Beam Stresses

The stresses along every beam element — axial, bending about each axis, the extreme direct stress, shear and torsion, and the Von Mises equivalent — recovered from the beam forces with elastic beam theory.

They depend on the section's shape, not its material: this is what a linear elastic section of that shape carries under those forces. To check them against a strength, use Utilisation.

🔧 Grasshopper component

Beam StressesAlpaca4d ▸ 08_NumericalOutput

Inputs

AlpacaModelModelThe analysed model.

HistoryBooleanfalseRead every recorded step instead of one.

The outputs become trees {step; element}. Step is ignored.

StepInteger0Analysis step.

ElementIdTextlist(empty)Read only some of the beams: an ElementId, a wildcard over them, a tag, or a regular expression.

Empty reads them all.

Outputs

All are data trees, one branch per beam element, keyed by its tag — the same layout as Beam Forces. Within a branch, the values are at the element's integration points, from the I end to the J end.

Stresses are in MPa (N/mm²) — the unit steel and timber strengths are tabulated in, rather than the model's kN/m², which would read a thousand times larger.

σNSigmaNNumbertreeAxial stress N/A. Positive is tension.

σMySigmaMyNumbertreeThe largest bending stress My causes on its own, at the fibre furthest along local z.

σMzSigmaMzNumbertreeThe largest bending stress Mz causes on its own, at the fibre furthest along local y.

σmaxSigmaMaxNumbertreeThe largest direct stress anywhere on the section — N, My and Mz together.

σminSigmaMinNumbertreeThe smallest, most compressive, direct stress anywhere on the section.

τVTauVNumbertreePeak shear stress from Vy and Vz.

τTTauTNumbertreePeak shear stress from torsion.

VonMisesNumbertree√(σ² + 3τ²), σ the larger of σmax and σmin in size and τ = τV + τT.

ElementElementlistThe beams read, in the order of the branches.

💡 How the stresses are worked out

Direct stress. σ = N/A − Mz·y/Iz + My·z/Iy, with y and z measured from the centroid — the convention OpenSees bends its fibre sections by, so a positive Mz compresses the +y side. It is checked at every corner of the shape, which is where a linear stress over a polygon has its extremes, and exactly for a round section. An I with unequal flanges or a pair of angles has its centroid away from the middle of the drawing, and the stresses are measured from it.

Shear and torsion. τ from Vy and Vz is the peak of the textbook (Jourawski) distribution, V·S / (I·b), worked out piece by piece so that each flange outstand of an I carries its own share. Torsion uses each shape's own formula:

Shape	τ from V	τ from T
Rectangular	1.5 V/A	Saint-Venant, within 0.5% of the tables
Circular, solid or tube	across a diameter: 4V/3A solid, → 2V/A thin	T·R / Ip
Rectangular Hollow	through both walls	Bredt, T / (2·Am·t)
I Section	mid-web for Vy, each flange outstand for Vz	T·t / J, J = Σ b·t³/3
Double L-Angle	both long legs for Vy, the short leg's root for Vz	T·t / J

Warning

Von Mises is conservative, mostly. It puts the worst σ with the worst τ, which rarely meet: bending peaks at the corners, where a free surface carries no shear, and shear peaks on the neutral axis, where bending is zero. Close to exact on slender members, conservative on short deep ones. It is not a strict bound: τV puts the peaks from Vy and Vz together as √(τy² + τz²), but where both act their shear flows can add along one wall of an I or a box, and the shear there is then higher than τV — by about a fifth on a box sheared equally both ways, and by a few percent under Vz alone on a box whose side walls are thinner than its flanges.

Note

A Generic Section is a list of properties with no shape behind it, so its beams give σN only; their other branches are empty, and the component says so.

📈 When to use it

Use it when

• You want the stress in a beam rather than the force, to compare members of different sections.
• You need σ and τ separately — a timber check, say, treats bending and shear apart.

Do not use it when

• You want a code check of steel members → Utilisation.
• You want to see the stresses → Model View, with Beam stresses ticked, colours the beams along their length — and, with Extruded on, paints the components that vary across the section point by point over it.

✅ How it is checked

The direct stresses are compared, fibre by fibre, with the stresses OpenSees integrates in elastic fibre sections — a round bar, a tube, an IPE 300, an unequal I and a pair of angles — to within the fibre mesh, a few hundredths of a percent. The shears and torsion match the published worked examples of Hibbeler's Mechanics of Materials to every digit printed.

Exact elasticity (a finite element solution of the cross-section) reads the shear from V up to 10% higher on thick round tubes, and about 4% on solid bars and thin tubes: Jourawski takes the shear as uniform across a cut, and on a circle it is not. That is beam theory, not a bug — it is what textbooks and EN 1993-1-1 6.2.6(4) use.