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chapter 4 2D problems in polar coordinates

The fourth chapter 2D problems in polar coordinates
Key points： (1) Solution of plane problems in polar coordinates： —— Balance equation、Geometric equation、Physical equation Compatibility equation、Boundary conditions。 (2) The solving method and application of 2D problems in polar coordinates. Application: Stress and deformation analysis of Disc、Ring、Thick Wall Cylinder、Wedge、Semi-infinite plate body

Content
§4-1 Differential equations of equilibrium in polar coordinates

§4-2 §4-3
§4-4 §4-5 §4-6 §4-7 §4-8 §4-9

Geometrical and physical equations in polar coordinates Stress function and compatibility equation in polar coordinates Coordinate transformation of stress components Axisymmetrial stresses and corresponding displacements Hollow cylinder subjected to uniform pressures
Pressure Tunnel Hole-edge stress concentration Concentrated normal load on a straight boundary of a semiinfinite plate

§4-10 Distribution force on a straight boundary of a semi-infinite plate

§4-1

Differential equations of equilibrium in polar coordinates

1. Micro unit in polar coordinates

O

Body force： Stress：
Face PA

k r , k?
rd ?
y

?

? ? , ? ?r Face PB ? r , ? r ?

?r ? r?
B

d?

r

P??r

?? x

(r ? dr )d? dr A kr ? r? ? ? ? r? dr ?r k
?

Face BC

Face BC

?? ? ?? ? d? ?? ? ? ?r ? ?r ? d? ?? ?? r ?r ? dr
?r ? ? r? ? dr ?r

C

? r?

Regulations of stress direction： Shear stress ——for the front side of r、θ，it is positive when it consist with the coordinate direction.

? ? ?r ?? ? d? ?? ? d ? ? ?r ? ?? ??

?? r ?r ? dr ?r

Normal stress—— pull is positive，push is negative；

2 Equilibrium differential equation. Considering the balance of micro unit（Take thickness for 1）：

O

?

? ? ?r B ? ? r rd ? ? ? ? r d r ? (? ? r ? d ? ) dr rd ? ? ?? ?? r ?? r C ? (? r ? dr )( r ? dr ) d ? ?r ? dr y ?r ?r ? ? ?? ? ?? ? ?r ? ? d? ? ? d? ? ?? ? ? d ? ? dr ?? ? d ? ?r ?? ?? 2 ? ? ?? d? ? k r rd rd ? ? 0 ? ? ? dr 2
Make an expansion： （infinitesimal of higher order，delete it）

?F

r

? 0,

?r ? r?

d?

r

P??r

?? x

(r ? dr )d? dr A kr ? r? ? ? ? r? dr ?r k

? ? r rd ? ?

? ? ?r ?? r ?? r d ? dr ? ? r rd ? ? rdrd ? ? ? r d r d ? ? ?dr ?2 d ? ?? ?r ?r d? ? ?? ? d? d? ? ? ? ? dr ? ? ? dr ?? d ? ? dr ? k r rd rd ? ? 0 2 2 2 ? ?? ?

O ? ? ?r ?? r ? rdrd ? ? d ? dr ? ? r d r d ? ?r ??

?

? ? ? d rd ? ? k r r d r d ? ? 0
Both sides at the same time divided by r d r d ? ：

rd ?

?r ? r?
B

d?

r

P??r

?