## 2.13. Axisymmetric Elements with Nonaxisymmetric Loads

An axisymmetric structure (defined with the axial direction along the global Y axis and the radial direction parallel to the global X axis) may 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. A special class of ANSYS axisymmetric elements (called harmonic elements) allows a nonaxisymmetric load. For these elements (PLANE25, SHELL61, PLANE75, PLANE78, FLUID81, and PLANE83) , the load is defined as a series of harmonic functions (Fourier series). For example, 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 + ...

Each term of the above series must be defined as a separate load step. A term is defined by the load coefficient (A  or B ), the number of harmonic waves (), and the symmetry condition (cos  θ or sin  θ). The number of harmonic waves, or the mode number, is input with the MODE command. Note that = 0 represents the axisymmetric term (A0).  θ is the circumferential coordinate implied in the model. The load coefficient is determined from the standard ANSYS boundary condition input (i.e., displacements, forces, pressures, etc.). Input values for temperature, displacement, and pressure should be the peak value. The input value for force and heat flow should be a number equal to the peak value per unit length times the circumference. The symmetry condition is determined from the ISYM value also input on the MODE command. The description of the element given in "Element Library" and in the appropriate sections of the Theory Reference for ANSYS and ANSYS Workbench should be reviewed to see which deformation shape corresponds to the symmetry conditions.

Results of the analysis are written to the results file. The deflections and stresses are output at the peak value of the sinusoidal function. The results may be scaled and summed at various circumferential (θ) locations with POST1. This may be done by storing results data at the desired θ location using the ANGLE argument of the SET command. A load case may be defined with LCWRITE. Repeat for each set of results, then combine or scale the load cases as desired with LCOPER. Stress (and temperature) contour displays and distorted shape displays of the combined results can also be made.

Caution should be used if the harmonic elements are mixed with other, nonharmonic elements. The harmonic elements should not be used in nonlinear analyses, such as large deflection and/or contact analyses.

The element matrices for harmonic elements are dependent upon the number of harmonic waves (MODE) and the symmetry condition (ISYM). For this reason, neither the element matrices nor the triangularized matrix is reused in succeeding substeps if the MODE and ISYM parameters are changed. In addition, a superelement generated with particular MODE and ISYM values must have the same values in the "use" pass.

For stress stiffened (prestressed) structures, the ANSYS program uses only the stress state of the most recent previous MODE = 0 load case, regardless of the current value of MODE.

Loading Cases - 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. The loading cases 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. In all cases illustrated, it is assumed that the nodal coordinate system is parallel to the global Cartesian coordinate system. The loading description may be extended to any number of modes. The harmonic thermal elements (PLANE75 and PLANE78) are treated the same as PLANE25 and PLANE83, respectively, with the following substitutions: UY to TEMP, and FY to HEAT. The effects of UX, UZ, ROTZ, FX, FZ and MZ are all ignored for thermal elements.

Case A: (MODE = 0, ISYM not used) - This is the case of axisymmetric loading (similar to the axisymmetric option of PLANE42, etc.) except that torsional effects are included. Figure 2.3: "Axisymmetric Radial, Axial, Torsion and Moment Loadings" shows the various axisymmetric loadings. 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.

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.

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. Nodes at the centerline (X = 0.0) should have UX and UZ (and ROTZ, for SHELL61) 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. Also, one node with a nonzero, positive X coordinate must have a specified or constrained value of UZ if applicable. When Case A defines the stress state used in stress stiffened analyses, torsional stress is not allowed.

Case B: (MODE = 1, ISYM=1) - An example of this case is the bending of a pipe. Figure 2.4: "Bending and Shear Loading (ISYM = 1)" shows the corresponding forces or displacements on a nodal circle. All functions are based on sin θ or cos θ. The input and output values of UX, FX, etc., represent the peak values of the displacements or forces. The peak values of UX, UY, FX and FY (and ROTZ and MZ for SHELL61) occur at  θ = 0°, whereas the peak values of UZ and FZ occur at  θ = 90°. Pressures and temperatures are applied directly as their peak values at θ = 0°. The thermal load vector is computed from Tpeak, where Tpeak is the input element or nodal temperature. 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. Gravity (g) acting in the global X direction should be input [ACEL] as ACELX = g, ACELY = 0.0, and ACELZ = -g. The peak values of  σx, σ y,  σz and σ xy occur at θ = 0°  , whereas the peak values of σ yz and σ xz occur at 90 °.

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. 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. These net forces are independent of radius so that they may be applied at any radius (including X = 0.0) for the same net effect.

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, equal magnitudes of UX and UZ (or FX and FZ) may be combined as shown in Figure 2.5: "Uniform Lateral Loadings".

When UX and UZ are input in this manner, the nodal circle moves in an uniform manner. When FX and FZ are input in this manner, a uniform load is applied about the circumference, but the resulting UX and UZ will not, in general, be the same magnitude. If it is desired to have the nodal circle moving in a rigid manner, it can be done by using constraint equations [CE] so that UX = -UZ.

Node points on the centerline (X = 0.0) should have UY specified as zero. Further, UX must equal -UZ at all points along the centerline, which may be enforced with constraint equations. In practice, however, it seems necessary to do this only for the harmonic fluid element, FLUID81, since this element has no static shear stiffness. To prevent rigid body motions, at least one value of UX or UZ, as well as one value of UY (not at the centerline), or ROTZ, should be specified or constrained in some manner. For SHELL61, if plane sections (Y = constant) are to remain plane, ROTZ should be related to UY by means of constraint equations at the loaded nodes.

Case C: (MODE = 1, ISYM = -1) - This case (shown in Figure 2.6: "Bending and Shear Loading (ISYM = -1)") represents a pipe bending in a direction 90° to that described in Case B.

The same description applying to Case B applies also to Case C, except that the negative signs on UZ, FZ, and the direction cosine are changed to positive signs. Also, the location of the peak values of various quantities are switched between the 0° and 90° locations.

Case D: (MODE = 2, ISYM = 1) - The displacement and force loadings associated with this case are shown in Figure 2.7: "Displacement and Force Loading Associated with MODE = 2 and ISYM = 1". All functions are based on sin 2 θ and cos 2 θ.

Additional Cases: There is no programmed limit to the value of MODE. Additional cases may be defined by the user.