13.61. SHELL61 - Axisymmetric-Harmonic Structural Shell

Matrix or VectorShape 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 TypeDistribution
Element TemperatureLinear through thickness and along length, harmonic around circumference
Nodal TemperatureConstant through thickness, linear along length, harmonic around circumference
PressureLinear along length, harmonic around circumference

Reference: Zienkiewicz([40])

13.61.1. Other Applicable Sections

Structures discusses fundamentals of linear elements. PLANE25 - Axisymmetric-Harmonic 4-Node Structural Solid has a discussion on temperature, applicable to this element.

13.61.2. Assumptions and Restrictions

The material properties are assumed to be constant around the entire circumference, regardless of temperature dependent material properties or loading.

13.61.3. Stress, Force, and Moment Calculations

Element output comes in two forms:

  1. 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)

    Figure 13.13: Stress Locations

    Stress Locations

    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)

  2. 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.

Figure 13.14: Element Orientations

Element Orientations

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)