Tapered and curved beams

The check that decides is rarely bending — it is tension perpendicular to the grain, EN 1995-1-1 §6.4.

Beam type

The laminations are bent during production, which costs strength through k_r. The apex zone spans the whole curved part.

Material

Geometry

b — width [mm] h — depth of the section checked [mm] alpha — taper angle [degrees]

Section on the tapered portion. §6.4.3(2) applies the rules of §6.4.2 there too. For a curved beam of constant section put alpha = 0 and the check degenerates to plain bending.

h_ap — depth at the apex [mm] alpha_ap — angle at mid apex zone [degrees] r_in — inner radius [mm] t — lamination thickness [mm]

Actions

M_d — moment in the section checked [kNm] M_ap,d — design moment at the apex [kNm] tau_d — shear stress in the apex zone [N/mm2] use expression (6.55) instead of (6.54)Stresses parallel to the tapered edge

Volumes

V is the stressed volume of the apex zone, capped at two thirds of the total beam volume. V0 = 0.01 m3.

V — stressed apex volume [m3] V_b — total beam volume [m3]

Design strengths

f_m,d f_v,d f_t,90,d f_c,90,d

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Tool information

What this page computes

Tapered and curved timber beams to EN 1995-1-1 §6.4, plus torsion from §6.1.8:

§6.4.2 — single taper: the factor k_m,α on the tapered edge;
§6.4.3 — double taper, curved beams, and pitched cambered beams: k_ℓ, k_r, k_p, k_vol, k_dis, and the interaction (6.53).

The check that decides is rarely bending

That is the central idea of this section, and it is easy to miss if you only read the formulae.

On an ordinary curved beam — glulam 140×1000, inner radius 5 m, M_ap,d = 200 kNm — bending at the apex comes out at 42% of capacity. That looks comfortable. But tension perpendicular to the grain comes out at 180%, and the interaction with shear at 212%.

The reason is simple and brutal: f_t,90,k is of the order of 0.5 N/mm², some fifty times smaller than the bending strength. Curvature converts part of the moment into tension across the grain, and there timber fails by splitting along its length, not by bending rupture. That is why the diagram draws the arrows pulling outwards, vertically: you see the beam coming apart, not snapping.

The asymmetry in §6.4.2, often missed

On the tapered edge the cut fibre no longer works in pure bending: shear and perpendicular-to-grain stress appear together. k_m,α gathers them into one factor. But the two expressions are not symmetric:

\(k_{m,\alpha} = \frac{1}{\sqrt{1 + \left(\dfrac{f_{m,d}}{0.75 f_{v,d}}\tan\alpha\right)^2 + \left(\dfrac{f_{m,d}}{f_{t,90,d}}\tan^2\alpha\right)^2}} \qquad (6.39)\)

For compression parallel to the tapered edge, the denominators become 1.5·f_v,d and f_c,90,d. Both changes pull the same way: shear is divided by twice as much, and the perpendicular term uses f_c,90,d instead of f_t,90,d — some seven times larger.

The result, with a test pinning it down: at 10° with ordinary material, k_m,α in compression is more than twice that in tension. That is not a nuance — it is the difference between a beam that works and one that does not.

Hence a practical rule: where the layout allows, put the taper on the compression side.

The four types differ by more than coefficients

type	laminations bent?	`k_r`	`k_dis`
double taper	no	1.0	1.4
curved	yes	(6.49)	1.4
pitched cambered	yes	(6.49)	1.7

k_r drops below 1 only when r_in/t < 240 — that is, when the laminations really were bent tightly in production. At r_in/t = 151 it gives 0.91: 9% of the bending strength is lost in the factory, before the beam sees any load.

k_vol pulls the same way, but harder: at V = 0.6 m³ it gives 0.44. The reference volume is V₀ = 0.01 m³, and the exponent 0.2 means a tenfold volume costs a factor of 0.63. For solid timber k_vol = 1.0 — the volume effect applies only to glulam and LVL.

A discrepancy in the text of the code

§6.4.3(5) in the Romanian copy reads: "For double tapered beams k_r = 1.6."

That cannot be right. k_r is defined in that same code as the factor accounting for the reduction in strength due to bending the laminations during production — a reduction factor cannot exceed 1. And (6.49), which gives k_r for the other cases, explicitly caps it at 1. In a double-tapered beam the laminations are not bent at all: they are cut.

The page uses 1.0, the physically coherent value, but leaves it editable and posts the note on every run. If you have a copy in another language, it is worth checking.

§6.1.8 — torsion

\(\tau_{tor,d} \le k_{shape}\, f_{v,d}, \qquad k_{shape} = \begin{cases} 1.2 & \text{circular} \\ \min(1 + 0.15\,h/b\,;\ 2.0) & \text{rectangular} \end{cases}\)

The code gives the condition and k_shape, but not how τ_tor,d follows from a torsional moment. That is classical mechanics, outside the Eurocode, and it is flagged as such.

The cap of 2.0 comes into play above h/b = 6.67.

What it does not cover

The calculation of τ_tor,d from T_Ed and geometry — it is entered directly.
The volume V of the apex zone — entered, not derived from geometry.
§6.5 — notched beams.
Solid timber in §6.4.3: the code restricts that clause to glulam and glued laminated veneer.
Stability — lateral torsional buckling of curved beams. The conditions of §6.3 are checked separately.

The texts of §6.1.8 and §6.4 are transcribed from EN 1995-1-1:2004, pages 42 and 46–49.

Timber member — the basic §6.1 and §6.2 checks on a constant section.
Timber deflections — serviceability, with creep weighted by ψ₂.
Dowel-type connections — the other large family in EN 1995.