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Structural Analysis

Eccentric Weld Group Simulator

Analyse what happens to a bracket connection made with two fillet welds when the load is applied off the centroid of the weld group — an eccentricity. Using the elastic vector method, this tool combines the direct shear and the torsional shear as vectors and finds the resultant peak stress at the most critical far corner in real time.

Parameters
Weld length (each) d
mm
Length of one vertical fillet weld
Weld horizontal spacing b
mm
Centre-to-centre distance between the two welds
Load P
kN
Vertical load applied to the bracket
Eccentricity e
mm
Distance from the weld-group centroid to the load
Weld size (leg length) s
mm
Leg length of the fillet weld. Throat = 0.707·s
Results
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Total weld length (mm)
—
Throat area (mm²)
—
Direct shear stress (N/mm²)
—
Torsional shear, peak point (N/mm²)
—
Resultant peak stress (N/mm²)
—
Strength verdict
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Weld group & stress vectors — elastic vector method

The two vertical welds, the weld-group centroid and the load P applied at the eccentricity are drawn. At the critical far corner, the direct-shear (blue), torsional-shear (orange) and resultant (red) vectors are shown.

Resultant peak stress vs eccentricity e
Resultant peak stress vs weld size s
Theory & Key Formulas

$$\tau_{direct}=\frac{P}{A},\qquad \tau_{torsion}=\frac{M\,r}{J},\qquad \tau_{result}=\sqrt{(\tau_{tV}+\tau_{direct})^{2}+\tau_{tH}^{2}}$$

Direct shear stress τ_direct (P: load, A: throat area), torsional shear stress τ_torsion (M = P·e: torsional moment, r: distance from the centroid, J: polar moment of inertia), and the resultant peak stress τ_result. The direct and torsional shears must be combined as vectors at the critical far corner (τ_tV: vertical torsion component, τ_tH: horizontal torsion component).

$$J = 2\left(\frac{a\,d^{3}}{12}\right)+2\,(a\,d)\left(\frac{b}{2}\right)^{2},\qquad a = 0.707\,s$$

Polar moment of inertia J of the weld group. The welds are treated as lines of effective throat thickness a, summing each weld's own term and the spacing term from the parallel-axis theorem. a: throat, d: weld length, b: weld spacing, s: weld size.

What is an Eccentric Weld Group?

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"Eccentrically loaded weld group" sounds intimidating. How is it different from an ordinary weld-strength calculation?
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The phrase sounds stiff, but the situation is extremely common in the field. Picture a bracket welded to a wall or a column, with a load hanging off it. If the load happened to pass straight through the centre of the welds, the welds would just carry plain shear and the problem would be easy. But in reality the load is almost always applied at an arm's length away from the weld group. That offset is the eccentricity.
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So the load is not directly below, but a bit off to the side. What does that offset do to the welds?
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Two things happen at once. First, the load itself still has to be carried — that produces a direct shear stress, spread uniformly over the whole throat area of the weld group. Second, and trickier, the load multiplied by the eccentric arm becomes a torsional moment that tries to twist the weld group about its centroid. That twist produces a torsional shear stress: zero at the centroid, growing linearly the farther out you go, reaching its maximum at the weld corner farthest from the centroid.
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So two kinds of stress act together. Can I just add the two to get the maximum stress?
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That is exactly the key point — you must not simply add them. The direct shear and the torsional shear point in different directions at each point of the weld group. The direct shear is uniform and downward like the load; the torsional shear is perpendicular to the radius vector from the centroid. So at the far corner the two arrows cross at an angle. You have to add them correctly as arrows — that is, as vectors. That is the elastic vector method: resolve into horizontal and vertical components, then resultant = sqrt((tau_tV + tau_direct)^2 + tau_tH^2).
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So you combine them by direction. Then if the resultant stress is too high, do I just make the welds longer?
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That helps, but the most efficient move is actually different. First, reduce the eccentricity e — bring the bracket or load point as close to the centroid as you can, and the torsional moment M = P·e drops directly. Next, spread the welds farther apart. Spacing feeds into the polar moment of inertia J as the square of the distance, so it is very efficient. Spreading welds left and right beats stretching them longer. Try moving the "weld horizontal spacing b" slider on the left and watch the chart below.
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Interesting — spreading the welds beats lengthening them. And there is a formula behind that?
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Exactly. The beauty of the elastic vector method is that the design lessons fall straight out of the formulas. "The peak stress is always at a far corner, never at the centre" and "reducing the eccentricity or spreading the welds works best" — keep those two in mind and you will not go badly wrong on a bracket weld. Then just use this tool to compare the resultant peak stress with the allowable (150 N/mm² here) and check the verdict.

Frequently Asked Questions

Use the elastic (vector) method, treating the welds as lines. First, find the direct shear stress P/A, where the load P is shared uniformly over the total effective throat area of the weld group. Next, the load multiplied by the eccentricity e gives a torsional moment M = P·e that twists the weld group about its centroid, producing a torsional shear stress M·r/J, where J is the polar moment of inertia of the weld group. Finally, at the corner farthest from the centroid, combine the two stresses as vectors (accounting for their directions) to get the resultant peak stress. This tool performs the whole calculation in real time.
Because the direct shear stress and the torsional shear stress point in different directions at each point of the weld group. The direct shear is uniform and acts in the direction of the load (downward in this tool), while the torsional shear is perpendicular to the radius vector joining the centroid to the point. At the most highly stressed far corner the two cross at an angle, so a plain numerical sum would either overestimate or underestimate the true value. The correct approach resolves them into horizontal and vertical components: resultant = sqrt((tau_tV + tau_direct)^2 + tau_tH^2).
The torsional shear stress grows linearly with the distance r from the centroid, so the maximum always occurs at the corner farthest from the centroid. At the centroid itself the torsional stress is zero. Among those far corners, the governing point is the one where the direct-shear vector and the torsional-shear vector add together in the same sense. In design checks, you verify the allowable stress against the resultant at this far corner, never at the centre.
The most effective move is to reduce the eccentricity e, that is, to bring the bracket or support point as close as possible to the centroid of the weld group, because the torsional moment M = P·e drops directly. The next most effective move is to spread the welds farther apart, which raises the polar moment of inertia J and lowers the torsional stress for the same moment. Making the welds longer also helps, but spreading them apart contributes more to J (it scales with the square of the distance) and is more efficient. Increasing the weld size (leg length) raises the throat area and lowers both the direct and the torsional shear.

Real-World Applications

Bracket connections in steel structures: In building and plant steelwork, gusset plates and support brackets are fillet-welded to columns and beams, and pipes, equipment and walkways are then hung from them. The load is always applied an arm's length away from the centroid of the weld group, so checking the eccentric torsion is unavoidable. The elastic vector method is the standard procedure for sizing and arranging the welds in such connections.

Support fittings on machine frames: The same problem arises when fittings that carry motors, cylinders or sensors are welded to the frame of a machine tool or a conveyor. When a load is applied at the tip of a cantilevered fitting, the weld group at its root carries direct shear and torsion at the same time. A quick calculation like this tool first confirms whether the resultant stress at the far corner is within the allowable.

Lifting lugs on cranes and hoists: When lifting pieces or lug plates are welded to a structure, the line of action of the lifted load rarely coincides with the centroid of the weld group. Ignoring the eccentric torsion lets cracks start from the most highly stressed far corner. Because lifting fittings are life-safety items, a check by the elastic vector method including eccentric torsion is mandatory.

Pre-study and sanity check before FEM: Before modelling a welded connection in detail with FEM, the elastic vector method gives a first read on "roughly how many times the allowable the resultant peak stress is". If the estimate is well over, you can revise the weld layout before investing in a mesh. Conversely, if the FEM result differs from this estimate by an order of magnitude, it is a sanity check that points to a load-point or restraint-input mistake.

Common Misconceptions and Pitfalls

The biggest pitfall is simply adding the direct shear and the torsional shear as numbers. The two stresses point in different directions at each point of the weld group, so adding their magnitudes directly underestimates where they align and overestimates where they diverge. The correct method resolves them into horizontal and vertical components at the far corner and combines them as vectors. That is why this tool uses resultant = sqrt((tau_tV + tau_direct)^2 + tau_tH^2). Conversely, "designing on the direct shear alone and ignoring the torsion" becomes a dangerous underestimate the larger the eccentricity gets.

Next, the belief that "making the welds longer makes the joint indefinitely stronger in torsion". In the polar moment of inertia J, the term from the weld spacing b scales with the square of the distance, whereas the term from the weld length d itself grows as d³/12 — but the far-corner radius r grows at the same time, so the reduction in torsional stress tends to plateau. In general, spreading the welds left and right (increasing b) gives a more torsion-resistant arrangement for the same amount of weld than stretching them vertically. Where space allows, the rule of thumb is to arrange the weld group "wide".

Finally, remember that the elastic vector method result is not the real strength of the actual joint. The elastic vector method is an idealised model that treats welds as lines of effective throat thickness; it ignores defects such as lack of penetration, undercut and porosity, the stress concentration at the weld toe, and fatigue under cyclic loading. In practice, allow a sufficient safety factor on the allowable stress, confirm with non-destructive testing that critical joints are free of defects, and for cyclic loading carry out a separate check based on the fatigue class of the weld detail. This tool is a static, elastic estimate for study purposes only.

How to Use

  1. Enter weld length (mm) for each fillet weld using the slider or numeric input; typical range 20–100 mm for bracket connections.
  2. Set weld spacing (mm) between the two parallel welds; values 30–150 mm are common for structural steel angles.
  3. Input applied load (kN) and eccentricity (mm)—the perpendicular distance from load line to weld group centroid.
  4. Simulator calculates direct shear stress (load divided by throat area), torsional moment stress (eccentricity × load / polar moment), and resultant peak stress at the critical point.
  5. Compare resultant stress against material strength; AWS D1.1 typically limits fillet weld shear to 0.6 × Fy for structural steel.

Worked Example

Two 6 mm fillet welds, each 80 mm long, spaced 60 mm apart, receiving 50 kN load at 40 mm eccentricity. Throat area = 2 × 80 × 0.707 × 6 = 678.7 mm². Direct shear stress = 50,000 N / 678.7 mm² = 73.7 N/mm². Polar moment of inertia ≈ 2 × (80³/12 + 80 × 30²) = 186,700 mm⁴. Torsional stress at peak = (50,000 × 40 × 50) / 186,700 = 535 N/mm². Resultant peak stress = √(73.7² + 535²) = 540 N/mm². For Grade 250 steel (Fy = 250 MPa), allowable = 0.6 × 250 = 150 N/mm²; connection requires strengthening.

Practical Notes

  1. Eccentricity dominates stress; moving load 10 mm off-centre on a 60 mm spaced pair can double torsional shear—always minimize e-distance in design.
  2. Larger weld spacing reduces stress concentration by increasing polar moment; 100 mm spacing versus 50 mm typically cuts peak torsional stress by 40%.
  3. Throat area = 0.707 × fillet size × weld length; undersized fillet (4 mm vs 6 mm) increases stress by 50%, often the failure mode in field welds.
  4. AWS D1.1 requires inspection when resultant stress exceeds 60% of electrode strength (e.g., 0.6 × 413 N/mm² for E70 rod).