Internal Pressure ■

We can find the hoop stress in the cylinder by using Eq. (11-56) with Rm = <

^ (6,899)0.0) _^^xlO6 N/m2(limit) " / (0.00295)

From Eq. (11-55), we see that the meridional (longitudinal) pressure stress is half this value.

Although these stresses are small, we must combine them with stresses from load factors when sizing for tensile strength. The pressure and load factors must be time-consistent (for example, do not combine lift-off loads with venting pressures that occur later in the ascent). In the case of stability, internal pressure can strengthen a shell. We can increase the reduction factor, y, slightly to account for the stiffening effect of the internal pressure. Lateral shear will tend to lower the buckling load.

Calculating the Mass

The mass of the cylinder is the product of the density, p, and volume, 2nRtL.

m = p2nRtL = (2.8 x 103)(2)(7t)(l .0)(0.002 95)(10.0) (11-72)

Any fasteners, attachments^and access doors would increase this mass somewhat, making allowances for material lost in drilled holes and cut outs.

Summary of Monocoque Options

The driving requirements for the monocoque cylinder are bending rigidity and compressive stability, which represent actual design conditions. Please note that the calculation for first natural frequency depends on a crude assumption of equally distributed mass. In this example, we want only to illustrate methods and clarify the need for iterative design. In an actual design, we would know the mass distribution and use computerized techniques to get a more realistic weight for the structure.

If we break the cylinder into several assemblies, such as an adapter on the bottom with a spacecraft bus on top, we could analyze each section separately. For cylinder sections closer to the base, Peq loads increase. Thus, we would want to analyze different sections for varying types of construction, each with its own applied loads. In this example, we could assume that the spacecraft adapter occupies the bottom 2 m of the cylinder, resulting in a preliminary mass of 519 x 2/10 = 103.8 kg.

Option 2—Skin-Stringer

Suppose we stiffen the cylinder with 12 longitudinal members, called stringers, and 11 circumferential rings, or frames. The cylinder's circumference is 6.28 m, so the 30-deg stringer spacing results in a stringer spacing of 0.5236 m, measured along the curved surface. The frames separate the cylinder into 10 sections, or bays, each with a height of 1.0 m. Figure 11 -45 identifies the stringers by number.

Fig. 11 -45. Stringer Arrangement and Geometry.

It is reasonable to assume that the presence of stiffening stringers and rings in this design allows us to reduce the skin's thickness. A designer's initial concern with a thinner skin is buckling; the concern is real and we will indeed check for this mode of failure. In addition, thin, external surfaces with large surface areas are also susceptible to the acoustic environment. Acoustically driven loads are based on many factors, including:

• The launch vehicle's acoustic environment

• Location of the structure within the payload fairing, or shroud

• Whether acoustic blankets are used to help diminish noise within the shroud

• Type of structure (as we said, large and thin surfaces are more affected)

• Whether the structure is an external or internal payload surface

• Boundary conditions of the surface edges

• Whether the surface is flat or curved

• The first resonant frequency of the surface (depends on size, shape, thickness, material's modulus of elasticity, and edge boundary conditions).

The calculations for acoustic loads are cumbersome; see Sec. 7.7 of Sarafin [1995] for an example of one technique. We will assume a starting standard gage skin thickness of 0.127 cm is adequate against acoustic noise for our design.

First, we must choose whether to design the skin to help sustain load or whether to allow it to buckle, forcing the stiffeners to take on more of the burden. In this example, we will design the skin not to buckle, as is usually done when performing preliminary sizing analysis. Chapter Cll of Bruhn [1973] provides details on how to analyze buckled skin.


Again, let's first size for stiffness. We already know from calculations for Option 1 that we need a skin thickness of 0.045 cm to meet the axial frequency requirement of 25 Hz. Therefore, the 0.127-cm-thick skin alone will be adequate for axial rigidity. In the bending case, the required area moment of inertia, /, of the cylinder's cross-section is 8.98 x 105 cm4. The skin will satisfy part of this:

Iskin = nlPt = tc(1.0)3(0.001 27) = 4.00 x 105 cm4 (i 1.73)

Therefore, the contribution to / from the 12 stringers must equal the remainder:

1st, = 8.98 x 105-4.00x105= 4.98 x 105 cm4 (i 1.74)

We can calculate the / of the 12 stringers in the cylinder using the parallel axis theorem, Ixx = h (Icm + Ad2). We can ignore the or I about each stringer's center of mass, because it will be very small compared to its Ad2 term. Therefore, the I of the stringer system is a function of stringer cross-sectional area, A, and d, the distance from the cylinder's neutral axis (Table 11-60).

Therefore, =4.98 x 105 cm4 = A x 60,000 cm2. This results in a required cross-sectional area of each stringer of 832 cm2. Hie cylinder area combines the skin and twelve stringers for a total area of 180.00 cm2. Note that both the skin and stringers must contribute to overall I to meet this requirement When we allow skin to buckle,

TABLE 11-60. Calculations for Moment of Inertia Based on Stringer Area.
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