In two dimensional problems there are two equations and three unknowns.Cylindrical coordinates were nominated as a good candidate for analyzing stress distributions from α variation since the number of significant coordinates for plane stress condition is reduced from two in rectangular coordinates (x, y) to just one in cylindrical (r). An example of a 3-D cylindrical structure is modeled and validated by means of a magnetostatic finite element analysis (FEA), and excellent agreement is found.

The other method is to derive the equation for the stress tensor for your situation directly in cylindrical coordinates. Use your intuition, while keeping track of the terms you are ignoring (check your assumptions at the end). 7 Relations in Curvilinear Cylindrical and Spherical Coordinates Appendix ЗВ Equations of Equilibrium Including Couple Stress and Body Couple Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory 3C-1 Material Derivative. They contain correction terms due to the The general solution of this first-order differential equation is Vr()=-a r + b where b is a constant. in polar coordinates of the equations of momentum conser- vation and the stress strain relations for an isotropic solid. 4 Force and Moment Equilibrium For all differential elements in the continuum of Figure 2, force and moment equilibrium leads respectively to the equilibrium equations and the symmetry of the stress tensor Fij. dinate systems are derived, and are then speciﬁed for the cases of cylindrical coordinates and spherical coordinates. Therefore wide beams are a special class of rectangular plates.

Transformation of stress Problem 68 (from previous lecture): The state of plane stress in indicated in the figure.

Morrison, Michigan Technological University Cartesian Coordinates L ì̃ ë ë ì̃ ë ì ì̃ ë í ì̃ ì ë ì̃ ì ì ì̃ ì í ì̃ í ë ì í ì ì̃ í í M ë ì í L ä É È È È Ç 2 ò R ë ò T ò R ë ò U E ò R ì ò T Stress Compatibility Equations in Cylindrical Coordinates Barrekette, Euval S. Airy stress Function Goodman approach Equilibrium equations in polar coordinates Hooke’s Law in polar coordinates √ Miner’s rule Crack Propagation √ √ A A Strain displacement Equations in Polar Coordinates Airy stress function in polar Coordinates Fracture mechanics √ von Mises effective stress: With reference to the two-dimensional equations or stress transformation [Equations (2. The derivation follows the procedure given in and the results are presented as in, although the 3-term expansion for the transverse dis- Figure 1. Cylindrical coordinates Many problems are such that it is advantageous to use cylindrical (r, O, z) instead of Cartesian (x, y, z) coordinates. 2) The two ﬁrst equations in both transformations simply deﬁne polar coordinates in the xy- Proof: Write the equations of equilibrium as.

Stress equilibrium equations in cylindrical coordinates CYLINDRICAL COORDINATES (Section 13.

Polar equation to rectangular equation