| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness Matrix; and Thermal and Pressure Load Vectors | Equation 11–39, Equation 11–40, and Equation 11–41. If extra shape functions are not included (KEYOPT(3) = 1): Equation 11–36, Equation 11–37, and Equation 11–38 | 3 along length |
| Mass and Stress Stiffness Matrices | Equation 11–33, Equation 11–34, and Equation 11–35 | Same as stiffness matrix |
| Load Type | Distribution |
|---|---|
| Element Temperature | Linear through thickness and along length, harmonic around circumference |
| Nodal Temperature | Constant through thickness, linear along length, harmonic around circumference |
| Pressure | Linear along length, harmonic around circumference |
Reference: Zienkiewicz([40])
Structures discusses fundamentals of linear elements. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid has a discussion on temperature, applicable to this element.
The material properties are assumed to be constant around the entire circumference, regardless of temperature dependent material properties or loading.
Element output comes in two forms:
Stresses as well as forces and moments per unit length: This printout is controlled by the KEYOPT(6). The thru-the-thickness stress locations are shown in Figure 13.13: Stress Locations. The stresses are computed using standard procedures as given in Structural Strain and Stress Evaluations. The stresses may then be integrated thru the thickness to give forces per unit length and moments per unit length at requested points along the length:
(13–102)
(13–103)
(13–104)
(13–105)
(13–106)
(13–107)
where:
Tx, Tz, Txz, Mx, Mz, Mxz = resultant forces and moments (output as TX, TZ, TXZ, MX, MZ, MXZ, respectively) t = thickness (input as TK(I), TK(J) on R command) σx, σy, σz, σxz = stresses (output as SX, SY, SZ, and SXZ, respectively) 
Forces and moments on a circumference basis: This printout is controlled by KEYOPT(4). The values are computed using:
(13–108)
where:
[TR] = local to global transformation matrix [Ke] = element stiffness matrix {ue} = nodal displacements 
Another difference between the two types of output are the nomenclature
conventions. Since the first group of output uses a shell nomenclature convention
and the second group of output uses a nodal nomenclature convention, Mz
and
represent moments in different
directions.
The rest of this subsection will describe some of the expected relationships between these two methods of output at the ends of the element. This is done to give a better understanding of the terms, and possibly detect poor internal consistency, suggesting that a finer mesh is in order. It is advised to concentrate on the primary load carrying mechanisms. In order to relate these two types of output in the printout, they have to be requested with both KEYOPT(6) > 1 and KEYOPT(4) = 1. Further, care must be taken to ensure that the same end of the element is being considered.
The axial reaction force based on the stress over an angle Δβ is:
 | (13–109) |
or
 | (13–110) |
where:
| Rc = radius at midplane |
| t = thickness |
The reaction moment based on the stress over an angle Δβ is:
 | (13–111) |
or
 | (13–112) |
Since SHELL61 computes stiffness matrices and load vectors using the
entire circumference for axisymmetric structures, Δβ = 2π.
Using this fact, the definition of
,
and Equation 13–102 and Equation 13–105, Equation 13–110 and Equation 13–112
become:
 | (13–113) |
 | (13–114) |
As the definition of φ is critical for these equations, Figure 13.14: Element Orientations is provided to show φ in all four quadrants.
In a uniform stress (σx) environment, a reaction moment will be generated to account for the greater material on the outside side. This is equivalent to moving the reaction point outward a distance yf. yf is computed by:
 | (13–115) |
Using Equation 13–113 and Equation 13–114 and setting Mx to zero gives:
 | (13–116) |