**3. Multi-Scale Modeling Strategy—Interface Connection**

#### *3.1. Combined Multi-Point Constraint (CMPC) of Multi-Scale Model*

### 3.1.1. Combine Multi-Point Constraint Relations

In the multi-scale model, the interface connection of different scale elements can be established by the constraint equations according to the degrees of freedom of interface nodes. The sketch of the interface connection of multi-scale model shown in Figure 3, where *Si* (*i* = 1, 2, 3 ... .) signifies micro element nodes with 3 degrees of freedom and *B* signifies macro element node with 6 degrees of freedom.

**Figure 3.** Sketch of the interface connection.

According to the coupling relation of degrees of freedom of nodes, the unified form of the constraint equations of the multi-scale interface connection is as follows:

$$\mathcal{L}(\mu\_{\rm B}, \mu\_{\rm Si}) = \mu\_{\rm B} - \mathbb{C}\mu\_{\rm Si} = 0 \tag{1}$$

where *u*<sup>B</sup> is the displacement vector of macro elements at the interface; *u*S*<sup>i</sup>* is the displacement vector of micro elements at the interface; *C* is the coefficient matrix of interface constraint equations.

The accuracy of multi-scale simulation depends on the rationality of coefficient matrix *C*. If the constraint equations can effectively simulate the actual deformation coordination, a better effect of the coupling can be obtained.

The solution of multi-point constraint equations is usually based on the single constraint relation such as displacement coordination [40] or energy balance [41]. For the multi-scale simulation of UHPFRC structure, due to the nonlinear characteristics of the interface stress and deformation relation in the plastic stage, a single multi-point constraint method will lead to the over-constraint in tangential direction and the requirement of stress iteration in plastic stage. Therefore, the combined multi-point constraint (CMPC) is established in this study through the simultaneous equations of displacement coordination and energy balance. The equations form are as follows:

⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ *u*1*Si* − *f*1*i*(*u*1B, *u*5B, *u*6B, *bi*, *hi*) = 0 *F*2*u*2B = ' *A* σ2*i*,*F*2*u*2*Si*d*A F*3*u*3B = ' *A* σ3*i*,*F*3*u*3*Si*d*A F*4*u*4B = ' *A* (σ2*i*,*F*4*u*2*Si* + σ3*i*,*F*4*u*3*Si*)d*A* (2)

where *u*1S*<sup>i</sup>* is the axial displacement of the node i of the micro element; *u*2S*i*, *u*3S*<sup>i</sup>* are the tangential displacements of the node i of the micro element; *u*1B is the axial displacement of the macro element node; *u*2B, *u*3B are the tangential displacements of the macro element node; *u*4B, *u*5B, *u*6B are the angular displacements of macro element node; *Fj* is the nodal force of macro element in the direction j; *bi*, *hi* are the distances from the node i of the micro element to the macro element node; σ*i*2,*Fj*, σ*i*3,*Fj* are the nodal tangential stress of the node i of the micro element caused by *Fj*.

In the Equation set (2), the first equation is the axial and rotational constraint equation, which is established by displacement coordination. The last three equations are tangential constraint equations, which are established by energy balance. The aim is to eliminate the limitation of the single constraint relation in the plastic stage and improve the simulation accuracy of the multi-scale model.

#### 3.1.2. Constraint of the Interface in Tangential Direction

Based on the multi-point constraint relation of displacement coordination, the tangential deformations of all micro element node at the interface are assumed to be consistent, and the displacement constraint relation of each node is established one by one. The deformation diagrams of displacement coordination are shown in Figure 4, and the constraint equations can be obtained as follows:

$$\begin{aligned} \mu\_{\text{2Si}} - f\_{\text{2i}}(\mu\_{\text{2B}}, \mu\_{\text{4B}}, b\_{i\text{}}, h\_{i}) &= 0\\ \mu\_{\text{3Si}} - f\_{\text{3i}}(\mu\_{\text{3B}}, \mu\_{\text{4B}}, b\_{i\text{}}, h\_{i}) &= 0 \end{aligned} \tag{3}$$

In the equations, tangential displacements (*u*2S*i*, *u*3S*i*) of the micro element node are calculated from *u*2B, *u*3B and *u*4B. There is no coupling relation between the degrees of freedom of different nodes of micro elements. Under this condition, when there is no nonzero tangential displacement or rotational displacement of macro element node, the tangential displacement of each node of micro elements along the interface is zero. It leads to the problem of over-constraint in tangential direction at the interface under the axial compression load.

**Figure 4.** Deformation diagram of displacement coordination. (**a**) Tangential direction; (**b**) Rotational direction.

In the CMPC equation set, the displacement constraint equation in the tangential direction of the interface nodes can be obtained after the stress is eliminated by substituting the formula for the shear stress distribution:

$$\mathfrak{u}\_{\rm B} = f\_2(\mathfrak{u}\_{\rm S1}, \mathfrak{u}\_{\rm S2} \cdots \mathfrak{u}\_{\rm Sn}) \tag{4}$$

where, the tangential displacements of each micro element node have a coupling relation with each other. When the tangential displacement of macro element node is zero, it can generate the relative displacements among micro element nodes and satisfy the constraint equation. The tangential deformation diagram of CMPC under the axial compression is shown in Figure 5.

**Figure 5.** Tangential deformation diagram of CMPC under the axial compression.

The interface deformation under axial compression at the plasticity stage of the multi-scale model is shown in Figure 6. According to the Poisson ratio of UHPFRC, the uniform longitudinal stress causes the transverse strain of the section. And there is obvious transverse expansion deformation in the middle of the micro element model. The micro element nodes at the interface of displacement coordination model only produce vertical displacement with the macro element node with no tangential displacement, which is over-constraint compared with the micro model. The CMPC equation eliminates the over-constraint in tangential direction at the interface and conforms to the deformation relation of the interface nodes under the actual stress state.

**Figure 6.** Interface deformation under axial compression. (**a**) Micro element model; (**b**) Multi-scale model of displacement coordination; (**c**) Multi-scale model of CPMC.

#### 3.1.3. Constraint of the Interface in Rotational Direction

The multi-point constraint relation based on energy balance is established by the virtual work principle. It is assumed that the nodal force of the macro element and the nodal forces of the micro elements do equal work at the interface in the rotational direction. The equation is as follows:

$$\begin{aligned} F\_5 \mu\_{5\text{B}} &= \int \sigma\_{1i, \text{F5}} \mu\_{1Si} \text{d}A \\ F\_6 \mu\_{6\text{B}} &= \int\_A \sigma\_{1i, \text{F6}} \mu\_{1Si} \text{d}A \end{aligned} \tag{5}$$

By substituting the formula for stress distribution under bending moment, the displacement constraint equation in rotational direction of the interface nodes can be obtained. Its precision depends on the rationality of the stress distribution of the stress formula.

The normal stress distribution of UHPFRC section under bending moment is shown in Figure 7. As in other studies [33,47], the stress distribution of UHPFRC section has underwent different stages. The first stage is the linear-elastic stage, in which the fiber and matrix show elasticity and the stress distribution is linear. With the increase of load, due to the strong bond between the high strength steel fiber and the matrix, the macro crack begins to expand slowly. The strain hardening phenomenon occurred is different from that of NSC, and the tensile stress is nonlinear distributed. This stage is called strain hardening stage and the formula for stress distribution in linear-elastic stage is no longer applicable. If the formula is not updated iteratively, the multi-point constraint equation at the interface based on energy balance will be distorted in the rotational direction in strain hardening stage.

**Figure 7.** Stress distribution under bending moment. (**a**) Linear-elastic stage; (**b**) Strain hardening stage.

The CMPC method establishes the multi-point constraint equation of the interface in rotational direction based on the displacement coordination. The axial displacement of each node of the micro elements can be obtained through the s constraint equation (see the first equation in Equation set (2)). After entering the strain-hardening stage, this multi-point constraint equation avoids the problem that the formula for stress distribution needs to be updated iteratively. Therefore, the CMPC method proposed in this paper combines the advantages of displacement coordination method and energy balance method. The multi-point constraint equations conform to the transfer relations of displacement and stress between the interface nodes. It can achieve good constraint effect in axial, tangential, and rotational directions. It is applicable to the analysis of UHPFRC components under complex loads.

#### *3.2. CMPC Equations of Multi-Scale Connection of Beam-Solid Element*

According to Equation set (2), displacement vector [*u*<sup>B</sup> *v*<sup>B</sup> *w*<sup>B</sup> θ*<sup>x</sup>* θ*<sup>y</sup>* θ*<sup>z</sup>* ] of beam element and displacement vector *u*S*<sup>i</sup> v*S*<sup>i</sup> w*S*<sup>i</sup>* of solid element are substituted, and the multi-point constraint equations can be expressed as:

$$\begin{cases} w\_{\rm Si} - f\_i'(w\_{\rm B}, \theta\_{\rm x}, \theta\_{y^\*} R\_{\rm xi}, R\_{\rm yi}) = 0 \\ F\_x u\_{\rm B} = \int \tau\_{\rm xi} u\_{\rm Si} \mathrm{d}A \\\ F\_y v\_{\rm B} = \int \tau\_{\rm yi} v\_{\rm Si} \mathrm{d}A \\\ T \theta\_{\rm x} = \int \left( \tau\_{\rm xi} u\_{\rm Si} + \tau\_{\rm yi} v\_{\rm Si} \right) \mathrm{d}A \end{cases} \tag{6}$$

where *Rxi*, *Ryi* are the distances between the node i of the solid element and the beam element node at the interface in the x and y direction, respectively; *Fx*, *Fy* are the shear forces acting on the beam element node in the x and y direction, respectively; *T* is the torque acting on the beam element node; τ*xi*, τ*yi* are the shear stresses of the node i of the solid element in the x and y direction, respectively. The multi-scale connection of beam - solid element is shown in Figure 8.

**Figure 8.** Multi-scale connection of beam - solid element.

In the Equation set (6), the first equation is the constraint equation of axial and rotational displacement, which can be solved according to the displacement coordination. The last three equations are the constraint equations of tangential displacements, which need to be solved by substituting the formula for stress distribution. For example, the formula for shear stress distribution of the rectangular section in the y direction is as follows:

$$\tau\_{yi} = \frac{3F\_y}{2bh} \left( 1 - \frac{4R\_{yi}}{h^2} \right) \tag{7}$$

where *b* and *h* are the width and height of the rectangular section, respectively.

After solving the Equation set (6), the following can be obtained:

$$\begin{cases} \boldsymbol{w\_{Si}} = \boldsymbol{w\_{B}} + \boldsymbol{R\_{xi}} \sin \theta\_{y} + \boldsymbol{R\_{yi}} \sin \theta\_{x} \\ \boldsymbol{u\_{B}} = \mathbb{C}\_{\nu 1} \boldsymbol{u\_{1}} + \mathbb{C}\_{\nu 2} \boldsymbol{u\_{2}} + \dots + \mathbb{C}\_{\nu \nu n} \boldsymbol{u\_{n}} \\ \boldsymbol{v\_{B}} = \mathbb{C}\_{\upsilon 1} \boldsymbol{v\_{1}} + \mathbb{C}\_{\upsilon 2} \boldsymbol{v\_{2}} + \dots + \mathbb{C}\_{\tau \nu} \boldsymbol{v\_{n}} \\ \boldsymbol{\theta\_{Z}} = (\boldsymbol{u\_{1} R\_{y1}} + \dots + \boldsymbol{u\_{n} R\_{yn}}) - (\boldsymbol{v\_{1} R\_{x1}} + \dots + \boldsymbol{v\_{2} R\_{x2}}) \end{cases} \tag{8}$$

where *Cui*, *Cvi* are the influence coefficients of the tangential displacements related to the section size and node position. An example of the Equation set (8) is given in Appendix A.

#### **4. Multi-Scale Models of Ultra-High-Performance Steel Fiber-Reinforced Concrete**

#### *4.1. Multi-Scale Models Built-Up*

The CMPC multi-scale modeling strategy with the same parameters selected above is adopted to establish the multi-scale models of reinforced UHPFRC components, as shown in Figures 9 and 10. Where (a) is the solid element model taken as the standard for comparison without experimental results, and (b), (c) and (d) are the multi-scale models of beam-solid element, whose interface connections are established by the displacement coordination method, the energy balance method and the CMPC method (Section 3.2. for the expressions) respectively. The height of the component is 3m, and the section size is 0.4 m × 0.4 m. The multi-scale interface is located at 1/3 height of the component with the height of 3 m. The parameters of the CDP model adopted for the UHPFRC solid elements are shown in Table 1.

**Figure 9.** UHPFRC parts of the models. (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

**Figure 10.** Reinforcement parts of the models. (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

In the micro element model, C3D8R elements are used to model UHPFRC with the calibrated CDP model which is the same as that in the second section. The reinforcement is simulated by T3D2 element. In the macro element model, B31 elements are used to simulate UHPFRC and the reinforcement net. The uniaxial stress-strain relationship of the material subroutine (UMAT) of UHPFRC is the same as that of the calibrated CDP model. The section size of the B31 element of the reinforcement net is calculated equivalent to total reinforcement area. The material model parameters of reinforcement are the same as those in Section 2. With a yield strength and ultimate strength of 525 and 625 MPa, respectively. The mesh size of the models is 0.05m. A fixed constraint is set at the bottom of the component and a loading point is set at the top. The number of model elements of the solid element model and the multi-scale model is shown in Table 4. It can be seen that the number of model elements in the multi-scale model is reduced by nearly 2/3 compared with that in the solid element model, which significantly improves the computational efficiency.


**Table 4.** Number of model elements.
