Two Dimensional Analysis – Plane Stress and Plane Strain

1 Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringTwo Dimensional Analysis Plane Stress and Plane StrainIn a large class of every day engineering problems certain approximations are made to simplify the structural Analysis of three Dimensional components. Recall that with the strong formulation there are 15 equations to solve in terms of 15 unknowns. One approach to reduce the computational effort in solving this system of equations is realizing that certain problems are really two Dimensional . This reduces the number of equations to types of two Dimensional idealizations are employed:1. Plane stress2. Plane strain3. AxisymmetryFor example, simplifying approximations can be made in analyzing deformations in a thin plate subjected to in- Plane forces. Consider a prismatic structural member with a very short length or thickness (h).

2 The mid Plane contains the xand ycoordinate axes and the thickness extends along the zaxis to +h/2and h/2. If the member is not loaded on surfaces perpendicular to the zaxis thenon these surfaces perpendicular the zaxis and through the thickness. Assuming the remaining Stress components are not dependent on z, then this is known as Plane yzxzz Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringHere the Stress matrix and the Strain matrix take the following formsFor Plane Strain one dimension (say along the zaxis) is exceedingly large relative to the other two dimensions. Applied forces act in the x yplane and do not vary along the zdirection. Some practical examples include dams and tunnels as well as bars that are compressed along their length. Hereon these surfaces perpendicular the zaxis. Assuming the remaining Strain components are not dependent on z, then this is known as Plane Strain .

3 20 yzxzz yxzyxyxyxyxyxyxEyxyxyxyxyxyxyxyx 000,,0,,0000,,0,,Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringNote that for Plane Strain the functional dependence of the displacements areThusThe axisymmetric formulation has the same geometric implication, , a two Dimensional problem with the additional wrinkle of posing the problem in a different coordinate system. 0,, wyxvvyxuu yxzyxyxyxyxyxyxyxyxyxyxyxyxyxyx 000,,0,,0000,,0,,Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringAxisymmetryWe have considered line elements and two Dimensional elements . We now turn our attention to axisymmetric elements . This is a special two Dimensional element utilized when there is a specific type of geometric symmetry and load symmetry present in the problem. We use cylindrical coordinates (r, , z) to describe all aspects of the problem, , displacements, strains and stresses.

4 The z-axis is the axis of of axisymmetric problems include, but are not limited to pressure vessels Cylindrical shafts Hertzian contact between spheresUse of axisymmetric elements provides computational efficiency relative to a full three Dimensional 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringQuasi-Axisymmetric NotationSome fundamental concepts are presented first. An axisymmetric element is essentially a triangular (or quadrilateral) torus. Each node traces a circular line which are depicted as dashed lines in the figure below on the left:The z-axis is the axis of symmetry. The component being modeled must have geometric as well as load symmetry with respect to this axis. These problems are best modeled in a polar coordinate system (r, , z). Corresponding displacements will be designated (u, v, w) and the two Dimensional Stress state is indicated in the figure on the 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringThe following two load applications are acceptable for an axisymmetric Analysis :The next two are not because the load is functionally dependent on.

5 The figure on the left represents the wind loads (windward and leeward) on a cooling 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringAs a previous figure indicates, we wish to use the notation, and to some extent the derivations, developed for the two Dimensional constant Strain element. That element will be spun around an axis of symmetry (the z-axis) and thereby generate an axisymmetric that context the two Dimensional axisymmetric element presents itself in an r-zplane much the same way the constant Strain element presents itself in the x-yplane. For the two Dimensional constant Strain and linear Strain element we ignored displacements in the zdirection. Here we do not. We could wave our hands over the constant Strain and linear Strain elements and mumble something regarding Plane Stress or Plane Strain and somewhat ignore displacements in the z direction.

6 We will rectify that when solid (tetrahedron) elements are developed. For axisymmetric elements the out of Plane displacements are a bit more term quasi-axisymmetric refers to fact that in truly axisymmetric problems the circumferential displacements (vor u depending on the notation used) are identically zero. Where stresses and strains are not functionally dependent on and this class of problems are labeled vSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringThe most familiar components that can be modeled in an axisymmetric fashion are thin walled pressure vessel (hot dog stresses). In fact any cylindrical pressure vessel , thin walled or otherwise, has the potential of being modeled as an axisymmetric problem. Consider the gun barrel for an Abrams M-1 tank:Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringA portion of the axisymmetric mesh used to model the tank barrel is depicted below:Note that the mesh is predominantly quadrilateral axisymmetric elements , with a triangular element furnishing a transition as the throat of the barrel thins out.

7 The point here is that there are a lot of elements in this mesh and they all trace out a torus when spun around the z-axis. If three Dimensional elements had been used to model the full barrel the problem would have been too big to transition elementSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringRecall that in Cartesian coordinates the Strain displacement relationships wereIn cylindrical coordinates the Strain displacement relationships are Strain Displacement Relationships Cylindrical Coordinatesywzvzwxwzuyvxvyuyuyzzxzyxyx wrzvzwrwzuvrrurvrvurruzzrzrrr111 Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringFor Plane Strain in the z-directionThe Stress and Strain matrices take the following formAny dependence upon z is suppressed for Plane Strain , and due to symmetry about the z-axis the strains in an axisymmetric component are independent of.

8 Thus all derivatives with respect z and vanishkeeping in mind that w = 0for Plane zrzyzxzz 000 zzrzrrrrurvrvru zrrr 0000 00000 rrrSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringFor Plane Stress in the z-directionThe Stress and Strain matrices take the following formDue to symmetry about the z-axis the strains in an axisymmetric component are independent of . Thus all derivatives with respect vanish. In addition, shear strains associated with the zero shear stresses are similarly zero0 yzxzzrzz rrzrrrErurvrvru 00000 rrr zrrr 000000 zrz Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringIn axisymmetric problems radial displacements produce circumferential strains, which in turn generate circumferential stresses on face EBDFof the differential element shown below (this is not a depiction of a finite element).

9 Line AB becomes longer as it moves from its original position, marked in red, to its new position, marked with a solid black line. This lengthening produces strains in the axis into the pagez axis out of the pageSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringThe Strain in the radial direction is defined by the change in length in the radial direction of line segment BDin the previous figure divided by the original length (dr).Looking at the change in length of line segment ABdivided by its original length we would obtain from the previous figureruudrruudrr 1 rurdrddur Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringIf we now look directly at face BDFEin the previous figure we would see the following (the original position of the face is marked with a dashed line)Focusing on line segment BE, the change in its length divided by its original length would yield the Strain in the z-direction, ,zwwdzzwwdzz 1 Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of Engineering mmjjiiwuwuwudNotation Three Node Triangular Axisymmetric ElementWe formulate the linear displacement function as followsThe nodal displacements are zaraazrwzaraazru654321,, Section 9.

10 AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringThe general displacement function can be expressed in matrix notation as 65432165432110000001aaaaaazrzrzaraazaraa wuSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringTo obtain the coefficients we substitute the coordinates of the nodes into the previous equations. This yieldsWe can solve for the first three coefficients and the last three coefficients from the following two systems of equations;mmmjjjiiimmmjjjiiizaraawzaraaw zaraawzaraauzaraauzaraau6546546543213213 21 321111aaazrzrzruuummjjiimji 654111aaazrzrzrwwwmmjjiimjiSection 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringInverting the first expression leads toThe method of cofactors is used to invert the 3 x 3matrix, ,where mjimmjjiiuuuzrzrzraaa1321111 mjimjimjimmjjiiAzrzrzr 211111mmjjiizrzrzrA1112 Section 9: AXISYMMETRIC ELEMENTSW ashkewicz College of EngineeringThis determinant isNote that Ais the area of the triangular element.