# Dynamic Response and Geogrid Strain Analysis of GRS Retaining Wall
## 4 Geogrid Strain Calculation Method
### Derivation of Calculation Formula  
As shown in Figure 14, for the analysis of the compression zone of the layer \(i\) geogrid, establishing a Cartesian coordinate system, the displacement equation \(y(x)\) is established based on the average relationship of the forces after deformation of the AB section. The component force \(T_{x},\ T_{y}\) of the geogrid tension \(T\) in the x-axis and y-axis directions is  
\[\big{\{}T_{x}(x),\ T_{y}(x)\big{\}}=\bigg{\{}F_{H},\ F_{H}\frac{dy}{dx}\bigg{\}}, \tag{2}\]  
where \(F_{H}\) is the design value of horizontal tension of the geogrid. The force analysis of one unit of the geogrid in the pressurized zone is shown in Figure 15.  
Figure 14: Geogrid deformation in pressurized zone. Figure 13: Deformation diagram of geogrid in pressurized zone. Figure 15: Force analysis of unit. According to the balance of force, the differential equation is obtained:  
\[\sum F_{y}=0,\ F_{H}\frac{d^{2}y}{dx^{2}}+q(x)=0. \tag{3}\]  
Two integrals are available:  
\[y=\frac{q}{2F_{H}}\cdot x\cdot(l-x)+\frac{c}{l}x. \tag{4}\]  
Using in the boundary conditions available, \(c=0\), the displacement equation of the AB section is  
\[y(x)=\frac{q}{2F_{H}}\cdot x\cdot(l-x). \tag{5}\]  
Bringing the point (\(x\), \(y\)) = (\(l/2\), \(r\)) into Equation (5), the expression of mid-span deflection \(r\) is  
\[r=\frac{ql}{8F_{H}}. \tag{6}\]  
By the Pythagorean theorem, the expression for the variation in the geogrid tension \(T\) with \(x\) can be found  
\[T=F_{H}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}=F_{H}\sqrt{1+\left(\frac{8f}{l ^{2}}\left(\frac{l}{2}-x\right)\right)^{2}}. \tag{7}\]  
For a general GRS retaining wall, the vertical stress on layer \(i\) of the geogrid is  
Figure 15.0  
where \(\gamma\) is the volumetric weight of the filler, \(z_{i}\) is the distance from the top of the wall to layer \(i\) of the geogrid, \(F\) is the peak load, and \(l_{i}\) is the length of the pressure zone of layer \(i\) of the geogrid, calculated according to the following formula. When \(z_{i}\leq b_{c}\),  
Figure 15.1  
When \(z_{i}>b_{c}\),  
Figure 15.2  
where \(l_{0}\) is loading plate width, \(b_{c}\) is distance between loading plate and facing. Tensile forces at the nodes of the geogrid are calculated with reference to the current relevant codes [51]:  
Figure 15.3  
In the formula, \(K\) is the additional coefficient of the peak tension of the geogrid, 1.5-2.0, \(p_{i}\) is the horizontal earth pressure, \(s_{x}\), \(s_{y}\) is the horizontal and vertical spacing between the geogrids. Bringing \(F_{H}\), \(q\) into Equation (5), and according to the geometric relationship, the elongation of the geogrid can be obtained:  
Figure 15.4  
Therefore, the geogrid strain \(\varepsilon\) is  
Figure 15.5  
A large number of large-scale tests and field monitoring results in the world show that increasing the tensile strength of the geogrid can reduce the settlement under external load. However, in the calculation method established in this paper, the deformation deflection \(r\) of the geogrid does not consider the elastic modulus of the geogrid, and the result thus obtained is obviously inconsistent with the actual situation. According to Equation (6), if we let r decrease, the tension \(F_{H}\) at the geogrid node should increase. Thus, it is found that the derivation of the specification for \(F_{H}\) is imperfect, \(F_{H}\) is the internal tension of the geogrid and the specification only considers the horizontal earth pressure there, and the correct derivation is as follows. The modulus of elasticity \(E\) of the geogrid is calculated by a tensile test of geosynthetic material. Figure 15.6  
where: \(E\) is the modulus of elasticity, \(\Delta F\) is the force per unit area on the cross section, \(\Delta L\) is the ratio of elongation to clamping length \(h\). The geogrid deformation can be found according to [PERSON]'s law that  
Figure 15.7  
where: \(E\) is the linear modulus of elasticity, \(t\) is the thickness of the geogrid, this paper takes t\({}_{0}\) = 0.2 cm.