Refer to Figure 17.5: Typical Stress Distribution for a graphical representation of stresses. The membrane values of the stress components are computed from:

(17–18)

where:

= membrane value of stress component i

t = thickness of section, as shown in Figure 17.4: Coordinates of Cross Section

σi = stress component i along path from results file (`total' stress)

xs = coordinate along path, as shown in Figure 17.4: Coordinates of Cross Section

Figure 17.5:  Typical Stress Distribution

The subscript i is allowed to vary from 1 to 6, representing σ_x, σ_y, σ_z, σ_xy, σ_yz, and σ_xz, respectively. These stresses are in global Cartesian coordinates. Strictly speaking, the integrals such as the one above are not literally performed; rather it is evaluated by numerical integration:

(17–19)

where:

The integral notation will continue to be used, for ease of reading.

The “bending” values of the stress components at node N₁ are computed from:

(17–20)

where:

= bending value of stress component i at node N1

The bending values of the stress components at node N₂ are simply

(17–21)

where:

= bending value of the stress component i at node N2

The “peak” value of stress at a point is the difference between the total stress and the sum of the membrane and bending stresses. Thus, the peak stress at node N₁ is:

(17–22)

where:

= peak value of stress component i at node N1

σi1 = value of total stress component i at node N1

Similarly, for node N₂,

(17–23)

At the center point (x = 0.0)

(17–24)

where:

= peak value of stress component i at center

σic = computed (total) value of stress component i at center

The axisymmetric case is the same, in principle, as the Cartesian case, except for the fact that there is more material at a greater radius than at a smaller radius. Thus, the neutral axis is shifted radially outward a distance x_f, as shown in Figure 17.6: Axisymmetric Cross-Section. The axes shown in Figure 17.6: Axisymmetric Cross-Section are Cartesian, i.e., the logic presented here is only valid for structures axisymmetric in the global cylindrical system. As stated above, the axisymmetric case is selected if ρ ≠ 0.0. ρ is defined as the radius of curvature of the midsurface in the X-Y plane, as shown in Figure 17.7: Geometry Used for Axisymmetric Evaluations. A point on the centerplane of the torus has its curvatures defined by two radii: ρ and the radial position R_c. Both of these radii will be used in the forthcoming development. In the case of an axisymmetric straight section such as a cylinder, cone, or disk, ρ = , so that the input must be a large number (or -1).

Figure 17.6:  Axisymmetric Cross-Section

Figure 17.7:  Geometry Used for Axisymmetric Evaluations

Each of the components for the axisymmetric case needs to be treated separately. For this case, the stress components are rotated into section coordinates, so that x stresses are parallel to the path and y stresses are normal to the path.

Starting with the y direction membrane stress, the force over a small sector is:

(17–25)

where:

The area over which the force acts is:

(17–26)

where:

Ay = area of small sector

R1 = radius to node N1

R2 = radius to node N2

Thus, the average membrane stress is:

(17–27)

where:

= y membrane stress

To process the bending stresses, the distance from the center surface to the neutral surface is needed. This distance is shown in Figure 17.6: Axisymmetric Cross-Section and is:

(17–28)

The derivation of Equation 17–28 is the same as for y_f given at the end of SHELL61 - Axisymmetric-Harmonic Structural Shell. Thus, the bending moment may be given by:

(17–29)

or

(17–30)

The moment of inertia is:

(17–31)

The bending stresses are:

(17–32)

where:

Combining the above three equations,

(17–33)

or

(17–34)

where:

= y bending stress at node N1

Also,

(17–35)

or

(17–36)

where:

= y bending stress at node N2

σ_x represents the stress in the direction of the thickness. Thus, σ_x1 and σ_x2 are the negative of the pressure (if any) at the free surface at nodes N₁ and N₂, respectively. A membrane stress is computed as:

(17–37)

where:

= the x membrane stress

The treatment of the thickness-direction "bending" stresses is controlled by K_B (input as KBR on PRSECT, PLSECT, or FSSECT commands). When the thickness-direction bending stresses are to be ignored (K_B = 1), bending stresses are equated to zero:

(17–38)

(17–39)

When the bending stresses are to be included (K_B = 0), bending stresses are computed as:

(17–40)

(17–41)

where:

= x bending stress at node N1

σx1 = total x stress at node N1

= x bending stress at node N2

σx2 = total x stress at node N2

and when K_B = 2, membrane and bending stresses are computed using Equation 17–27, Equation 17–34, and Equation 17–36 substituting σ_x for σ_y.

The hoop stresses are processed next.

(17–42)

where:

= hoop membrane stress

Fh = total force over small sector

Δφ = angle over small sector in the meridional (y) direction

σh = hoop stress

Ah = area of small sector in the x-y plane

r = radius of curvature of the midsurface of the section (input as RHO)

x = coordinate thru cross-section

t = thickness of cross-section

Equation 17–42 can be reduced to:

(17–43)

Using logic analogous to that needed to derive Equation 17–34 and Equation 17–36, the hoop bending stresses are computed by:

(17–44)

and

(17–45)

where:

(17–46)

for hoop-related calculations of Equation 17–44 and Equation 17–45.

An xy membrane shear stress is computed as:

(17–47)

where:

= xy membrane shear stress

σxy = xy shear stress

Since the shear stress distribution is assumed to be parabolic and equal to zero at the ends, the xy bending shear stress is set to 0.0. The other two shear stresses (σ_xz, σ_yz) are assumed to be zero if K_B = 0 or 1. If K_B = 2, the shear membrane and bending stresses are computing using Equation 17–27, Equation 17–34, and Equation 17–36 substituting σ_xy for σ_y

All peak stresses are computed from

(17–48)

where:

At this point it is important to mention one exceptional configuration related to the y-direction membrane and bending stress calculations above. For paths defined on the centerline (X = 0), R_c = 0 and cosΦ = 0, and therefore Equation 17–27, Equation 17–28, Equation 17–34, and Equation 17–36 are undefined. Since centerline paths are also vertical (φ = 90°), it follows that R = R_c, and R_c is directly cancelled from stress Equation 17–27, Equation 17–34, and Equation 17–35. However, x_f remains undefined. Figure 17.8: Centerline Sections shows a centerline path from N₁ to N₂ in which the inside and outside wall surfaces form perpendicular intersections with the centerline.

Figure 17.8:  Centerline Sections

For this configuration it is evident that cos φ = R_c/ρ as φ approaches 90° (or as approaches N₁ - N₂). Thus for any paths very near or exactly on the centerline, Equation 17–28 is generalized to be:

(17–49)

The second option of Equation 17–49 applied to centerline paths is an accurate representation for spherical/elliptical heads and flat plates. It is incorrect for axisymmetric shapes that do not form perpendicular intersections with the centerline (e.g., conical heads). For such shapes (as shown in Figure 17.9: Non-Perpendicular Intersections) centerline paths must not be selected.

Figure 17.9:  Non-Perpendicular Intersections