Resistance of the steel lattice members in Shapemaker

Shapemaker is a tool used to evaluate the utilization of telecom structures. The most commonly used structural type in the mobile network industry is the lattice tower, typically composed of three or four truss faces and multiple vertically stacked sections.

These 3D frameworks are made up of members arranged in triangular configurations. As such, the members are generally considered to be effectively pin-connected in accordance with Clause 5.2.2 of EN 1993-3-1, meaning they do not resist bending moments.

Based on this assumption, the structural check of individual members involves verifying that the axial force, obtained from elastic global analysis across various load combinations, does not exceed the following resistances:

• Tension resistance, as defined in Clause 6.2.3 of EN 1993-1-1
• Compression resistance, as defined in Clause 6.2.4 of EN 1993-1-1
• Flexural buckling resistance, as defined in Clause 6.3.1 of EN 1993-1-1, including slenderness evaluation in accordance with Annexes G and H of EN 1993-3-1

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Computation of Cross-Sectional Properties

Shapemaker performs automatic calculation of cross-sectional properties, including:

• Cross-sectional area $\left({A}\right)$
• Moments of inertia $\left({I}\right)$ about principal axes: parallel to the tower face, perpendicular to the tower face, and about the weak axis
• Radii of gyration $\left({i}\right)$ corresponding to the above axes

The software supports the following cross-sectional profiles:

• Equal and unequal leg angle sections, example cross-section ID: L120x10 or L120x80x8
• Tubular hollow sections, example cross-section ID: CHSC114.3x5.0 or CHSH114.3x5.0 (‘C’ stands for cold-formed and ‘H’ stands for hot rolled)
• Circular solid sections, example cross-section ID: D30 
• Square and rectangular hollow sections, example cross-section ID: SHSC80x5 or RHS100x80x4 
• 60° equal angle sections, example cross-section ID: VL120x10
• Custom built-up sections composed of closely spaced elements in various configurations using the above standard profiles (see next paragraph)

These properties are computed in accordance with standard geometric formulations and are used as input for resistance checks and global analysis.

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Definition of built-up cross-sections

Shapemaker has the capability to use built-up profiles that fall within the definition of closely built-up cross-sections of EN 1993-1-1 6.4.4.
This type of cross-section can now be utilised during the tower modelling process in Tower Builder.
The following built-up cross-section configurations are currently supported:

• Star arrangement with Equal Angle + Equal Angle (type `XL`), example cross-section ID: L120x10+L120x10@XL:10

• Star arrangement with 60° Equal Angle + 60° Equal Angle (type `VX`), example cross-section ID: VL120x10+VL120x10@VX:10

• Back-to-back arrangement with Equal Angle + Equal Angle (type `EL`), example cross-section ID: L120x10+L120x10@EL:10

• Back-to-back arrangement with Unequal Angle + Unequal Angle connected via the longer leg (type `LL`), example cross-section ID: L120x80x8+L120x80x8@LL:10

• Back-to-back arrangement with Unequal Angle + Unequal Angle connected via the shorter leg (type `SL`), example cross-section ID: L120x80x8+L120x80x8@SL:10

• Aligned leg arrangement with 2x 60° Equal Angle (type `VV`), example cross-section ID: VL120x10@VV:10

• Equal Angle clamped onto a Circular member (type `OL`), example cross-section ID: CHSC114.3x5+L120x10@OL:10

In the cross-section name, the first element always represents the continuous member along the full length, while the second element may either be continuous along the member length or discontinuous as not connected with the adjacent members through the joint.
The parameter that defines the continuity is called ‘composite action’ and should be selected only when full continuity of both chords is present.

A composite cross-section may also include a parameter that specifies the gap between the components, described after the colon (:) in the cross-section name as the final value.

Another required parameter is the spacing between connectors (battens or bolts) that join the components along their length.

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Effective Parameters for Class 4 Cross-Sections

Cross-sections are classified into one of four classes based on width-to-thickness (or diameter-to-thickness) ratios and steel grade, in accordance with Table 5.2 of EN 1993-1-1.

This classification directly impacts the approach used for calculating member buckling resistance.
For cross-section classes 1 to 3, standard section properties are used in design. However, when a section falls into Class 4, an additional assessment is required to determine effective cross-sectional properties $\left({A_{eff}}, {I_{eff}}, {i_{eff}}\right)$.

As per Clause 4.3 of EN 1993-1-5, a Class 4 section must be subdivided into individual plate elements. For each plate, a reduction factor $\left({\rho}\right)$ is determined using expressions (4.2) or (4.3), depending on the boundary conditions and loading.

The procedure for each plate is as follows:
\[\overline{\lambda}_{p} = \frac{\frac{b}{t}}{28.4 \times \varepsilon \times \sqrt{k_{\sigma}}}\]

where:

${b}$ - is adequate portion of the plate width defined in table 5.2 of EN 1993-1-1

${t}$ - plate thickness

$\varepsilon = \sqrt{\frac{235 MPa}{f_y}}$

${k_\sigma}$ - buckling factor (for the cases when no bending occurs (${\Psi = 1}$), can be defined as 0.43 for outstand plates and 4 for internal plates

The reduction factor is then computed as one value from the below:

• Outstand plates:
  
  • $\varrho = 1 \quad \text{for} \quad \overline{\lambda}_{p} \leq 0.673$
  • $\varrho = \frac{\overline{\lambda}_{p} - 0.055 \times (3 + \Psi)}{\overline{\lambda}_{p}^2} \quad \text{for} \quad \overline{\lambda}_{p} > 0.673, \quad \text{where} \quad \Psi \geq -3$
• Internal plates:
  
  • $\varrho = 1 \quad \text{for} \quad \overline{\lambda}_{p} \leq 0.748$
  • $\varrho = \frac{\overline{\lambda}_{p} - 0.188}{\overline{\lambda}_{p}^2} \quad \text{for} \quad \overline{\lambda}_{p} > 0.748$

This reduction factor is then applied to the plate widths to account for the effects of local buckling. Portions of the plate—typically the free edges of outstand elements or internal parts are excluded from the effective section. The modified geometry is then used to recalculate the effective section properties used in subsequent strength and stability verifications.

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Net cross-section area

For members connected using bolted joints, the cross-sectional area must be reduced to account for bolt holes when evaluating tension resistance, in accordance with Clause 6.2.2.2 of EN 1993-1-1. The net area $\left({A_{net}}\right)$ represents the minimum effective cross-sectional area along the member, considering the material removed by the bolt holes.

When the fastener holes are staggered the net area is calculated as the lowest from:

• ${A_{net}}$  in any cross-section of the member
• $A - t \times \left[ n \times d_0 - \sum \left( \frac{s^2}{4p} \right) \right]$

where:

${t}$ - is the member thickness

${n}$ - is the number of holes extending in any diagonal or zig-zag line across the member

${d_0}$ - diameter of the hole

${s}$ - spacing between 2 rows of staggered bolts

${p}$ - spacing between the columns of the bolt hole

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Tension resistance algorithm

For the sections with the holes the tension resistance $\left({N_{t,Rd}}\right)$ is taken from formulas 6.5 and 6.6 of EN 1993-1-1 as a smaller of the:

• $N_{\mathrm{pl,Rd}} = \frac{A \times f_y}{\gamma_{\mathrm{M0}}}$
• $N_{u,\mathrm{Rd}} = \frac{0.9 \times A_{\mathrm{net}} \times f_u}{\gamma_{\mathrm{M2}}}$

where:

${A}$ - cross-section area

${A_{net}}$ - net cross-section area

${f_y}$ - yield strength of the steel from table 3.1 of EN 1993-1-1

${f_u}$ - ultimate strength of the steel from table 3.1 of EN 1993-1-1

${\gamma_{M0}}$ - partial factor due to member yielding from 6.1 of EN 1993-3-1

${\gamma_{M2}}$ - partial factor due to member net section at bolt holes from 6.1 of EN 1993-3-1

If no bolt holes are present in the cross-section, then the first formula defines the tension resistance of the member.

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Compression resistance algorithm

The compression with no buckling effect resistance is calculated as per formulas 6.10 for class 1 to 3 members and 6.11 for class 4 members for EN 1993-1-1.
No fastener holes are assumed to not to be oversized and filled with the bolts, so the total cross-section area is considered in the checks, however in the class 4 members the ${A_{eff}}$ with reduced cross-section size is used.

Compression resistance is defined as:

\[N_{c,\mathrm{Rd}} = \frac{A \times f_y}{\gamma_{\mathrm{M0}}}\]where:

${f_y}$ - yield strength of the steel from table 3.1 of EN 1993-1-1

${\gamma_{M0}}$  - partial factor due to member yielding from 6.1 of EN 1993-3-1

${A}$ is replaced with ${A_{eff}}$ for the class 4 members

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Flexural Buckling Resistance Algorithm

Flexural buckling resistance is often the governing criterion in member utilization checks, particularly for slender members subject to axial compression. Long members with relatively small cross-sections are especially susceptible to buckling, and their axial load-carrying capacity is significantly reduced as slenderness increases.

In 3D structural systems such as lattice towers, members must be evaluated for potential buckling in multiple directions. Due to the triangulated geometry—where legs and bracing elements mutually restrain each other—the buckling length of a member depends on its orientation and the stiffness and positioning of adjacent elements.

Shapemaker automatically determines the buckling lengths in the following directions:

• In the plane of the tower face
• In the plane normal to the tower face
• About the weakest axis of the cross-section (i.e., axis with the smallest radius of gyration)

The effective slenderness factor $\left({k}\right)$ is also computed automatically, based on Table G.1 of EN 1993-3-1, taking into account:

• The cross-section type and bracing pattern for tower legs
• The number of bolts in connections for bracing members, 
  
  • Assessment in circular members:
    
    • ${k  = 0.7}$ when no bolts are specified
    • ${k = 0.95}$ when the single bold is used
    • ${k = 0.85}$ (in plane) or ${k = 0.95}$ (out of plate), when more than a single bolt is used to connect the member
  • Assessment in angular members:
    
    • Table G.2 (a) of EN 1993-3-1 is used. No bolts or more than a single bolt are considered a welded connection.

The buckling resistance for uniform members in compression is calculated using the procedure outlined in Clause 6.3.1 of EN 1993-1-1, while the slenderness evaluation follows the guidance provided in Annex G of EN 1993-3-1.

The algorithm for evaluating buckling resistance under a single buckling mode proceeds as follows:

Member slenderness:

\[\lambda = \frac{l}{i}\]

where:

${l}$ - considered buckling length

${i}$ — radius of gyration about the considered axis $\left(i_{\mathrm{eff}} \text{ for class 4 members}\right)$

Non-dimensional slenderness

\[\overline{\lambda} = \frac{\lambda}{\pi \times \sqrt{\frac{E}{f_y}}}\]

Non-dimensional effective slenderness:

\[\overline{\lambda_{eff}} = k \times \overline{\lambda}\]

Reduction factor:

\[\chi = \frac{1}{\Phi + \sqrt{\Phi^{2} - \overline{\lambda_{eff}}^{2}}}\]

or ${1}$ whichever is less

where:

$\Phi = 0.5 \times \left[ 1 + \alpha \left( \overline{\lambda_{eff}} - 0.2 \right) + \overline{\lambda_{eff}}^2 \right]$

${\alpha}$ - imperfection factor from buckling curve, while the buckling curve comes from table 6.2 of EN 1993-1-1 based on the cross-section shape.

Finally the buckling resistance of the member is defined as:

\[N_{b,\mathrm{Rd}} = \frac{\eta \times \chi \times A \times f_y}{\gamma_{M1}}\]

where:

${\gamma_{M1}}$ - partial factor due to member buckling

${\eta}$ - reduction factor for single angle members connected with 1 bolt (0.8 for member connected with 1 bolt at each end, 0.9 for member connected with 1 bolt at one end and continuous at the other end, while buckling length is defining the member ends)

${A}$ is replaced with ${A_{eff}}$ for the class 4 members

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Structural capacity assessment of built-up cross-sections

Case with no composite action

If the additional chord is not continuous, only the original member (the first in the section name) is considered for axial compression and tension resistance.

The flexural buckling capacity is calculated as the lowest value from:

• The original chord’s buckling capacity, accounting for moments of inertia about the buckling axes in the tower face, perpendicular to the tower face, and about the weakest member axis, is defined as the sum of the corresponding moments of inertia of both chords about their local axes.
• The buckling capacity of the original chord under the buckling length, defined as the connector spacing and effective slenderness factor ${k = 1}$.

Composite action case

If the cross-section is continuous, the parameter has composite action must be ticked.

According to EN 1993-1-1, cross-sections that meet the requirements of Table 6.9 can be considered as an integral member.

The requirements are expressed as follows:

• For elements connected by bolts (assumed to include types of EL, LL, and SL) or intermittent welds, the minimum connector spacing must be ${15 × {i_{min}}}$
• For elements connected by plates (assumed to include types of XL, VX, VV, OL), the minimum connector spacing must be ${70 × {i_{min}}}$

where ${i_{min}}$ stands for the minimum radius of gyration of the single chord.

While the connectors are applied close enough and the requirement is met, the member’s resistance is determined as if it were a solid section, using the combined cross-sectional parameters. The calculation is carried out just like for single-member elements, and according to Table G.1 of EN 1993-3-1, the effective slenderness factor ${k = 1}$.

If the minimum connector spacing condition is not satisfied, the calculations proceed as follows:

• Compression and tension resistance are checked using the parameters of the integral composite cross-section
• Buckling resistance is assessed as the minimum value of:
  
  • The sum of each component buckling resistance, which is calculated with a buckling length equal to the connector spacing and ${k = 1}$.
  • The buckling resistance of the unified (solid) member, as defined when the minimum connector spacing requirement is met.

Compiling Results and Assessing Member Utilisation

After calculating all resistance values, they are compared to the maximum tension and compression forces obtained from the FEA analysis for each load combination. The ratio of applied force to corresponding resistance defines the utilisation of each member.

These forces, associated resistances, and resulting utilisation ratios are organised by member type in the "Maximum Utilisation of Structural Members (ULS)" section of the full report.

Overutilised members in the panel are highlighted in red.