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Loads and Dynamics

10 tools in the Structures area.

V-n Diagram and Load Factors

Maneuver and gust envelope of an airplane from the airworthiness rules: the limit maneuvering load factors of 14 CFR 23.337 (normal, utility, acrobatic and commuter categories, 2016 edition, Amendment 23-62) or 25.337 (transport), the design speeds V_A, V_B, V_C and V_D with the minima of 23.335 printed from the rule text, the stall curves from C_Lmax or the stall speeds, the gust load factors of 23.341 with the mass ratio and the alleviation factor K_g of Pratt and Walker (NACA Report 1206) for the 50 / 25 ft/s gusts at V_C and V_D and the 66 ft/s rough-air gust at V_B, reduced with altitude as the rule allows, air density by the 1976 U.S. Standard Atmosphere, the combined envelope with every corner tabulated at limit and ultimate level, and the V-n diagram drawn. The default is a Part 23 normal-category airplane of 2,400 lb and 160 ft^2 at sea level, for which the page gives n = +3.8 / -1.52 by maneuver and -1.66 by the V_C gust.

1 worked example reproduced, listed on the tool page.

Random Vibration Response

Response of a single-degree-of-freedom component to a base random vibration given as an acceleration power spectral density: the input PSD from breakpoints (frequency and g²/Hz, or a dB/octave slope from the previous point) pasted, typed in a table or chosen from the standard profiles (GSFC-STD-7000 GEVS qualification, acceptance and workmanship levels, NASA-HDBK-7005 USAF workmanship, MIL-STD-810H Method 514.8 minimum integrity, highway truck and jet aircraft cargo exposures, NAVMAT P-9492) with the exact log-log area of every segment and the overall g rms; the response of the component (natural frequency and Q or damping ratio) by Miles' equation and by integration of the transmissibility over the actual PSD, with the response PSD, the spectral moments, the 1σ, 2σ and 3σ levels, the relative displacement, the equivalent static load on a mass, the rates of zero crossings and peaks, the Rayleigh peak distribution, the expected largest peak and the time to a chosen exceedance probability, a stress check with margins of safety (F_tu, F_cy at ultimate, F_ty, F_cy at limit) from a stress per g, and a table comparing Miles' estimate with the exact integral. The default case is the GEVS qualification level (14.1 g rms) on a component at 100 Hz with Q = 10: 15.74 g rms response by integration against 15.85 g by Miles' equation.

1 worked example reproduced, listed on the tool page.

Shock Response Spectrum

Shock response spectrum (SRS) of a transient base acceleration: a classical pulse (half-sine, haversine, terminal-peak or initial-peak sawtooth, trapezoidal, rectangular, decaying sinusoid) from its amplitude and duration, or a pasted two-column time history with the sample rate detected, run through Smallwood's ramp-invariant recursive filters for the absolute acceleration and the relative displacement of a single-degree-of-freedom oscillator at Q or ζ over a logarithmic frequency grid. Reports the maximax, positive and negative primary and residual acceleration spectra, the pseudo-velocity and relative-displacement spectra, the velocity change of the pulse, the peak of the spectrum and its frequency, the high-frequency convergence to the peak input, the MIL-STD-810H crossover frequency and effective duration, the undamped closed forms of the classical pulses beside the digital result, a comparison with a pasted specification SRS or with the MIL-STD-810H Method 516.8 functional and crash-hazard test spectra against the standard's tolerance, and margins of safety (F_tu, F_cy at ultimate, F_ty, F_cy at limit) for a component of known natural frequency from a stress per g. The default case is Irvine's 50 g, 11 ms half-sine at Q = 10, which gives 55, 82 and 70 g at 30, 80 and 140 Hz.

1 worked example reproduced, listed on the tool page.

Wing Span Loads

Spanwise lift distribution of a wing by Schrenk's approximation (NACA TM 948, 1940) and the shear, bending moment and torque it produces along the span with inertia relief: the planform chord as a trapezoid or a typed chord table, the elliptical chord of equal area, the additional lift as the mean of the two (c c_l = ½ (c + c_ell) C_L), the basic lift of a linear twist as half the chord-proportional distribution about the chord-weighted mean twist, Schrenk's lift-curve slope correction with κ from his Figure 1, the section pitching moment, the distributed wing and fuel weight (proportional to the chord, to the chord squared or uniform) and concentrated masses at the load factor, the running loads and the shear V, bending moment M and torque T about the elastic axis integrated from the tip, with the elastic-axis sweep resolved into the torque and the bending about the structural axis, a station table, a planform figure and charts of every distribution. The default is a light-airplane wing of 36 ft span tapered from 5.5 to 3.5 ft carrying 2,400 lb at n = 3.8 with 150 lb of structure and 150 lb of fuel per side and a 300 lb mass at 6 ft: root shear 2,280 lb, root bending moment 20,870 lb-ft, root torque 1,340 lb-ft nose up.

1 worked example reproduced, listed on the tool page.

Inertia and Gyroscopic Loads

Inertia loads of the mass items of an airplane under a load-factor and angular-rate condition, and the gyroscopic couples of its engines. Each item's load is the rigid-body result F = −m (a_cg + ω̇ × r + ω × (ω × r)) in the airplane's body axes, written component by component for the first item and tabulated for all, with the totals about the centre of gravity; the standard conditions of 14 CFR 23.371 (yaw 2.5 rad/s, pitch 1.0 rad/s, normal load factor 2.5 at maximum continuous rpm and thrust) and the side load of 23.363 / 25.363 are offered as a fill. The engines' rotating parts give the couple M = H × ω with H = I_p Ω along the shaft, printed with the right-hand rule and the sense of each component, and the engine-mount load cases are listed as the rules combine them: the 23.371 combinations with the inertia force and thrust, the 23.361 limit torque (mean torque × 1.25, 1.33, 2, 3 or 4) with 75 % and 100 % of flight condition A, and the 23.363 side load alone. The equations are those of NACA TN 587 (Donely 1936) carried in full vector form. The default is a twin-engine light airplane with four mass items under the yaw condition: each engine loads its mount with 1,125 lb down, 612 lb outboard and 423 lb forward plus a 24,270 lb-in nose-down gyroscopic couple.

1 worked example reproduced, listed on the tool page.

Landing Gear Loads

Landing-impact loads of an airplane landing gear from the drop-test energy balance of 14 CFR Part 23 (2016 edition, Amendment 23-62) or Part 25: the limit descent velocity of § 23.473(d), V = 4.4 (W/S)^(1/4) ft/s between 7 and 10 ft/s, or the 10 ft/s of § 25.473, the wing lift the rule allows during the impact, the kinetic and potential energy absorbed by the shock strut and the tire at their efficiencies, the ground-reaction and inertia load factors, the drop-test form of § 23.725 with the effective weight, the reserve-energy case of § 23.723(b) / § 25.723(b), every landing and ground-handling condition of §§ 23.479–23.509 or §§ 25.479–25.509 (level, three-point, tail-down, one-wheel, side and drift loads, spin-up and spring-back drag, braked roll, the § 25.493(e) dynamic braking nose load, turning, towing) as per-gear vertical, drag and side loads, the trunnion loads of a cantilever strut with rake and trail, the Appendix D wheel spin-up drag, the polytropic air curve of the strut, and the generalized orifice-strut impact solution of Milwitzky and Cook (NACA TN 2755 / Report 1154). The default is a 2,400 lb Part 23 airplane with 160 ft^2 of wing, two main gears with a 6 in stroke at 80 % efficiency and 2 in of tire deflection at 47 %, for which the page gives V = 8.66 ft/s, a ground reaction factor of 2.90 and an inertia load factor of 3.57.

1 worked example reproduced, listed on the tool page.

Pressurization Loads

Cabin pressurization loads on a fuselage. The differential pressure follows from the cabin and flight altitudes by the 1976 U.S. Standard Atmosphere (or is entered), the design pressures from 14 CFR 25.365 printed from the rule text: the relief-valve setting combined with flight loads, and 1.33 times that setting (1.67 above 45,000 ft) omitting other loads, carried to ultimate with the factor of safety of § 25.303. The skin takes N_θ = p r and N_x = p r/2; the frame load follows from the interaction between shell and rings of NACA TN 2612 (Flügge 1952), the beam-on-elastic-foundation bay between frames with the ring's own stiffness, printed beside the smeared share and the rigid-ring bound; a double-bubble section gives the floor tie T = p (r_1 cos θ_1 + r_2 cos θ_2); the bulkhead is a spherical dome, an ellipsoidal dome (with the compressive hoop zone flagged) or a flat web on beams; a door cutout gives the stop, edge-frame and sill loads by statics; margins through the entered allowables and the pressure-cycle count for the fatigue tools. The default is a 74 in cabin at 8,000 ft cabin altitude and 41,000 ft, skin 0.040 in, frames 0.30 in² at 20 in, relief valve 8.6 lb/in²: differential 8.31 lb/in², ultimate design pressure 17.16 lb/in², hoop stress 15.4 ksi operating and 31.7 ksi ultimate, frame load 2,100 lb.

1 worked example reproduced, listed on the tool page.

Modal Estimates

Natural frequencies and mode shapes of simple structures for a first dynamic check: uniform Bernoulli-Euler beams with the six classical end-condition pairs from the exact frequency equations (the roots beta L = 1.8751, 4.6941, 7.8548 of the cantilever, 4.7300 and 7.8532 clamped-clamped, n pi pinned-pinned, checked against the tables of NASA SP-160), beams with a tip mass or a mid-span mass and stepped beams by the exact transfer-matrix solution (the tip-mass frequency equation of NACA MR L4K30), Rayleigh's estimate with the static deflection as the assumed shape beside the exact value with its error, Dunkerley's lower bound for the beams with masses, uniform shafts in torsion and axial rods with an optional end inertia, rectangular plates (SSSS closed form, clamped and cantilever plates from Leissa's frequency parameters) and lumped spring-mass chains of up to six degrees of freedom solved by Jacobi iteration, with the first three mode shapes drawn. The default is an aluminium cantilever 30 x 2 x 0.25 in with a 1 lbm tip mass: 4.568 Hz exact, 4.572 Hz by Rayleigh's method and 4.555 Hz by Dunkerley's, against 8.817 Hz for the bare beam.

1 worked example reproduced, listed on the tool page.

Stick Model Lumping

Lumped-mass and stiffness stick model of a wing or fuselage from distributed data, with a Myklestad bending pass and a Holzer torsion pass for the uncoupled natural frequencies: the span stations with EI, GJ, running weight, the chordwise centroid offset and radius of gyration of the running mass and concentrated weights with their offsets are lumped to stations by the tributary-length rule (half the segment on each side) or the statically equivalent lever rule, every segment gets the harmonic-mean EI and GJ (exact for the linear variation between rows), the station table lists the lumped weight, mass, chordwise static moment and the mass moment of inertia about the elastic axis, the totals and the centre of gravity are checked against the distributed input, the Myklestad transfer pass (deflection, slope, moment and shear across massless segments and point masses, the tip boundary determinant swept in frequency and its sign changes refined by bisection) gives the first bending modes beside the flexibility-matrix eigenvalues of the same model, the Holzer pass gives the torsion modes, the mode shapes are charted, the stick is drawn, and a paste-ready finite-element table (GRID / CONM2 / CBAR) is written. The default is a 180 in tapered wing in ten rows with EI from 10^9 to 10^8 lb-in^2, GJ from 6 × 10^8 to 6 × 10^7 lb-in^2, 4.0 to 1.3 lb/in of structure and a 300 lb engine at station 3; checked against NACA TN 1522 (Houbolt and Anderson 1948), the exact uniform cantilever and an independent numpy transfer-matrix and finite-element pass.

1 worked example reproduced, listed on the tool page.

Fatigue Spectrum Generation

A flight-by-flight fatigue load sequence from a mission profile and exceedance data, ready to paste into the Spectrum Fatigue Life and Crack Growth tools: mission segments with their duration, speed and 1-g stress; gust exceedance curves pasted as breakpoints (per 1,000 flight hours or per mile) or taken from the generalized gust distribution of NACA TN 4332 at the segment's altitude; a maneuver curve with positive and negative increments; the curves stepped into discrete levels between an omission (truncation) level and a clipping level by differencing the exceedances, as the USAF Damage Tolerant Design Handbook describes; the occurrences per flight and the integer occurrence table of a block of flights; the sequence itself, with the cycles of each flight in random order (seeded, reproducible) or low-high-low, the taxi cycles and the landing impact forming the ground-air-ground cycle; its peak-count check against the input exceedances and an ASTM E1049 rainflow count of the block; charts of the first flights, of the exceedance curves with their levels and of the rainflow range spectrum. The default is the three-segment operation of NASA Memorandum 1-17-59L (a four-engine transport on a northwestern United States to Alaska route, 263 flights, 672.9 hours): its Table I gust accelerations by climb, en route and descent and its Table II operational maneuvers, with illustrative 1-g stresses of 13, 12 and 11 ksi, a 100-flight block, 0.3 g omission and 0.8 g clipping.

1 worked example reproduced, listed on the tool page.

Other Structures sections

Materials · Composites · Frames and Trusses · Beams · Joints · Stress and Strain · Stability · Plates and Shells · Thin-Walled Structures · Contact · Stress Concentrations · Fatigue · Damage Tolerance