Fiber Direction and Elastic Stiffness

Fiber direction changes elastic response

A fiber-reinforced composite combines a reinforcement with a surrounding matrix. In a unidirectional layer, the reinforcement is continuous and parallel. The layer is much stiffer along the fibers than across them, so its elastic response depends on direction.

Stress is force divided by loaded area. This lesson compares patches under the same uniform axial stress, so each square millimeter carries the same applied force. Strain is deformation divided by original dimension. Elastic stiffness connects stress to strain in the reversible small-strain model.

The illustrative material uses E1 = 40 GPa along the fibers, E2 = 10 GPa across them, G12 = 5 GPa, and ν12 = 0.30. These round teaching values do not describe a named composite and are not design allowables.

Review Plastics as Materials for reinforcement, matrix, thermosets, and thermoplastics. Review Plastic Forming Processes for hand layup and other forming routes.

Off-axis extension can also create shear strain

At 0° and 90°, the loading direction matches a principal material direction. The patch stretches and narrows without extension–shear coupling. At an off-axis angle, an axial stress can also make the free patch skew. Turning +45° to −45° mirrors that skew while leaving the axial stiffness unchanged.

The interactive drawing exaggerates every strain by the same factor of 120. A larger drawn change therefore means a larger calculated strain; the display does not secretly rescale each angle to fill the frame.

How the axial stiffness is calculated

Let m = cos θ and n = sin θ. For this plane-stress orthotropic layer, the compliance in the load direction is S̄11 = m⁴/E1 + n⁴/E2 + m²n²(1/G12 − 2ν12/E1). Then Ex = 1/S̄11 and εx = σx/Ex. The full stress and strain transformation supplies transverse and shear strain.

Stiffness does not predict strength

A smaller strain under the same stress means greater elastic stiffness. It does not say when the material fails, which direction is safest, or which laminate should be built. Strength needs failure data and criteria; a laminate also needs its complete stacking sequence, ply thicknesses, bonding, environment, loading, and boundary conditions.

This lesson models one thin, continuous-fiber lamina in small-strain linear elasticity and plane stress. It omits failure, damage, buckling, bending, temperature, moisture, manufacturing defects, grips, and interaction between multiple plies.

Primary mechanics source and model boundary

NASA Reference Publication 1351, Basic Mechanics of Laminated Composite Plates, sections I and II, defines isotropic and orthotropic behavior, develops plane-stress compliance and off-axis transformation, and identifies extension–shear coupling. Its stated scope separates elastic response from strength. The constants and examples on this page are original illustrative teaching inputs, not values from that report.

Practice axial strain Practice angle symmetry Practice model limits

Symmetric, balanced and quasi-isotropic are three different properties

• Symmetric: the plies above the midplane are a mirror image of the plies below it. Then the coupling matrix B is zero, so in-plane loads and bending stay separate. [0/90]s is symmetric; [0/90] is not.
• Balanced: every +θ ply has a −θ ply of the same thickness somewhere in the stack (0° and 90° plies need no partner). Then A16 = A26 = 0: there is no extension–shear coupling in the flat-held in-plane response, so a pull along x or y held flat does not shear the laminate in its plane. A balanced and symmetric stack also has B = 0, so it does not shear when free either; a balanced unsymmetric stack such as [0/+45/−45/90] can still shear and twist when free to curve.
• Quasi-isotropic: the in-plane stiffness matrix A is the same in every direction, so the rose is a circle. Equal numbers of 0°, +45°, −45° and 90° plies do this, and because A ignores the order of the plies, [0/+45/−45/90]s and [+45/0/−45/90]s both qualify. It describes in-plane stiffness only; bending stiffness D still depends on order and direction.

An unsymmetric laminate does not bend just because B is not zero. With no applied force and no applied moment, this model predicts no deformation at all. B only links a load to a response: pull an unsymmetric panel in its plane and it also curls or twists, even though no moment is applied. A symmetric panel under the same pull stays flat: it stretches (and, if unbalanced, also shears in its plane) but does not curl or twist.

Flat-held and free stiffness differ for unsymmetric laminates

The two stiffness numbers for an unsymmetric laminate answer different questions. Flat-held (curvature-constrained) stiffness uses A alone and assumes something stops the panel from curving. Free stiffness lets it curve with zero applied moment and solves the full relation including B and D; it is never higher, and in most directions lower, because the panel can curl as well as stretch. For a symmetric laminate the two are identical.

Real unsymmetric panels can also warp as they cool after curing, because each ply shrinks differently in different directions. That temperature effect is not modeled here, and the curled panel in the interactive is not a picture of it: every curl drawn above is the response to the stated in-plane load.

How the laminate stiffness and the free-panel response are calculated

Each ply's reduced stiffness Q (MPa) comes from E₁, E₂, G₁₂ and ν₁₂, and is rotated to the ply angle to give Q̄, using engineering shear strain γ_xy. With z measured up from the midplane and each ply running from z_bottom to z_top (mm):

A = Σ Q̄ (ztop − zbottom) in N/mm; B = Σ Q̄ (ztop² − zbottom²)/2 in N; D = Σ Q̄ (ztop³ − zbottom³)/3 in N·mm.

The force resultants N (N/mm) and moment resultants M (N·mm per mm) relate to midplane strain ε⁰ and curvature κ (1/mm) by N = A ε⁰ + B κ and M = B ε⁰ + D κ. The interactive applies N with M = 0 and solves all six equations together. The drawn surface is w = −(κxx² + κyy² + κxyxy)/2, exaggerated by the stated factor.

For the rose, a pull N_θ along direction θ has components (cos²θ, sin²θ, sinθ cosθ) Nθ. Flat-held: Ex(θ) = 1/(h · vᵀA⁻¹v) with v = (cos²θ, sin²θ, sinθ cosθ) and laminate thickness h. Free: the same, with the in-plane block of the inverted six-by-six A-B-D matrix in place of A⁻¹.

Reversing the ply order (the bottom ply listed first) leaves A and D unchanged and reverses the sign of every B term, so the same pull makes the opposite curl. A single ply is symmetric about its own midplane, so it reproduces the one-ply result above: 40 GPa along the fibers and 10 GPa across.

This is linear, small-deflection classical laminate theory with perfectly bonded plies. It does not predict strength, first-ply failure, delamination, buckling, temperature or moisture effects, or how a real panel behaves once its deflection is comparable to its thickness.

Primary mechanics source and model boundary

NASA Reference Publication 1351, Basic Mechanics of Laminated Composite Plates (A. T. Nettles, 1994) defines balanced and symmetric laminates, derives the A, B and D matrices and the laminate constitutive equations, shows that a symmetric laminate has B = 0, and works an unsymmetric laminate example in which a tensile load alone bends the plate. It also notes that unsymmetric laminates can warp from processing temperature changes, which this lesson excludes. The ply constants, ply thickness, loads and panel size here are original illustrative teaching inputs, not values from that report and not design data.