Calculation Example of Rigid End Plate Connection According to EN 1993-1-8

Technical Article

In this example, design resistance of an end plate according to EN 1993-1-8 [1] is to be determined; the other components are not described here. To check the results, the dimensions of the connection IH 3.1 B 30 24 of Typified Connections [2] were used. The material S 235 and bolts with the strength 10.9 are used.

Figure 01 - Calculation Example

Determining Effective Lengths

First, it is necessary to determine effective lengths of stub flanges according to Table 6.6. The bottom bolt row has barely an effect on the compression flange due to the very small lever arm, and is therefore neglected. Since both upper bolt rows are divided by the tension flange of the beam, the bolt rows must only be considered separately. Thus, you can avoid group failure of the bolt rows. The calculation of the effective lengths requires the parameters e, m, ex, mx, m2, bp, w.

Figure 02 - Parameters for Effective Lengths (Source: [1])

For this example, the following values have been calculated:
$\begin{array}{l}\mathrm e\;=\;75\;\mathrm{mm}\\\mathrm m\;=\;\frac{300\;-\;13.5}2\;-\;0.8\;\cdot\;\sqrt2\;=\;62.6\;\mathrm{mm}\\\;{\mathrm e}_\mathrm x\;=\;35\;\mathrm{mm}\\\;{\mathrm m}_\mathrm x\;=\;50\;-\;0.8\;\cdot\;9\;\cdot\;\sqrt2\;=\;39.8\;\mathrm{mm}\\\;{\mathrm m}_2\;=\;125\;-\;50\;-\;24\;-\;0.8\;\cdot\;9\;\cdot\;\sqrt2\;=\;40.8\;\mathrm{mm}\\\;{\mathrm b}_\mathrm p\;=\;300\;\mathrm{mm}\\\mathrm w\;=\;150\;\mathrm{mm}\end{array}$

In the case of the effective lengths, a distinction is made between circular and non-circular yield line patterns. The linear yield line pattern requires the value α from Figure 6.11. The input values for this are based on the relation of the lever arms to the girder web (λ1) or to the girder flange (λ2) to the total width of the T-stub flange. The values for α between two diagrams in Figure 6.11 may be linearly interpolated.

$\begin{array}{l}{\mathrm\lambda}_1\;=\;\frac{62.6}{62.6\;+\;75}\;=\;0.45\\{\mathrm\lambda}_2\;=\;\frac{40.8}{62.6\;+\;75}\;=\;0.30\\\mathrm\alpha\;\approx\;6.65\end{array}$

Figure 03 - Determining Value α (Source: [1])

Using these input values, the effective lengths are determined according to Table 6.6 as follows.

Circular yield line pattern for the outer bolt row:
${\mathrm l}_{\mathrm{eff},\mathrm{cp},\mathrm a}\;=\;\min\;\begin{Bmatrix}2\;\cdot\;\mathrm\pi\;\cdot\;39.8\\\mathrm\pi\;\cdot\;39.8\;+\;150\\\mathrm\pi\;\cdot\;39.8\;+\;2\;\cdot\;75\end{Bmatrix}\;=\;250.1\;\mathrm{mm}$

Circular yield line pattern for the inner bolt row:
leff,cp,i = 2 ∙ π ∙ 62.6 = 393.3 mm

Non-circular yield line pattern for the outer bolt row:
${\mathrm l}_{\mathrm{eff},\mathrm{nc},\mathrm a}\;=\;\min\;\begin{Bmatrix}4\;\cdot\;39.8\;+\;1.25\;\cdot\;35\\75\;+\;2\;\cdot\;39.8\;+\;0.625\;\cdot\;35\\0.5\;\cdot\;300\\0.5\;\cdot\;150\;+\;2\;\cdot\;39.8\;+\;0.625\;\cdot\;35\end{Bmatrix}\;=\;150.0\;\mathrm{mm}$

Non-circular yield line pattern for the inner bolt row:
leff,nc,i = 6.65 ∙ 62.6 = 416.3 mm

To determine the design resistance in Failure Mode 1, that is with pure flange yielding, the shorter length of both yield line patterns is used. When determining the design resistance in Failure Mode 2, that is bolt failure with simultaneous flange yielding, the non-circular yield line pattern can only occur.

This results in the following effective lengths.

Outer bolt row:
leff,1,a = 150 mm
leff,2,a = 150 mm

Inner bolt row:
leff1,i = 393.3 mm
leff2,i = 416.3 mm

Check if Prying Forces May Develop

Before the design resistance of the end plate in Failure Mode 1 is determined, it must be checked whether prying forces may develop. Since this allows you to achieve higher design resistance values, the dimensions and the thickness of the grip package should always be chosen or changed in the way that the equation Lb < Lb* is fulfilled. Lb is the bolt elongation length, taken equal to the grip length (total thickness of material and washers), plus half the sum of the height of the bolt head and the height of the nut.

${\mathrm L}_\mathrm b\ast\;=\;\frac{8.8\;\cdot\;\mathrm m^3\;\cdot\;{\mathrm A}_\mathrm s}{{\mathrm l}_{\mathrm{eff}1}\;\cdot\;{\mathrm t}_\mathrm f³}$

The grip length, assuming that a symmetrical beam joint is applied, results in:
Lb = 2 ∙ 25 + 2 ∙ 4 + 0.5 ∙ 19 + 0.5 ∙ 15 = 75 mm

Lb* must be determined separately for the outer and the inner bolt row.

Outer bolt row:
${\mathrm L}_\mathrm b\ast\;=\;\frac{8.8\;\cdot\;39.8^3\;\cdot\;353}{150\;\cdot\;25³}\;=\;83.6\;\mathrm{mm}$

Inner bolt row:
${\mathrm L}_\mathrm b\ast\;=\;\frac{8.8\;\cdot\;62.6^3\;\cdot\;353}{393.3\;\cdot\;25³}\;=\;124\;\mathrm{mm}$

Therefore, the prying forces may develop in both bolt rows.

Design Resistance of T-Stub Flanges

For the failure mode 'complete yielding of the flange', Method 1 of EN 1993-1-8 is used in this example. Tension resistance of both T-stub flanges is determined as follows.

$\begin{array}{l}{\mathrm F}_{\mathrm T,1,\mathrm{Rd}}\;=\;\frac{4\;\cdot\;{\mathrm M}_{\mathrm{pl},1,\mathrm{Rd}}}{\mathrm m}\\\mathrm{where}\\\;{\mathrm M}_{\mathrm{pl},1,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;{\mathrm l}_{\mathrm{eff},1}\;\cdot\;{\mathrm t}_\mathrm f²\;\cdot\;{\mathrm f}_\mathrm y}{{\mathrm\gamma}_{\mathrm M0}}\\\mathrm m\;=\;39.8\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\mathrm m\;=\;62.6\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\\\\\;{\mathrm M}_{\mathrm{pl},1,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;15.0\;\cdot\;2.52\;\cdot\;23.5}{1.0}\;=\;550.78\;\mathrm{kNcm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm M}_{\mathrm{pl},1,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;39.33\;\cdot\;2.52\;\cdot\;23.5}{1.0}\;=\;1,444.15\;\mathrm{kNcm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm F}_{\mathrm T,1,\mathrm{Rd}}\;=\;\frac{4\;\cdot\;550.78}{3.98}\;=\;553.55\;\mathrm{kN}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm F}_{\mathrm T,1,\mathrm{Rd}}\;=\;\frac{4\;\cdot\;1,444.15}{6.26}\;=\;922.78\;\mathrm{kN}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\end{array}$

Failure mode 'Bolt failure with yielding of the flange':

$\begin{array}{l}{\mathrm F}_{\mathrm T,2,\mathrm{Rd}}\;=\;\frac{2\;\cdot\;{\mathrm M}_{\mathrm{pl},2,\mathrm{Rd}}\;+\;\mathrm n\;\cdot\;{\mathrm{ΣF}}_{\mathrm t,\mathrm{Rd}}}{\mathrm m\;+\;\mathrm n}\\\mathrm{where}\\\;{\mathrm M}_{\mathrm{pl},2,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;{\mathrm l}_{\mathrm{eff},2}\;\cdot\;{\mathrm t}_\mathrm f²\;\cdot\;{\mathrm f}_\mathrm y}{{\mathrm\gamma}_{\mathrm M0}}\\{\mathrm{ΣF}}_{\mathrm t,\mathrm{Rd}}\;=\;\mathrm{ΣTension}\;\mathrm{resistance}\;\mathrm{of}\;\mathrm{bolts}\\\mathrm n\;=\;{\mathrm e}_\min\;<\;1.25\;\cdot\;\mathrm m\\\;{\mathrm e}_\min\;=\;35\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm e}_\min\;=\;75\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\\\mathrm m\;=\;39.8\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\mathrm m\;=\;62.6\;\mathrm{mm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\\\\{\mathrm{ΣF}}_{\mathrm t,\mathrm{Rd}}\;=\;\frac{2\;\cdot\;{\mathrm k}_2\;\cdot\;{\mathrm f}_\mathrm{ub}\;\cdot\;{\mathrm A}_\mathrm S}{{\mathrm\gamma}_{\mathrm M2}}\;=\;\frac{2\;\cdot\;0.9\;\cdot\;100\;\cdot\;3.53}{1.25}\;=\;508.32\;\mathrm{kN}\\\mathrm{The}\;\mathrm{punching}\;\mathrm{force}\;\mathrm{has}\;\mathrm{been}\;\mathrm{checked}\;\mathrm{but}\;\mathrm{is}\;\mathrm{not}\;\mathrm{governing}.\\\\\;{\mathrm M}_{\mathrm{pl},2,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;15.0\;\cdot\;2.52\;\cdot\;23.5}{1.0}\;=\;550.78\;\mathrm{kNcm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm M}_{\mathrm{pl},2,\mathrm{Rd}}\;=\;\frac{0.25\;\cdot\;41.63\;\cdot\;2.52\;\cdot\;23.5}{1.0}\;=\;1,528.60\;\mathrm{kNcm}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm F}_{\mathrm T,2,\mathrm{Rd}}\;=\;\frac{2\;\cdot\;550.78\;+\;3.5\;\cdot\;508.32}{3.98\;+\;3.5}\;=\;385.12\;\mathrm{kN}\;\mathrm{for}\;\mathrm{the}\;\mathrm{outer}\;\mathrm{bolt}\;\mathrm{row}\\\;{\mathrm F}_{\mathrm T,2,\mathrm{Rd}}\;=\;\frac{2\;\cdot\;1,528.60\;+\;7.5\;\cdot\;508.32}{6.26\;+\;7.5}\;=\;499.24\;\mathrm{kN}\;\mathrm{for}\;\mathrm{the}\;\mathrm{inner}\;\mathrm{bolt}\;\mathrm{row}\end{array}$

Governing Design Resistance of T-Stub Flanges

For both bolt rows, the Failure Mode 2 is governing.

Outer bolt row: 385.12 kN

Inner bolt row: 499.24 kN

Moment Resistance of Joint

The calculated design resistance values of the individual bolt rows must now be multiplied by the respective lever arm to the compression point.

The lever arms are
438 mm for the outer bolt row,
313 mm for the inner bolt row.

Thus, the design moment resistance of the joint results in
MRd = 385.12 ∙ 0.438 + 499.24 ∙ 0.313 = 324.95 kNm.

Figure 04 - Design Resistance Values of Bolt Rows and Related Lever Arms

Comparison of Results

If this joint is calculated as a rigid frame joint in RF-/FRAME-JOINT Pro, the resulting design resistance of the end plate is 319.79 kNm. According to Typified Connections [2], the design resistance is 331.3 kNm, which relatively accurately corresponds with the manual calculation.

Figure 05 - Design Resistance in RF-/FRAME-JOINT Pro

Reference

[1]  Eurocode 3: Design of steel structures - Part 1-8: Design of joints; EN 1993-1-8:2005 + AC:2009
[2]  Weynand, K. & Oerder, R. (2013). Typisierte Anschlüsse im Stahlhochbau nach DIN EN 1993-1-8. Düsseldorf: Stahlbau.

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