The specimen is a narrow beam: only the bending moment M_x acts (M_y = M_xy = 0, no axial force). Laminate theory with the bending stiffness matrix D gives the curvature, the strain ε = ε⁰ + zκ through the thickness and the stress in every ply.

3-point (Procedure A): M = PL/4, V = P/2. 4-point (Procedure B, load span L/2): M = PL/8 between the noses, V = P/2 outside them.

Flexural stress (ASTM D7264): σ = 3PL/(2bh²) for 3-point, σ = 3PL/(4bh²) for 4-point.

Surface strain from the midspan deflection δ: ε = 6δh/L² for 3-point, ε = 4.36δh/L² for 4-point.

Deflection, bending plus shear: 3-point δ = PL³/(48EI) + PL/(4κGA); 4-point δ = 11PL³/(768EI) + PL/(8κGA); κ = 5/6, A = bh.

EI comes from the laminate (EI = b/d₁₁, d = D⁻¹). The fibers are stiffer in tension than in compression (E₁ 158.5 against 138.2 GPa for IM7/8552, NCAMP), so the plies on the compression side use the compression modulus and the neutral axis moves toward the tension side; each ply is split in 8 layers and the stiffness of each follows the sign of its fiber strain. The transverse shear modulus G is the through-thickness (series) average of the plies: G₁₃ along the fibers, G₂₃ across them.

Interlaminar shear from equilibrium: τ_xz(z) = −V ∫_−h/2^z σ_x/M dz (and τ_yz from σ_xy). A homogeneous beam gives the parabola with its peak 1.5V/(bh) = 0.75P/(bh).

Ply failure, from the shared failure engine used on every test page: fiber σ₁/X_T or −σ₁/X_C; matrix √[(σ₂/Y)² + (τ₁₂/S)²] with the in-situ strengths of the ply's block (Camanho et al. 2006) and Y = Y_C in compression; delamination √(τ_xz² + τ_yz²)/S_ILS.

Each ply's stress is the bending stress plus the cure residual stress of the laminate (cool-down from the stress-free temperature, the same as on the Tension page; none for an estimated material). The bending stress chart shows the bending part only. Ply properties also come from the shared engine.

Ultimate rule: the beam fails when any ply breaks its fibers (σ₁ reaches X_T or −X_C), or a delamination runs, whichever load is lower. Only when no ply lies within 15° of 0° can matrix failure also end the test: of the ±45° plies (shear), or, without those, of any ply. This is the same rule as on the other test pages. Matrix cracks in other plies are drawn and marked on the curve but do not end the test. The beam is linear elastic up to failure; the load drop after it and the damage drawings are illustrative. Not modelled: contact stresses under the nose, large deflection, cure residual stresses, free-edge stresses.

Delamination, two mechanisms. On short spans the shear force is large compared with the moment, and the beam delaminates near mid-thickness by interlaminar shear, checked against the measured short-beam strength S_ILS. A delamination can also grow from matrix cracks in a block of off-axis plies, the mechanism of the tension and open-hole tests; for it every test page uses the same fitted form σ_d = √(E_x G_IIc / (0.47 t_block)), with the constant 0.47 fitted to the ply-block tension tests of Wisnom, Khan and Hallett (2008) and t_block the thickest same-angle off-axis block. Here it is checked on the tension side of the beam: a run of off-axis plies with two or more angles delaminates once one of its plies has cracked and the laminate strain at the run's outer edge reaches σ_d/E_x. It is not checked on the compression side, where the matrix cracks are closed, nor for an estimated material, which has no G_IIc. The fitted form comes from uniform tension; using it with the bending strain at the run is this page's assumption.

Material data

Sources

• ASTM D7264/D7264M-21, flexural properties of polymer matrix composites. doi:10.1520/D7264_D7264M-21
• ASTM D2344/D2344M-22, short-beam strength of polymer matrix composites. doi:10.1520/D2344_D2344M-22
• NCAMP report NCP-RP-2009-028 Rev B, 8552 IM7 unidirectional prepreg qualification statistics (2019): ply properties, SBS (Table 4-12), SBS1 (Table 5-25), E_1c (Table 4-7). Per-specimen data: NCAMP report CAM-RP-2009-015 Rev B.
• NCAMP report NCP-RP-2010-008 Rev D, 8552 AS4 unidirectional prepreg qualification statistics: ply properties, SBS (Table 4-12), LSBS (Table 5-25), E_1c (Table 4-7). Per-specimen data: NCAMP report CAM-RP-2010-002 Rev A.
• M.R. Wisnom, B. Khan and S.R. Hallett, Size effects in unnotched tensile strength of unidirectional and quasi-isotropic carbon/epoxy composites, Compos. Struct. 84 (2008) 21.
• J.M. Whitney, C.E. Browning and A. Mair, Analysis of the flexure test for laminated composite materials, ASTM STP 546 (1974) 30. doi:10.1520/STP34856S