## 4 Prediction for Load Capacity  
### Crack Capacity  
Cracks in beam-column connections have an impact on the seismic performance of structural strength, deformation, and durability [35]. Based on the crake capacity model of monolithic concrete connections, the crack capacity model of MRPC assembled beam-column connections is deduced.  
The model makes the following assumptions:  
1. The MRPC is considered to be in the elastic stage before crack initiation;
2. The shear effect of reinforcement is not considered, and it is considered that the shear stress is fully borne by MRPC before crack initiation;
3. The beam axial force is neglected.  
When the principal tensile stress of the oblique section of the connections reaches the initial crack strength of the MRPC, it will be in a critical state of cracking. The horizontal shear stress is obtained as follows:  
\[\tau=\frac{V_{\mathrm{jcr}}}{b_{\mathrm{j}}h_{\mathrm{j}}} \tag{4}\]  
where \(V_{\mathrm{jcr}}\) is the shear force at the initial crack of the beam-column connections, and \(b_{\mathrm{j}}\) and \(h_{\mathrm{j}}\) are the effective width and height of the section of connections, respectively.  
The beam-column connection is subjected to the vertical pressure transmitted by the column. The axial stress is obtained as follows:  
\[\sigma=\frac{N_{\mathrm{c}}}{b_{\mathrm{c}}h_{\mathrm{c}}} \tag{5}\]  
where \(N_{\mathrm{c}}\) is axial force, and \(b_{\mathrm{c}}\) and \(h_{\mathrm{c}}\) are the width and height of the section of the column, respectively.  
The principal tensile stress of the connection is obtained as follows:  
\[\sigma_{1}=-\frac{\sigma}{2}+\sqrt{\left(\frac{\sigma}{2}\right)^{2}+\tau^{2}} \tag{6}\]  
When the connection has not yet cracked, the principal tensile stress satisfies Equation (7). Considering that the connection is in a critical state of cracking, substituting Equations (4) and (5) into Equation (7), the shear force of the theoretical is obtained as follows:  
Figure 16: Cumulative energy consumption.  
\[\sigma_{1}\leq f_{\rm{tk}} \tag{7}\]  
\[V_{\rm{jcr}}=f_{\rm{tk}}b_{j}h_{\rm{j}}\sqrt{1+\frac{N_{c}}{f_{\rm{tk}}b_{c}h_{c}}} \tag{8}\]  
where \(f_{\rm{tk}}\) is the initial cracking stress of the uniaxial tensile test of the MRPC.  
The sheer force of the test can be calculated as follows [26]:  
\[V_{\rm{jt}}=\frac{\sum M_{\rm{b}}}{h_{0}-a_{\rm{s}}^{\prime}}\left(1-\frac{h_{0 }-a_{\rm{s}}^{\prime}}{H_{\rm{c}}-h_{\rm{b}}}\right) \tag{9}\]  
where \(V_{\rm{jt}}\) is the cracking test value of the beam-column connections, \(M_{\rm{b}}\) is the sum of the bending moments at the beam ends, \(H_{\rm{c}}\) is the distance between the inflection points of the upper and lower columns, \(a_{\rm{s}}^{\prime}\) is the protective layer thickness, \(h_{0}\) is the effective height of the beam section, and \(h_{\rm{b}}\) is the height of the beam section.  
The crack capacity is similar to the material strength of the specimen in Section 2.2. Short steel fibers can better suppress the generation and development of microcracks and significantly improve the initial cracking strength of connections. Because of the poor tensile capacity of MRPC-SF-4, the initial crack load of SFZP-4 is significantly lower than that of other specimens, as shown in Table 9. The end-bend steel fiber has the strongest restraining effect on the matrix. Considering that the reinforcement ratio of the connection is the same, it may be that the damage accumulated after the positive loading of the specimens leads to the low crack capacity under negative loading. The test value of connections (\(V_{\rm{jt}}\)) is generally smaller than the theoretical value (\(V_{\rm{jcr}}\)); thus, the Chinese code overestimated the shear contribution of MRPC. If \(\eta\) is defined as the safety factor, it is determined to be 0.80 according to the test data.  
\[V_{\rm{jcr}}=f_{\rm{tk}}b_{j}h_{\rm{j}}\sqrt{1+\frac{N_{c}}{f_{\rm{tk}}b_{c}h_ {c}}} \tag{10}\]