Weld Static Strength and Weld-Group Analysis

The Throat Stress and the Weld Allowable

The static strength of a fillet weld is assessed on the throat plane — the plane through the weld throat, which is the shortest distance from the root to the face. The throat area is the throat dimension times the weld length: A_throat = 0.707 × leg × L for an equal-leg fillet. The force on the weld is resolved into components on the throat plane: a shear force parallel to the throat and a normal force perpendicular to the throat. The shear stress is the shear force divided by the throat area; the normal stress is the normal force divided by the throat area. The weld allowable is the weld metal strength — typically the shear strength is taken as 0.6 × the tensile strength (the von Mises relationship) for the directional method, or a specific weld shear allowable from the welding code (AWS D1.1, Eurocode 3, BS 5950). The allowable includes a factor of safety — typically 2.0–3.0 on ultimate for static loading, depending on the code and the application. The weld must also be checked against the parent metal strength: the weld can be stronger than the parent material (overmatching), in which case the parent material fails first, or weaker (undermatching), in which case the weld fails first. The HAZ can also be the critical region — the HAZ can be softer than the parent material (over-aged or annealed) and fail before the weld or the parent material. The engineer should check all three: the weld metal, the parent metal and the HAZ.

  Fillet weld throat area:

  A_throat = 0.707 × leg × L_weld   (equal-leg fillet)

  Throat shear stress:

  τ = F_shear / A_throat

  Throat normal stress:

  σ = F_normal / A_throat

  Allowable (directional method, AWS):

  τ_allow = 0.30 × F_EXX   (F_EXX = electrode tensile strength)
  σ_allow = 0.60 × F_EXX

  →  Check weld metal, parent metal and HAZ
  →  Factor of safety per the applicable code

Axial Load, Shear and the Simple Weld Check

For a simple weld — a single fillet weld carrying a single load component — the check is straightforward. For a fillet weld in shear (load parallel to the weld length), the throat shear stress is the shear force divided by the throat area: τ = F / (0.707 × leg × L). The stress is uniform along the weld (for a short weld) and is compared with the allowable shear stress. For a fillet weld in tension (load perpendicular to the weld length, through the throat), the throat normal stress is the tensile force divided by the throat area: σ = F / (0.707 × leg × L). The tension can also produce a peel component — the load perpendicular to the throat plane — which is more severe than the in-plane shear. For a fillet weld subject to both shear and tension, the combined stress is assessed by the directional method (checking each component against its allowable) or by the equivalent stress method (computing the von Mises equivalent and checking against the weld tensile strength). The simple weld check is the basis for all weld strength assessment — the weld-group analysis is an extension that handles the combined loading and the multiple welds.

For a simple fillet weld, the throat stress (shear or normal) is the force divided by the throat area. For combined shear and tension, use the directional method (check each component) or the equivalent stress method (von Mises). The simple weld check is the basis for the weld-group analysis.

Bending and the Weld Group under Moment

When a weld group is subject to a bending moment, the stress distribution in the welds is analogous to the bending stress in a beam — the stress varies linearly from the neutral axis of the weld group to the extreme weld. The weld group is treated as a line (for a linear weld) or an area (for a distributed weld), and the section properties — the moment of inertia and the section modulus — are computed for the weld group treated as a line (the throat area per unit length). The bending stress at the extreme weld is σ = M × c / I, where M is the moment, c is the distance from the neutral axis to the extreme weld, and I is the moment of inertia of the weld group. The bending stress is a normal stress on the throat — it acts perpendicular to the throat plane. The extreme weld (the weld farthest from the neutral axis) has the highest stress and is the critical weld. The bending stress is combined with any direct stress (from axial load) and any shear stress (from transverse shear) at the same point. For a T-joint with double-sided fillet welds subject to a bending moment on the web, the weld group is the two parallel welds, the neutral axis is at the centre of the web, and the extreme welds are at the top and the bottom of the web. The bending stress is highest at the extreme welds and zero at the neutral axis.

  Weld group under bending (treated as a line):

  σ_bend = M × c / I_weld_group

  where  M    = applied bending moment
         c    = distance from neutral axis to extreme weld
         I    = moment of inertia of the weld group (throat area)

  →  Stress varies linearly from the neutral axis
  →  Extreme weld has the highest stress
  →  Combine with direct stress and shear at the same point

Torsion and the Instantaneous Centre Method

When a weld group is subject to a torsional moment — a moment about an axis perpendicular to the plane of the weld group — the load distribution is more complex. The torsion produces a shear stress in each weld that is proportional to the distance from the centre of rotation to the weld and perpendicular to the radius. For a symmetric weld group (a circular pattern of welds, or a symmetric rectangular pattern), the centre of rotation is the centroid of the weld group, and the shear stress at each weld is τ = T × r / J, where T is the torsional moment, r is the distance from the centroid to the weld, and J is the polar moment of inertia of the weld group. The shear stress is perpendicular to the radius (the torsional shear direction). For an eccentrically loaded weld group — a load applied with an offset from the centroid of the weld group — the load produces both a direct shear (through the centroid) and a torsional moment (the load times the eccentricity). The direct shear is distributed uniformly among the welds; the torsional shear is distributed proportional to the distance from the centroid. The total shear at each weld is the vector sum of the direct shear and the torsional shear. The critical weld is the one with the highest vector sum — typically the weld farthest from the centroid in the direction perpendicular to the load. For an asymmetric weld group or a group with welds of different lengths, the instantaneous centre of rotation method is used: the centre of rotation is not at the centroid but at a point determined by the equilibrium of the weld group (the point about which the weld forces balance the applied load). The instantaneous centre is found by iteration — the method is more accurate than the simple centroid method for asymmetric groups.

Torsion produces a shear stress proportional to the distance from the centre of rotation, perpendicular to the radius. For eccentric loading, the total shear at each weld is the vector sum of the direct shear and the torsional shear. For asymmetric groups, the instantaneous centre of rotation method iterates to find the equilibrium centre.

Combined Loading and the Equivalent Stress

A weld group in a real structure is typically subject to combined loading: axial load, shear in two directions, bending about two axes, and torsion. The stress at each point in the weld is the sum of the stresses from each load component. The combination is done at a specific point — the critical point — which is the point in the weld group with the highest resultant stress. The critical point is typically the extreme weld under bending (the highest normal stress from bending) combined with the maximum shear from torsion and direct shear. The stresses at the critical point are: the direct stress from axial load, the bending stress from the bending moments, the shear stress from the transverse shear, and the torsional shear stress from the torsional moment. These are combined into an equivalent stress for comparison with the weld allowable. The directional method (AWS D1.1) checks each component separately: the shear stress (from all sources) against the shear allowable, and the normal stress (from all sources) against the normal allowable. The equivalent stress method (Eurocode 3) computes the von Mises equivalent: σ_eq = √(σ² + 3τ²), and checks it against the weld metal tensile strength divided by a factor. The two methods give similar results for most cases, but they differ in the treatment of combined tension and shear. The engineer should use the method specified by the applicable design code.

  Equivalent stress (Eurocode 3 / von Mises):

  σ_eq = √(σ² + 3τ²)

  where  σ = total normal stress on the throat (axial + bending)
         τ = total shear stress on the throat (transverse + torsional)

  Check:  σ_eq ≤ f_u / (β_w × γ_M2)

  where  f_u   = weld metal or parent metal tensile strength
         β_w    = correlation factor (per code)
         γ_M2   = partial safety factor

  →  Combined at the critical point in the weld group
  →  Check against weld metal, parent metal and HAZ

For combined loading, the stresses from each component are summed at the critical point and combined into an equivalent stress. The directional method (AWS) checks each component separately; the equivalent stress method (Eurocode) uses the von Mises equivalent. Use the method specified by the applicable code.

Weld Metal, Parent Metal and HAZ Failure

The static strength of a welded joint is governed by the weakest of the three regions: the weld metal, the parent metal and the HAZ. The weld metal strength depends on the electrode and the welding process — the electrode is specified by its tensile strength (e.g., E70XX for 70 ksi ultimate in AWS, or an electrode matching the parent material in Eurocode). The weld metal can be overmatching (stronger than the parent material), matching (equal), or undermatching (weaker). Overmatching weld metal is common for structural steel — the weld is stronger than the parent material, and the failure occurs in the parent material, not the weld. Undermatching weld metal is used for high-strength steels where a matching electrode would be too expensive or too susceptible to cracking — the weld is designed to be weaker, and the joint capacity is limited by the weld. The parent metal strength is the specified minimum yield and tensile strength of the plate. The HAZ strength depends on the thermal cycle — the HAZ can be harder and stronger (for quenched-and-tempered steels that are re-heated) or softer and weaker (for heat-treated steels that are over-aged or annealed by the welding heat). The HAZ is typically the critical region for high-strength steels and for heat-treated alloys. The engineer must check all three regions and design the joint so that the governing region has adequate strength with the required factor of safety. The check is: weld metal stress ≤ weld metal allowable; parent metal stress ≤ parent metal allowable; HAZ stress ≤ HAZ allowable. The lowest margin governs.

The static strength is governed by the weakest of the weld metal, the parent metal and the HAZ. Check all three. Overmatching weld metal is common for structural steel (parent fails first). Undermatching is used for high-strength steels (weld governs). The HAZ is critical for heat-treated alloys — it can be softer than the parent material.

Classical Calculation versus FE Load Recovery

The relationship between the classical weld-group calculation and the FE analysis is complementary, not exclusive. The classical calculation (the elastic method, the vector method, the instantaneous centre method) computes the stress in the weld from the applied loads and the weld group geometry — it is a hand calculation or a spreadsheet calculation that is transparent, quick and verifiable. The FE analysis computes the forces at the weld (the throat forces) from the global model — it captures the load distribution, the stiffness effects and the load path that the classical calculation cannot. The two are used together: the FE provides the throat forces (the input), and the classical calculation converts the throat forces into the equivalent throat stress (the output for the strength check). The FE does not replace the classical calculation because the FE model, unless it has explicit weld geometry (solid weld, beam weld), does not produce the throat stress directly — it produces the forces or the plate stresses, which must be converted. The classical calculation provides the conversion. The engineer should extract the throat forces from the FE model (as described in Article 23), apply them to the weld-group calculation, and compare the resulting throat stress with the allowable. The classical calculation also serves as a verification of the FE — the hand calc should give a result in the same range as the FE, and any discrepancy should be explained. A model that gives a weld force significantly different from the hand calc should be investigated — the discrepancy may indicate a modelling error, a load path difference, or a genuine effect that the hand calc did not capture.

The classical weld-group calculation and the FE analysis are complementary. The FE provides the throat forces; the classical calculation converts them to the throat stress for the strength check. The FE does not replace the classical calculation — it provides the input. The hand calc also verifies the FE — a significant discrepancy should be investigated.

Local Yielding and Plastic Redistribution

The classical weld-group methods (elastic method, instantaneous centre) assume elastic behaviour — the stress distribution is linear, and the maximum stress at the critical point is compared with the allowable. In reality, the weld metal and the parent material can yield plastically before the ultimate strength is reached. The yielding redistributes the load: the yielded weld or the yielded region has a lower stiffness, and the load shifts to the less-stressed welds or regions. The plastic redistribution can increase the ultimate capacity of the weld group by 20–50% above the elastic prediction, depending on the ductility and the geometry. The plastic (ultimate) method — used in AWS D1.1 for eccentrically loaded weld groups — accounts for this by using the ultimate strength of the weld and a load factor, rather than the elastic stress distribution. The plastic method gives a higher capacity than the elastic method, reflecting the redistribution. However, the plastic method assumes adequate ductility — the weld metal and the parent material must be able to yield and deform without brittle fracture. For weld metals with low toughness (high-strength steels, high-carbon deposits) or for dynamic loading (seismic, impact), the plastic redistribution may not be reliable, and the elastic method with a higher factor of safety is more appropriate. The engineer should use the method specified by the applicable code and should verify that the material has adequate ductility for the plastic method if it is used.

The classical elastic methods assume linear stress distribution. In reality, local yielding redistributes the load and can increase the ultimate capacity by 20–50%. The plastic (ultimate) method in AWS D1.1 accounts for this. However, the plastic method requires adequate ductility — for low-toughness weld metals or dynamic loading, use the elastic method with a higher factor of safety.

Key takeaways

• The static strength of a fillet weld is governed by the throat stress — the shear stress on the throat plane. The throat area is the throat dimension times the weld length, and the allowable is the weld metal shear strength with an appropriate factor of safety.
• For a weld group under combined loading (axial, shear, bending, torsion), the stress at each point in the weld is the vector sum of the stresses from each load component. The critical point is the point with the highest resultant stress — typically the extreme fibre of the weld group under bending or the extreme weld under torsion.
• The classical weld-group methods (elastic method, vector method, instantaneous centre method) differ in how they distribute the load among the welds. The elastic method assumes a linear distribution; the instantaneous centre method accounts for the rotation of the weld group and gives a more realistic distribution for eccentrically loaded groups.
• The relationship between classical calculations and FE load recovery is complementary: the FE model provides the forces at the weld (the throat forces), and the classical calculation converts those forces into the equivalent throat stress. The FE does not replace the weld-group calculation — it provides the input to it.