# Point Deflection in Topological Interlocking Plates
## 3 Rectangular Block Model of Topological Interlocking Beams  
In line with the reasoning of the previous section, we consider a 2D model of the deflection of a TI beam as a beam consisting of \(n\) rectangular blocks of length \(a\); its thickness is \(2b\). The beam length is \(2l=na\), Figure 8a. The beam is loaded by point force \(Q\). The role of the peripheral constraint is played by an internal post-tensioning cable of stiffness  
Figure 7: Relative horizontal displacement and rotation, \(\theta=\Delta\omega_{z}\), of topological interlocking blocks in the process of deflection initiated by the interaction of the block corners (elbowing): (**a**) interaction of equal symmetrical blocks; (**b**) interaction of blocks of alternating shapes; and (**c**) equivalent rectangular blocks (in (**a,b**) these blocks are shown by broken lines).  
loaded by initial force \(F_{0}\) (post-tensioning here refers to applying tension to the cables after the blocks are assembled).  
Since topological interlocking does not resist tension, the behavior of a TI beam is controlled by a post-tensioning cable or external constraint. Appendix A shows that a post-tensioning cable and an external constraint frame affect the bending in a similar way. Macroscopically, the cable behaves as a string (Figure 8b). (Obviously, this deflection shape produces the smallest cable length increase, resulting in the minimum elastic energy of the cable.) The tension in the string is produced by elbowing [52], which produces an additional deformation that is an increase in the distance between the block centers due to their relative rotation.  
We name the deflection of this geometry _point deflection_ in order to distinguish it from classical beam deflection. We use Figure 8b as a (macroscale) continuum model of the point deflection of the fragmented beams shown in Figure 8c,d. The discrete (microscopic) interpretations of the fragmented beam deflection will have the shapes shown in Figure 8c,d for odd and even numbers of blocks, respectively. Obviously, in the macroscopic model the length, \(a\), of a block is not distinguishable. Subsequently, Figure 8 presents a mechanism of beam deflection localization [49].  
Following this logic, one can also construct the fragmented beam deflection under a couple of point loads of equal, Figure 9a, or different, Figure 9b, magnitudes.  
Figure 8: Fragmented beam model; the broken lines indicate the holes in the blocks for the post-tensioning cable (see Appendix A for the equivalence between the post-tensioning cable and an external constraint frame): (**a**) fragmented beam with \(n\) rectangular blocks of width \(a\) and height \(2b\) as a model of a TI beam under point force \(Q_{i}\) (**b**) point deflection of the post-tensioning cable; this configuration will be used as a representation of a macroscale continuum model of the bending of a fragmented beam; and (**c**,d**) discrete (microscale) interpretations of the continuum model of the point deflection of a fragmented beam.  
Figure 9: Fragmented beam under two point loads: (**a**) equal point loads; (**b**) different point loads.