nu_{12} \nu_{21} }}\varepsilon_{2} $$

(8)

$$ \bar{\varepsilon }_{\ell } = \bar{\varepsilon }_{2} \cos^{2} \alpha_{2} + \bar{\varepsilon }_{1} \sin^{2} \alpha_{2} + \frac{{\gamma_{12} }}{2}2\sin \alpha_{2} \cos \alpha_{2} $$

(9)

$$ \bar{\varepsilon }_{t} = \bar{\varepsilon }_{2} \sin^{2} \alpha_{2} + \bar{\varepsilon }_{1} \cos^{2} \alpha_{2} - \frac{{\gamma_{12} }}{2}2\sin \alpha_{2} \cos \alpha_{2} $$

(10)

The uniaxial strains \( \overline{{\varepsilon_{1} }} ,\overline{{\varepsilon_{2} }} ,\overline{{\varepsilon_{l} }} \) and \( \overline{\varepsilon }_{t} \) can be calculated by Eqs. (7) to (10), then the stresses \( \sigma_{1}^{c} \), \( \sigma_{2}^{c} \), \( \tau_{12}^{c} \), \( f_{\ell } \) and \( f_{t} \) in Eqs. (1) to (3) can be obtained using the uniaxial constitutive laws.

Under monotonic shear stresses, two Hsu/Zhu ratios of panels can be given in Eqs. (11) and (12) **(**Zhu and Hsu 2002).

$$ \nu _{{12}} = \left\{ {\begin{array}{*{20}c} {0.2 + 850\varepsilon _{{sf}} ,} & {\varepsilon _{{sf}} \le \varepsilon _{y} } \\ {1.9,} & {\varepsilon _{{sf}} > \varepsilon _{y} } \\ \end{array} } \right. $$

(11)

Where, \( \varepsilon_{sf} \) is the average (smeared) tensile strain of steel rebar in the \( \ell - \) or the \( t - \) direction, whichever yields first, taken to calculate the Hsu/Zhu ratio \( \nu_{12} \).

### 2.3 Constitutive Laws of Materials

#### 2.3.1 Uniaxial Constitutive Laws of Prestressed Steel Fiber Concrete

The constitutive model for PSFC along with the factors that will affect PSFC are summarized in this section. The results are plotted in Fig. 3. Note that the tensile stress is applied in 1-direction and the compressive stress in 2-direction. Development of these constitutive relationships has been reported by Hoffman (2010). These proposed constitutive laws of PSFC takes consideration on the effect of presence of the steel fibers in the concrete.

#### 2.3.2 SFC in Tension

The relationships of \( \sigma_{1}^{c} \) and the uniaxial strain \( \overline{\varepsilon }_{1} \) of prestressed SFC are given as follows:

Stage UC:

$$ \sigma_{1}^{c} = E^{\prime}_{c} \bar{\varepsilon }_{1} + \sigma_{ci} ,\bar{\varepsilon }_{1} \le \left( {\bar{\varepsilon }_{cx} - \bar{\varepsilon }_{ci} } \right) $$

(13a)

Stage T1:

$$ \sigma_{1}^{c} = E^{\prime\prime}_{c} (\bar{\varepsilon }_{1} + \bar{\varepsilon }_{ci} ),\left( {\bar{\varepsilon }_{cx} - \bar{\varepsilon }_{ci} } \right) < \bar{\varepsilon }_{1} \le \left( {\varepsilon_{cy} - \bar{\varepsilon }_{ci} } \right) $$

(13b)

Stage T2:

$$ \sigma_{1}^{c} = E^{\prime\prime\prime}_{c} (\bar{\varepsilon }_{1} + \bar{\varepsilon }_{ci} ),\left( {\bar{\varepsilon }_{cy} - \bar{\varepsilon }_{ci} } \right) < \bar{\varepsilon }_{1} \le \left( {\varepsilon_{cult} - \bar{\varepsilon }_{ci} } \right) $$

(13c)

Stage T3:

$$ \sigma_{1}^{c} = E_{c}^{IV} (\bar{\varepsilon }_{1} + \bar{\varepsilon }_{ci} ),\bar{\varepsilon }_{1} > \left( {\varepsilon_{cult} - \bar{\varepsilon }_{ci} } \right) $$

(13d)

Where,

\( E^{\prime}_{c} \) = decompression modulus of concrete given as \( \frac{{2f^{\prime}_{c} }}{{\varepsilon_{0} }} \)

\( \sigma_{ci} \) = initial stress in SFC

\( \bar{\varepsilon }_{ci} \) = initial strain in concrete due to prestress

\( \bar{\varepsilon }_{cx} \) = extra concrete strain after decompression, taken as \( \bar{\varepsilon }_{ci} - \frac{{\sigma_{ci} }}{{E^{\prime}_{c} }} \)

\( \varepsilon_{c\hbox{max} } \) = SFC maximum strain calculated by 0.04-\( \varepsilon_{pi} \), where, \( \varepsilon_{pi} \) = initial uniaxial strain of prestressing tendons

\( \varepsilon_{cult} \) = SFC strain under ultimate stress, calculated by 0.01—\( \varepsilon_{pi} \)

\( f_{cult} \) = SFC ultimate stress, taken as \( (0.2FF + 12\rho_{l} )\sqrt {f^{\prime}_{c} } \), where, *FF* = fiber factor, \( \rho_{l} \) = longitudinal steel ratio

\( \varepsilon_{cy} \) = SFC yield strain taken as 0.0005

\( f_{cy} \) = SFC effective yield stress, taken as \( 0.4 * FF * CF\sqrt {f^{\prime}_{c} } \), (\( f^{\prime}_{c} \) and \( \sqrt {f^{\prime}_{c} } \) are in *MPa*), *CF* = 1 for SFC tensile volume confined (sandwiched) by two or more tendons, or *CF* = ½ for SFC tensile volume unconfined by tendons

\( E^{\prime\prime}_{c} \) = modulus of SFC, taken as \( \frac{{f_{cy} }}{{\varepsilon_{cy} - \bar{\varepsilon }_{cx} }} \)

\( E^{\prime\prime\prime}_{c} \) = modulus of SFC, taken as \( \frac{{f_{cult} - f_{cy} }}{{\varepsilon_{cult} - \varepsilon_{cy} }} \)

\( E_{c}^{IV} \) = modulus of SFC, taken as \( \frac{{ - f_{cult} }}{{\varepsilon_{\hbox{max} } - \varepsilon_{cult} }} \)

#### 2.3.3 SFC in Compression

The average (smeared) constitutive laws of SFC compression stress \( \sigma_{2}^{c} \) and the uniaxial compression strain \( \bar{\varepsilon }_{2} \) are taken as follows:

$$ \sigma_{2}^{c} = \zeta f^{\prime}_{c} \left[ {2\left( {\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }}} \right) - \left( {\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }}} \right)^{2} } \right],\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }} \le 1, $$

(14a)

or

$$ \sigma_{2}^{c} = \zeta f^{\prime}_{c} \left[ {1 - \left( {\frac{{{{\bar{\varepsilon }_{2} } \mathord{\left/ {\vphantom {{\bar{\varepsilon }_{2} } {\zeta \varepsilon_{0} }}} \right. \kern-0pt} {\zeta \varepsilon_{0} }} - 1}}{{{4 \mathord{\left/ {\vphantom {4 \zeta }} \right. \kern-0pt} \zeta } - 1}}} \right)^{2} } \right],\frac{{\bar{\varepsilon }_{2} }}{{\zeta \varepsilon_{0} }} > 1 $$

(14b)

Where, \( \zeta \) is the softening coefficient, which can be calculated as follows:

$$ \zeta = f\left( {f^{\prime}_{c} } \right)f\left( {\bar{\varepsilon }_{1} } \right)f\left( \beta \right)W_{p} Wf \le 0.9 , $$

(15)

Where,

$$ f\left( {f^{\prime}_{c} } \right) = \frac{5.8}{{\sqrt {f^{\prime}_{c} } }} \le 0.9 ( f^{\prime}_{c} \, in \, MPa), $$

(16)

$$ f\left( {\bar{\varepsilon }_{1} } \right) = \frac{1}{{\sqrt {1 + 400\bar{\varepsilon }_{1} } }} , $$

(17)

$$ f\left( \beta \right) = 1 - \frac{\left| \beta \right|}{{24^{ \circ } }},\beta = \frac{1}{2}\tan^{ - 1} \left[ {\frac{{\gamma_{21} }}{{\left( {\varepsilon_{2} - \varepsilon_{1} } \right)}}} \right], $$

(18)

$$ W_{p} = 1.15 + \frac{{\left| \beta \right|\left( {0.09\left| \beta \right| - 1} \right)}}{6} , $$

(19)

and

$$ Wf = 1 + 0.2FF $$

(20)

#### 2.3.4 SFC in Shear

In the 1–2 coordinates, the relationship between the concrete stress (\( \tau_{12}^{c} \)) and the strain (\( \gamma_{12} \)) is reported by Zhu et al. (2001) and shown in Eq. (21).

$$ \tau_{12}^{c} = \frac{{\sigma_{1}^{c} - \sigma_{2}^{c} }}{{2(\varepsilon_{1} - \varepsilon_{2} )}}\gamma_{12} $$

(21)

where \( \sigma_{1}^{c} \) and \( \sigma_{2}^{c} \) are the average (smeared) concrete stresses; \( \varepsilon_{1} \) and \( \varepsilon_{2} \) are the biaxial smeared strains in the 1- and 2- directions of the principal applied stresses, respectively.

#### 2.3.5 Prestressing Tendons Embedded in SFC

The prestressing tendons are embedded in SFC. The average (smeared) stress–strain relationship of PSFC is given as follows:

$$ f_{ps} = E_{ps} \bar{\varepsilon }^{\prime}_{s} \varepsilon_{s} < \frac{{0.7f_{pu} }}{{E_{ps} }}, $$

(22a)

or

$$ f_{ps} = \frac{{E^{\prime\prime}_{ps} \bar{\varepsilon }^{\prime}_{s} }}{{\left[ {1 + \left( {\frac{{E^{\prime\prime}_{ps} \bar{\varepsilon }^{\prime}_{s} }}{{f^{\prime}_{pu} }}} \right)^{5} } \right]^{{\frac{1}{5}}} }},\bar{\varepsilon }^{\prime}_{s} \ge \frac{{0.7f_{pu} }}{{E_{ps} }}, $$

(22b)

Where,

\( E_{ps} \) = elastic modulus of prestressing strands, 200GPa

\( \bar{\varepsilon }^{\prime}_{s} \) = \( \bar{\varepsilon }_{s} \) + \( \varepsilon_{dec} \), uniaxial steel bar strain

\( f_{pu} \) = ultimate strength of prestressing strands, 1862 MPa

\( E^{\prime\prime}_{ps} \) = modulus of prestressing strands in plastic area (Eq. (22)), 209 GPa

\( f^{\prime}_{pu} \) = modified strength of prestressing strands, 1793 MPa

In the above equations, *ps* can be exchanged by \( \ell p \) and \( t{\text{p}} \) for the longitudinal tendons and the transverse tendons respectively (Fig. 4).

#### 2.3.6 Mild Steel Embedded in SFC

The mild steel bars are installed in concrete as those in SMM. The average (smeared) stress–strain relationships can be expressed as follows:

Stage 1:

$$ f_{s} = E_{s} \bar{\varepsilon }_{s} ,\bar{\varepsilon }_{s} \le \bar{\varepsilon }_{n} $$

(23)

Stage 2:

$$ f_{s} = f_{y} \times \left[ {(1 - 0.096FF)(0.91 - 2B) + (0.2FF + 1) \times (0.02 + 0.25B)\frac{{\bar{\varepsilon }_{s}^{\prime } }}{{\varepsilon _{y} }}} \right],\bar{\varepsilon }_{s} {\text{ > }}\bar{\varepsilon }_{n} $$

(24)

Stage 3 (unloading):

$$ f_{s} = f_{p} - E_{s} (\bar{\varepsilon }_{p} - \bar{\varepsilon }_{s} ),\bar{\varepsilon }_{s} < \bar{\varepsilon }_{p} $$

(25)

Where,

$$ \bar{\varepsilon }_{n} = \varepsilon_{y} (0.93 - 2B) $$

(26a)

$$ B = \frac{1}{\rho }\left( {\frac{{f_{cr} }}{{f_{y} }}} \right)^{1.5} $$

(26b)

\( \varepsilon_{cr} \) = 0.00008, concrete cracking strain

\( f_{cr} \) = \( 0.31\sqrt {f^{\prime}_{c} } \) (\( f^{\prime}_{c} \) and \( \sqrt {f^{\prime}_{c} } \) are in MPa), concrete cracking stress (Fig. 5)

### 2.4 Experimental Verification

The SMM-PSFC was used to realize the experimental shear behavior of PSFC membrane elements with different steel grid orientations and steel percentages (Laskar et al. 2014). Figure 6 shows the validity of SMM-PSFC in predicting the behavior of PSFC panels under pure shear. The figure shows the comparison between the analytical and the measured shear stress–strain curves of a PSFC panel by Hoffman (2010) with results predicted by SMM-PSFC. It can be seen that the predictions from SMM-PSFC are in good agreement.