ElasticityPolars_03_AxiSymmetric

ElasticityPolars_03_AxiSymmetric - Section 4.3 4.3 Plane...

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Section 4.3 Solid Mechanics Part II Kelly 66 4.3 Plane Axisymmetric Problems In this section are considered plane axisymmetric problems . These are problems in which both the geometry and loading are axisymmetric. 4.3.1 Plane Axisymmetric Problems Some three dimensional (not necessarily plane) examples of axisymmetric problems would be the thick-walled (hollow) cylinder under internal pressure, a disk rotating about its axis 1 , and the two examples shown in Fig. 4.3.1; the first is a complex component loaded in a complex way, but exhibits axisymmetry in both geometry and loading; the second is a sphere loaded by concentrated forces along a diameter. Figure 4.3.1: axisymmetric problems A two-dimensional (plane) example would be one plane of the thick-walled cylinder under internal pressure, illustrated in Fig. 4.3.2 2 . Figure 4.3.2: a cross section of an internally pressurised cylinder It should be noted that many problems involve axisymmetric geometries but non- axisymmetric loadings, and vice versa . These problems are not axisymmetric. An example is shown in Fig. 4.3.3 (the problem involves a plane axisymmetric geometry). 1 the rotation induces a stress in the disk 2 the rest of the cylinder is coming out of, and into, the page
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Section 4.3 Solid Mechanics Part II Kelly 67 Figure 4.3.3: An axially symmetric geometry but with a non-axisymmetric loading The important characteristic of these axisymmetric problems is that all quantities, be they stress, displacement, strain, or anything else associated with the problem, must be independent of the circumferential variable θ . As a consequence, any term in the differential equations of §4.2 involving the derivatives 2 2 / , / , etc. can be immediately set to zero. 4.3.2 Governing Equations for Plane Axisymmetric Problems The two-dimensional strain-displacement relations are given by Eqns. 4.2.4 and these simplify in the axisymmetric case to = = = r u r u r u r u r r r rr θθ ε 2 1 (4.3.1) Here, it will be assumed that the displacement 0 = u . Cases where 0 u but where the stresses and strains are still independent of are termed quasi-axisymmetric problems ; these will be examined in a later section. Then 4.3.1 reduces to 0 , , = = = r r r rr r u r u (4.3.2) It follows from Hooke’s law that 0 = σ r . The non-zero stresses are illustrated in Fig. 4.3.4. axisymmetric plane representative of feature
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Section 4.3 Solid Mechanics Part II Kelly 68 Figure 4.3.4: stress components in plane axisymmetric problems 4.3.3 Plane Stress and Plane Strain Two cases arise with plane axisymmetric problems: in the plane stress problem, the feature is very thin and unloaded on its larger free-surfaces, for example a thin disk under external pressure, as shown in Fig. 4.3.5. Only two stress components remain, and Hooke’s law 4.2.5a reads [] rr rr rr E E νσ σ ε θθ = = 1 1 or rr rr rr E E νε ν + = + = 2 2 1 1 (4.3.3) with () 0 , = = + = θ z zr rr zz E and 0 = zz .
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.

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ElasticityPolars_03_AxiSymmetric - Section 4.3 4.3 Plane...

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