Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1...

 

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Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1 r 20 1000 + ar 20 + do. 0 1 + - (arr - 000) 0 re 20 20 You Press + 20 10 - 0 T Ar Get ar 201 Or Ar L Show that the two equilibrium equations in polar coordinates for plane stress (see Ar + 1 r 20 1000 + ar 20 + do. 0 1 + - (arr - 000) 0 re 20 20 You Press + 20 10 - 0 T Ar Get ar 201 Or Ar L

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The image you have provided contains two equilibrium equations for an element under plane stress in polar coordinates and a diagram showing the stresses acting on an infinitesimal element in polar coordinates The two equations are 1 rrr 1r r 1r rr 0 2 rr 1r 2r r 0 These equations are derived from the equilibrium conditions on an infinitesimal element Lets break down the derivation For equilibrium in any system the sum of forces must be zero This must also be true for an infinitesimal element of a material undergoing deformation When dealing with a plane stress problem we consider an infinitesimal element in the shape of a sector of an annulus with internal radius r external radius rdr central angle d and thickness t into the screen The stresses on the element are rr radial stress circumferential or hoop stress r shear stress in the r direction and the opposite r shear stress in r direction To derive the first equation from your image 1 Write down the forces in the radial direction r direction Consider the stresses multiplied by the area over which they act t is the thickness into the page rr on the inside face of the sector acts over an area of r d t rr rrr dr on the outside face acts over an area of rdr d t accounting for the change in rr over dr r on the bottom face acts over an area of dr t r r d on the top face acts over an area of dr t accounting for the angular change in r on both the left and right faces will have vertical components that cancel out and horizontal components that contribute to the radial force balance 2 Consider the fact that the sum of the radial forces should equal zero meaning they are in equilibrium write an equation and divide by the differential elements and t to get the forces per unit length Incoming radial force rr r d t minus the outgoing force rr rrr dr rdr d t plus the contributions from the shear stress and the radial component of hoop stress should combine to zero 3 Expand the equation cancel out the common terms and divide through by dr d r to simplify leading to the provided differential equation rrr 1r r 1r rr 0 Similarly for the second equation performing the force balance in the direction tangential to the radius and following a similar process of considering the forces and moments acting on the infinitesimal element taking changes over dr and d into account and equilibrating them will yield rr 1r 2r r 0 This process utilizes the concepts of static equilibrium under plane stress conditions infinitesimal calculus and the assumption that the element considered is small enough for the stress ... View the full answer