# Harmonic Elements

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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

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 except that torsional effects are included. Figure 2.3 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 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.

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) 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. All functions are based on sin 2 θ and cos 2 θ.

Figure 2.7 Displacement and Force Loading Associated with MODE = 2 and ISYM = 1

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

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