**2. Method of Determining the Resistance of Fillet Welds**

Welded joints made of hollow sections should be made using fillet or butt welds, or combination of the two, laid on the perimeter of the profile. In overlapped joints, the covered part of the member need not be welded, when the components of axial forces in the brace members perpendicular to the chord axis do not differ by more than 20% [18].

According to Dexter and Lee [19], the resistance of the overlapped CHS joints with the hidden part welded was about 10% higher.

To simplify the calculation of the welds in hollow section structures European standard recommends the design of the welds in such a way, that the weld resistance per unit length of the member perimeter should not be less than the resistance of that member also calculated per unit length of perimeter. This condition is met when butt or fillet welds which are used have such thickness, that their resistance is equal to the resistance of connected members. Methods of estimating the thickness of fillet welds that meet this requirement are given in [16].

The European standard [20] suggests, in cases where the design of a full butt weld or the adequate fillet weld is not required, that the thickness of the weld may be reduced, on condition that the resistance of such weld and its rotation capacity are checked, considering only the weld effective lengths.

In Figure 1, using the IIW guidance [13] the layout of fillet welds in the K-type overlapped joint between the RHS braces and the H chord is shown. An assumption was made that the value of the component force perpendicular to the chord transferred directly through the welds connecting the brace members is equal to (Figure 1a):

$$\begin{aligned} \Delta K\_i &= \alpha K\_i \sin \theta\_i \text{ when } 25\% \le \lambda\_{\text{av}} \le 80\%, \\ \Delta K\_i &= K\_i \sin \theta\_i \text{ when } \lambda\_{\text{av}} > 100\%. \end{aligned} \tag{1}$$

where α = *q*/*p* and 0.25 ≤ α ≤ 0.80, λ*ov* = (*q*/*p*) · 100% in %, *q*—the overlapping surface of braces projected on the face of the chord, *p*—length of contact area between the overlapping brace and the chord (the procedure of assessment of welds resistance in K-type joints made of hollow section is presented in [21]).

**Figure 1.** The connection of hollow section brace members to I- or H-section chords: (**a**) The standard joint: (**b**) The layout of fillet welds joining the rectangular hollow section (RHS) braces to the H chords when 25% ≤ λ*ov* ≤ 80% and the hidden part of the connection is welded; (**c**) the layout of fillet welds joining the RHS braces to the H chords when 25% ≤ λ*ov* ≤ 80% and the hidden part of the connection does not have to be welded; (**d**) layout of fillet welds joining the RHS braces to the H chords when λ*ov* > 100%.

The remaining part of the load component perpendicular to the chord is

$$\begin{aligned} redd\Delta K\_j &= K\_j \sin \theta\_j - \alpha K\_l \sin \theta\_l \text{ when } 25\% \le \lambda\_{\text{OV}} \le 80\%\_\prime\\ redd\Delta K\_j &= K\_j \sin \theta\_j - K\_l \sin \theta\_l \text{ when } \lambda\_{\text{OV}} > 100\%\_\prime\\ redd\Delta K\_j &= 0 \text{ in the case of no external load applied to the joint.} \end{aligned} \tag{2}$$

The values of effective lengths of welds are determined as follows (Figure 2):

**Figure 2.** Effective lengths of welds: (**a**) the layout of fillet welds joining the RHS braces to the H chords when 25% ≤ λ*ov* ≤ 80% and the hidden part of the connection is welded; (**b**) the layout of fillet welds joining the RHS braces to the H chords when 25% ≤ λ*ov* ≤ 80% and the hidden part of the connection is not welded; (**c**) the layout of fillet welds joining the RHS braces to the H chords when λ*ov* > 100%; (**d**) welds between the braces in the case of the partial overlap; (**e**) welds between the braces in the case of the full overlap.

1. In connections of braces with the flange of the chord (Figure 2a–c):

$$\begin{aligned} l\_1 &= h\_j / \sin \theta\_{j\prime} \quad b\_{j,rad} = b\_j - 2a\_{w\prime} \\ l\_2 &= p\_{j\varepsilon f\prime} = t\_w + 2r + \mathcal{T}t\_f f\_{y0} / f\_{yj\prime} \text{ but } p\_{j\varepsilon f\prime} \le b\_{j\prime} \end{aligned}$$

$$\begin{aligned} l\_3 &= h\_{i,rad} / \sin \theta\_{i} = (1 - \alpha)h\_i / \sin \theta\_{i\prime} \\ l\_4 &= p\_{i\varepsilon f\prime} = t\_w + 2r + \mathcal{T}t\_f f\_{y0} / f\_{yj\prime} \text{ but } p\_{i\varepsilon f\prime} \le b\_i \end{aligned} \tag{3}$$

2. In the direct connection between braces (Figure 2d,e):

• In the case of the partial overlap:

$$l\_5 = \frac{q}{\left(1 + t g \theta\_j / t g \theta\_i \right) \cos \theta\_j}, \quad l\_6 = b\_i \tag{4}$$

• In the case of the complete overlap:

$$l\_7 = h\_i / \sin\left(\theta\_i + \theta\_j\right) \\ l\_6 = b\_i \tag{5}$$

where θ*i*, θ*j*—inclination angles of the overlapping and overlapped braces in relation to the chord, *fy0* —the yield strength of the chord, *bi*, *hi*—respectively, the width and the height of the section of the overlapping brace, *bj*, *hj*—respectively, the width and the height of the section of the overlapped brace, *ti*—the wall thickness of the overlapping brace, *tj*—the wall thickness of the overlapped brace, *tf*—the flange thickness of the I-section, *tw*—the web thickness of I-section, *r* the radius of the I-section.

Areas of cross-sections of effective lengths of welds are (Figure 1): *A*<sup>1</sup> = *l*1*aw*1, *A*<sup>2</sup> = *l*2*aw*2, *A*<sup>3</sup> = *l*3*aw*3, *A*<sup>4</sup> = *l*4*aw*4, *Aj*,*red* = *bj*,*redawb*, *A*<sup>5</sup> = *l*5*aw*5,*A*<sup>6</sup> = *l*6*aw*6, *A*<sup>7</sup> = *l*7*aw*7, where *aw*1, *aw*2, *aw*3, *aw*<sup>4</sup> *awb*, *aw*5, *aw*6—thicknesses of welds.

In the case of 25% ≤ λ*ov* ≤ 80% design situation, all welds are made (also in hidden part) in the place of the splice of braces with the chord. The sections of fillet welds are loaded by the component forces in braces parallel to the chord (Figure 3a).

**Figure 3.** Component loads in the welds of braces: (**a**) parallel to the chord; (**b**) perpendicular to the chord.

Loads for individual effective lengths are

$$P\_1 \,' = \left(K\_{\bar{j}} \cos \theta\_{\bar{j}} + K\_{\bar{i}} \cos \theta\_{\bar{i}}\right) A\_1 / \sum A\_{\prime} \tag{6a}$$

$$P\_2 \prime = \left(K\_j \cos \theta\_j + K\_i \cos \theta\_i\right) A\_2 \prime \sum A\_\prime \tag{6b}$$

$$P\_3 ' = \left(K\_{\dot{\jmath}} \cos \theta\_{\dot{\jmath}} + K\_{\dot{\imath}} \cos \theta\_{\dot{\imath}}\right) A\_3 / \sum A\_{\prime} \tag{6c}$$

$$P\_4{}^{\prime} = \left(K\_j \cos \theta\_j + K\_i \cos \theta\_i\right) A\_4 / \sum A\_{\prime} \tag{6d}$$

$$P\_b{}^{\prime} = \left(K\_{\bar{j}} \cos \theta\_{\bar{j}} + K\_{\bar{i}} \cos \theta\_{\bar{i}}\right) A\_{\bar{j}, \text{rad}} / \sum A\_{\prime} \tag{68}$$

where *Kj* and *Ki* are the designed axial loads acting respectively in overlapped and overlapping braces and

$$
\sum A = 2A\_1 + A\_2 + 2A\_3 + A\_4 + A\_{j, rad}.\tag{7}
$$

In the same design situation, the fillet weld sections are loaded by load components in braces perpendicular to the chord, as shown in Figure 3b. Forces loading effective lengths can be calculated based on equations:

$$P\_1'' = 0,\tag{8a}$$

$$P\_2^{\prime\prime} = red\Delta K\_{\circ} \cdot A\_2 / \left(A\_2 + A\_{\text{j.rad}}\right) \tag{8b}$$

$$P\_{\mathfrak{J}}^{\prime\prime} = (1 - \alpha)\Delta K\_{\mathfrak{i}} \cdot A\_{\mathfrak{J}} / (2A\_{\mathfrak{J}} + A\_{\mathfrak{4}}) ,\tag{8c}$$

$$P\_4' = (1 - \alpha)\Delta K\_i \cdot A\_4 / (2A\_3 + A\_4) \, \tag{8d}$$

$$P\_{\rm b}^{\prime\prime} = red\Delta K\_{\rm j} \cdot A\_{\rm j,red} / \left(A\_2 + A\_{\rm j,red}\right) \tag{8e}$$

Components of loads in welds directly between the braces, in the case of the partial overlap are determined from equations (Figure 2d):

• Loads parallel to the overlapped brace axis:

$$P\_{\mathbb{S}}' = \Delta \mathcal{K}\_{i} \cdot A\_{\mathbb{S}} \sin \theta\_{j} / (2A\_{\mathbb{S}} + A\_{\mathbb{6}}),\tag{9}$$

$$P\_6' = \Delta \mathcal{K}\_i \cdot A\_6 \sin \theta\_j / (2A\_5 + A\_6). \tag{10}$$

• Loads perpendicular to the overlapped brace axis:

$$P\_{\mathfrak{F}}^{\prime\prime} = \Delta K\_i \cdot A\_{\mathfrak{F}} \cos \theta\_j / (2A\_{\mathfrak{F}} + A\_{\mathfrak{G}}),\tag{11}$$

$$P\_{\theta}^{\prime\prime} = \Delta K\_i \cdot A\_6 \cos \theta\_j / (2A\_5 + A\_6). \tag{12}$$

Components of loads in welds directly between the braces, in the case of the full overlap are determined from equations (Figure 2e):

• Loads parallel to the overlapped brace axis:

$$P'\_6 = \Delta \mathcal{K}\_i \cdot A\_6 \sin \theta\_j / (2A\_6 + 2A\gamma),\tag{13}$$

$$P\_{\mathcal{T}}' = \Delta \mathbb{K}\_{i} \cdot A\_{\mathcal{T}} \sin \theta\_{\circ} / (2A\_{6} + 2A\_{7}). \tag{14}$$

• Loads perpendicular to the overlapped brace axis:

$$P\_{6}^{\prime\prime} = \Delta K\_{i} \cdot A\_{6} \cos \theta\_{j} / (2A\_{6} + 2A\_{7}) \tag{15}$$

$$P\_{7}^{\prime\prime} = \Delta K\_{\circ} \cdot A\_{7} \cos \theta\_{\circ} / (2A\_{6} + 2A\_{7}). \tag{16}$$

Stresses in welds caused by the force parallel to the chord at the partial overlap of 25% ≤ λ*ov* ≤ 80%: 1. In longitudinal welds (Figure 4a):

• In the case of welds placed by the overlapped brace's walls:

$$
\sigma' = 0,\ \sigma'\_{\perp} = \tau'\_{\perp} = 0,\ \tau'\_{II} = P'\_1 / \left(a\_{w1} \cdot l\_1\right). \tag{17}
$$

• In the case of welds placed by the overlapping brace's walls:

$$
\sigma' = 0,\ \sigma'\_{\perp} = \tau'\_{\perp} = 0,\ \tau'\_{II} = \mathcal{P}'\_3 / (a\_{w3} \cdot l\_3) \tag{18}
$$

**Figure 4.** Stresses in welds caused by the load parallel to the chord: (**a**) in longitudinal welds; (**b**) on the not fully cooperating overlapped brace's transverse length; (**c**) on the fully cooperating transverse length of the overlapped brace; (**d**) on the not fully cooperating transverse length of the overlapping brace.

2. On the not fully cooperating overlapped brace's transverse length (Figure 4b):

$$
\sigma' = \mathbf{P}'\_2 / (a\_{w2} l\_2), \ \sigma'\_{\perp} = \sigma' \sin(\theta\_j / 2), \ \tau'\_{\perp} = \sigma' \cos(\theta\_j / 2), \ \tau'\_{\Pi} = 0. \tag{19}
$$

3. On the fully cooperating transverse length of the overlapped brace (Figure 4c):

$$
\sigma \nu = \mathbf{P}'\_b / \left( a\_{wb} b\_{j,rd} \right), \ \sigma'\_{\perp} = \sigma' \cos \left( \theta\_j / 2 \right), \tau'\_{\perp} = \sigma' \sin \left( \theta\_j / 2 \right), \ \tau'\_{ll} = 0. \tag{20}
$$

4. On the not fully cooperating overlapping brace 's transverse length (Figure 4d):

$$
\sigma \nu = P\_4'/(a\_{w4}l\_4), \ \sigma\_\perp' = \sigma' \sin(\theta\_i/2), \ \tau\_\perp' = \sigma' \cos(\theta\_i/2), \ \tau\_{II}' = 0. \tag{21}
$$

Stresses in welds of load perpendicular to the chord at the partial overlap of 25% ≤ λ*ov* ≤ 80%: 1. In longitudinal welds (Figure 5a):

• In the case of welds placed by the walls of the overlapped brace:

$$
\sigma'' = \frac{P\_1 \prime}{a\_{w1} l\_1}, \; \sigma\_\perp \prime = \frac{\sigma \prime}{\sqrt{2}}, \tau\_\perp \prime = -\frac{\sigma \prime}{\sqrt{2}}, \tau\_{\parallel \parallel} \prime = 0. \tag{22}
$$

• I the case of welds placed by the walls of the overlapping brace:

, σ⊥-- <sup>=</sup> <sup>−</sup> <sup>σ</sup>--

σ-- <sup>=</sup> *<sup>P</sup>*<sup>3</sup> --

√

, τ⊥-- <sup>=</sup> <sup>σ</sup>--

√

, τ*II*--

= 0. (23)

**Figure 5.** Stresses in welds of the load perpendicular to the chord: (**a**) in longitudinal welds; (**b**) on the not fully cooperating the overlapped brace's transverse weld; (**c**) on the fully cooperating the overlapped brace's transverse weld; (**d**) On the not fully cooperating the overlapping brace's transverse weld.

2. On the not fully cooperating the overlapped brace's transverse weld (Figure 5b):

$$
\sigma'' = \frac{P\_2''}{a\_{\text{w2}}!\_2}, \sigma\_\perp'' = -\sigma'' \cos \frac{\theta\_j}{2}, \tau\_\perp'' = \sigma'' \sin \frac{\theta\_j}{2}, \tau\_{II}'' = 0. \tag{24}
$$

3. On the fully cooperating the overlapped brace's transverse weld (Figure 5c):

$$
\sigma'' = \frac{P\_b \prime}{a\_{wb} b\_{j, rad}}, \sigma\_\perp \prime = \sigma \prime \cos \frac{\theta\_j}{2}, \tau\_\perp \prime = -\sigma \prime \sin \frac{\theta\_j}{2}, \tau\_\Pi \prime = 0. \tag{25}
$$

4. On the not fully cooperating the overlapping brace's transverse weld (Figure 5d):

$$
\sigma'' = \frac{P\_4 \prime}{a\_{\text{n4}} l\_4}, \; \sigma\_\perp \prime = -\sigma \prime \cos \frac{\theta\_\text{i}}{2}, \; \tau\_\perp \prime = \sigma \prime \sin \frac{\theta\_\text{i}}{2}, \; \tau\_{\text{II}} \prime = 0. \tag{26}
$$

Stresses in the welds made directly between the brace members at the partial overlap of 25% ≤ λ*ov* ≤ 80% from the force parallel to the overlapped brace axis:

1. In longitudinal welds (Figure 6a):

$$
\sigma' = 0, \ \sigma'\_{\perp} = \tau'\_{\perp} = 0, \ \tau\_{II}{}' = \frac{P\_5{}}{a\_{w5} l\_5}. \tag{27}
$$

**Figure 6.** Stresses in welds between the brace members from the force parallel to the overlapped brace: (**a**) in longitudinal welds; (**b**) in the fully cooperating transverse weld.

2. In the fully cooperating transverse weld (Figure 6b):

$$
\sigma' = \frac{\mathbf{P}\_6'}{a\_{\text{w}6} l\_6}, \\
\sigma\_\perp ' = -\sigma' \cos \frac{\theta\_\mathbf{i} + \theta\_\mathbf{j}}{2}, \\
\tau\_\perp ' = \sigma' \sin \frac{\theta\_\mathbf{i} + \theta\_\mathbf{j}}{2}, \\
\tau\_{\text{II}} ' = 0. \tag{28}
$$

Stresses in welds placed between the brace members at the partial overlap of 25% ≤ λ*ov* ≤ 80% from the force perpendicular to the overlapped brace:

1. In longitudinal welds (Figure 7a):

$$
\sigma'' = \frac{P\_{5''}}{a\_{w\overline{5}l5}}, \ \sigma\_\perp '' = -\frac{\sigma''}{\sqrt{2}}, \tau\_\perp '' = \frac{\sigma''}{\sqrt{2}}, \tau\_{ll'} '' = 0. \tag{29}
$$

**Figure 7.** Stresses in welds between the brace members from the force perpendicular to the overlapped brace: (**a**) in longitudinal welds; (**b**) in the fully cooperating transverse weld.

2. In the fully cooperating transverse weld (Figure 7b):

$$
\sigma'' = \frac{P\_6''}{a\_{w\theta} l\_6}, \; \sigma\_\perp'' = \sigma'' \cos \frac{\theta\_j + \theta\_i}{2}, \; \tau\_\perp'' = -\sigma'' \sin \frac{\theta\_j + \theta\_i}{2}, \; \tau\_{\text{II}}'' = 0. \tag{30}
$$

The procedure of checking of a design situation with the full overlap of braces λ*ov* = 100% is analogous. In that case, the stresses in the transverse weld located near the connection of the overlapped brace with the chord should be examined (Figure 8), using the loads expressed by the Equations (24) and (25). The components of stress are:

**Figure 8.** Stresses in the transverse weld with full overlap: (**a**) in the transverse weld from the force parallel to the overlapped brace; (**b**) in the transverse weld from the force perpendicular to the overlapped brace.

1. In the transverse weld from the force parallel to the overlapped brace (Figure 8a):

$$
\sigma' = \frac{P\_6 '}{a\_{w6} l\_6 '}, \\
\sigma\_\perp ' = -\sigma' \sin \frac{\theta\_j + \theta\_i}{2}, \\
\tau\_\perp ' = -\sigma'' \cos \frac{\theta\_j + \theta\_i}{2}, \\
\tau\_{II} ' = 0. \tag{31}
$$

2. In the transverse weld from the force perpendicular to the overlapped brace (Figure 8b):

$$
\sigma'' = \frac{P\_6''}{a\_{w6} l\_6}, \sigma\_\perp ' = -\sigma' \cos \frac{\theta\_j + \theta\_i}{2}, \tau\_\perp '' = \sigma'' \sin \frac{\theta\_j + \theta\_i}{2}, \tau\_{\Pi} '' = 0. \tag{32}
$$

The component stresses occurring in cross-section of the welds should be added using the formulas:

$$
\pi \mu = \pi\_{\ll}' + \pi\_{\ll}'' ; \sigma\_{\perp} = \sigma\_{\perp}' + \sigma\_{\perp}'' ; \pi\_{\perp} = \pi\_{\perp}' + \pi\_{\perp}''. \tag{33}
$$

Standardized formulas for checking the safety of fully or partially cooperating transverse and longitudinal welds are:

$$\left[\sigma\_{\perp}^{2} + 3\left(\tau\_{\perp}^{2} + \tau\_{II}^{2}\right)\right] \le f\_{\text{u}} / \left(\beta\_{\text{w}} \gamma\_{\text{M2}}\right),\tag{34}$$

$$
\sigma\_{\perp} \le 0.9 f\_{\text{u}} / \gamma\_{M2\nu} \tag{35}
$$

where: β*w*—the correlation coefficient, *fu*—the tensile strength of steel, γ*M*<sup>2</sup> = 1.25—the safety factor.

### **3. Conclusions**

RHS joints are generally semi-rigid, mainly because of the preferred technologies for their production, i.e., the direct welding of members. It implements a significant load from braces to the relatively slender front walls of the chord. The basic information for calculating joint resistance is given in many standards and references, but European standards contain a general recommendation on the calculation of capacity of welded joints and do not provide detailed design guidelines. The information contained is random and concerns only Y, X, K, and N joints with the gap. Additionally, in the case of the K and N types of joints with partially overlapped brace members, there is no indication of how to calculate the capacity of welds between the members.

This paper presents the method of assessment of the welded K and N type overlapped joints between RHS brace members and I or H section chords. This method comprises determining the stress components in welds in different load cases based on their effective lengths.

### **Design Example:**

Check the resistance of welds of an overlap K joint with a chord made of HEB 120 and SHS braces (Figure 9). Steel grade of S355H, *fy* = 355 N/mm2, *fu* = 490 N/mm2. Chord: *h*<sup>0</sup> = 120 mm, *bf* = 120 mm, *tw* = 6.5 mm, *tf* = 11.0 mm, *r* = 12.0 mm, *A*<sup>0</sup> = 34.0 cm2, *Wpl.y,*<sup>0</sup> = 165.2 cm3, *N*<sup>0</sup> = <sup>−</sup>159.9 kN. Brace loaded with compressive force: *h*<sup>2</sup> = 80 mm, *b*<sup>2</sup> = 60 mm, *t*<sup>2</sup> = 4 mm; *N*2*.Ed* = −136.1 kN. Brace loaded

with tension force: *h*<sup>1</sup> = 60 mm, *b*<sup>1</sup> = 50 mm, *t*<sup>1</sup> = 3 mm, *N*1*.Ed* = + 103.2 kN. The angles: θ<sup>1</sup> = 50.34◦ > 30◦, θ<sup>2</sup> = 40.02◦ > 30◦ (sin θ<sup>1</sup> = 0.7698, cos θ<sup>1</sup> = 0.6382, sin θ<sup>2</sup> = 0.6431, cos θ<sup>2</sup> = 0.7658, sin (θ<sup>1</sup> + θ2) 1.0). The overlapped transverse weld is done −*cs* = 2.

**Figure 9.** The overlap joint made of H-section chord and SHS braces.
