Elasticity review

With the loads in hand, Arjun's next step is to turn them into stresses and compare those stresses with what the material can take. Marcus asks him to start at a single point: a spot on the lower wing skin, just outboard of the root, where the skin carries tension from wing bending, a little chordwise stress from the ribs and the fuel pressure, and shear from the wing's torsion. Getting that one point right, with the correct combination of stresses and the correct comparison with the material, is the foundation for everything that follows. This lesson reviews the parts of elasticity that aircraft stress analysis uses every day.

Stress and strain at a point

At any point inside a loaded body, the internal forces acting on a small cube of material can be described by nine stress components: three normal stresses, which pull or push on the faces, and six shear stresses, which slide them. Because the cube must not spin, the shear stresses come in equal pairs, so only six components are independent: \(\sigma_x, \sigma_y, \sigma_z, \tau_{xy}, \tau_{yz}, \tau_{zx}\). The matching strains are the normal strains \(\varepsilon\), which measure stretching, and the shear strains \(\gamma\), which measure the change in a right angle.

For a linear elastic isotropic material, such as an aluminum alloy below its yield point, Hooke's law links the two through the Young's modulus \(E\), Poisson's ratio \(\nu\) and the shear modulus \(G\). Only two of these are independent, because \(G = E/[2(1 + \nu)]\).

Aircraft skins are thin. The stress through the thickness, perpendicular to the skin surface, is tiny compared with the stresses in the plane of the skin, so it is usually taken as zero. This is the plane stress condition, and it applies to most of the skin, spar webs and frame webs in an airframe. Three stresses then describe the state at a point: \(\sigma_x\), \(\sigma_y\) and \(\tau_{xy}\). The exceptions, where three-dimensional stress matters, are thick fittings, lugs and joints, which are analyzed later with detailed models.

Equilibrium and compatibility

Two sets of equations govern any elastic body besides Hooke's law. Equilibrium requires that the stresses on every small element balance, including any body forces. Compatibility requires that the strains fit together without gaps or overlaps, so that a continuous body stays continuous. Together with the stress-strain law and the boundary conditions, these make up the theory of elasticity. Exact solutions exist only for simple shapes, which is why aircraft structural analysis relies on simplified theories: beam theory for wings and fuselages, thin-walled section theory for their cross-sections and plate theory for the skin panels. Each simplification keeps equilibrium exactly and treats compatibility approximately. Knowing what each one assumes, and checking that the assumptions hold, is a core skill of a stress engineer.

An important consequence is the difference between statically determinate and indeterminate structures. In a determinate structure, equilibrium alone fixes all the internal forces. In an indeterminate structure, such as a wing box with several spars or a fuselage with many frames and stringers, the internal forces depend on the relative stiffness of the parts as well, and compatibility must be used to find them. Most of an airframe is highly indeterminate, which is one reason finite element models, the topic of Lesson 12, are used so widely.

Transforming stress: Mohr's circle

The stresses on a point depend on the orientation of the faces you choose. Rotating the axes by an angle \(\theta\) gives new components, and among all the orientations there are two principal directions in which the shear is zero and the normal stresses reach their largest and smallest values. These principal stresses are what usually matter for yielding and fracture.

Mohr's circle draws these relations. Plot the stresses on the x face as one point and those on the y face as another, with opposite shear signs. The line joining them is a diameter; its center is the average normal stress and its radius is the maximum in-plane shear stress. The ends of the horizontal diameter are the principal stresses. Turning the element by \(\theta\) moves the points around the circle by \(2\theta\). The circle is quick to sketch, and it gives a feel for how a combination of tension and shear produces a larger principal stress than either alone.

Failure criteria for ductile metals

A material test gives a yield strength from a uniaxial tensile coupon. Real structure carries combined stresses, so a criterion is needed to compare the two. For ductile metals such as aluminum alloys and titanium, the von Mises criterion works well. It says yielding begins when the distortion energy reaches the value it has at yield in a tensile test, which gives an equivalent stress that can be compared directly with the yield strength.

The older Tresca criterion, based on the maximum shear stress, is slightly more conservative and easier to use by hand. For brittle materials and for composites, quite different criteria are needed, as Lesson 11 shows. Whatever criterion is used, the result is expressed in the aerospace industry as a margin of safety: the allowable divided by the applied stress (or load), minus one. A margin of zero means the part is exactly adequate; a negative margin means it fails the requirement. The factor of safety is already inside the applied ultimate load, so a margin of +0.10 means 10 percent of reserve beyond the requirement, not 10 percent beyond the expected service load.

Materials for airframes

Most of the Tern's structure, like that of most small and medium aircraft, is aluminum alloy. Two families dominate. The 2000-series alloys, with copper as the main alloying element, have good fatigue and crack growth resistance and are used where tension and fatigue govern, such as lower wing skins and fuselage skins. The 7000-series alloys, with zinc, are stronger and are used where compression and static strength govern, such as upper wing skins, stringers and spar caps. Titanium appears where heat, high strength in a small space, or contact with carbon fiber composites is involved. Composites, introduced in Lesson 11, are increasingly used for control surfaces, fairings and, on newer aircraft, primary structure.

# Plane stress at a point on the lower wing skin: principal stresses, Mohr's circle and margins of safety
import math
mats = {  # typical sheet values, MPa (A-basis allowables come from the approved data source, not from this table)
    "2024-T3 aluminum": (73100, 0.33, 324, 434),
    "7075-T6 aluminum": (71700, 0.33, 469, 538),
    "Ti-6Al-4V titanium": (113800, 0.34, 880, 950),
}
for k, (E, nu, Fty, Ftu) in mats.items():
    print(f"{k:20s} E = {E/1000:5.1f} GPa  nu = {nu}  yield {Fty} MPa  ultimate {Ftu} MPa")
sx, sy, txy = 182.0, 24.0, 58.0       # MPa at limit load: spanwise bending, chordwise, shear from torsion
c = (sx + sy) / 2; R = math.hypot((sx - sy) / 2, txy)
s1, s2 = c + R, c - R
theta = 0.5 * math.degrees(math.atan2(2 * txy, sx - sy))
vm = math.sqrt(sx**2 - sx * sy + sy**2 + 3 * txy**2)
print(f"\nMohr center {c:.1f} MPa, radius {R:.1f} MPa")
print(f"Principal stresses {s1:.1f} and {s2:.1f} MPa at {theta:.1f} deg from the span axis")
print(f"Maximum in-plane shear {R:.1f} MPa; von Mises stress {vm:.1f} MPa")
E, nu, Fty, Ftu = mats["2024-T3 aluminum"]
ex = (sx - nu * sy) / E; ey = (sy - nu * sx) / E; gxy = 2 * (1 + nu) * txy / E
print(f"Strains: ex = {ex*1e6:.0f}, ey = {ey*1e6:.0f}, gxy = {gxy*1e6:.0f} microstrain")
ms_yield = Fty / vm - 1                 # limit load against yield
ms_ult = Ftu / (1.5 * vm) - 1           # ultimate load (1.5 x limit) against ultimate strength
print(f"Margin of safety at limit (yield): {ms_yield:+.2f}")
print(f"Margin of safety at ultimate:      {ms_ult:+.2f}")

2024-T3 aluminum     E =  73.1 GPa  nu = 0.33  yield 324 MPa  ultimate 434 MPa
7075-T6 aluminum     E =  71.7 GPa  nu = 0.33  yield 469 MPa  ultimate 538 MPa
Ti-6Al-4V titanium   E = 113.8 GPa  nu = 0.34  yield 880 MPa  ultimate 950 MPa

Mohr center 103.0 MPa, radius 98.0 MPa
Principal stresses 201.0 and 5.0 MPa at 18.1 deg from the span axis
Maximum in-plane shear 98.0 MPa; von Mises stress 198.6 MPa
Strains: ex = 2381, ey = -493, gxy = 2111 microstrain
Margin of safety at limit (yield): +0.63
Margin of safety at ultimate:      +0.46

The point carries 182 MPa of spanwise tension, 24 MPa chordwise and 58 MPa of shear at limit load. The principal stress is 201 MPa, about 10 percent higher than the spanwise stress, and the von Mises stress is 199 MPa. Against the yield strength of 2024-T3 sheet that leaves a margin of +0.63 at limit load. At ultimate load, 1.5 times larger, the margin against the ultimate strength is +0.46. Both are positive, but Marcus reminds Arjun that this is a static check only: the lower skin is a fatigue-critical area, and Lesson 10 will show that fatigue, not static strength, usually sets its thickness.

The property values in the program are typical published figures, used here for teaching. Real design uses allowables from an approved source, such as the industry handbook of metallic material properties or the company's own qualified test data, chosen for the correct product form, thickness and grain direction. Using a value for thin sheet on a thick plate, or a longitudinal value for a part loaded across the grain, is a classic source of error that design reviews are set up to catch.

Thermal and residual stresses

Not all stresses come from external loads. Temperature changes cause materials to expand or contract, and where parts of different materials or different temperatures are joined, the mismatch causes thermal stresses. An aluminum skin bolted to a titanium fitting, or a wing exposed to sun on top and cold fuel below, develops such stresses. Manufacturing also leaves residual stresses: machining thick plate releases some, cold working fastener holes deliberately adds compressive ones that slow fatigue cracking. A complete stress analysis includes these effects where they matter, usually in joints and fittings.