2.12.Harmonic Axisymmetric Elements An axisymmetric structure can be represented by a plane (X,Y) finite-element model. The use of an axisymmetric model greatly reduces the modeling and analysis time compared to that of an equivalent 3-D model. ANSYS recommends general axisymmetric elements SOLID272 and SOLID273 for such applications because they can accept any type of load and can support nonlinear analyses. However, you can also use a special class of axisymmetric elements called harmonic elements: PLANE25, SHELL61, PLANE75, PLANE78, FLUID81, and PLANE83. The harmonic elements allow a nonaxisymmetric load and support linear analyses. Axisymmetric elements are modeled on a 360° basis, so all input and output nodal heat flows, forces, moments, fluid flows, current flows, electric charges, magnetic fluxes, and magnetic current segments must be input in this manner. Similarly, input real constants representing volumes, convection areas, thermal capacitances, heat generations, spring constants, and damping coefficients must also be input in on a 360° basis. Unless otherwise stated, the model must be defined in the Z = 0.0 plane. The global Cartesian Y-axis is assumed to be the axis of symmetry. Further, the model is developed only in the +X quadrants. Hence, the radial direction is in the +X direction. The boundary conditions are described in terms of the structural elements. The forces (FX, FY, etc.) and displacements (UX, UY, etc.) for the structural elements are input and output in the nodal coordinate system. All nodes along the y-axis centerline (at x = 0.0) should have the radial displacements (UX if not rotated) specified as zero, unless a pinhole effect is desired. At least one value of UY should be specified or constrained to prevent rigid body motions. Torsion, while axisymmetric, is available only for a few element types. If an element type allows torsion, all UZ degrees of freedom should be set to 0.0 on the centerline, and one node with a positive X coordinate must also have a specified or constrained value of UZ. Pressures and temperatures may be applied directly. Acceleration, if any, is usually input only in the axial (Y) direction. Similarly, angular velocity, if any, is usually input only about the Y axis. For more information, see Harmonic Axisymmetric Elements with Nonaxisymmetric Loads. 2.12.1. Harmonic Axisymmetric Elements with Nonaxisymmetric Loads the ANSYS program uses only the stress state of the most recent previous MODE = 0 load case. etc. a superelement generated with particular MODE and ISYM values must have the same values in the "use" pass. θ is the circumferential coordinate implied in the model. etc. The number of harmonic waves.e. Caution should be used if the harmonic elements are mixed with other. such as large deflection and/or contact analyses. forces. UY. Input values for temperature. The deflections and stresses are output at the peak value of the sinusoidal function. The harmonic elements should not be used in nonlinear analyses. The results may be scaled and summed at various circumferential (θ) locations with POST1. For these elements (PLANE25.. Each term of the above series must be defined as a separate load step. displacements. FY.).The following cases are provided to aid the user in obtaining a physical understanding of the MODE parameter and the symmetric (ISYM=1) and antisymmetric (ISYM=-1) loading conditions.An axisymmetric structure (defined with the axial direction along the global Y axis and the radial direction parallel to the global X axis) can be represented by a plane (X. neither the element matrices nor the triangularized matrix is reused in succeeding substeps if the MODE and ISYM parameters are changed. the number of harmonic waves ( ). The forces (FX. displacement.. For example. The symmetry condition is determined from the ISYM value also input on the MODE command. a load F is given by: F(θ) = A0 + A1 cos θ + B1 sin θ + A2 cos 2 θ + B2 sin 2 θ + A3 cos 3 θ + B3 sin 3 θ + . PLANE78. The element matrices for harmonic elements are dependent upon the number of harmonic waves (MODE) and the symmetry condition (ISYM). is input with the MODE command. PLANE75.) and displacements (UX. Stress (and temperature) contour displays and distorted shape displays of the combined results can also be made. For this reason. Y) finite-element model. The input value for force and heat flow should be a number equal to the peak value per unit length times the circumference. Results of the analysis are written to the results file. etc. SHELL61. and the symmetry condition (cos θ or sin θ). A load case may be defined with LCWRITE. then combine or scale the load cases as desired with LCOPER. The loading cases are described in terms of the structural elements. The ANSYS harmonic axisymmetric elements allow nonaxisymmetric loads. A term is defined by the load coefficient (A or B ). For stress stiffened (prestressed) structures. regardless of the current value of MODE. Note that = 0 represents the axisymmetric term (A0). This may be done by storing results data at the desired θ location using the ANGLE argument of the SET command. The use of an axisymmetric model greatly reduces the modeling and analysis time compared to that of an equivalent 3-D model. Repeat for each set of results. or the mode number. The load coefficient is determined from the standard ANSYS boundary condition input (i. In addition. and PLANE83) . Loading Cases . nonharmonic elements. FLUID81. the load is defined as a series of harmonic functions (Fourier series). and pressure should be the peak value. The description of the element given in Element Library and in the appropriate sections of the Theory Reference for the Mechanical APDL and Mechanical Applications should be reviewed to see which deformation shape corresponds to the symmetry conditions.) for the structural elements are input and output in the . pressures.. Case B: (MODE = 1. ISYM=1) . with the following substitutions: UY to TEMP. Figure 2. The input and output values of UX. for SHELL61) specified as zero. Similarly. Case A: (MODE = 0. FX and FY (and ROTZ and MZ for SHELL61) occur at θ = 0°. UZ. and FY to HEAT. FX. FZ and MZ are all ignored for thermal elements. is usually input only about the Y axis. ROTZ. Figure 2. positive X coordinate must have a specified or constrained value of UZ if applicable.An example of this case is the bending of a pipe. Acceleration. At least one value of UY should be specified or constrained to prevent rigid body motions. Nodes at the centerline (X = 0. if any. Pressures and temperatures are applied . The loading description may be extended to any number of modes.This is the case of axisymmetric loading except that torsional effects are included. FX. etc. represent the peak values of the displacements or forces.3 Axisymmetric Radial. torsional stress is not allowed. In all cases illustrated. All functions are based on sin θ or cos θ. respectively. The peak values of UX.4 shows the corresponding forces or displacements on a nodal circle. is usually input only in the axial (Y) direction. unless a pinhole effect is desired. Axial. one node with a nonzero. The total applied moment (M) due to a tangential input force (FZ) acting about the global axis is: where FZ is on a full 360° basis. Calculated reaction forces are also on a full 360° basis and the above expressions may be used to find the total force. whereas the peak values of UZ and FZ occur at θ = 90°. Pressures and temperatures may be applied directly.3 shows the various axisymmetric loadings. The harmonic thermal elements (PLANE75 and PLANE78) are treated the same as PLANE25 and PLANE83. The effects of UX. Also.0) should have UX and UZ (and ROTZ. Torsion and Moment Loadings The total force (F) acting in the axial direction due to an axial input force (FY) is: where FY is on a full 360° basis. Figure 2. it is assumed that the nodal coordinate system is parallel to the global Cartesian coordinate system. angular velocity. ISYM not used) . UY. if any. When Case A defines the stress state used in stress stiffened analyses.nodal coordinate system.. An applied moment (M) due to an axial input force (FY) for this case can be computed as follows: An additional applied moment (M) is generated based on the input moment (MZ): If it is desired to impose a uniform lateral displacement (or force) on the cross section of a cylindrical structure in the global X direction.5.4 Bending and Shear Loading (ISYM = 1) The total applied force in the global X direction (F) due to both an input radial force (FX) and a tangential force (FZ) is: where FX and FZ are the peak forces on a full 360° basis. σ y. whereas the peak values of σ yz and σ xz occur at 90 °. The peak values of σx. ACELY = 0.0. where Tpeak is the input element or nodal temperature. Gravity (g) acting in the global X direction should be input (ACEL) as ACELX = g.directly as their peak values at θ = 0°. . Figure 2. These net forces are independent of radius so that they may be applied at any radius (including X = 0. σz and σ xy occur at θ = 0° . Calculated reaction forces are also the peak values on a full 360° basis and the above expression may be used to find the total force. equal magnitudes of UX and UZ (or FX and FZ) may be combined as shown in Figure 2. The thermal load vector is computed from Tpeak. and ACELZ = -g.0) for the same net effect. The reference temperature for thermal strain calculations (TREF) is internally set to zero in the thermal strain calculation for the harmonic elements if MODE > 0. should be specified or constrained in some manner.6) represents a pipe bending in a direction 90° to that described in Case B.6 Bending and Shear Loading (ISYM = -1) The same description applying to Case B applies also to Case C. at least one value of UX or UZ.7. but the resulting UX and UZ will not. When FX and FZ are input in this manner. Also. Node points on the centerline (X = 0. Case C: (MODE = 1. it can be done by using constraint equations (CE) so that UX = -UZ. in general. or ROTZ. If it is desired to have the nodal circle moving in a rigid manner. Case D: (MODE = 2.The displacement and force loadings associated with this case are shown in Figure 2.0) should have UY specified as zero. Figure 2. except that the negative signs on UZ.5 Uniform Lateral Loadings When UX and UZ are input in this manner. however. FZ. a uniform load is applied about the circumference. since this element has no static shear stiffness. Further. which may be enforced with constraint equations. In practice. UX must equal -UZ at all points along the centerline. All functions are based on sin 2 θ and cos 2 θ.Figure 2. FLUID81. it seems necessary to do this only for the harmonic fluid element. the location of the peak values of various quantities are switched between the 0° and 90° locations. as well as one value of UY (not at the centerline). the nodal circle moves in an uniform manner. and the direction cosine are changed to positive signs. To prevent rigid body motions.This case (shown in Figure 2. For SHELL61. if plane sections (Y = constant) are to remain plane. be the same magnitude. ROTZ should be related to UY by means of constraint equations at the loaded nodes. ISYM = 1) . ISYM = -1) . . .7 Displacement and Force Loading Associated with MODE = 2 and ISYM = 1 Additional Cases: There is no programmed limit to the value of MODE.Figure 2. Additional cases may be defined by the user.