The MechCalc Clause 7: Fracture Assessment Calculator follows the assessment procedure of BS 7910:2019 Clause 7 to decide whether a known flaw is acceptable (BS 7910:2019, §7.2, §7.3).

Scope

What it can calculate

  • Edition: BS 7910:2019 and its Corrigendum No. 1, BS 7910:2019+C1:2025.
  • Type of analysis: initiation analysis, with one assessment point for the given flaw size (BS 7910:2019, §7.2.12 a)).
  • Failure assessment line (FAL): Option 1 (BS 7910:2019, §7.3.3, Eq. (7.26)–(7.28)) and Option 2 (BS 7910:2019, §7.3.4, Eq. (7.34), (7.35)); both can be used with continuous or discontinuous yielding.
  • Fracture ratio: $K_r=(K_I^P+K_I^S)/K_{mat}+\rho$ (BS 7910:2019, §7.3.6, Eq. (7.39)).
  • Primary stress intensity factor $K_I^P$: calculated with Annex M, or entered directly (for example from a finite element model) (BS 7910:2019, Annex M, §M.1).
  • Secondary stress intensity factor $K_I^S$: no secondary stress, calculated from $Q_m$ and $Q_b$ with Annex M (BS 7910:2019, Annex M, §M.1, Eq. (M.5)), or entered directly.
  • Plasticity interaction factor $\rho$: calculated with Annex R (BS 7910:2019, Annex R, §R.2.2, Eq. (R.1), (R.2)), entered directly, or zero.
  • Fracture toughness $K_{mat}$: entered directly.
  • Reference stress $\sigma_{ref}$: calculated with Annex P (BS 7910:2019, Annex P).
  • Stress magnifiers: $k_{tm}$, $k_{tb}$, $M_{km}$, $M_{kb}$ and $k_m$ are entered directly and act on $K_I^P$ through Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.4)).
  • 19 crack types; each label in the drop-down list gives the Annex M and Annex P clauses used (BS 7910:2019, Annex M, Annex P):
    • Plate, 7 types: surface semi-elliptical (M.4.1 / P.5.1), through-thickness (M.3.1 / P.4.1), embedded elliptical (M.4.5 / P.5.3), edge through-thickness (M.3.2 / P.4.2), surface, polynomial stress (M.4.2 / P.5.1), extended surface flaw (M.4.3 / P.5.2), extended surface flaw, polynomial stress (M.4.4 / P.5.2).
    • Cylinder, axial, 6 types: through-thickness (M.7.2.1 / P.8.1), internal surface (M.7.2.2 / P.8.2), extended internal surface flaw (M.7.2.3 / P.8.3), external surface (M.7.2.4 / P.8.4), extended external surface flaw (M.7.2.5 / P.8.5), embedded flaw (M.7.2.6 / P.9.6).
    • Cylinder, circumferential, 6 types: through-thickness (M.7.3.1 / P.9.1), internal surface (M.7.3.2 / P.9.2), fully circumferential internal surface flaw (M.7.3.3 / P.9.3), external surface (M.7.3.4 / P.9.4), fully circumferential external surface flaw (M.7.3.5 / P.9.5), embedded flaw (M.7.3.6 / P.9.6).
  • Two-point assessment: the 6 semi-elliptical surface flaw types (the plate surface semi-elliptical flaw and the plate surface flaw with polynomial stress, and the internal and external surface flaws in axial and circumferential cylinders) are assessed at the deepest point and at the surface point, and the flaw is acceptable only if both points are acceptable (cf. BS 7910:2019+C1:2025, Annex B, §B.5.6); the other types are assessed at one point.

What it cannot calculate

Each item below is not implemented in this calculator; this does not mean that the standard does not cover it.

  • The Option 3 FAL (BS 7910:2019, §7.3.5, Eq. (7.36), (7.37)).
  • Ductile tearing analysis (BS 7910:2019, §7.3.8).
  • The fracture ratio of Eq. (7.38), with $V$ applied to $K_I^S$ (BS 7910:2019, §7.3.6, Eq. (7.38); Annex R, §R.2.2, Eq. (R.3)).
  • $K_{mat}$ estimated from Charpy impact energy (BS 7910:2019, Annex J), converted from a J-integral or CTOD value (BS 7910:2019, §7.1.4.6, Eq. (7.15)–(7.17)), or found from weld specimen data (BS 7910:2019, Annex L).
  • $M_{km}$ and $M_{kb}$ found from the weld-toe geometry (BS 7910:2019, Annex M, §M.11.1.2, Eq. (M.43)), and $k_m$ found from the misalignment geometry (BS 7910:2019, Annex D, Eq. (D.1)).
  • Constraint correction (BS 7910:2019, Annex N) and strength mismatch correction (BS 7910:2019, Annex I).
  • $K_I^S$ from a welding residual stress distribution through weight functions (BS 7910:2019, Annex Q; Annex M, §M.4.2, §M.4.4).
  • Corner flaws (BS 7910:2019, Annex P, §P.6.1), corner flaws at holes (BS 7910:2019, Annex P, §P.6.2, §P.6.3), through-thickness flaws in spheres (BS 7910:2019, Annex P, §P.11), flaws in round bars and bolts (BS 7910:2019, Annex P, §P.12), and the limit load solutions for thick-walled cylinders (BS 7910:2019, Annex P, §P.8.6–§P.8.8, §P.10).
  • Flaw characterization and interaction (BS 7910:2019, §7.1.2): this calculator assesses the flaw size as entered.
  • Assessment with the $K_J^s$ of R6 when secondary stress acts alone (BS 7910:2019, Annex R, §R.1).
  • Creep assessment (BS 7910:2019, Clause 9). Fatigue crack growth (BS 7910:2019, Clause 8) is calculated by the Clause 8: Fatigue Crack Growth Calculator (E2) .

Input cards

Card Title When shown
— Assessment Option (FAD) Always; “Strain-hardening $n$ Source” only for Option 2, with $n$ shown when “Direct n value” is selected; “LĂŒders Strain Source” only for discontinuous yielding, with $\Delta\varepsilon$ shown when “Direct ΔΔ value” is selected
1 1. Crack / Flaw Type Always; $c$ is not shown for through-thickness, extended and fully circumferential types; $p$ only for the 3 embedded flaw types
2 2. Component Geometry Always; $W$ is not shown for the 6 circumferential cylinder types and the 2 plate extended surface flaw types; $r$ only for cylinders; “Bending Restraint ($\sigma_{ref}$)” only for the plate surface semi-elliptical flaw, the plate surface flaw with polynomial stress and the 2 extended surface flaw types
3 3. Stress & Polynomial Coefficients Always; the 2 polynomial stress types show $\sigma_0$–$\sigma_5$ and “$\sigma_{ref}$ Linearization Method” and do not show $P_m$ and $P_b$; $P_{m,b}$ only for the 6 circumferential cylinder types; $Q_m$ and $Q_b$ only when “Secondary (residual/thermal) load” is set to “Enter Q_m / Q_b stresses (→ K_I^S via Annex M)” and the type is not a polynomial stress type; the direct $K_I^P$ and $K_I^S$ appear with their sources, and the surface point fields only for the 6 two-point types
4 4. Stress Magnifiers (Weld Toe / Misalignment) Always; “Misalignment Bending Treated As” only when $k_m>1$; “Direct $K_I^P$ Includes Misalignment” only when $K_I^P$ is entered directly and $k_m>1$, and the calculation cannot run until it is answered; when “Stresses Include the Weld Toe” is “Yes — M_k taken as 1”, $M_{km}$ and $M_{kb}$ are greyed out and taken as 1
5 5. Material Always
6 6. Fracture Toughness Always
7 7. Plasticity Interaction Always; the direct $\rho$ appears with its source, and the surface point field only for the 6 two-point types

Inputs

Crack Type

The crack type sets both the component shape and the flaw type. The calculator uses the matching Annex M solution for $K_I$ and Annex P solution for $\sigma_{ref}$; the clause numbers of both are shown in brackets in each option.

Where to get the value

  • First pick the group under “Geometry” (flat plate, axial or circumferential flaw in a cylinder), then the type under “Flaw”.
  • Semi-elliptical surface flaws (in a flat plate, and on the inside or outside of a cylinder) are assessed at the deepest point and at the surface point; embedded flaws are assessed at the point nearer the surface; for a through-thickness flaw in a cylinder, $K_I^P$ is the larger of the values at the inside and outside surfaces.
  • The validity of the Annex M solutions is limited strictly to the ranges stated; no extrapolation outside these limits should be carried out (BS 7910:2019, Annex M, §M.1). When Annex P, or Annex M used to calculate $K_I^P$, is out of range, the calculator refuses to calculate and names the parameter, its limit and the clause.

Standard reference: BS 7910:2019, Annex M, §M.1

Crack Depth (a) [mm]

Half flaw length for a through-thickness flaw, flaw height for a surface flaw, or half height for an embedded flaw (BS 7910:2019, §7.0).

Where to get the value

The calculator does not carry out flaw characterization or flaw interaction (BS 7910:2019, §7.1.2); it assesses the value of $a$ as entered.

  • The calculator accepts only $a \ge 0.01$ mm.

Standard reference: BS 7910:2019, §7.0

Crack Half-length (c) [mm]

Half flaw length for a surface or embedded flaw (BS 7910:2019, §7.0).

Where to get the value

Through-thickness flaws, extended surface flaws and fully circumferential flaws do not use $c$; the field is hidden for these types.

  • The calculator accepts only $c \ge 0.01$ mm.

Standard reference: BS 7910:2019, §7.0

Wall Thickness (B) [mm]

Section thickness in plane of flaw (BS 7910:2019, §7.0).

Where to get the value

The calculator uses the same $B$ in the Annex M solution for $K_I$ and in the Annex P solution for $\sigma_{ref}$.

  • The calculator accepts only $B \ge 0.1$ mm.

Standard reference: BS 7910:2019, §7.0

Component Width (W) [mm]

Specimen or structural width (BS 7910:2019, §7.0).

Where to get the value

  • For an axial flaw in a cylinder, enter the length along the axis, marked $W$ in Figure M.14 and Figure M.15 (BS 7910:2019, Annex M, §M.7.2.1, Figure M.14; §M.7.2.2, Figure M.15).

  • For a surface flaw in a plate, Annex P uses the local solution when $W\ge2(c+B)$ and the global solution when $W\lt 2(c+B)$ (BS 7910:2019, Annex P, §P.5.1).

  • Not shown for circumferential flaws in a cylinder or for extended surface flaws in a plate.

  • The calculator accepts only $W \ge 1$ mm.

Standard reference: BS 7910:2019, §7.0; Annex M, §M.7.2.1, Figure M.14; §M.7.2.2, Figure M.15; Annex P, §P.5.1

Embedded Ligament Distance (p) [mm]

Shortest distance from surface to embedded flaw (BS 7910:2019, §7.0).

Where to get the value

  • For an embedded flaw, Annex M uses the effective thickness $B'=2a+2p$ (BS 7910:2019, Annex M, §M.4.5.1).

  • Shown only for embedded flaws.

  • The calculator accepts only $p \ge 0.01$ mm.

Standard reference: BS 7910:2019, §7.0; Annex M, §M.4.5.1

Geometric Radius (r) [mm]

For cylinder crack types, enter the internal radius $r_i$ in this field (BS 7910:2019, Annex M, §M.0).

Where to get the value

Shown only for flaws in cylinders. For circumferential flaws the calculator takes $W=2\pi r_m$ with $r_m=r_i+B/2$.

  • The calculator accepts only $r \ge 0.1$ mm.

Standard reference: BS 7910:2019, Annex M, §M.0

Primary Membrane Stress (P_m) [MPa]

Primary membrane stress (BS 7910:2019, §7.0). Primary stresses come from loads that contribute to plastic collapse (BS 7910:2019, §7.2.2).

Where to get the value

  • For circumferential flaws in a cylinder, enter only the membrane stress from axial load and pressure; enter the membrane stress from global bending in “Membrane Stress due to Global Bending ($P_{m,b}$)”.

  • Not shown for the two polynomial stress crack types, which use $\sigma_0$ to $\sigma_5$ instead.

  • The button “Linearize primary stress” next to “Get from:” fills $P_m$ and $P_b$ from the stress linearization calculator.

  • The calculator accepts only $P_m \ge 0$ MPa.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ by Eq. (M.4) and higher $\sigma_{ref}$, so higher $K_r$ and $L_r$ (BS 7910:2019, Annex M, §M.1, Eq. (M.4); §7.3.7, Eq. (7.40)).

Standard reference: BS 7910:2019, §7.0; §7.2.2; Annex M, §M.1, Eq. (M.4); §7.3.7, Eq. (7.40)

Membrane Stress due to Global Bending (P_m,b) [MPa]

Primary membrane stress due to global bending moments (BS 7910:2019, Annex P, §P.0).

Where to get the value

  • Shown only for circumferential flaws in a cylinder. Enter the tensile stress at the flaw from global bending; enter 0 if the flaw is in compression.

  • From a global moment: $P_{m,b}=4M_{global}(r_i+a)/\{\pi[(r_i+B)^4-r_i^4]\}$ (BS 7910:2019, Annex M, §M.7.3.2, Eq. (M.29)).

  • Annex M includes a global bending moment by adding $P_{m,b}$ to $P_m$ (BS 7910:2019, Annex M, §M.7.3.2). In Annex P, Eq. (P.22) has a separate $P_{m,b}$ term (BS 7910:2019, Annex P, §P.9.1, Eq. (P.22)), and in P.9.2 $P_m$ is the total membrane stress due to external bending, axial loads and pressure (BS 7910:2019, Annex P, §P.9.2).

  • The calculator accepts only $P_{m,b} \ge 0$ MPa.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ and $\sigma_{ref}$ (BS 7910:2019, Annex M, §M.7.3.2; Annex P, §P.9.1, Eq. (P.22)).

Standard reference: BS 7910:2019, Annex P, §P.0; Annex M, §M.7.3.2, Eq. (M.29); Annex P, §P.9.1, Eq. (P.22); §P.9.2

Primary Bending Stress (P_b) [MPa]

Primary bending stress (BS 7910:2019, §7.0).

Where to get the value

  • $P_b$ may be negative. The signed $P_b$ is used for $K_I$, and $|P_b|$ for $\sigma_{ref}$.
  • Sign convention of the calculator:surface flaws: tension at the surface containing the flaw is positive;
  • embedded flaws: tension at the surface nearer the flaw is positive;
  • through-thickness and edge flaws in a plate: tension on the $z=0$ side is positive, and the calculator evaluates Annex M at $z=0$;
  • through-thickness flaws in a cylinder: tension on the outside surface is positive, as in the Annex M definition of through-wall bending (tensile on outside surface, compressive on inside surface) (BS 7910:2019, Annex M, §M.7.2.1).

If the primary stresses give $K_I^P\lt 0$, the calculator refuses to calculate.

Standard reference: BS 7910:2019, §7.0; Annex M, §M.7.2.1

Bending Restraint (σ_ref)

Annex P gives two equations for the reference stress of a surface flaw in a plate: Eq. (P.9) for normal bending restraint (e.g. fixed-grip tension plus bending) and Eq. (P.10) for negligible bending restraint (e.g. pin jointed) (BS 7910:2019, Annex P, §P.5.1, Eq. (P.9), (P.10)). This field selects which equation the calculator uses.

Where to get the value

  • Shown for surface flaws and extended surface flaws in a plate; for extended surface flaws both equations use $a''=a/B$ (BS 7910:2019, Annex P, §P.5.2).
  • The default is “Restrained (Eq. P.9, fixed-grip)”.

Effect on the results

  • Pin-jointed: Eq. (P.10) replaces $P_b$ in Eq. (P.9) by $P_b+3P_ma''$, so for the same flaw and stresses it gives a higher $\sigma_{ref}$ and $L_r$ (BS 7910:2019, Annex P, §P.5.1, Eq. (P.9), (P.10)).

Standard reference: BS 7910:2019, Annex P, §P.5.1, Eq. (P.9), (P.10); §P.5.2

Secondary Membrane Stress (Q_m) [MPa]

Secondary membrane stress (BS 7910:2019, §7.0). Secondary stresses come from loads that do not contribute to plastic collapse (BS 7910:2019, §7.2.2).

Where to get the value

  • Thermal and residual stresses act as primary stresses when the flaw is small compared with their zone of influence, for example with large elastic follow-up, and are then included in $K_r$ and $L_r$ (BS 7910:2019, §7.1.8). Where it is not clear whether elastic follow-up is present, the loading is classified as load-controlled and so primary (BS 7910:2019, §7.1.11). The calculator includes such stresses in $L_r$ when they are entered in $P_m$ and $P_b$.
  • Treating all loads as primary is a simplified analysis; it underestimates margins on load provided that the secondary stresses are tensile (BS 7910:2019, §7.2.2).
  • Thermal stresses (primary and secondary) should be multiplied by the appropriate stress concentration and misalignment factors $k_t$ and $k_m$ (BS 7910:2019+C1:2025, §6.4.3); secondary stresses are not influenced by structural discontinuities, except thermal stresses treated as primary (BS 7910:2019+C1:2025, §6.4.4.1).
  • The calculator uses $(Y\sigma)_s=M_mQ_m+M_bQ_b$, which contains no $k_t$ or $k_m$ (BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)), so $Q_m$ and $Q_b$ are entered as the local stresses at the flaw from a stress analysis of the unflawed structure (BS 7910:2019+C1:2025, §6.4.1):local values from a finite element model of the unflawed structure with the discontinuity: enter as they are;
  • a nominal thermal stress with the flaw in a structural discontinuity: first multiply by the $k_t$ for the thermal load (and by $k_m$ for a thermal membrane force across a joint); without this the result is non-conservative;
  • welding residual stress: enter the local value at the flaw.

“⊞ Break down secondary stress” splits $Q_m$ and $Q_b$ into a non-weld part and a weld residual part and adds them up.

Effect on the results

  • ↑ Increase: Higher $K_I^S$ by Eq. (M.5), so higher $K_r$ by Eq. (7.39); $L_r$ does not change (BS 7910:2019, Annex M, §M.1, Eq. (M.5); §7.3.6, Eq. (7.39); §7.3.7).

Standard reference: BS 7910:2019, §7.0; §7.1.8; §7.1.11; §7.2.2; BS 7910:2019+C1:2025, §6.4.1; §6.4.3; §6.4.4.1; Annex M, §M.1, Eq. (M.5)

Secondary Bending Stress (Q_b) [MPa]

Secondary bending stress (BS 7910:2019, §7.0).

Where to get the value

Enter it in the same way as “Secondary Membrane Stress ($Q_m$)”, as the local value at the flaw. The calculator uses it in the term $M_bQ_b$ of Eq. (M.5) and does not multiply it by $k_{tb}$ or $k_m$ (BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)).

Effect on the results

  • ↑ Increase: Higher $K_I^S$, so higher $K_r$; $L_r$ does not change (BS 7910:2019, Annex M, §M.1, Eq. (M.5); §7.3.6, Eq. (7.39); §7.3.7).

Standard reference: BS 7910:2019, §7.0; BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Non-weld secondary membrane [MPa]

The membrane part of the non-weld secondary stress, shown after “⊞ Break down secondary stress”. The calculator adds it to “Weld residual membrane” and uses the sum as $Q_m$.

Where to get the value

A blank field counts as 0. The button “Linearize secondary stress” next to “Get from:” takes the membrane and bending parts from the stress linearization calculator.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Non-weld secondary bending [MPa]

The bending part of the non-weld secondary stress, shown after “⊞ Break down secondary stress”. The calculator adds it to “Weld residual bending” and uses the sum as $Q_b$.

Where to get the value

A blank field counts as 0. The button “Linearize secondary stress” of “Non-weld secondary membrane” also fills this field.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Weld residual membrane [MPa]

The membrane part of the welding residual stress, shown after “⊞ Break down secondary stress”. The calculator adds it to “Non-weld secondary membrane” and uses the sum as $Q_m$.

Where to get the value

A blank field counts as 0. The button “Weld Residual (Annex Q)” next to “Get from:” takes the membrane and bending parts from the Annex Q residual stress calculator.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Weld residual bending [MPa]

The bending part of the welding residual stress, shown after “⊞ Break down secondary stress”. The calculator adds it to “Non-weld secondary bending” and uses the sum as $Q_b$.

Where to get the value

A blank field counts as 0. The button “Weld Residual (Annex Q)” of “Weld residual membrane” also fills this field.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Membrane Stress Concentration (k_tm)

Membrane stress concentration factor at a gross structural discontinuity, such as a pressure vessel nozzle or a major intersection in a tubular structure; the applied stress is multiplied by it for calculating the crack driving force $K_I$ (BS 7910:2019, §6.4.4.2).

Where to get the value

  • The calculator applies it to the membrane term of $K_I^P$ by Eq. (M.4) and does not use it for $\sigma_{ref}$ or $K_I^S$ (BS 7910:2019, Annex M, §M.1, Eq. (M.4)). The default is 1.0.

  • For a flaw within the stress concentration of a local structural discontinuity (hole, notch, sharp corner), the effect on both fracture and plastic collapse should be taken into account, and the peak stress may be calculated by multiplying the applied stress by $k_t$ (BS 7910:2019, §6.4.4.4); the calculator counts such peak stresses for plastic collapse when they are entered in $P_m$ and $P_b$.

  • The calculator accepts only $k_{tm} \ge 1$.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ and $K_r$; $L_r$ does not change (BS 7910:2019, §6.4.4.2; Annex M, §M.1, Eq. (M.4)).

Standard reference: BS 7910:2019, §6.4.4.2; §6.4.4.4; Annex M, §M.1, Eq. (M.4)

Bending Stress Concentration (k_tb)

Bending stress concentration factor (BS 7910:2019, §7.0) at a gross structural discontinuity, used for calculating the crack driving force $K_I$ (BS 7910:2019, §6.4.4.2).

Where to get the value

The calculator applies it to the bending term $P_b+(k_m-1)P_m$ of $K_I^P$ by Eq. (M.4) and does not use it for $\sigma_{ref}$ or $K_I^S$ (BS 7910:2019, Annex M, §M.1, Eq. (M.4)). The default is 1.0.

  • The calculator accepts only $k_{tb} \ge 1$.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ and $K_r$; $L_r$ does not change (BS 7910:2019, §6.4.4.2; Annex M, §M.1, Eq. (M.4)).

Standard reference: BS 7910:2019, §7.0; §6.4.4.2; Annex M, §M.1, Eq. (M.4)

Stresses Include the Weld Toe

Whether $P_m$ and $P_b$ come from an analysis that already includes the weld toe geometry, such as a finite element model with the weld toe.

Where to get the value

  • Stress concentration effects due to weld toes do not contribute to plastic collapse, but they increase the stress intensity factor (BS 7910:2019, §6.4.4.4).
  • With “Yes — M_k taken as 1”, the calculator uses $M_{km}=M_{kb}=1$, shows both fields greyed out at 1, and notes this in the calculation steps and the report.
  • The default is “No”.

Standard reference: BS 7910:2019, §6.4.4.4

Weld Toe Membrane Factor (M_km)

Stress intensity magnification factor for a flaw at the weld toe, membrane component (BS 7910:2019, §7.0).

Where to get the value

The calculator applies it to the membrane term of $K_I^P$ by Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.4)). The default is 1.0; with “Yes — M_k taken as 1” in “Stresses Include the Weld Toe”, it is fixed at 1.

  • The calculator accepts only $M_{km} \ge 1$.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ and $K_r$; $L_r$ does not change (BS 7910:2019, §6.4.4.4; Annex M, §M.1, Eq. (M.4)).

Standard reference: BS 7910:2019, §7.0; §6.4.4.4; Annex M, §M.1, Eq. (M.4)

Weld Toe Bending Factor (M_kb)

Stress intensity magnification factor for a flaw at the weld toe, bending component (BS 7910:2019, §7.0).

Where to get the value

The calculator applies it to the bending term $P_b+(k_m-1)P_m$ of $K_I^P$ by Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.4)). The default is 1.0; with “Yes — M_k taken as 1” in “Stresses Include the Weld Toe”, it is fixed at 1.

  • The calculator accepts only $M_{kb} \ge 1$.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ and $K_r$; $L_r$ does not change (BS 7910:2019, §6.4.4.4; Annex M, §M.1, Eq. (M.4)).

Standard reference: BS 7910:2019, §7.0; §6.4.4.4; Annex M, §M.1, Eq. (M.4)

Misalignment Factor (k_m)

Stress magnification factor due to misalignment (BS 7910:2019, Annex D, §D.0): $k_m=1+\sigma_s/P_m$, where $\sigma_s$ is the bending stress due to misalignment (BS 7910:2019, Annex D, §D.1, Eq. (D.1)).

Where to get the value

The calculator adds $(k_m-1)P_m$ to the bending term of $K_I^P$ by Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.4)). The default is 1.0; for $k_m\gt 1$, “Misalignment Bending Treated As” is shown.

  • The calculator accepts only $k_m \ge 1$.

Effect on the results

  • ↑ Increase: Higher $K_I^P$ by Eq. (M.4); with the default primary treatment, also higher $\sigma_{ref}$ (BS 7910:2019, Annex M, §M.1, Eq. (M.4)).

Standard reference: BS 7910:2019, Annex D, §D.0; §D.1, Eq. (D.1); Annex M, §M.1, Eq. (M.4)

Misalignment Bending Treated As

How the misalignment bending stress $(k_m-1)P_m$ is treated. “Primary stress (in K_I^P and σ_ref)”: it enters $K_I^P$ by Eq. (M.4) and $\sigma_{ref}$, where Annex P uses the bending stress $P_b+(k_m-1)P_m$. “Secondary stress (in K_r only, not in L_r)”: its K contribution is added to $K_I^S$ and it does not enter $\sigma_{ref}$.

Where to get the value

  • It is conservative to treat stresses due to misalignment as primary, although in some cases they act as secondary, i.e. they affect $K_r$ but not $L_r$ (BS 7910:2019, Annex D, §D.1).
  • The bending stress due to misalignment is given as a function of the applied membrane stress $P_m$ (BS 7910:2019+C1:2025, Annex D, §D.1); with “Secondary stress (in K_r only, not in L_r)”, its K contribution still grows in proportion to the primary load when the reserve factor on load is found. When angular misalignment is combined with axial misalignment, the angular component reduces the overall stress (BS 7910:2019+C1:2025, Annex D, §D.1, Table D.1); the net bending then grows faster than the load, and this proportional growth is non-conservative.
  • Shown only for $k_m\gt 1$.

Standard reference: BS 7910:2019, Annex D, §D.1; BS 7910:2019+C1:2025, Annex D, §D.1, Table D.1; Annex M, §M.1, Eq. (M.4)

Direct K_I^P Includes Misalignment

Whether the directly entered $K_I^P$ (both points for a semi-elliptical surface flaw) already includes the misalignment bending, for example from a finite element model of the misaligned geometry. Shown only when $K_I^P$ is entered directly and $k_m\gt 1$, and it must be chosen.

Where to get the value

  • “Yes — the direct value includes the misalignment bending”: no misalignment term is added.
  • “No — add it from Annex M”: the calculator works out Annex M twice, with the entered $k_m$ and with $k_m=1$, and takes the difference as the misalignment K increment $\Delta K_{km}$, the contribution of the $(k_m-1)P_m$ term of Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.4)); it adds $\Delta K_{km}$ to $K_I^P$ for primary treatment or to $K_I^S$ for secondary treatment. If Annex M is out of range, the calculator refuses to calculate.
  • With “Yes” and secondary treatment, the misalignment contribution stays in the direct $K_I^P$ and does not enter the Annex R ratio $K_I^S/(K_I^P/L_r)$, so $\rho$ is slightly lower than with it in $K_I^S$, which is non-conservative; $\rho$ is limited by $0\le\rho\le\rho_1\le0.25$ (BS 7910:2019, Annex R, §R.2.2, Eq. (R.1), (R.2)).
  • Until a choice is made, the calculation cannot start.

Standard reference: BS 7910:2019, Annex M, §M.1, Eq. (M.4); Annex R, §R.2.2, Eq. (R.1), (R.2)

Polynomial Stress σ_0 [MPa]

$\sigma_i$ ($i=0$ to $5$) are the stress components that define the through-wall stress distribution $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$ for $0\le x\le B$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)); $\sigma_0$ is the constant term. The calculator measures $x$ from the surface at the crack mouth.

Where to get the value

  • Shown only for the two polynomial stress crack types. $K_I$ is calculated from all $\sigma_i$ by Eq. (M.13) or Eq. (M.17) (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.13); §M.4.4, Eq. (M.17)); $\sigma_{ref}$ is calculated from the $P_m$ and $P_b$ obtained by linearizing this distribution, and the $P_m$ and $P_b$ fields are not used.
  • The button “Fit from stress profile” next to “Get from:” fits $\sigma_0$ to $\sigma_5$ to a stress profile in the stress linearization calculator.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.13), (M.14); §M.4.4, Eq. (M.17)

Polynomial Stress σ_1 [MPa]

In $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$, $\sigma_1$ is the coefficient of $x/B$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)).

Where to get the value

Shown and used in the same way as $\sigma_0$.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)

Polynomial Stress σ_2 [MPa]

In $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$, $\sigma_2$ is the coefficient of $(x/B)^2$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)).

Where to get the value

Shown and used in the same way as $\sigma_0$.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)

Polynomial Stress σ_3 [MPa]

In $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$, $\sigma_3$ is the coefficient of $(x/B)^3$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)).

Where to get the value

Shown and used in the same way as $\sigma_0$.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)

Polynomial Stress σ_4 [MPa]

In $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$, $\sigma_4$ is the coefficient of $(x/B)^4$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)).

Where to get the value

Shown and used in the same way as $\sigma_0$.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)

Polynomial Stress σ_5 [MPa]

In $\sigma(x)=\sum_{i=0}^{5}\sigma_i(x/B)^i$, $\sigma_5$ is the coefficient of $(x/B)^5$ (BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)).

Where to get the value

Shown and used in the same way as $\sigma_0$.

Standard reference: BS 7910:2019, Annex M, §M.4.2.2, Eq. (M.14)

σ_ref Linearization Method

The calculator linearizes the polynomial stress distribution into $P_m$ and $P_b$ and uses them in Annex P for $\sigma_{ref}$; $K_I$ still uses the full polynomial. This field selects the linearization method. Any linearized distribution of stress is acceptable provided that it is greater than or equal to the magnitude of the real distribution over the flaw surface (BS 7910:2019+C1:2025, §6.4.1, NOTE).

Where to get the value

  • “Integral method (work-equivalent)”: $P_m=\frac{1}{B}\int_0^B\sigma\,dx$ and $P_b=\frac{6}{B^2}\int_0^B\sigma\left(\frac{B}{2}-x\right)dx$. This is the default.
  • “Clause 6 linearization (Fig 6.1)”: $P_m=(\sigma_1+\sigma_2)/2$ and $P_b=(\sigma_1-\sigma_2)/2$ (BS 7910:2019, §6.4.1, Figure 6.1); here $\sigma_1$ and $\sigma_2$ are the two stresses set by “Clause 6 Span”, not polynomial coefficients.
  • With either method, if the linearized stress is below the polynomial at any point of the flaw surface ($0\le x\le a$), the results show a warning with the largest shortfall and where it occurs; the calculation goes on.
  • A linearized $P_m$ below 0 is taken as 0.

Standard reference: BS 7910:2019, §6.4.1, Figure 6.1; BS 7910:2019+C1:2025, §6.4.1, NOTE

Clause 6 Span

Linearization may be carried out across the flaw (Figure 6.1a) or across the section containing the flaw (Figure 6.1b) (BS 7910:2019, §6.4.1, Figure 6.1). This field sets which two stresses $\sigma_1$ and $\sigma_2$ the Clause 6 method uses.

Where to get the value

  • “Across full thickness (σ at x=0 and x=B)”: $\sigma_1=\sigma(0)$ and $\sigma_2=\sigma(B)$. This is the default.
  • “Across flaw (fit over crack depth a, Fig 6.1a)”: the calculator fits a straight line over $0\le x\le a$ by least squares, shifts it up until it is not below the polynomial, and extends it over the full thickness.
  • Shown only with “Clause 6 linearization (Fig 6.1)”.

Standard reference: BS 7910:2019, §6.4.1, Figure 6.1

Yield Strength (σ_Y) [MPa]

Lower yield strength or 0.2% proof strength (BS 7910:2019, §7.0).

Where to get the value

  • Option 1: $R_{p0.2}$, $R_{p1.0}$, $R_{eL}$ or $R_{t0.5}$; where a steel product standard defines yield strength as the upper yield strength $R_{eH}$, $\sigma_Y=0.95R_{eH}$. Option 2: $R_{p0.2}$ for ferritic steels, $R_{p1.0}$ for austenitic steels (BS 7910:2019, §7.1.3.1, Table 7.1).

  • Mean minus two standard deviation tensile properties are used for $K_r$ and $L_r$ of the assessment point, and mean tensile properties for the FAL and $L_{r,max}$ (BS 7910:2019, §7.1.3.2; §7.3.2). The calculator uses one set of $\sigma_Y$ and $\sigma_U$ for $L_r$, the FAL and $L_{r,max}$.

  • The calculator accepts only $\sigma_Y \ge 1$ MPa.

Effect on the results

  • ↑ Increase: Lower $L_r=\sigma_{ref}/\sigma_Y$ (BS 7910:2019, §7.3.7, Eq. (7.40)).

Standard reference: BS 7910:2019, §7.0; §7.1.3.1, Table 7.1; §7.1.3.2; §7.3.2; §7.3.7, Eq. (7.40)

Tensile Strength (σ_U) [MPa]

Tensile strength (BS 7910:2019, §7.0); it corresponds to $R_m$ (BS 7910:2019, §7.1.3.1, Table 7.1).

Where to get the value

  • $L_{r,max}=(\sigma_Y+\sigma_U)/(2\sigma_Y)$; for the cut-off, mean rather than minimum tensile properties should be used (BS 7910:2019, §7.3.2, Eq. (7.25)).

  • The calculator refuses to calculate when $\sigma_U\le\sigma_Y$: Eq. (7.7) and $N=0.3(1-\sigma_Y/\sigma_U)$ need $\sigma_U\gt\sigma_Y$ (BS 7910:2019, §7.3.3).

  • The calculator accepts only $\sigma_U \ge 1$ MPa.

Effect on the results

  • ↑ Increase: Higher $L_{r,max}$ (BS 7910:2019, §7.3.2, Eq. (7.25)).

Standard reference: BS 7910:2019, §7.0; §7.1.3.1, Table 7.1; §7.3.2, Eq. (7.25); §7.3.3

Elastic Modulus (E) [GPa]

Elastic modulus (BS 7910:2019, §7.0).

Where to get the value

  • Entered in GPa; the calculator converts it to MPa.

  • The calculator uses it in $\mu=\min(0.001E/\sigma_Y,0.6)$ for Option 1 (BS 7910:2019, §7.3.3, Eq. (7.26)–(7.28)), in $\lambda=1+E\Delta\varepsilon/R_{eL}$ for discontinuous yielding (BS 7910:2019, §7.3.3, Eq. (7.33)) and in Eq. (7.34) for Option 2 (BS 7910:2019, §7.3.4, Eq. (7.34)).

  • The calculator accepts only $E \ge 10$ GPa.

Standard reference: BS 7910:2019, §7.0; §7.3.3, Eq. (7.26)–(7.28), (7.33); §7.3.4, Eq. (7.34)

Fracture Toughness (K_mat) [MPa√m]

Characteristic material fracture toughness determined in terms of stress intensity factor (BS 7910:2019, §7.0).

Where to get the value

The fracture toughness should correspond to the material containing the crack tip (BS 7910:2019, §7.2.3).

  • The calculator accepts only $K_{mat} \ge 0.1$ MPa√m.

Effect on the results

  • ↑ Increase: Lower $K_r$ by Eq. (7.39) (BS 7910:2019, §7.3.6, Eq. (7.39)).

Standard reference: BS 7910:2019, §7.0; §7.2.3; §7.3.6, Eq. (7.39)

FAD Option

Each option has a failure assessment line (FAL) given by the equation of a curve, $K_r=f(L_r)$, and a cut-off value $L_{r,max}$ (BS 7910:2019, §7.3.1). This field selects which FAL the calculator uses.

Where to get the value

  • “Option 1 (Section 7.3.3)”: the FAL is given by Eq. (7.26)–(7.28) with $\mu=\min(0.001E/\sigma_Y,0.6)$ and $N=0.3(1-\sigma_Y/\sigma_U)$ (BS 7910:2019, §7.3.3, Eq. (7.26)–(7.28)).
  • “Option 2 (Section 7.3.4, Ramberg-Osgood)”: $f(L_r)=\left(E\varepsilon_{ref}/(L_r\sigma_Y)+L_r^3\sigma_Y/(2E\varepsilon_{ref})\right)^{-0.5}$ (BS 7910:2019, §7.3.4, Eq. (7.34)); the standard asks for the mean uniaxial tensile true stress-strain curve at the assessment temperature (BS 7910:2019, §7.3.4). The calculator takes $\varepsilon_{ref}$ as the strain of the stress-strain curve of Eq. (7.4) at $\sigma_{ref}=L_r\sigma_Y$, and uses this engineering strain without conversion to true strain (BS 7910:2019, §7.1.3.5, Eq. (7.4)).
  • The default is “Option 1 (Section 7.3.3)”.

Standard reference: BS 7910:2019, §7.3.1; §7.3.3, Eq. (7.26)–(7.28); §7.3.4, Eq. (7.34); §7.1.3.5, Eq. (7.4)

Yielding Type

Whether the stress-strain curve of the material shows a yield discontinuity (LĂŒders plateau) (BS 7910:2019, §7.1.3.6). For materials with discontinuous yielding, the Option 1 FAL uses Eq. (7.29)–(7.32) instead of Eq. (7.26)–(7.28) (BS 7910:2019, §7.3.3).

Where to get the value

  • Table 7.4 gives guidance on whether rolled steel plate yields continuously or discontinuously; if there is uncertainty or data are not available, a LĂŒders plateau should be assumed for $L_r\ge1$ (BS 7910:2019, §7.1.3.6, Table 7.4).
  • With “Discontinuous (LĂŒders plateau)”, the Option 1 FAL drops vertically at $L_r=1$ from Eq. (7.29) to Eq. (7.30) (BS 7910:2019, §7.3.3), and the calculator uses the lower value for $L_r\ge1$; for Option 2 the calculator adds the LĂŒders strain $\Delta\varepsilon$ to $\varepsilon_{ref}$ (for $L_r\ge1$) (BS 7910:2019, §7.1.3.6).
  • $R_{eL}$ in Eq. (7.33) is taken as the value of “Yield Strength ($\sigma_Y$)”.
  • The default is “Continuous (smooth, no plateau)”.

Standard reference: BS 7910:2019, §7.1.3.6, Table 7.4; §7.3.3, Eq. (7.29)–(7.33)

Primary SIF (K_I^P) Source

Where $K_I^P$ comes from: the Annex M formula, or a value found by another method and entered directly. Stress intensity factors may also be found from handbooks, numerical analysis (e.g. finite elements) or weight functions, but the basis of the method and the results should be fully documented (BS 7910:2019+C1:2025, Annex M, §M.1).

Where to get the value

  • With “Enter K_I^P directly (e.g. from a finite element model)”, the calculator still works out the Annex M value and lists it next to the direct value in the calculation steps; if the Annex M solution is out of range, this is only recorded in the results and the direct value is used.
  • The results and the report mark a direct $K_I^P$ as a user input.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1

Primary SIF (K_I^P, direct) [MPa√m]

Stress intensity factor due to primary stress (BS 7910:2019, §7.0), entered directly: the deepest-point value for a semi-elliptical surface flaw, the single value for other flaw types (for a through-thickness flaw, the value at the controlling crack front).

Where to get the value

  • If $k_m\gt 1$, also choose “Direct $K_I^P$ Includes Misalignment” in card 4; the calculation cannot start until it is chosen.

  • With “Annex R (calculated) — default”, $\rho$ is found from this value (including any misalignment K increment added to it).

  • The calculator accepts only $K_I^P \ge 0$ MPa√m.

Standard reference: BS 7910:2019, §7.0

Primary SIF at Surface Point (K_I^P, direct) [MPa√m]

The directly entered surface-point $K_I^P$ for a semi-elliptical surface flaw. For these flaws Annex M gives $K_I$ at the deepest point and at the surface point (BS 7910:2019, Annex M, §M.4.1.2.3; §M.4.1.3.3; §M.7.3.2, Table M.10).

Where to get the value

  • The calculator builds one assessment point at the deepest point and one at the surface point, with the same $L_r$; if either point lies on or outside the FAL, the flaw is unacceptable, following by analogy the two-point rule of B.5.6 (BS 7910:2019+C1:2025, Annex B, §B.5.6).

  • Shown only for semi-elliptical surface flaws.

  • The calculator accepts only $K_I^P \ge 0$ MPa√m.

Standard reference: BS 7910:2019, Annex M, §M.4.1.2.3; §M.4.1.3.3; §M.7.3.2, Table M.10; BS 7910:2019+C1:2025, Annex B, §B.5.6

Secondary (residual/thermal) load

$K_I^S$ is the stress intensity factor due to secondary stress (BS 7910:2019, §7.0). This field selects where it comes from.

Where to get the value

  • “None (no secondary stress)”: $K_I^S=0$.
  • “Enter Q_m / Q_b stresses (→ K_I^S via Annex M)”: $K_I^S=(M_mQ_m+M_bQ_b)\sqrt{\pi a}$ (BS 7910:2019, Annex M, §M.1, Eq. (M.1), (M.5)), with the same Annex M solution and the same assessment point as $K_I^P$.
  • “Enter K_I^S directly”: the value of “Secondary SIF ($K_I^S$, direct)” is used.
  • If $K_I^S$ is negative, $K_I^S$ and $\rho$ are set to zero (BS 7910:2019, §7.3.6).
  • The equations of R.2 are not applicable when there are secondary stresses only; the relevant stresses may then be conservatively classified as primary (BS 7910:2019, Annex R, §R.1). The calculator refuses to calculate when, at any assessment point, $K_I^P=0$ and $K_I^S\gt 0$.

Standard reference: BS 7910:2019, §7.0; §7.3.6; Annex M, §M.1, Eq. (M.1), (M.5); Annex R, §R.1

Secondary SIF (K_I^S, direct) [MPa√m]

Stress intensity factor due to secondary stress (BS 7910:2019, §7.0), entered directly: the deepest-point value for a semi-elliptical surface flaw, the single value for other flaw types.

Where to get the value

The value is taken as not including the misalignment bending; when the misalignment bending is treated as secondary, the calculator adds the misalignment K increment from Annex M to $K_I^S$.

Effect on the results

  • ↑ Increase: Higher $K_r$ by Eq. (7.39); $L_r$ does not change (BS 7910:2019, §7.3.6, Eq. (7.39); §7.3.7).

Standard reference: BS 7910:2019, §7.0; §7.3.6, Eq. (7.39)

Secondary SIF at Surface Point (K_I^S, direct) [MPa√m]

The directly entered surface-point $K_I^S$ for a semi-elliptical surface flaw. Eq. (M.5) uses a separate pair of $M_m$, $M_b$ at the deepest point and at the surface point (BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)).

Where to get the value

Shown only for semi-elliptical surface flaws; like the deepest-point value, it is taken as not including the misalignment bending.

Standard reference: BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)

Plasticity Interaction (ρ) Source

$\rho$ is the plasticity interaction factor (BS 7910:2019, §7.0), defined in Annex R as a function of both the primary and secondary loads (BS 7910:2019, §7.3.6). This field selects where $\rho$ comes from.

Where to get the value

  • “Annex R (calculated) — default”: $\rho$ is found from $K_I^P$, $K_I^S$ and $L_r$ by Eq. (R.1) and (R.2) (BS 7910:2019, Annex R, §R.2.2, Eq. (R.1), (R.2)), once at each assessment point; with no secondary stress it gives $\rho=0$.
  • “Direct ρ value”: the value of “Plasticity Interaction ($\rho$, direct)” is used.
  • “Zero (no secondary stress only)”: $\rho=0$; if $K_I^S\gt 0$, the results show a non-conservative warning.
  • For $K_I^S/(K_I^P/L_r)\gt 4$, the values of $\rho$ from R.2.2 could possibly be over-conservative (BS 7910:2019, Annex R, §R.2.1); the results then show a warning.

Standard reference: BS 7910:2019, §7.0; §7.3.6; Annex R, §R.2.1; §R.2.2, Eq. (R.1), (R.2)

Plasticity Interaction (ρ, direct)

Plasticity interaction factor (BS 7910:2019, §7.0), entered directly: the deepest-point value for a semi-elliptical surface flaw, the single value for other flaw types.

Where to get the value

  • $0\le\rho\le\rho_1$ (BS 7910:2019, Annex R, §R.2.2, NOTE 1), and $\rho_1$ is at most 0.25 (BS 7910:2019, Annex R, §R.2.2, Eq. (R.1)).

  • At an assessment point where $K_I^S$ is set to zero, $\rho$ is also set to zero (BS 7910:2019, §7.3.6).

  • The calculator accepts only $0 \le \rho \le 0.25$.

Effect on the results

  • ↑ Increase: Higher $K_r$ by Eq. (7.39); $L_r$ does not change (BS 7910:2019, §7.3.6, Eq. (7.39); §7.3.7).

Standard reference: BS 7910:2019, §7.0; §7.3.6, Eq. (7.39); Annex R, §R.2.2, Eq. (R.1), NOTE 1

Plasticity Interaction at Surface Point (ρ, direct)

The directly entered surface-point $\rho$ for a semi-elliptical surface flaw. $\rho$ depends on $K_I^S/(K_I^P/L_r)$ and on $L_r$ (BS 7910:2019+C1:2025, Annex R, §R.2.2).

Where to get the value

Shown only for semi-elliptical surface flaws; the range is the same as for the deepest-point value.

  • The calculator accepts only $0 \le \rho \le 0.25$.

Standard reference: BS 7910:2019+C1:2025, Annex R, §R.2.2

Strain-hardening n Source

$n$ is the strain-hardening exponent (BS 7910:2019, §7.0). This field sets where the $n$ used by Option 2 comes from.

Where to get the value

  • “Estimate from σ_Y/σ_U (Eq. 7.7)”: $n=1/[0.3(1-\sigma_Y/\sigma_U)]$ (BS 7910:2019, §7.1.3.5, Eq. (7.7)). Eq. (7.7) was derived from ferritic steels with yield/proof strengths from 300 MPa to 1000 MPa and $0.65\lt\sigma_Y/\sigma_U\lt 0.95$; it is a lower bound estimate of strain and underestimates $n$ (BS 7910:2019, §7.1.3.5, NOTE 1). Outside this range the calculation steps show a warning.
  • “Direct n value”: the value of “Strain-hardening Exponent ($n$)” is used.
  • The default is “Estimate from σ_Y/σ_U (Eq. 7.7)”.

Standard reference: BS 7910:2019, §7.0; §7.1.3.5, Eq. (7.7), NOTE 1

Strain-hardening Exponent (n)

Strain-hardening exponent (BS 7910:2019, §7.0), the exponent in Eq. (7.4), $\varepsilon=\sigma/E+(A_r\sigma_Y/E)(\sigma/\sigma_Y)^n$ (BS 7910:2019, §7.1.3.5, Eq. (7.4)).

Where to get the value

  • $n$ can be found from Eq. (7.6), which uses the uniform elongation $\varepsilon_U$ (BS 7910:2019, §7.1.3.5, Eq. (7.6)).

  • The calculator takes $A_r=0.002E/\sigma_Y$ from Eq. (7.5) with $\varepsilon_Y=\sigma_Y/E+0.002$, so $\varepsilon_{ref}=L_r\sigma_Y/E+0.002L_r^n$ (BS 7910:2019, §7.1.3.5, Eq. (7.5)).

  • Shown only for Option 2 with “Direct n value”.

  • The calculator accepts only $n \ge 1.01$.

Standard reference: BS 7910:2019, §7.0; §7.1.3.5, Eq. (7.4)–(7.6)

LĂŒders Strain Source

$\Delta\varepsilon$ is the extent of LĂŒders strain (BS 7910:2019, §7.0). This field sets where the $\Delta\varepsilon$ used for discontinuous yielding comes from.

Where to get the value

  • “Estimate (Eq. 7.8)”: in the absence of detailed stress-strain data, $\Delta\varepsilon$ may be estimated from $\Delta\varepsilon=0.0375(1-0.001\sigma_Y)$, for materials where $\sigma_Y\lt 1000$ N/mmÂČ (BS 7910:2019, §7.3.3; §7.1.3.6, Eq. (7.8)). With this choice the calculator refuses to calculate when $\sigma_Y\ge1000$ MPa.
  • “Direct ΔΔ value”: the value of “LĂŒders Strain ($\Delta\varepsilon$)” is used.
  • The default is “Estimate (Eq. 7.8)”.

Standard reference: BS 7910:2019, §7.0; §7.1.3.6, Eq. (7.8); §7.3.3

LĂŒders Strain (ΔΔ)

The increase in strain at stress $R_{eL}$ without any increase in stress (BS 7910:2019, §7.3.3).

Where to get the value

  • Option 1 uses it in $\lambda=1+E\Delta\varepsilon/R_{eL}$ (BS 7910:2019, §7.3.3, Eq. (7.33)); Option 2 adds it to $\varepsilon_{ref}$ (for $L_r\ge1$).

  • Shown only with “Discontinuous (LĂŒders plateau)” and “Direct ΔΔ value”.

  • The calculator accepts only $0 \lt \Delta\varepsilon \le 0.099$.

Effect on the results

  • ↑ Increase: Larger $\lambda$, so the FAL value at $L_r=1$ from Eq. (7.30), $f=(\lambda+1/(2\lambda))^{-0.5}$, becomes lower (BS 7910:2019, §7.3.3, Eq. (7.30), (7.33)).

Standard reference: BS 7910:2019, §7.3.3, Eq. (7.30), (7.33); §7.1.3.6

Calculation steps

The “Steps” tab lists each step by number with its formula, substituted values, result and clause; for the 6 semi-elliptical surface flaw types, steps 2, 3, 7, 8, 10 and 11 are listed once for the deepest point and once for the surface point.

Step What it does Where it shows in the results
Step 1 Classes $P_m$, $P_{m,b}$ and $P_b$ as primary stresses and $Q_m$ and $Q_b$ as secondary stresses (BS 7910:2019, §7.2.2)
Step 2pre When “Stresses Include the Weld Toe” is “Yes — M_k taken as 1”, takes $M_{km}=M_{kb}=1$ (BS 7910:2019, §6.4.4.4) Notes to the conclusions in the report
Step 2 Finds $K_I^P$ with Annex M, with $k_{tm}$, $k_{tb}$, $M_{km}$, $M_{kb}$ and $k_m$ applied through Eq. (M.4) (BS 7910:2019, Annex M, §M.1, Eq. (M.1), (M.4)); when $K_I^P$ is entered directly, the assessment uses the entered value and the Annex M value is shown for comparison only “Stress Intensity” card; two-point table
Step 2a For through-thickness flaws in axial and circumferential cylinders, takes the larger $K_I$ of the inner and outer crack fronts as $K_I^P$ (BS 7910:2019, Annex M, §M.7.2.1, §M.7.3.1)
Step 2b When $K_I^P$ is entered directly and the Annex M formula is outside its range, lists the parameter, value and range (BS 7910:2019, Annex M, §M.1) Out-of-range notice at the top of the results
Step 2c When $K_I^P$ is entered directly and $k_m>1$: with “Yes — the direct value includes the misalignment bending”, no misalignment term is added; with “No — add it from Annex M”, Annex M is run once with the entered $k_m$ and once with $k_m=1$, the difference is the misalignment K increment $\Delta K_{km}$, and it is added to $K_I^P$ when the misalignment bending is treated as primary (BS 7910:2019, Annex M, §M.1, Eq. (M.4); Annex D, §D.1) $\Delta K_{km}$ row
Step 3 Finds $K_I^S$ from $Q_m$ and $Q_b$ with Eq. (M.5), without $k_{tm}$, $k_{tb}$, $M_{km}$, $M_{kb}$ or $k_m$ (BS 7910:2019, Annex M, §M.1, Eq. (M.5)); if $K_I^S<0$, $K_I^S$ and $\rho$ are set to zero (BS 7910:2019, §7.3.6) “Stress Intensity” card; two-point table
Step 3a When $K_I^P$ of a through-thickness flaw is taken at the outer front, $K_I^S$ is taken at the same front
Step 3b When the misalignment bending is treated as secondary, finds $\Delta K_{km}$ with Annex M and adds it to $K_I^S$ (BS 7910:2019, Annex D, §D.1) $\Delta K_{km}$ row
Step 3c When $K_I^P$ is entered directly, Annex M is out of range and $Q_m=Q_b=0$, takes $K_I^S=0$ (BS 7910:2019, Annex M, §M.1, Eq. (M.5))
Step 4 Takes the entered $K_{mat}$; the fracture toughness should correspond to the material containing the crack tip (BS 7910:2019, §7.2.3) “Stress Intensity” card
Step 5a For the 2 polynomial stress types: linearizes the stress distribution given by $\sigma_0$–$\sigma_5$ into $P_m$ and $P_b$ for Annex P, with the method set in “$\sigma_{ref}$ Linearization Method” (BS 7910:2019+C1:2025, §6.4.1, Figure 6.1)
Step 5a check Gives a warning when the linearized distribution is below the polynomial stress distribution at any point over the flaw; any linearized distribution of stress is acceptable provided that it is greater than or equal to the real distribution over the flaw surface (BS 7910:2019+C1:2025, §6.4.1 NOTE) Warning bar in the results
Step 5 Finds $\sigma_{ref}$ with Annex P, and $L_r=\sigma_{ref}/\sigma_Y$ (BS 7910:2019, §7.3.7, Eq. (7.40)); when the misalignment bending is treated as primary, the bending stress passed to Annex P is $P_b+(k_m-1)P_m$ “Plastic Collapse” card
Step 6 $L_{r,max}=(\sigma_Y+\sigma_U)/(2\sigma_Y)$ (BS 7910:2019, §7.3.2, Eq. (7.25)) “Plastic Collapse” card
Step 7 $\rho$ is calculated with Annex R, taken from the input, or zero (BS 7910:2019, Annex R, §R.2.2, Eq. (R.1), (R.2)) “FAD Assessment” card; two-point table
Step 8 $K_r=(K_I^P+K_I^S)/K_{mat}+\rho$ (BS 7910:2019, §7.3.6, Eq. (7.39)) “Clause 7 Assessment-Point Assembly” card
Step 9 $f(L_r)$: Option 1 uses Eq. (7.26)–(7.28) for continuous yielding and Eq. (7.29)–(7.33) for discontinuous yielding (BS 7910:2019, §7.3.3); Option 2 uses Eq. (7.34), with the reference strain taken from a Ramberg-Osgood curve built from $\sigma_Y$, $E$ and $n$, where $n$ is entered directly or estimated from $\sigma_Y/\sigma_U$ with Eq. (7.7) (BS 7910:2019, §7.3.4, Eq. (7.34); §7.1.3.5, Eq. (7.4)–(7.7)); for Option 2 with discontinuous yielding, the reference strain for $L_r \ge 1$ includes the LĂŒders strain $\Delta\varepsilon$ (BS 7910:2019, §7.1.3.6, Eq. (7.8)) “FAD Assessment” card
Step 10 Verdict (BS 7910:2019, §7.3.1): $L_r \ge L_{r,max}$ (plastic collapse) is checked first, then $K_r \ge f(L_r)$ (fracture) Verdict banner
Step 11 Finds the margin on $K_r$, $F_{Kr}$, and the reserve factor on load $F_{load}$; see Reading the results “FAD Assessment” card

Engineering topics: A Concise Guide to BS 7910 Fracture Assessment , choosing between Option 1 / 2 / 3 , continuous and discontinuous yielding , Annex M stress intensity factor , Annex P reference stress .

Reading the results

Where to look first

  1. Out-of-range notice: it can appear only when $K_I^P$ is entered directly. It sits at the top of the results under the title “Formula value outside the range of its solution” and lists, line by line, the quantity, building block, clause, parameter, value and range. The assessment uses the $K_I^P$ you entered; when stress intensity factors are derived by methods such as finite elements, the basis of the method and the results should be fully documented (BS 7910:2019, Annex M, §M.1).
  2. Verdict banner: shows “ACCEPTABLE (PASS)” or “NOT ACCEPTABLE (FAIL)” with $L_r$ and $K_r$; when the flaw is not acceptable, it states whether the failure mode is plastic collapse ($L_r \ge L_{r,max}$) or fracture ($K_r \ge f(L_r)$).
  3. Warning bars: NOTE-type warnings of the standard passed on from Annex M and Annex P, and the linearization warning; they do not change the verdict. When the Annex R ratio $K_I^S/(K_I^P/L_r)>4$, a further warning appears under the “Clause 7 Assessment-Point Assembly” card: the values of $\rho$ given in R.2.2 could then be over-conservative (BS 7910:2019, Annex R, §R.2.1).
  4. Criterion badges and utilization strip: the two badges are the fracture criterion (BS 7910:2019, §7.3.1) and the plastic collapse cut-off (BS 7910:2019, §7.3.2, Eq. (7.25)); the strip lists the “$K_r$ ratio” $K_r/f(L_r)$ and the “$L_r$ ratio” $L_r/L_{r,max}$, and the larger one governs.

Criterion

If the assessment point $(L_r,\ K_r)$ lies within the area bounded by the axes, the FAL and the vertical line $L_r=L_{r,max}$, the flaw is acceptable; if it lies on or outside the line, the flaw is unacceptable (BS 7910:2019, §7.3.1). The calculator checks $K_r\lt f(L_r)$ and $L_r\lt L_{r,max}$; when $L_r \ge L_{r,max}$ the failure mode is plastic collapse, otherwise $K_r \ge f(L_r)$ gives fracture.

No inherent safety factors are included; sensitivity studies with realistic variations in the input parameters are used to assess the effect on margins (BS 7910:2019, §7.3.1). The margin between the specified loading conditions and the limiting loading condition is judged by a sensitivity analysis (BS 7910:2019, §7.2.12).

Output quantities

Result Meaning
Primary SIF $K_I^P$ Annex M value or direct value; with a direct value, misalignment added by the calculator and misalignment bending treated as primary, it includes $\Delta K_{km}$. For the two-point assessment this is the deepest-point value; the surface-point value is in the two-point table
Misalignment K increment $\Delta K_{km}$ Difference between Annex M run with the entered $k_m$ and with $k_m=1$; shown only when the calculator adds it
Secondary SIF $K_I^S$ Set to 0 when negative (BS 7910:2019, §7.3.6); includes $\Delta K_{km}$ when the misalignment bending is treated as secondary
Fracture Toughness $K_{mat}$ The $K_{mat}$ used
Reference Stress $\sigma_{ref}$ Annex P value
Load Ratio $L_r$ (x-axis) $\sigma_{ref}/\sigma_Y$ (BS 7910:2019, §7.3.7, Eq. (7.40)); the same for both points in the two-point assessment
Cut-off $L_{r,max}$ $(\sigma_Y+\sigma_U)/(2\sigma_Y)$ (BS 7910:2019, §7.3.2, Eq. (7.25))
Plasticity Interaction $\rho$ Set to 0 when $K_I^S<0$ (BS 7910:2019, §7.3.6)
Fracture Ratio $K_r$ (y-axis) $(K_I^P+K_I^S)/K_{mat}+\rho$ (BS 7910:2019, §7.3.6, Eq. (7.39))
FAD Limit $f(L_r)$ FAL value at the $L_r$ of the assessment point
Assessment Result, Failure Mode Acceptable, brittle/ductile fracture or plastic collapse
Margin on $K_r$, $F_{Kr}$ Defined in this calculator: $F_{Kr}=f(L_r)/K_r$, the margin along the $K_r$ direction at fixed $L_r$
Reserve Factor on Load $F_{load}$ The factor by which the primary load is scaled so that the assessment point just reaches the FAL; a value above 1 means the primary load can still be raised $F_{load}$ times before the point reaches the FAL

The source tags next to $K_I^P$, $K_I^S$, $K_{mat}$, $\rho$ and $\sigma_{ref}$ say whether each value comes from a “formula” or from “user input”.

How $F_{load}$ is found: for an initiation analysis under a single primary load, the standard gives a graphical construction for the load factor $F^L$, taken where the locus of assessment points for different loads meets the failure assessment curve (BS 7910:2019+C1:2025, Annex N, §N.2.4.6, Figure N.3). With or without secondary stress, this calculator always solves the equation below in the interval $[10^{-3},\ L_{r,max}/L_r]$, with the part of $K_I^S$ that does not change with load held constant:

$$\frac{F_{load}\,(K_I^{P}+\Delta K_{km}^{S})+K_I^{S,0}}{K_{mat}}+\rho(F_{load})=f\!\left(F_{load}\,L_r\right)$$

Here $\Delta K_{km}^{S}$ is the misalignment K increment added to $K_I^S$ when the misalignment bending is treated as secondary (otherwise 0), and $K_I^{S,0}=K_I^S-\Delta K_{km}^{S}$; when $\rho$ comes from Annex R it is recalculated from the scaled $K_I^P$, $K_I^S$ and $L_r$, and when it is entered directly or zero it is constant. The bending stress due to misalignment is a function of $P_m$ (BS 7910:2019+C1:2025, Annex D, §D.1), and this calculator scales $\Delta K_{km}^{S}$ linearly with the primary load.

Controlling points in the two-point assessment

  • Both points have the same $L_r$; each flaw has one $\sigma_{ref}$.
  • If either point lies on or outside the FAL, the flaw is not acceptable; the failure mode is the less favourable of the two points (plastic collapse, then fracture, then acceptable, from least to most favourable).
  • $F_{Kr}$ and $F_{load}$ each take the smaller of the two points, and the result rows and the banner state whether each comes from the deepest point or the surface point. When the share of secondary stress differs between the two points, the two reserve factors can come from different points.
  • The two-point table lists $K_I^P$, $\Delta K_{km}$, $K_I^S$, $\rho$, $K_r$ and the verdict (PASS / FAIL) of both points side by side.

Failure assessment diagram

  • The horizontal axis is the load ratio $L_r$ and the vertical axis is the fracture ratio $K_r$; the curve is “FAL f(Lr)” and the vertical line is “Lr,max cut-off”.
  • Green is the acceptable region (under the FAL and $L_r\lt L_{r,max}$) and the orange band is the plastic collapse cut-off; an assessment point is green when acceptable and red when not, and its label gives the coordinates $(L_r,\ K_r)$.
  • The dotted vertical line runs from the controlling point along the $K_r$ direction to the FAL and ends with $F_{Kr}$; the dashed line joins the controlling point and the point where the FAL is reached with the primary load scaled by $F_{load}$, and ends with $F_{load}$. The dashed line only joins these two end points.
  • In the two-point assessment, the circle is the deepest point and the diamond is the surface point; the larger marker is the point that gives $F_{load}$.
  • For discontinuous yielding, a vertical line at $L_r=1$ joins the points of Eq. (7.29) and Eq. (7.30) (BS 7910:2019, §7.3.3); where the stress-strain curve has a discontinuity at the lower yield strength, the Option 2 curve has a discontinuity at the matching $L_r$, which is often $L_r=1$ (BS 7910:2019, §7.3.4).

Schematic, steps and report

  • “Schematic” tab: the flaw cross-section and the through-wall stress distribution; the input in focus is highlighted in the figures.
  • “Steps” tab: lists each formula with the substituted values.
  • “Download Report (PDF)”: contains the inputs (only the fields shown on screen), the key results, the assessment points (two-point assessment) table and the sources of values table; the out-of-range notice, the warnings and the weld-toe note are printed as notes to the conclusions.

Limits

  • Section containing the flaw: structural collapse is governed by failure of the section containing the flaw, and the possibility of premature collapse elsewhere in the structure should be investigated separately (BS 7910:2019, §7.3.7).
  • Material properties: the mean minus two standard deviation tensile properties are used for $K_r$ and $L_r$ of the assessment points, and the mean tensile properties for the FAL and $L_{r,max}$ (BS 7910:2019, §7.1.3.2); for the cut-off, mean rather than minimum tensile properties should be used (BS 7910:2019, §7.3.2). This calculator has one set of $\sigma_Y$ and $\sigma_U$, used for $L_r$, the FAL and $L_{r,max}$ alike.
  • Out of range: the validity of the Annex M solutions is limited strictly to the ranges stated, with no extrapolation (BS 7910:2019, Annex M, §M.1). When a $K_I^P$, $K_I^S$, $\Delta K_{km}$ or $\sigma_{ref}$ calculated by formula is outside the range of its building block, the calculator does not calculate, and the error message names the clause and the range; when the reason is Annex M, the message adds that the stress intensity factor can be found by finite elements, a weight function or a handbook solution and entered directly, and that the basis of the method and the results should be fully documented (BS 7910:2019+C1:2025, Annex M, §M.1). The only exception is a directly entered $K_I^P$: an Annex M value outside its range is only recorded in the out-of-range notice, and the calculation goes on.
  • Alternative solutions: where the standard gives an alternative solution, the building block uses it; for example, a surface flaw with $a/2c>1.0$ uses the solution for $a/2c=1.0$ (BS 7910:2019, Annex M, §M.4.1.2.3 c)), and a circumferential internal surface flaw in a cylinder with $B/r_i<0.1$ uses the flat plate solution (BS 7910:2019, Annex M, §M.7.3.2).
  • Magnifying secondary stresses: thermal stresses (primary and secondary) should be multiplied by the relevant stress concentration factor $k_t$ and misalignment factor $k_m$ (BS 7910:2019+C1:2025, §6.4.3); secondary stresses are not affected by structural discontinuities, except thermal stresses treated as primary (BS 7910:2019+C1:2025, §6.4.4.1). This calculator finds $K_I^S$ with Eq. (M.5), does not multiply $Q_m$ and $Q_b$ by $k_{tm}$, $k_{tb}$ or $k_m$, and uses the entered $Q_m$ and $Q_b$ as the local stresses at the flaw position (BS 7910:2019+C1:2025, Annex M, §M.1, Eq. (M.5)).
  • Secondary stress only: the equations of R.2 do not apply when there are secondary stresses only, and the relevant stresses may be conservatively classified as primary (BS 7910:2019, Annex R, §R.1); when $K_I^P=0$ and $K_I^S>0$ at any assessment point, the calculator does not calculate and asks you to enter these stresses in $P_m$ and $P_b$ and assess again. With $K_I^P=K_I^S=0$ it calculates as normal.
  • $K_I^P<0$: the calculator does not calculate; the message says that the primary stress closes the flaw, so this fracture assessment does not apply, and that plastic collapse can still be checked with the Annex P calculator ($L_r$).
  • Material inputs: the calculator does not calculate when $\sigma_U \le \sigma_Y$, and the Option 1 exponent is $N=0.3(1-\sigma_Y/\sigma_U)$ (BS 7910:2019, §7.3.3); with discontinuous yielding and the LĂŒders strain estimated by Eq. (7.8), it does not calculate when $\sigma_Y \ge 1000$ MPa, as Eq. (7.8) is for materials with $\sigma_Y<1000\ \mathrm{N/mm^2}$ (BS 7910:2019, §7.3.3).
  • Option 2 stress-strain curve: the standard requires the mean uniaxial tensile true stress-strain curve at the assessment temperature (BS 7910:2019, §7.3.4). This calculator builds the curve from $\sigma_Y$, $E$ and $n$ with the Ramberg-Osgood equation (BS 7910:2019, §7.1.3.5, Eq. (7.4), (7.5)) and uses the engineering strain of this curve as the reference strain, with no conversion to true stress and strain. Eq. (7.7) was derived from ferritic steels with yield or proof strengths of 300–1000 MPa and $0.65\lt \sigma_Y/\sigma_U<0.95$, and is a lower bound estimate of strain that underestimates $n$ (BS 7910:2019, §7.1.3.5 NOTE 1); when $n$ is estimated with Eq. (7.7) and $\sigma_Y$ or $\sigma_Y/\sigma_U$ is outside this range, step 9 gives a warning.
  • Assessment point position: through-thickness flaws in cylinders take the larger $K_I$ of the inner and outer fronts; embedded flaws take the Annex M point nearest the surface; extended surface and fully circumferential flaws take the deepest point.
  • Sign of $P_b$: Annex P uses $|P_b|$ for $\sigma_{ref}$; for plate through-thickness and edge through-thickness flaws, $P_b$ is positive when the $z=0$ side is in tension, and the calculator finds $K_I$ at $z=0$.
  • Misalignment: it is conservative to treat stresses due to misalignment as primary, although in some cases they act as secondary, that is, they affect $K_r$ but not $L_r$ (BS 7910:2019, Annex D, §D.1). When the direct $K_I^P$ includes the misalignment and the misalignment bending is treated as secondary, the misalignment contribution stays in $K_I^P$ and does not enter the Annex R ratio $K_I^S/(K_I^P/L_r)$.

Examples

The “Verification examples” drop-down list has 13 built-in examples; after loading one, click “Run Calculation”. The examples with links below can also be loaded directly from the link.

The point-by-point check against published examples is in the verification manual of this calculator .

🧼 Online calculator: Clause 7: Fracture Assessment Calculator — Decides with the failure assessment diagram of BS 7910:2019 Clause 7 whether a known flaw is acceptable.