Loads and Transverse Stresses around a Radius

Loads Around a Radius

What this part of the tool does

A curved corner of a laminate, such as the radius between the web and the flange of an angle, a clip or a frame, receives an axial load, a shear load and a moment from the flat part it joins. Those three loads change as they travel around the bend. The tool takes the loads per unit width at the tangency point, where the flat part meets the radius, and finds the axial load, shear load and moment at any angle, then the two sections that matter most: the one with the largest moment and the one with the largest shear.

Geometry and sign convention

The radius has outer radius R_o, thickness t, inner radius R_i = R_o − t and mean radius R_m = (R_i + R_o)/2. The angle θ is measured from the tangency face, where θ = 0, around the bend to the full angle φ of the radius (90° for a right-angle corner). All loads are per unit width of the part.

The signs are the ones drawn in the figure. N_o is positive in tension, V_o is positive when it acts radially outward on the tangency face, and M_o is positive when it opens the radius, that is, when it puts the inner surface in tension. The loads N_θ, V_θ and M_θ on the cut at θ are positive as drawn there.

Equilibrium of the cut piece

Cutting the bend at θ and summing forces along and across the cut, and moments about the middle of the cut, gives

N_θ = N_o cos θ + V_o sin θ

V_θ = −N_o sin θ + V_o cos θ

M_θ = M_o − N_o R_m (1 − cos θ) + V_o R_m sin θ

The loads are taken to act at the mean radius. Two consequences are worth knowing. First, dM_θ/dθ = R_m V_θ and dV_θ/dθ = −N_θ, the curved-beam forms of the familiar beam relations. Second, M_θ − N_θ R_m = M_o − N_o R_m at every angle: the moment of the loads about the center of the radius does not change around the bend. The stress equations use that constant.

Where the moment and the shear are largest

The moment is stationary where the shear is zero, tan θ = V_o/N_o, and the shear is stationary where the axial load is zero, tan θ = −N_o/V_o. Each of these equations has roots 180° apart, and a root is useful only when it lies inside the bend, between 0 and φ. When it does not, the largest value is at one end of the bend. With N_o and V_o of the same sign, for example, the shear has no stationary point inside a 90° bend and its largest magnitude is at θ = 0 or θ = φ, whichever is greater.

The tool therefore compares the stationary point, when it is inside the bend, with the two ends and reports the loads at the governing section for moment and for shear.

What it assumes

The bend is a circular arc of constant thickness, loaded only through its two ends: no pressure or other load is applied along the radius itself. The loads are uniform across the width. A part with pressure on the radius, or with a load that varies across the width, needs those effects added before the stresses are checked.

Transverse Stresses in a Curved Laminate

Why a curved laminate has through-the-thickness stress

In a flat laminate, bending puts stress only along the plies. In a curved one the circumferential stresses on the two sides of a small piece are not parallel, so they have a resultant across the thickness. A moment that opens the radius pulls the inner plies in tension and pushes the outer plies in compression, and both resultants act to separate the plies: the result is radial tension, which a laminate resists only with its resin and its ply interfaces. A moment that closes the radius gives radial compression instead. The shear load on the section adds a transverse shear stress between the plies, as in a straight beam. Both stresses are zero on the inner and outer surfaces and largest near the middle of the thickness.

The simple equations

Treating the section as a straight beam of thickness t at the mean radius R_m gives quick estimates of the peaks:

radial stress: f_r = 3 M / (2 t R_m), or with the geometric mean radius, f_r = 3 M / (2 t √(R_i R_o))

transverse shear stress: f_rθ = 3 V / (2 t)

circumferential stress at the surfaces: f_θ = N / t ± 6 M / t²

These take the stresses as symmetric about the mid-thickness. They are good for thin, gently curved laminates and lose accuracy as t/R_m grows, when the peaks move toward the inner radius and the circumferential stress on the inner surface rises above the straight-beam value.

The Lekhnitskii solution

Lekhnitskii solved the curved bar of a material with different stiffness around the bend and through the thickness: circumferential modulus E_θ, radial modulus E_r, shear modulus G_rθ and major Poisson's ratio ν_θr. The solution has two parts, which are added.

End-force part. A force applied at the end of the bar, acting through the center of the radius. It carries the axial load N_θ and the shear load V_θ of the section. With c = R_i/R_o,

β = [1 + E_θ/E_r − 2 ν_θr + E_θ/G_rθ]^1/2,   g₁ = (2/β)(1 − c^β) + (1 + c^β) ln c

Pure-bending part. A moment that is the same at every section. Because the end force acts through the center of the radius, it already produces a moment N_θ R_m at the middle of the cut, so the pure-bending part carries only the remainder,

M_c = M_θ − N_θ R_m = M_o − N_o R_m,

which is the same at every angle. Its constants are

k = (E_θ/E_r)^1/2,   P₁ = (1 − c^k+1)/(1 − c^2k),   P₂ = (1 − c^k−1)/(1 − c^2k)

g = (1 − c²)/2 − [k/(k + 1)] (1 − c^k+1)²/(1 − c^2k) + [k c²/(k − 1)] (1 − c^k−1)²/(1 − c^2k)

With x = r/R_o, the stresses at radius r are

f_r = [N_θ/(g₁ r)] [x^β + c^β x^−β − 1 − c^β] − [M_c/(R_o² g)] [1 − P₁ x^k−1 − P₂ c^k+1 x^−(k+1)]

f_θ = [N_θ/(g₁ r)] [(1 + β) x^β + (1 − β) c^β x^−β − 1 − c^β] − [M_c/(R_o² g)] [1 − k P₁ x^k−1 + k P₂ c^k+1 x^−(k+1)]

f_rθ = [V_θ/(g₁ r)] [x^β + c^β x^−β − 1 − c^β]

Using the end-force part alone, without the pure-bending part, overstates the radial stress, because it assigns the section a moment N_θ R_m whatever the real moment is.

Where the stresses are evaluated

The radial and transverse shear stresses of the end-force part peak at the radius where the bracket divided by r is largest:

(r/R_o)^β = [1 + c^β − ((1 + c^β)² + 4 c^β(β² − 1))^1/2] / [2 (1 − β)]

This radius lies slightly inside the mid-thickness. The tool evaluates the radial stress there with the loads of the maximum-moment section, the transverse shear stress there with the loads of the maximum-shear section, and the circumferential stress on the inner and outer surfaces of the maximum-moment section. The figures at the end of the results show the same equations over the whole bend, so a peak elsewhere would be visible.

Margins of safety

The margins compare the Lekhnitskii stresses with the interlaminar strengths: the radial (interlaminar tension) strength F_r and the transverse (interlaminar) shear strength F_rθ. Both are strength checks, so they are written at ultimate load: when limit loads are entered the stresses are multiplied by the factor of safety, and when ultimate loads are entered they are used as computed.

M.S. = F / f − 1

No radial margin is written when the radial stress is compressive, since the radial strength entered is a tension strength. The two stresses are checked separately; no interaction between radial tension and transverse shear is applied.

These are static margins only. A laminated composite develops interlaminar cracks and fails if it is operated close to its ultimate interlaminar shear and radial tension strengths. For durability, cycled interlaminar stresses need to be much lower, on the order of 500 to 1,000 psi (3.5 to 7 MPa).

Material properties

The circumferential modulus is the in-plane laminate modulus in the direction that runs around the bend. A saved laminate supplies it together with the thickness and the lowest interlaminar shear strength of its plies. The radial modulus, the transverse shear modulus, the Poisson's ratio and the radial strength are through-the-thickness properties that laminate theory does not give. They can be stored with each lamina (E₃, G₁₃, G₂₃, ν₁₃, ν₂₃ and the interlaminar tension strength F_3t), and the saved laminate then fills them as well:

radial modulus: 1/E_r = Σ (t_k/t) / E_3,k

transverse shear modulus: 1/G_rθ = Σ (t_k/t) (cos²α_k/G_13,k + sin²α_k/G_23,k), with α_k the ply angle

major Poisson's ratio: ν_θr = −(thickness strain of the stack)/(strain around the bend) under a load around the bend, each ply contributing −(ν₁₃/E₁) σ₁ − (ν₂₃/E₂) σ₂

radial strength: the lowest F_3t of the plies; transverse shear strength: the lowest F₁₃

The two moduli are series averages, which treat the plies as free to strain independently in their own plane; they are estimates, and the stresses of this tool are not sensitive to them. The radial strength is usually well below the in-plane transverse strength of a ply and depends strongly on processing, so it should be a value measured on curved specimens.

What it assumes

The laminate is treated as one homogeneous orthotropic material with the same properties through the thickness. Ply-by-ply variations of stress, which matter when stiff and soft plies alternate, are not found. The bend is a circular arc of constant thickness under loads that are uniform across the width, away from free edges, ply drops and fasteners, where local stresses are higher. The solution is for a section in a plane; the Poisson's ratio term has little effect on the result.

When E_θ = E_r the pure-bending constants take a limiting form; the tool's printed equations divide by k − 1, so enter moduli that differ slightly in that case.