The stresses and displacements in a soil mass due to applied loading are considered in this chapter. Many problems can be treated by analysis in two dimensions, i.e. only the stresses and displacements in a single plane need to be considered. The total normal stresses and shear stresses in the x and z directions on an element of soil are shown in Figure 5.1, the stresses being positive as shown; the stresses vary across the element. The rates of change of the normal stresses in the respective directions are ∂ σ x / ∂ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq426.tif"/> and ∂ σ z / ∂ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq427.tif"/> the rates of change of the shear stresses are ∂ τ x z / ∂ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq428.tif"/> and ∂ τ z x / ∂ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq429.tif"/> . Every such element in a soil mass must be in static equilibrium. By equating moments about the centre point of the element, and neglecting higher-order differentials, it is apparent that τ x z = τ z x . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq430.tif"/> By equating forces in thex and z directions the following equations are obtained: () ∂ σ x ∂ x + ∂ τ z x ∂ z − X = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq431.tif"/> () ∂ τ x z ∂ x + ∂ σ z ∂ z − Z = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429209307/b43df8c6-a58e-4416-be47-a3da8aac7534/content/eq432.tif"/>