*4.2. Unidirectional Load Cases*

#### 4.2.1. Axial Compression Load Case

Under the axial compression load, the stress distributions of UHPFRC of the multi-scale models and the connection interface are shown in Figure 11. By comparison, it can be seen that there is the phenomenon of stress concentration at the connection interface of the displacement coordination model whose stress distribution is different from that of the solid model. The overall stress distribution of the energy balance model is also different from that of the solid model, and the stress distribution at the interface connection is not uniform. The stress distribution obtained by the CMPC method is highly consistent with the solid element model. The constraint effect of the CMPC method is obviously better than that of the single multi-point constraint method.

**Figure 11.** Stress distributions under the axial compression load (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

#### 4.2.2. Bending Load Case

Unidirectional concentrated moment loads is applied to the loading point at the top of the models under the bending load case. The stress distribution and tensile damage distribution of UHPFRC of the models are shown in Figures 12 and 13. By comparison, it can be seen that the stress distribution at the connection interface in the plastic stage is nonlinear. The UHPFRC on the tensile side enters the strain hardening stage with tensile damage. Compared with the solid element model, the results of the displacement coordination model and the energy balance model show obvious stress distortion at the connection interface. As UHPFRC on the tension side enters the strain hardening stage, the formula for stress distribution of the energy balance method is no longer applicable. The original constraint equation needs to be balanced by over increasing the strain on the tension side, resulting in the distortion of the damage distribution at the connection interface. Due to the over-constraint in tangential direction mentioned above, the results of the displacement coordination model at the connection interface look distorted. The simulation results of CMPC model are highly consistent with that of the solid element model, and the stress and damage distribution of UHPFRC at the connection interface are simulated accurately.

**Figure 12.** Stress distributions under the bending load (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

**Figure 13.** Tensile damage distributions under the bending load (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

#### 4.2.3. Shear Load Case

In shear load case, a concentrated horizontal force is applied to the loading point at the top of the models. The stress distribution and tensile damage distribution of UHPFRC of the models are shown in Figures 14 and 15. It can be seen that after entering the strain hardening stage of UHPFRC, the stress distribution at the connection interface presents nonlinear. The multi-point constraint equation derived from the formula for elastic stress distribution is no longer applicable, and the simulation results of the stress distribution and damage distribution at the connection interface of the energy balance model are inaccurate. At the same time, due to the tangential over-constraint, the stress concentration occurs at the connection interface of the displacement coordination model. The damage distribution is distorted. However, the simulation results of the CMPC model still have good accuracy. The stress and damage distribution of UHPFRC at the connection interface are simulated accurately.

**Figure 14.** Stress distributions under the shear compression load (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

**Figure 15.** Stress distributions under the shear compression load (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model.

#### *4.3. Multidirectional Composite Load Case*

Under unidirectional load cases, the multi-scale model of UHPFRC component established according to the proposed multi-scale modeling strategy achieved good accuracy. The performance of this multi-scale model under multidirectional composite load case will be studied below.

In this load case, the axial compression force, the bidirectional moments and the bidirectional horizontal forces are applied composited at the loading point at the top of the models. The diagram of multidirectional composite load is shown in Figure 16. Where, the red arrow represents the bidirectional horizontal forces, the yellow arrow represents the axial compression force, and the purple double arrow represents the bidirectional moments.

**Figure 16.** The diagram of multidirectional composite load.

Under the multidirectional composite load case, the stress distributions of UHPFRC of the multi-scale models and the connection interface are shown in Figure 17.

Through comparison, it can be seen that the CPMC equations established based on the proposed multi-scale modeling strategy achieve good connection effect under the multidirectional composite load case. The calculation accuracy of the CMPC model for UHPFRC is consistent with that of the solid element model, which is better than the displacement coordination model and the energy balance model. The multi-scale modeling strategy proposed in this study can be effectively applied to the multi-scale finite element analysis of UHPFRC structures with accuracy and efficiency.

**Figure 17.** Stress distributions under the multidirectional composite loads case (unit: Pa). (**a**) Solid element model; (**b**) Displacement coordination model; (**c**) Energy balance model; (**d**) CMPC model; (**e**) Interface location of the solid element model; (**f**) Interface of the displacement coordination model; (**g**) Interface of the energy balance model; (**h**) Interface of CMPC model.

#### **5. Conclusions**

This study proposed a novel multi-scale modeling strategy for ultra-high-performance steel fiber-reinforced concrete (UHPFRC) structures. The main work and conclusions are summarized as follows:


4. The simulation results of the multi-scale model under each load case show that the multi-scale model established by the CMPC method can significantly reduce the number of model elements and improve the calculation efficiency. The CMPC models have good simulation accuracy in the analysis of each load case compared with the displacement coordination model and the energy balance model. In the strain-hardening stage of UHPFRC, the CMPC method can still accurately simulate the stress distribution and damage distribution of the connection interface. It can be applied to multi-scale finite element analysis of UHPFRC structures with accuracy and efficiency.

**Author Contributions:** Conceptualization, Z.L. and J.T.; software, Z.P.; validation, Z.P.; writing—original draft preparation, Z.P.; writing—review and editing, Z.L.; supervision, Z.L.; project administration, J.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was funded by the National Key Research and Development Program of China under grant number 2016YFC0701102, the National Natural Science Foundations of China under grant number 51538003 and 51978224, China Major Development Project for Scientific Research Instrument under grant number 51827811, and Shenzhen Technology Innovation Program under grant number JCYJ20170811160003571 and JCYJ20180508152238111.

**Conflicts of Interest:** The authors declare that there are no conflicts of interests regarding the publication of this article.

### **Appendix A**

An example of the multi-scale connection interface is established with section size 0.4 m × 0.4 m and mesh size 0.1 m shown in Figure A1, where the solid element node number on the connection interface is 1–25, and the beam element node number is 26.

**Figure A1.** An example of the multi-scale connection interface.

The Equation set (8) of the multi-scale connection interface shown in Figure A1 can be obtained as follow. The multi-scale connection of CMPC model can be established by coding the '\*EQUATION' keyword commands to input the constrained Equation set (8) into the '\*.inp' file of ABAQUS. Similarly, this strategy can be implemented in the other FEA software by setting the multi-point constraint equations.

```
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
  wSi = wB + Rxi sin θy + Ryi sin θx
  1024u26 = [7(u25 + u1 + u5 + u21) + 14(u20 + u15 + u10 + u16 + u11 + u6) + 34(u24 + u22 + u4 + u2)+
     68(u19 + u14 + u9 + u17 + u12 + u7) + 46(u23 + u3) + 92(u8 + u13 + u18)]
  1024v26 = [7(v25 + v1 + v5 + v21) + 14(v24 + v22 + v4 + v2 + v23 + v3) + 34(v20 + v16 + v10 + v6)+
     68(v19 + v17 + v9 + v7 + v18 + v8) + 46(v15 + v11) + 92(v14 + v12 + v13)]
  10θz26 = 2(u1 + u2 + u3 + u4 + u5 + v5 + v10 + v15 + v20 + v25)+
      2(−u21 − u22 − u23 − u24 − u25 − v1 − v6 − v11 − v16 − v21)+
      (u6 + u7 + u8 + u9 + u10 + v4 + v9 + v14 + v19 + v24)+
      (−u16 − u17 − u18 − u19 − u20 − v2 − v7 − v12 − v17 − v22)
  w1 = w26 − 0.2 sin θx26 + 0.2 sin θy26
  w2 = w26 − 0.2 sin θx26 + 0.1 sin θy26
  w3 = w26 − 0.2 sin θx26
  w4 = w26 − 0.2 sin θx26 − 0.1 sin θy26
  w5 = w26 − 0.2 sin θx26 − 0.2 sin θy26
  w6 = w26 − 0.1 sin θx26 + 0.2 sin θy26
  w7 = w26 − 0.1 sin θx26 + 0.1 sin θy26
  w8 = w26 − 0.1 sin θx26
  w4 = w26 − 0.1 sin θx26 − 0.1 sin θy26
  w10 = w26 − 0.1 sin θx26 − 0.2 sin θy26
  w11 = w26 + 0.2 sin θy26
  w12 = w26 + 0.1 sin θy26
  w13 = w26
  w14 = w26 − 0.1 sin θy26
  w15 = w26 − 0.2 sin θy26
  w16 = w26 + 0.1 sin θx26 + 0.2 sin θy26
  w17 = w26 + 0.1 sin θx26 + 0.1 sin θy26
  w18 = w26 + 0.1 sin θx26
  w19 = w26 + 0.1 sin θx26 − 0.1 sin θy26
  w20 = w26 + 0.1 sin θx26 − 0.2 sin θy26
  w21 = w26 + 0.2 sin θx26 + 0.2 sin θy26
  w22 = w26 + 0.2 sin θx26 + 0.1 sin θy26
  w23 = w26 + 0.2 sin θx26
  w24 = w26 + 0.2 sin θx26 − 0.1 sin θy26
  w25 = w26 + 0.2 sin θx26 − 0.2 sin θy26
                                                                                                                             (A1)
```
## **References**


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