Worked examples

Assignment 1 Q3: one-sided rectangular weld group

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Determine the fillet-weld leg for the bracket’s four welded edges. One side plate carries characteristic dead $140 kN$ and imposed $200 kN$ . Steel S355, electrode Class 42; rectangular group $340 mm$ wide by $560 mm$ high; load eccentricity $420 mm$ from the group centreline.

Original source: Assignment/AY2627s 1-CON4334-Assignment 1.pdf — p. 2. Values tagged given are in the question or diagram; lookup values come from a named table; calculated values follow from the working; assumptions are stated explicitly.

Read the diagram and collect the data

Quantity	Source/type
$P$	Derived ultimate $1.4 \times 140 + 1.6 \times 200$ ; loads labelled dead/live are characteristic despite the general introductory wording “design load”.
Geometry	GivenB340,H560,e 420 mm; centroid offsetsx±170,y±280 mm calculated by rectangle symmetry.
Weld model	Use the course elastic line-weld method: uniform throat around a continuous closed rectangle. The S355/Class 42 table gives $p_{w} = 250 \frac{N}{{mm}^{2}}$ .
Unprovided inputs	Bracket thickness and column-flange thickness are not stated; minimum/maximum weld-leg detailing cannot be numerically certified.

Before calculating: recognition and strategy

Combine direct vertical force per unit length with the torsional line force caused byPe. Torsion acts tangentially; at the most heavily loaded corner one vertical component adds to direct shear and the horizontal component combines by Pythagoras. Use line inertias in ${mm}^{3}$ , not plate inertias in ${mm}^{4}$ .

1. Centroid, effective line length and corner coordinates

B = 340 mm

，

H = 560 mm

. Four-side line length:

L = 2 B + 2 H = 680 + 1,120 = 1,800 mm

By symmetry, the centroid is at half-width and half-height.

x_{corner} = \pm \frac{B}{2} = \pm 170 mm

y_{corner} = \pm \frac{H}{2} = \pm 280 mm

The course rectangular-group model uses the continuous perimeter as its effective line. Do not subtract $2 s$ separately at each corner of a continuous closed weld. If fabrication instead provides four separate terminated runs, their actual effective lengths and centroid/inertias require a new model.

Animation labA weld group is a set of lines1 concept · 1 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

The centre is empty: resistance comes from the weld lines, not a solid plate filling the group.
Use line lengths for centroid weighting. Line second moments have units $mm^{3}$ ; add the parallel-axis terms.
Direct force per length is $\frac{P}{L}$ . The eccentric moment adds tangential flow proportional to distance from the centroid.
Combine signed components at every candidate corner, then compare the maximum with throat resistance per length.

2. Ultimate load and eccentric moment

P = 1.4 \times 140 + 1.6 \times 200 = 196 + 320 = 516 kN

M = P e = 516 \times 420 = 216,720 kN \cdot mm

Direct vertical line force:

q_{s} = \frac{P}{L} = \frac{516}{1,800} = 0.286666667 \frac{kN}{mm}

The $420 mm$ eccentricity is already measured from the centroid. Adding $170 mm$ would double-count a geometric offset that is not present in this drawing.

Animation labFrom characteristic to design load2 concepts · 1 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

G is permanent load; Q is imposed load. A surface load and a line load also have different units.
This illustration uses the course gravity case $1.4 G + 1.6 Q$ . Other combinations in the original text retain their own factors.
For illustrative $G = 5 \frac{kN}{m ^{2}}$ , change Q and watch each separate contribution.
Do not carry this ULS total automatically into deflection. Follow the stated SLS load case.

3. Derive the line polar inertia

Simple explanation: Only the weld lines resist as weld

Imagine a wire rectangle: its empty middle is not more wire.

Count only the weld runs actually shown.
Use their total length and line second moments.
Combine direct and torsional forces at the critical location.

Remember: Line second moments have units ${mm}^{3}$ ; plate-area moments use ${mm}^{4}$ .

Related concept and full method

About the horizontal centroid axis, the two vertical runs contribute $\frac{2 H^{3}}{12}$ , and the two horizontal runs at $\pm \frac{H}{2}$ contribute $2 B {(\frac{H}{2})}^{2}$ . SwapB/ $H$ for the other axis.

I_{x, line} = 2 [\frac{H^{3}}{12} + B {(\frac{H}{2})}^{2}]

= 2 [\frac{560^{3}}{12} + 340 \times 280^{2}]

= 2 [14,634,666.666667 + 26,656,000]

= 82,581,333.333333 {mm}^{3}

I_{y, line} = 2 [\frac{B^{3}}{12} + H {(\frac{B}{2})}^{2}]

= 2 [\frac{340^{3}}{12} + 560 \times 170^{2}]

= 2 [3,275,333.333333 + 16,184,000]

= 38,918,666.666667 {mm}^{3}

J_{line} = I_{x} + I_{y} = 121,500,000 {mm}^{3}

The unit check is essential: $M$ ( $kN \cdot mm$ )×distance( $mm$ )/ $J$ ( ${mm}^{3}$ ) gives $\frac{kN}{mm}$ , which can be combined withP/L.

Animation labA weld group is a set of lines1 concept · 1 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

The centre is empty: resistance comes from the weld lines, not a solid plate filling the group.
Use line lengths for centroid weighting. Line second moments have units $mm^{3}$ ; add the parallel-axis terms.
Direct force per length is $\frac{P}{L}$ . The eccentric moment adds tangential flow proportional to distance from the centroid.
Combine signed components at every candidate corner, then compare the maximum with throat resistance per length.

4. Corner resultant and required fillet leg

Simple explanation: Add arrows before taking the magnitude

Walking east and walking north do not point in the same direction.

Choose signed horizontal and vertical directions.
Add contributions along each direction separately.
For perpendicular components, use the right-triangle resultant.

Remember: Check the corner where direct and torsional components reinforce each other.

Related concept and full method

At a corner where torsion adds to downward shear:

q_{torsion, vertical} = \frac{M \times 170}{J}

= \frac{216,720 \times 170}{121,500,000} = 0.303229630 \frac{kN}{mm}

q_{vertical, total} = 0.286666667 + 0.303229630 = 0.589896296 \frac{kN}{mm}

| q_{horizontal} | = \frac{M \times 280}{J}

= \frac{216,720 \times 280}{121,500,000} = 0.499437037 \frac{kN}{mm}

Resultant:

q = \sqrt{{0.589896296}^{2} + {0.499437037}^{2}}

= 0.772926254 \frac{kN}{mm}

Weld resistance per unit length:

\frac{0.7 s p_{w}}{1,000} = \frac{0.7 s \times 250}{1,000} = 0.175 s \frac{kN}{mm}

Required weld leg:

s \geq \frac{0.772926254}{0.175} = 4.416721 mm

For strength, trial a

5 mm

fillet weld; resistance:

0.175 \times 5 = 0.875 \frac{kN}{mm}

Utilisation:

\frac{0.772926254}{0.875} = 0.883344 < 1

Strength passes.

For a $5 mm$ equal-leg fillet the throat is $0.7 \times 5 = 3.5 mm$ . The smallest side length $340 mm$ greatly exceeds the $40 mm$ minimum effective-run scale; the continuous group model remains the assumed geometry.

Animation labA weld group is a set of lines2 concepts · 4 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

The centre is empty: resistance comes from the weld lines, not a solid plate filling the group.
Use line lengths for centroid weighting. Line second moments have units $mm^{3}$ ; add the parallel-axis terms.
Direct force per length is $\frac{P}{L}$ . The eccentric moment adds tangential flow proportional to distance from the centroid.
Combine signed components at every candidate corner, then compare the maximum with throat resistance per length.

Compact exam answer

One plate: $P = 516 kN$ , $e = 420 mm$ . Four-sided $340 \times 560$ weld group: $L = 1,800 mm$ , $J = 121.5 \times 10^{6} {mm}^{3}$ . Governing corner $q_{vertical} = 0.5898963$ , $q_{horizontal} = 0.4994370$ , resultant $0.7729263 \frac{kN}{mm}$ . S355／Class 42: $0.175 s \geq 0.7729263$ , giving $s \geq 4.41672 mm$ ; $5 mm$ passes strength. Minimum/maximum sizes and connected-part checks still require the unspecified plate/flange thickness.

Mistakes to avoid

Do not halve a load explicitly carried by one side plate.
Do not add half the weld width to $420 mm$ .
Do not add perpendicular force components arithmetically.
Do not use a plane-area polar inertia for a line-weld group.

Procedure for an unfamiliar variant

Read whether load is on one or two plates.
Locate the weld centroid and actual eccentricity.
Calculate line length and both centroidal line inertias.
Resolve torsion and direct force at the governing corner.
Convert the resultant to throat/leg and then check minimum/maximum detailing with actual member thicknesses.

Independent self-check

Try it yourself. Invented variant: keepP516 kN but increase eccentricity to $500 mm$ . Is the $5 mm$ strength trial still enough?

Reveal answer and reasoning

M258,000 kNmm. Vertical total= $\frac{516}{1,800} + \frac{258,000 \times 170}{121,500,000} \approx 0.647654 \frac{kN}{mm}$ ; horizontal= $\frac{258,000 \times 280}{121,500,000} \approx 0.594568$ . Resultant $\approx 0.87918 \frac{kN}{mm}$ exceeds $0.875$ , so $5 mm$ narrowly fails strength. Recompute the exact required leg and choose a larger permitted size; missing detailing inputs still remain.

Animation labA weld group is a set of lines1 concept · 1 source expressions

Supplement to the original lesson. Enable JavaScript to play, step through calculations and rotate 3D models. The following explanation remains readable offline.

The centre is empty: resistance comes from the weld lines, not a solid plate filling the group.
Use line lengths for centroid weighting. Line second moments have units $mm^{3}$ ; add the parallel-axis terms.
Direct force per length is $\frac{P}{L}$ . The eccentric moment adds tangential flow proportional to distance from the centroid.
Combine signed components at every candidate corner, then compare the maximum with throat resistance per length.