Pressurization Loads: Method

Pressurization Loads: Method, Conventions and Limits

What the tool does

A pressurized cabin is a thin-walled pressure vessel that is also the primary structure of the airplane. The tool follows the cabin pressure from the operating schedule to the loads it puts into the structure: the differential pressure between the cabin and the ambient air, the design pressures the airworthiness rule prescribes from the relief-valve setting, the hoop and longitudinal running loads in the skin, the share of the pressure that the frames pick up through the bending of the skin between them, the tension in the floor of a double-bubble section, the membrane forces in a spherical or ellipsoidal bulkhead or the bending of a flat one on beams, the loads a door cutout throws onto its edge frames and sills, margins of safety against entered allowables, and the pressure cycle per flight that the fatigue and damage-tolerance tools need. The closed forms are those of NACA Technical Note 2612, Flügge's 1952 survey of the stress problems of pressurized cabins, which remains the standard reference, and of the membrane theory of shells.

The differential pressure and the standard atmosphere

The cabin is held at the ambient pressure of its cabin altitude while the airplane flies at its flight altitude; the differential is the difference of the two ambient pressures. The tool evaluates both with the 1976 U.S. Standard Atmosphere (NOAA, NASA and USAF): a sea-level pressure of 101,325 Pa and temperature of 288.15 K, a lapse rate of −6.5 K/km to the tropopause at 11 km geopotential, an isothermal stratosphere at 216.65 K to 20 km and the warmer layers above, with the geometric altitude converted to geopotential altitude by H = r₀ Z / (r₀ + Z), r₀ = 6,356,766 m. In each layer the pressure follows the hydrostatic law, p = p_b (T_b / T)^{g₀ / (R L)} in a layer with a lapse rate and p = p_b exp(−g₀ (H − H_b) / (R T_b)) in an isothermal one. The function reproduces the printed Table IV rows to the five digits of the table (752.71 mb at 8,000 ft, 179.43 mb at 41,000 ft). The alternative is to enter the operating differential directly, as a specification often does. An entered external suction (the local aerodynamic pressure on the fuselage in flight, which § 25.365(b) says must be accounted for) adds to every differential.

14 CFR 25.841(a) limits the cabin pressure altitude to 8,000 ft at the maximum operating altitude under normal conditions, and 25.841(b)(1) requires two pressure relief valves that limit the positive differential to a predetermined value: that value, the maximum relief-valve setting p_rv, is the pressure the structure is designed for. It is set a little above the schedule's differential so that the valves do not open in normal operation; the tool flags a setting below the schedule.

The design pressures of 14 CFR 25.365

Paragraph (a) requires the structure to withstand the flight loads combined with pressure differential loads from zero up to the maximum relief-valve setting; paragraph (d) requires it to withstand the pressure differential loads corresponding to the maximum relief-valve setting multiplied by 1.33 for airplanes approved for operation to 45,000 ft, or by 1.67 for operation above 45,000 ft, omitting other loads. Neither paragraph names the level of the load. By § 25.301(a) prescribed loads are limit loads unless otherwise provided, and § 25.303 applies a factor of safety of 1.5 to them, so the tool reads the (d) pressure as a limit load: the limit design pressure alone is 1.33 p_rv and the ultimate design pressure alone is 1.5 × 1.33 p_rv ≈ 2.0 p_rv (2.5 p_rv above 45,000 ft). This is the historical "twice the relief setting" of the pressure-cabin requirements, and it governs over the pressure part of the combined case, 1.5 p_rv. The factor of safety is an input: an analyst who treats 1.33 p_rv as an ultimate pressure enters 1.0. The page prints every level in a table and names the governing limit pressure (the yield check, § 25.305(a): no detrimental permanent deformation at limit load) and the governing ultimate pressure (the strength check, § 25.305(b)). The flight loads of the combined case, the landing case of paragraph (c) and the decompression loads of paragraphs (e) to (g) are outside the tool; it does print the opening size H_o = P A_s of paragraph (e)(2), with P = A_s/6240 + 0.024 and A_s the maximum cross-sectional area in square feet, which sizes the sudden release of pressure the compartments must survive.

Skin loads

A thin circular shell of radius r and thickness t under the differential p carries it by the boiler formulas (TN 2612 eq. 1): the hoop running load N_θ = p r and, because the pressure on the bulkheads, p π r², is carried by the circumference 2π r, the longitudinal running load N_x = p r/2. The stresses are the loads divided by the skin thickness; stringers share the longitudinal load, so the longitudinal skin stress of a stiffened shell is lower than p r/(2t) and the hoop stress a little lower than p r/t, which the tool neglects on the safe side for the skin. The free radial expansion of a closed shell is w₀ = (1 − ν/2) p r²/(E t), the ν/2 term being the Poisson contraction from the longitudinal stress. For a double bubble each lobe carries p times its own radius; the margin uses the larger.

Frames: the interaction between shell and rings

The frames are stiffer in hoop than the skin and expand less, so the skin between two frames bulges outward relative to them ("pillowing") and hands part of its pressure load to each frame through the bending of the skin. TN 2612 solves this exactly for an unstiffened or stringer-stiffened skin on equally spaced rings. The additional deflection w of the skin over a bay of length l obeys the equation of a beam on an elastic foundation,

E I w'''' + (E t / r²) w = 0,

where I is the longitudinal bending inertia per unit circumference (t³/(12(1 − ν²)) for the skin alone; the stringer's inertia with its attached skin over the stringer pitch when stringers are given) and E t / r² is the hoop stiffness of the skin. Measured from mid-bay, the solution symmetric about the bay centre is w = C₁ cosh βx cos βx + C₂ sinh βx sin βx with β = [t / (4 r² I)]^1/4, which for the skin alone is the familiar [3(1 − ν²) / (r² t²)]^1/4. The slope vanishes at the frame by symmetry across it, and the frame's own radial expansion under the line load 2T it receives from the two adjacent bays, 2T r²/(E A_f), must equal the skin's deflection there. Eliminating the constants gives the shear of one bay into the frame, TN 2612 eq. (14):

T = (p/2) [(1 − ν²)/t − ν (1 − 2ν)/(2 t_x)] / [1/A_f + (βl/2) (cosh(βl/2) sinh(βl/2) + cos(βl/2) sin(βl/2)) / (l t (cosh²(βl/2) − cos²(βl/2)))],

with t_x the longitudinal area per unit circumference (t plus the stringer area over the pitch). The frame hoop force is N_R = 2 T r and its stress N_R/A_f; the skin hoop force at the frame is p r + E t (w_R − w₀)/r, much below p r because the frame holds the skin in; the longitudinal bending moment at the frame is E I w'' there, which for a long bay is T/(2β) and gives the surface bending stress 6M/t² of an unstiffened skin, the pillowing stress that is a classic fatigue driver at frame stations; and the hoop force overshoots p r by e^−π times the hoop drop at the frame, about 3 %, where the decaying wave reverses at β(l/2 − x) = π. The tool prints these, draws the hoop force, deflection and moment along the bay, and checks the solution against the closed form of eq. (14) in its test script.

The parameter βl/2 tells which simple picture applies (TN 2612 fig. 10). When it is large (frames far apart, light stringers; 7.4 in the default case), each frame acts alone on a long shell: the frame load tends to the rigid-ring bound 2 p r (1 − ν/2)/β and the mid-bay skin carries the full p r. When it is small (frames close together, heavy stringers), skin and frames strain alike and the frame carries the smeared share p r l (1 − ν/2) A_f/(A_f + l t_x). The naive estimate that the frame carries the pressure over its pitch, p r l, is an upper bound that the solution never approaches for an aircraft fuselage; the page prints all three beside the exact value. Flügge's remark on the floating skin follows from eq. (14): a flexible link between the frame and the skin adds to the 1/A_f term and lowers T, down to zero for a frame that merely touches the skin. The model's line load at the frame makes the skin bending stress there an upper bound; a frame flange of finite width spreads it. The structure of NACA TN 993 (a Lockheed test section: r 65 in, skin 0.032 in, rings of 0.238 in² at 18.4 in, 72 stringers) gives a frame stress of 5,400 psi at 5 lb/in² by this model with the stringers' inertia and area, against 5,490 psi by that report's energy analysis and 4,400 psi measured; the skin alone, without the stringers' bending stiffness, would give 2,000 psi, which shows that the stringer inertia matters whenever it is known.

The floor tie of a double-bubble section

A section made of two circular arcs of radii r₁ and r₂ meeting at the floor line (TN 2612 figs 5 and 6) carries the pressure in each lobe as a hoop tension p r; at the joint the two tensions pull along their tangents, their vertical components cancel because both arcs meet the floor at the same half-width, and their horizontal components must be balanced by the floor acting as a tie: T = p (r₁ cos θ₁ + r₂ cos θ₂) per unit length of fuselage, θ the angle of each joint radius from the vertical through its own centre. Since r cos θ is the vertical distance from each centre to the floor, this is simply T = p × (vertical distance between the two centres): a full circle, whose lobes share a centre, needs no tie, and the tie is in tension when the lower-lobe centre lies below the upper one. The tool takes the upper radius, the floor's distance below the upper centre and the depth of the lower lobe below the floor, from which the lower radius r₂ = (w² + h_L²)/(2 h_L) and its centre follow (w the floor half-width). Each floor beam, one per frame pitch, carries T × pitch in tension on top of the cabin floor loads and the decompression differential of § 25.365(e). Flügge notes that for the same inside space any such section weighs the same as the circle to first order, provided the tie is in tension.

Bulkheads

A shell of revolution under pressure carries it by the membrane forces N_φ = p r₂/2 along the meridian and N_θ = p r₂ (1 − r₂/(2 r₁)) around it (TN 2612 eqs 18a and 18b), r₁ the radius of curvature of the meridian and r₂ the length of the normal to the axis. A spherical dome of radius R_b has r₁ = r₂ = R_b and a uniform N = p R_b/2; where it meets the cylinder at the angle φ_e (sin φ_e = r/R_b) its axial component N sin φ_e = p r/2 is exactly the cylinder's longitudinal load, while its radial component N cos φ_e must be carried by the junction ring in compression; a hemisphere has none, and Flügge's advice is that the meridian of any bulkhead should end tangent to the generators. An ellipsoidal dome with the semi-axes a = r across and b along the axis has r₂ = a²/√(a² sin²φ + b² cos²φ) and r₁ = r₂³ b²/a⁴; at the crown both forces are p a²/(2b), at the equator N_φ = p a/2 and N_θ = p a (1 − a²/(2b²)), which vanishes for b = 0.707 a and is compressive for a flatter dome (−p a for the 2:1 head), a zone that needs a stiffening ring and a buckling check; the page draws both forces along the meridian and flags the compressive region. A flat bulkhead is a web on parallel beams: the web strip between beams at the spacing b_b is a long plate clamped on the beams with M = p b_b²/12 per unit width and σ = p b_b²/(2 t_b²), and the longest beam spans the diameter under w = p b_b with M = w (2r)²/8, delivering w r into the ring at each end; the tool reports the beam stress when a section modulus is entered. Flügge's judgement stands: a flat bulkhead results in a heavy construction and is to be avoided where possible. The discontinuity stresses at the junction of the dome and the cylinder are the subject of the Membrane Shell Stresses tool.

Door cutouts

Across a door opening the skin carries nothing, so the hoop load it would have carried, p r w_d, and the longitudinal load, p r h_d/2, must go round it, and the pressure on the door itself, p w_d h_d, goes to the stops along its edges. The tool gives the statics: a stop line load of p w_d/2 per unit height on each vertical edge frame, half the intercepted hoop load to each edge frame on top of its own share from the frame model, and a quarter of p r h_d to the sill and to the header. TN 2612 shows that the sill and header cannot in fact carry the intercepted hoop load as beams (they would deflect and shed it) and that the shell itself takes it over a zone either side of the opening, so the edge-frame figure is an upper bound and the real distribution needs the shell analysis of that report or a finite-element model; the frame bending from the stop loads is not computed.

Margins and fatigue

Margins are allowable over applied minus one: the skin hoop stress at the ultimate pressure against F_tu (ULT) and at the limit pressure times the yield factor FS_yld (default 1.15, LIM) against F_ty, the frame, bulkhead, floor-beam and edge-frame stresses at the ultimate pressure against F_tu. One material is assumed for all of them; take the allowables from the MIL-HDBK-5J tool for the alloy and form. Every flight loads the cabin from zero to the operating differential once, a stress cycle at R = 0 whose count is the number of design flights; the page gives the operating hoop, longitudinal, frame and frame-station bending stresses to carry into the Spectrum Fatigue Life tool, with the joint's stress concentration, and into the Crack Growth tool for the damage-tolerance evaluation of § 25.571.

Conventions, units and limits

Pressure differentials are positive when the cabin pressure exceeds the ambient; tension is positive; r is the skin mid-surface radius; θ₁, θ₂ are measured from the vertical through each lobe's centre. Altitudes, lengths, pressures, moduli, areas and allowables pass through the unit dictionary (ft, in, lb/in², Msi, in², ksi or m, mm, kPa, GPa, mm², MPa); every computed quantity is stored in SI and shown in the chosen system. Not covered: the flight loads combined with pressure (fuselage bending and shear), landing pressurized, decompression loads on floors and partitions, the junction discontinuity stresses, window-belt and door-frame bending, stringer stresses and the quilting of the skin between stringers (NACA TN 993), thick walls (r/t below 20 is refused), and non-circular lobes.