4.1 Cylindrical and Polar Coordinates

Transcription

1 4.1 Cylindical and Pola Coodinates Geometical Axisymmety A lage numbe of pactical engineeing poblems involve geometical featues which have a natual axis of symmety, such as the solid cylinde, shown in Fig The axis of symmety is an axis of evolution; the featue which possesses axisymmety (axial symmety) can be geneated by evolving a suface (o line) about this axis. axis of symmety Figue 4.1.1: a cylinde ceate cylinde by evolving a suface about the axis of symmety Some othe axisymmetic geometies ae illustated Fig ; a fustum, a disk on a shaft and a sphee. Figue 4.1.2: axisymmetic geometies Some featues ae not only axisymmetic they can be epesented by a plane, which is simila to othe planes ight though the axis of symmety. The hollow cylinde shown in Fig is an example of this plane axisymmety. 55

2 axisymmetic plane epesentative of featue Figue 4.1.3: a plane axisymmetic geometies Axially Non-Symmetic Geometies Axially non-symmetic geometies ae ones which have a natual axis associated with them, but which ae not completely symmetic. Some examples of this type of featue, the cuved beam and the half-space, ae shown in Fig ; the half-space extends to infinity in the axial diection and in the adial diection below the suface it can be thought of as a solid half-cylinde of infinite adius. One can also have plane axially nonsymmetic featues; in fact, both of these ae examples of such featues; a slice though the objects pependicula to the axis of symmety will be epesentative of the whole object. Figue 4.1.4: a plane axisymmetic geometies Cylindical and Pola Coodinates The above featues ae best descibed using cylindical coodinates, and the plane vesions can be descibed using pola coodinates. These coodinates systems ae descibed next. Stesses and Stains in Cylindical Coodinates Using cylindical coodinates, any point on a featue will have specific (,, ) coodinates, Fig : 56

3 the adial diection ( out fom the axis) the cicumfeential o tangential diection ( aound the axis counteclockwise when viewed fom the positive side of the = 0 plane) the axial diection ( along the axis) = 0 plane Figue 4.1.5: cylindical coodinates The displacement of a mateial point can be descibed by the thee components in the adial, tangential and axial diections. These ae often denoted by u u, v and w u u espectively; they ae shown in Fig Note that the displacement v is positive in the positive diection, i.e. the diection of inceasing. w v u Figue 4.1.6: displacements in cylindical coodinates The stesses acting on a small element of mateial in the cylindical coodinate system ae as shown in Fig (the nomal stesses on the left, the shea stesses on the ight). 57

4 σ σ σ σ σ σ σ σ σ Figue 4.1.7: stesses in cylindical coodinates The nomal stains, and ae a measue of the elongation/shotening of mateial, pe unit length, in the adial, tangential and axial diections espectively; the shea stains, and epesent (half) the change in the ight angles between line elements along the coodinate diections. The physical meaning of these stains is illustated in Fig stain at point o o C A B = unit elongation of oa = unit elongation of ob = unit elongation of oc = ½ change in angle AoB = ½ change in angle BoC = ½ change in angle AoC Figue 4.1.8: stains in cylindical coodinates Plane Poblems and Pola Coodinates The stesses in any paticula plane of an axisymmetic body can be descibed using the, shown in Fig two-dimensional pola coodinates ( ) 58

5 Figue 4.1.9: pola coodinates Thee ae thee stess components acting in the plane = 0 : the adial stess σ, the cicumfeential (tangential) stess σ and the shea stess σ, as shown in Fig Note the diection of the (positive) shea stess it is conventional to take the axis out of the page and so the diection is counteclockwise. The thee stess components which do not act in this plane, but which act on this plane ( σ, σ and σ ), may o may not be eo, depending on the paticula poblem (see late). σ σ σ σ σ σ Figue : stesses in pola coodinates 59