6.3.2 Gravity generation method
Assume the boundaries of the excavation profile are all rollers（this is a temporary assumption to compute the initial stresses.When starting analysis，it should be set to what it should be）.Assign suitable E and v to each element.Considering the lateral strain of each element in the initial state is 0，according to the theories of elasticity，we can infer that v＝K0／（1＋K0）.E can be an arbitrary large number.Then have the body forces act throughout the whole area and use the finite element method to solve the initial stresses for each element.
Considering that the values of E and v are not easily determined，a modification can be made by determining the stresses for each element with direct input method first and then by having the body forces act throughout all elements.The stresses of an element can be transformed into the equivalent internal nodal forces and the body force can be transformed into the equivalent external nodal force.The difference between the internal nodal force and the external nodal force can be expressed as follows：
where
｛R｝＝residual force matrix；
［B］＝strain⁃displacement matrix；
{σ}＝initial stress matrix；
[f ]＝displacement shape function matrix；
{γ}＝unit weight of soil.
The stresses of the elements are solved repeatedly with the residual forces acting at the nodes until the residual forces｛R｝equal to 0.The stresses are then the initial stresses.