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# Orthogonal Coordinate Systems

The laws of tensor transformation [Equation (2.13) or Mohr's Circle] allow us to describe stress or strain in any orientation of coordinate axes once we have solved for the tensor in one particular orthogonal set.

Other Orthogonal Coordinate Systems

The laws of tensor transformation [Equation (2.13) or Mohr's Circle] allow us to describe stress or strain in any orientation of coordinate axes once we have solved for the tensor in one particular orthogonal set. In many physical situ-ations, the field equations, which have been derived in a global, rectangular coordinate system, can be more easily solved in some other coordinate system.

Rather than re-derive them from consideration of stress and geometry changes for a new differential element in each coordinate system, it is easier to present the most general form and show how it can be reduced in each specific case.** The differential equations of equilibrium relative to general

in which (?, ?, ?) are metric coefficients which are functions of the coordi-nates (x, y, z) defined by

where ds is the differential diagonal of the element (Figure 2.12a) with edge   lengths ?dx, ?dy, ?dz. In Cartesian coordinates ?= ?=?=1 and Equations (2.45) reduce to Equation (2.43).

The strain-displacement equations for the element are:

where u,v,w are the projections of the displacement vector of point x,y,z on the tangents to the coordinate lines at the point.

Cylindrical Coordinates (rz)

Let x = r, y , z = z as shown in Figure 2.12b. Then:

Spherical Coordinates (r, , )

Let x = ry = , z  where   is the latitude and  the longitude as shown in Figure 2.12c. The equilibrium equations become:

The strain-displacement relations in spherical coordinates from Equation (2.47) become:

Plane Polar Coordinates (r, )

Polar coordinates for plane problems are, it will turn out, equally as useful as Car-tesian coordinates. They can be considered a special case of cylindrical coordi-nates where rz =z = (d/dz) = 0 and from Equation (2.49), equilibrium requires

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Civil : Principles of Solid Mechanics : Strain and Stress : Orthogonal Coordinate Systems |