## 6 Numerical Simulation Results
### Parametric Analysis of Loading Plate Diameter on Plate Load Test Results  
From a continuum mechanics point of view, the plate load test mechanism and associated stress distribution can be considered as an elastic half-space and a rigid foundation problem. The stress distribution underneath the plate is fairly complex and difficult to characterize. The loading system-rock interface properties control the stress distribution beneath the loading plate. The contact stress distribution is complex and can become singular at some points on the half-space surface. The geometry of the problem can be illustrated as in Figure 13.  
Assuming a constant displacement distribution, the stress beneath the plate can be expressed by the following equation which its resultant is equal to the applied load:  
\[\int_{0}^{2\pi}\;\int_{0}^{a}P(r)rdrd\theta=P \tag{6}\]  
Similarly, the vertical displacement beneath the plate can be determined by integrating the appropriate equation provided that the correct stress distribution is known at the depth  
Figure 12: The sensitivity of the deformation modulus to measurement depth in the plate load test. Figure 13: Free diagram of a rigid loading plate problem. of interest. [PERSON] was the first person which provided the stress distribution beneath the plate as below:  
\[P(r)=\frac{P}{2\pi a}\frac{1}{\sqrt{a^{2}-r^{2}}}\qquad for\qquad 0\leq r\leq a \tag{7}\]  
With regard to the above equation, from a theoretical point of view, the stress is minimum at \(r=0\) and increases to infinity as r approaches a. However, in practice, rock plastic deformations and failures at the plate edge lead to stress drop and redistribution at the plate boundaries. Moreover, Equation (6) can be verified by the necessary integrations. The stress distribution given in Equation (7) will result in a constant settlement beneath the plate. To see how displacement is related to load (\(P\)), we can use [PERSON]'s solution and integrate the given stress distribution. The displacement beneath the plate can be calculated at the plate center from the following equation [22]:  
\[w(r=0)=\iint\ \frac{P(r)rdrd\theta(1-v)}{2\pi Gr}=\int_{0}^{2\pi}\ \int_{0}^{a}\frac{P(1-v)}{4\pi^{2}Ga} \frac{drd\theta}{\sqrt{a^{2}-r^{2}}}=\frac{P(1-v)}{4 Ga} \tag{8}\]  
With regard to the above equation, displacement beneath the plate depends only on the applied load, \(P\), the plate radius, \(a\), and the elastic properties of the half-space. In order to determine displacement analytically at higher depths below the plate, a much more complicated integration must be carried out. The complexity of the stress distribution below the plate is a function of plate-rock interface frictional properties which is very difficult to characterize and measure. Therefore, numerical modeling is a viable alternative to assess the stress and displacement fields as a function of depth below the loading plate. In common practices of plate load testing, it is seen that the loading plate diameter is changed by contractors without knowing its effect on the calculated displacements and, thus, the determined in-situ moduli. In order to shed some light on the effects of plate diameter and displacement measurement depth on plate load test results, numerical simulations were carried out. For this purpose, considering three different plate diameters of 60, 80, and 100 cm, three sets of 3D simulations were carried out to investigate the effect of plate diameter on the calculated rock mass moduli. The displacement for larger diameters should be greater than that of smaller diameters in the same load at the surface of the plate. The contours of horizontal stress distribution on the test gallery wall for two plate diameters of 80 and 100 cm are plotted in Figure 14. The stresses were evaluated at the PLT measurement points and it was observed that for the same load at the surface of the plate, at a given point, the stress value for larger plate diameters is greater than the stress value for smaller plate diameters (see Figure 14).