## 5 Results
### Contact Force  
The distribution of the contact force along the interface of the two sub-beams is presented in Figure 7 for the CZM and VCCT models for values of the external load equal to 5 ((a), (b), (c)), 15 ((d), (e), (f)), and 25 ((g), (h), (i)) kN/m as indicative values within the considered range. Furthermore, note that (a), (d), and (g) represent the results for the case without a thermal load, (b), (e), and (h) represent the results for the case with +135 degrees, and (c), (f), and (i) represent the results for the case with \(-135\) degrees. Each dot on the lines of CZM and VCCT represents one nodal value of the contact force with the x coordinate counted from the left end tip of the composite laminate, as depicted in Figure 4c, and refers to the upper sub-beam. The coefficient of friction for the results presented here was considered to be 0.01 to limit the influence of the friction effects and focus only on the contact force. Figure 7 illustrates that the distribution of the contact force was not identical for the three cases. In particular, the results of the +135-degree case and the case without thermal loads demonstrated increased effects at the edge with larger magnitudes than the \(-135\)-degree case. The \(-135\)-degree case further presented a spread of the values over a larger area. This area moved towards the left edge of the laminate as the external load increased. This result is physically explained by the deformation caused by the residual thermal stresses on the structure. The application of +135 degrees of temperature difference resulted in an inward rotation of the lower sub-beam, leading to a point contact of the edge of the lower sub-beam and the upper sub-beam. On the other hand, a \(-135\)-degree temperature difference resulted in an outward rotation of the lower sub-beam, leading to the contact of a more extended area of the lower and upper sub-beam, located further away from the edge  
Figure 7: Distribution of the contact force along the interface of the two sub-beams for the cases without thermal load (**a**,**d**,**g**), and with +135-degree (**b**,**e**,**h**) and \(-135\)-degree (**c**,**f**,**i**) temperature differences. tip due to the rotation. However, an increase in external load led to an inward rotation of the lower sub-beam, moving the area towards the edge. In both cases, the upper sub-beam remained unaffected by the residual thermal stresses, being symmetric with respect to its middle plane. The case without thermal stresses did not present any additional rotation, leading to a point contact of the edge of the lower sub-beam and the upper sub-beam. Figure 8 presents the contact force versus the external load for the cases without thermal loads (a), and with +135-degree (b) and \(-\)135-degree (c) temperature differences. The values of the VCCT and CZM models represent the sum of all contact nodal forces acting on the upper sub-beam. It is evident that the flexible joint model better followed the results of the CZM and the VCCT, which could be attributed to the release of the translation degree of freedom. The rigid and semi-rigid joint models presented a slightly higher estimation of the contact force. Moreover, the comparison of the analytical theories with the numerical results revealed that the contact force for the case without thermal loads (a) and a +135-degree (b) degree temperature difference was almost perfect for all the range of the considered load. However, for a \(-\)135-degree (c) temperature difference, there was a higher difference for the lower loads, still remaining relatively low. This implies that the location of contact plays an important role in the evaluation of the contact force. Nevertheless, as explained below, this difference did not result in large deviations in the prediction of the ERR. Additionally, it is evident that the contact force was not equal to the reaction force of the left end, which was half of that of the external load \(P\).