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 *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 (***r***, ****, ***z***)**

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

**Spherical Coordinates (***r,*** ***,*** ****)**

Let *x* = *r*, *y* = , *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/d*z)* = 0 and from Equation (2.49), equilibrium requires

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

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