# Analytical Model of Two-Directional Cracking Shear-Friction Membrane for Finite Element Analysis of Reinforced Concrete

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Significance

## 3. Brief Description of the Shear-Friction Model

_{cr}and γ

_{cr}) are the crack separations and slips divided by the average crack spacing. This definition of smeared crack strains only works if the crack-opening path is simplified to a linear relationship, otherwise the crack shear and normal stresses would not be uniquely determined by the crack strains. Crack slip can be compared with pushing a block up a slope with a rough surface, in which the coefficient of friction relative to the crack-opening path can be defined. The friction coefficient relative to the surface of the slope is defined by Equation (1) [15]:

^{up}and μ

^{down}are defined by Equations (2) and (3) for either the upslope or downslope direction as functions of the single-crack-path friction coefficient μ’ and the angle of the crack opening path θ (both should remain constant) [5]. The detailed process of the one-crack formulation is referred to in [16].

## 4. Proposed Orthogonal-Cracking Model Based on Shear Friction

_{cop}ε. The crack-path model is a constitutive model for concrete subjected to uniaxial compression that predicts axial and transverse strains as functions of the applied axial stress. It assumes the presence of closely spaced slip surfaces that occur at all possible planar orientations that form an angle approximately equal to 0.45 radians with respect to the axis of loading. Strain is the sum of linear elastic strain in the material between slip surfaces and nonlinear strain due to slip and separation of the surfaces. For convenience, the slip surfaces can be thought of as closely spaced cracks. Shear and normal strain equal the slip and separation divided by the average crack spacing, respectively. Slip causes the crack to separate due to crack-surface roughness. The relationship between slip and separation is referred to as the “crack-opening path” for convenience. The slope of the crack-opening path can flatten due to stress normal to the crack surface.

_{cop}is a crack-opening-path slope that can be estimated from shear-friction test and a parametric study. See [15,16] for more details. From the crack-opening path, the effective strain

^{e}ε can be defined. For the initial formulation of the model, it is assumed that the total strains are almost all crack strains. Total shear slip will be tracked as it will be equal to the crack slip. Material properties and cracks are also smeared across the element. Once two cracks have formed, it will be difficult to determine how incremental shear strain is distributed between the two crack directions. Another important assumption will be that only one crack can slip or be “active” at a time.

#### 4.1. Stress/Strain Formulations

#### 4.1.1. Effective Strain

#### 4.1.2. Normal Concrete Stress

_{1}and σ

_{2}. Positive values of effective strain represent tension across the crack surface, while negative values represent compression. A tension-stiffening curve is used to model the reduced stiffness in the tensile strain region, and the full elastic modulus of concrete is used for simplicity in the compressive strain regions. Equations (6) and (7) [5] for normal stress in the concrete are listed below, formulated for Crack 1. Equations for Crack 2 can be obtained by replacing “1” with “2” for the stress and strain notations.

- (a)
^{e}ε_{1}≤ ε_{cr}(the cracking strain of concrete):$${\sigma}_{1}={E}_{c}{}_{}{}^{e}\epsilon _{1}={E}_{c}\left({\epsilon}_{1}-\frac{\left|{\gamma}_{12}\right|}{{a}_{cop}}\right)$$- (b)
^{e}ε_{1}> ε_{cr}:$${\sigma}_{1}=\frac{{f}_{cr}}{\sqrt{1+200{}_{}{}^{e}\epsilon _{1}}}$$

#### 4.1.3. Maximum and Minimum Shear Stresses

- Tension across the Crack

_{1}is tensile, shear stress is calculated as follows, assuming that this is the shearing stress resisted by dowel action of the reinforcement crossing the crack:

- Compression across the Crack

_{1}is compressive and the crack surfaces are in contact, shear stress is determined by the equations shown in Table 1.

_{12}> 0) and negative shearing strain (γ

_{12}< 0), the three cases considered are shown in Table 1: if the crack surface is slipping down a slope, “if the crack surface is not slipping and if the crack surface is slipping up a slope.

_{12}, shear stress τ

_{12}is determined from the “not slipping” equation, but is limited by the minimum and maximum shear stresses, τ

^{a}

_{12}and τ

^{c}

_{12}, as shown below:

_{1}with the normal stress relative to Crack 2, σ

_{2}.

#### 4.1.4. Secant Stiffness Matrix Formulation

**Case 1: Crack Direction 1 Active**

- (a)
- If the normal stress is tensile, the stress–strain relationship is as follows for positive and negative shearing strain, γ
_{12}:$$\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{1}& 0& \frac{\pm {E}_{1}}{a}\\ 0& {E}_{2}& 0\\ 0& 0& \frac{{\beta}^{\prime}G}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$$_{1}is the secant stiffness of concrete defined as the normal stress (Equation (7)) divided by the current effective strain value,^{e}ε_{1}. E_{2}can be obtained in the same manner as E_{1}by replacing “1” with “2” in the stress and strain notations. - (b)
- If normal stress is compressive, the secant stiffness matrix is defined based on normal compressive stress (Equation (6)) and the various cases of shear stress. For positive total shear strain (γ
_{12}> 0) and negative total shearing strain (γ_{12}< 0), the equations and secant stiffness matrices can be found in Table 2 and Table 3.

**Case 2: Crack Direction 2 Active**

- (a)
- If normal stress is tensile, the stress–strain relationship is as follows for positive and negative shearing strain, γ
_{12}:$$\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{1}& 0& 0\\ 0& {E}_{2}& \frac{\pm {E}_{2}}{a}\\ 0& 0& \frac{{\beta}^{\prime}G}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$$ - (b)
- If normal stress is compressive, the secant stiffness matrix is defined based on the normal compressive stress and the various cases of shear stress. For positive total shear strain (γ
_{12}> 0) and negative total shearing strain (γ_{12}< 0), secant stiffness matrices can be found in Table 4.

**Active Crack Criteria**

_{12}) is split between the cracks. However, this case is unlikely for most realistic loading situations. If Crack 1 is in tension and Crack 2 is in compression, then the active crack is defined as Crack 1. This is because slip would tend to be focused on the open crack, as it can only be resisted by tension in the reinforcement crossing the crack. If Crack 2 is in tension and Crack 1 is in compression, however, the active crack will be Crack 2. The last possible case is both cracks being in compression. To determine the active crack for this situation requires the maximum and minimum shear-stress limits. The upper and lower limits on shear stress (τ

^{a}and τ

^{c}) represent the stress necessary to overcome the force of friction across the crack and initiate slip.

^{a}

_{1}, τ

^{a}

_{2}, τ

^{c}

_{1}, and τ

^{c}

_{2}). The active crack will be defined as the crack that has reached the limits first, because slip would be initiated in this crack first, thus controlling the behavior. The shear-stress limits on a one-dimensional plot are shown in Figure 4. A summary of the active crack criteria is presented in Table 5.

#### 4.2. Uniform State of Stress and Strain

#### Shear Stress across the Crack Surface

## 5. Developmental Study of the Finite-Element Model

#### 5.1. Phase 1: Nonlinear Cyclical Model Framework

#### 5.2. Phase 2: Initial Shear-Friction Implementation

#### 5.3. Phase 3: Proportional Load Vector (PLV)

_{R}= 0), plugging Equation (14) into Equation (15) and solving for a yields:

#### 5.4. Phase 4: Constant Vertical Load (CVL)

_{F}and plugged into Equation (19). Now a can be solved, such that:

#### 5.5. Phase 5: Modified Newton–Raphson Method (MNRM)

^{old}and F

^{old}, respectively. This point “0” lies on a representative curve of the force–displacement relationship, which includes all of the model’s properties contained within the global stiffness matrix (S).

## 6. Results and Discussions

## 7. Conclusions

- A deflection controlled analysis was successfully obtained in all phases of model development, which was necessary to produce useful and comparable results for arbitrary loading, such as cyclical loading.
- A uniform state of stress and strain was successfully obtained in all phases of model development. It was first obtained by applying equal incremental displacements to the top two nodes. The top two nodes’ vertical displacements were then linked together. In later phases of the model development, a uniform state of stress and strain was obtained by a proportional load vector, ensuring this state while still allowing a displacement-controlled analysis.
- Shear stress was successfully obtained across the surface of the crack. This was a major consideration, as the crack directions were defined by the orientation of the principal tensile stress. By definition, there is no shear in the direction of the principal stresses. Without shear being transferred across the surface of the crack, the consideration of shear friction for shear transfer becomes irrelevant. In order to obtain shear across the crack surface, a vertical compressive load was required. This was accomplished by means of a proportional load vector and later a constant vertical load combined with a proportional load vector.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

a | shape vector multiplier |

a_{cop} | a parameter that defines the slope of the crack opening path |

f_{cr} | concrete crack stress |

E_{c} | modulus of elasticity of concrete |

G | concrete shear modulus |

N | concrete normal strength |

V | concrete shear strength |

β′ | dowel action shear retention factor |

ε_{cr} | concrete crack strain |

^{e}ε | effective concrete crack strain |

^{e}ε_{1} | effective concrete crack strain in the 1 direction |

^{e}ε_{2} | effective concrete crack strain in the 2 direction |

^{e}ε^{A}_{cr} | effective concrete crack strain at Point A |

ε_{1} | concrete crack strain in the 1 direction |

ε_{2} | concrete crack strain in the 2 direction |

γ_{12} | shear strain |

γ_{12,}_{ prev} | previously converged shear strain |

µ′ | friction coefficient defined relative to the crack opening path |

µ^{up} | friction coefficient defined relative to crack surface in the upslope direction |

µ^{down} | friction coefficient defined relative to crack surface in the downslope direction |

σ_{1} | concrete stress in the 1 direction |

σ_{2} | concrete stress in the 2 direction |

σ_{xy, prev} | previously converged concrete stress in the x direction |

τ_{12} | shear stress |

τ^{a}_{12} | minimum shear stress |

τ^{b}_{12} | outside upper limit shear stress |

τ^{c}_{12} | maximum shear stress |

τ_{12, }_{prev} | previously converged shear stress |

A | global actions or forces |

S | global stiffness matrix |

D | nodal displacements |

$\xc3$,$\tilde{\mathrm{B}}$ | shape vectors |

## References

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**Figure 2.**(

**a**) Bilinear rectangular element, dimensions, DOFs and boundary conditions. (

**b**) Element coordinate systems.

**Figure 3.**(

**a**) Crack definitions and (

**b**) Schematic of the slip–separation relationship of the active crack.

**Figure 7.**(

**a**) Partitioned DOFs for the PLV. (

**b**) Shape vector applied to the element. (

**c**) Shape vector and vertical load, v, applied to element.

**Figure 12.**(

**a**) Concrete shear stress (Crack 1) vs. shear strain (PLV); (

**b**) shape vector multiplier vs. prescribed displacement (PLV); (

**c**) concrete normal stress (Crack 1) vs. effective strain (CVL); (

**d**) concrete shear stress (Crack 1) vs. shear strain (MNRM); (

**e**) interaction between concrete and steel shearing stresses (MNRM).

Shearing Strain | Crack Surface Slipping Down a Slope | Crack Surface Is Not Slipping | Crack Slipping Up a Slope |
---|---|---|---|

positive total shearing strain (γ_{12} > 0) | ${\tau}_{12}^{a}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{down}{\sigma}_{1}}{\left(1+{\beta}^{\prime}\right)}$ | ${\tau}_{12}^{b}={\tau}_{12,prev}+G\left({\gamma}_{12}-{\gamma}_{12,prev}\right)$ | ${\tau}_{12}^{c}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{up}{\sigma}_{1}}{\left(1+{\beta}^{\prime}\right)}$ |

negative total shearing strain (γ _{12} < 0) | ${\tau}_{12}^{c}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{down}{\sigma}_{1}}{\left(1+{\beta}^{\prime}\right)}$ | ${\tau}_{12}^{b}={\tau}_{12,prev}+G\left({\gamma}_{12}-{\gamma}_{12,prev}\right)$ | ${\tau}_{12}^{a}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{up}{\sigma}_{1}}{\left(1+{\beta}^{\prime}\right)}$ |

Shearing Strain | Crack Surface Slipping Down a Slope | Crack Surface Is Not Slipping | Crack Slipping Up a Slope |
---|---|---|---|

positive total shearing strain (γ_{12} > 0) | ${\tau}_{12}^{a}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{down}\left({E}_{c}{\epsilon}_{eff}\right)}{\left(1+{\beta}^{\prime}\right)}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{down}{E}_{c}\left({\epsilon}_{1}-\frac{{\gamma}_{12}}{a}\right)}{\left(1+{\beta}^{\prime}\right)}$ | ${\tau}_{12}^{b}={\tau}_{12,prev}+G\left({\gamma}_{12}-{\gamma}_{12,prev}\right)$ | ${\tau}_{12}^{c}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{up}\left({E}_{c}{\epsilon}_{eff}\right)}{\left(1+{\beta}^{\prime}\right)}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{up}{E}_{c}\left({\epsilon}_{1}-\frac{{\gamma}_{12}}{a}\right)}{\left(1+{\beta}^{\prime}\right)}$ |

negative total shearing strain (γ _{12} < 0) | ${\tau}_{12}^{c}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{down}\left({E}_{c}{\epsilon}_{eff}\right)}{\left(1+{\beta}^{\prime}\right)}=\frac{{\beta}^{\prime}G{\gamma}_{12}-{\mu}^{down}{E}_{c}\left({\epsilon}_{1}+\frac{{\gamma}_{12}}{a}\right)}{\left(1+{\beta}^{\prime}\right)}$ | ${\tau}_{12}^{b}={\tau}_{12,prev}+G\left({\gamma}_{12}-{\gamma}_{12,prev}\right)$ | ${\tau}_{12}^{a}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{up}\left({E}_{c}{\epsilon}_{eff}\right)}{\left(1+{\beta}^{\prime}\right)}=\frac{{\beta}^{\prime}G{\gamma}_{12}+{\mu}^{up}{E}_{c}\left({\epsilon}_{1}+\frac{{\gamma}_{12}}{a}\right)}{\left(1+{\beta}^{\prime}\right)}$ |

Shearing Strain | Crack Surface Slipping Down a Slope | Crack Surface Is Not Slipping | Crack Slipping Up a Slope |
---|---|---|---|

positive total shearing strain (γ_{12} > 0) | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& -\frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ \frac{{\mu}^{down}{E}_{c}}{1+{\beta}^{\prime}}& 0& \frac{{\beta}^{\prime}G-{\mu}^{down}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& -\frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ 0& 0& \frac{{\tau}_{12}^{\mathrm{b}}}{{\gamma}_{12}^{\mathrm{old}}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& -\frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ \frac{-{\mu}^{up}{E}_{c}}{1+{\beta}^{\prime}}& 0& \frac{{\beta}^{\prime}G+{\mu}^{up}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ |

negative total shearing strain (γ _{12} < 0) | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& \frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ \frac{-{\mu}^{down}{E}_{c}}{1+{\beta}^{\prime}}& 0& \frac{{\beta}^{\prime}G-{\mu}^{down}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& \frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ 0& 0& \frac{{\tau}_{12}^{\mathrm{b}}}{{\gamma}_{12}^{\mathrm{old}}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& \frac{{E}_{c}}{a}\\ 0& {E}_{c}& 0\\ \frac{{\mu}^{up}{E}_{c}}{1+{\beta}^{\prime}}& 0& \frac{{\beta}^{\prime}G+{\mu}^{up}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ |

Shearing Strain | Crack Surface Slipping Down a Slope | Crack Surface Is Not Slipping | Crack Slipping Up a Slope |
---|---|---|---|

positive total shearing strain (γ _{12} > 0) | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& -\frac{{E}_{c}}{a}\\ 0& \frac{{\mu}^{down}{E}_{c}}{1+{\beta}^{\prime}}& \frac{{\beta}^{\prime}G-{\mu}^{down}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& -\frac{{E}_{c}}{a}\\ 0& 0& \frac{{\tau}_{12}^{\mathrm{b}}}{{\gamma}_{12}^{\mathrm{old}}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& -\frac{{E}_{c}}{a}\\ 0& \frac{-{\mu}^{up}{E}_{c}}{1+{\beta}^{\prime}}& \frac{{\beta}^{\prime}G+{\mu}^{up}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ |

negative total shearing strain (γ_{12} < 0) | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& \frac{{E}_{c}}{a}\\ 0& \frac{-{\mu}^{down}{E}_{c}}{1+{\beta}^{\prime}}& \frac{{\beta}^{\prime}G-{\mu}^{down}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& \frac{{E}_{c}}{a}\\ 0& 0& \frac{{\tau}_{12}^{\mathrm{b}}}{{\gamma}_{12}^{\mathrm{old}}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ | $\left\{\begin{array}{c}{\sigma}_{1}\\ {\sigma}_{2}\\ {\tau}_{12}\end{array}\right\}=\left[\begin{array}{ccc}{E}_{c}& 0& 0\\ 0& {E}_{c}& \frac{{E}_{c}}{a}\\ 0& \frac{{\mu}^{up}{E}_{c}}{1+{\beta}^{\prime}}& \frac{{\beta}^{\prime}G+{\mu}^{up}{E}_{c}/a}{1+{\beta}^{\prime}}\end{array}\right]\left\{\begin{array}{c}{\epsilon}_{1}\\ {\epsilon}_{2}\\ {\gamma}_{12}\end{array}\right\}$ |

^{e}ε_{1} | ^{e}ε_{2} | Crack 1 Surface State | Crack 2 Surface State | Active Crack |
---|---|---|---|---|

+ | + | Tension | Tension | Split—both cracks |

+ | − | Tension | Compression | Crack 1 |

− | + | Compression | Tension | Crack 2 |

− | − | Compression | Compression | Requires analysis of shear-stress limits |

Material Properties and Parameters | Values |
---|---|

Concrete Compressive Strength | 41.4 MPa |

Concrete Modulus of Elasticity (E_{c}) | 27.6 Gpa |

Concrete Shear Modulus (G) | 13.8 Gpa |

Shear Retention Factor (β′) | 0.05 |

Steel Yield Strength | 414 Mpa |

Steel Modulus of Elasticity (E_{s}) | 206.8 Gpa |

Friction Coefficients (µ_{up}) | 1.2 |

Friction Coefficients (µ_{down}) | 0.2 |

Crack Opening Path (a_{cop}) | 2.0 |

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**MDPI and ACS Style**

Mitchell, J.P.; Cho, B.-Y.; Kim, Y.-J.
Analytical Model of Two-Directional Cracking Shear-Friction Membrane for Finite Element Analysis of Reinforced Concrete. *Materials* **2021**, *14*, 1460.
https://doi.org/10.3390/ma14061460

**AMA Style**

Mitchell JP, Cho B-Y, Kim Y-J.
Analytical Model of Two-Directional Cracking Shear-Friction Membrane for Finite Element Analysis of Reinforced Concrete. *Materials*. 2021; 14(6):1460.
https://doi.org/10.3390/ma14061460

**Chicago/Turabian Style**

Mitchell, Jeffrey P., Bum-Yean Cho, and Yoo-Jae Kim.
2021. "Analytical Model of Two-Directional Cracking Shear-Friction Membrane for Finite Element Analysis of Reinforced Concrete" *Materials* 14, no. 6: 1460.
https://doi.org/10.3390/ma14061460