# Considering Elastic Slip Modulus of Timber Connection

### Tips & Tricks

If a timber connection is designed as shown in Figure 01, the torsional spring stiffness resulting from the connection can be considered. This can be determined by using the modulus of displacement of the fastener and the polar moment of inertia of the connection, neglecting the area of the fastener.

#### Polar moment of inertia

The polar moment of inertia for the connection shown in Figure 01 results in:

Polar moment of inertia

$$Ip = ∑xi2i=1n + ∑yi2i=1n$$

 Ip Polar moment of inertia without component of fastener surfaces xi Distance from the centroid of the fastener group to the fastener in the x-direction yi Distance from the centroid of the fastener group to the fastener in the y-direction

Ip = 75 2 + 75 2 + 225 2 +225 2 = 112,500 mm 2

#### Determination of the modulus of displacement for the serviceability limit state

The modulus of displacement for the serviceability limit state can be calculated according to [1] Table 7.1. For fitting bolts with a diameter of 20 mm in softwood C24, this results in per shear plane as follows:

Modulus of displacement per shear plane

$$Kser = ρm1,5 · d23$$

 Kser Modulus of displacement per shear plane ρm Mean value of the bulk density in kg/m³ d Diameter of the fastener

Kser = 420 1.5 20/23 = 7,485 N/mm = 7,485 kN/m

Thus, there are two shear planes for an internal steel plate. In addition, the modulus of displacement should be multiplied by the factor 2.0 for steel plate-timber connections according to [1], Chapter 7.1 (3). The modulus of displacement for the fitting bolt can thus be determined as follows:

Kser = 2 ⋅ 2 ⋅ 7,485 kN/m = 29,940 kN/m

#### Determination of the modulus of displacement for the ultimate limit state

According to [1] , the modulus of displacement of a connection in the ultimate limit state, Ku, has to be assumed as follows:

Initial modulus of displacement

$$Ku = 23 · Kser$$

 KU Initial modulus of displacement Kser Displacement modulus of a fastener

Ku = 2/3 ⋅ 29,940 kN/m = 19,960 kN/m

In [2] and [3] , it is required to consider the design value of the modulus of displacement of a connection.

Design value of the modulus of displacement

$$Kd = KuγM$$

 Kd Design value of the modulus of displacement KU Initial modulus of displacement γM Partial safety factor for connections according to [1] Table 2.3

Kd = 19,960 kN/m/1.3 = 15,354 kN/m

#### Determination of the torsional spring stiffness

For the ultimate limit state design, you must use the design value of the slip modulus for calculation, and the mean value for the serviceability limit state design, and therefore you obtain two torsional spring rigidities.

Torsional spring stiffness for the serviceability limit state

$$Cφ,SLS = Kser · Ip$$

 Cφ, SLS Torsional spring stiffness for the serviceability limit state Kser Displacement modulus of a fastener Ip Polar moment of inertia without component of fastener surfaces

Cφ, SLS = 29,940 N/mm 112,500 mm 2 = 3,368 kNm/rad

Torsional spring stiffness for the ultimate limit state

$$Cφ,ULS = Kd · Ip$$

 Cφ, ULS Torsional spring stiffness for the ultimate limit state Kd Design value of the modulus of displacement Ip Polar moment of inertia without component of fastener surfaces

Cφ, ULS = 15,354 N/mm ⋅ 112,500 mm 2 = 1,727 kNm/rad

To take into account both rigidities, activate the ‘Modify Stiffness’ subtab (select the corresponding check box in the ‘Calculation Parameters’ subtab of the ‘Load Combinations’ tab in the ‘Edit Load Combinations and Calculations’ dialog box). Thus, as in this example, the torsional spring stiffness for all ULS combinations can be multiplied bythe factor C φ, SLS/Cφ, ULS. The value of C φ, SLS is entered in the support or hinge conditions. Thus, a torsional spring stiffness of 1,727 kNm/rad is used for all ULS combinations and 3,368 kNm/rad for all SLS combinations. The video shows the procedure.

In this example, the elastic foundation rotation is considered to be infinite and is not taken into account.

#### Determination of the torsional spring stiffness using the RF-/JOINTS Timber - Steel to Timber add -on module

When calculating the connection with RF-/JOINTS Timber - Steel to Timber, the results of the torsional spring stiffnesses are also displayed (see Figure 02). In RSTAB, these must then be transferred manually to the support or hinge conditions. In RFEM, this can be done automatically. The connections are automatically created in RFEM and the stiffness is adopted accordingly. The video shows the procedure.

#### Dipl.-Ing. (FH) Gerhard Rehm

Product Engineering & Customer Support

Mr. Rehm is responsible for the development of products for timber structures, and provides technical support for customers.

#### Reference

 [1] Eurocode 5: Design of timber structures - Part 1-1: General - Common rules and rules for buildings; EN 1995-1-1:2010-12 [2] National Annex - Eurocode 5: Design of timber structures - Part 1-1: General - Common rules and rules for buildings; DIN EN 1995-1-1/NA:2013-08 [3] Eurocode 5: Bemessung und Konstruktion von Holzbauten - Teil 1‑1: Allgemeines - Allgemeine Regeln und Regeln für den Hochbau - Nationale Festlegungen zur Umsetzung der OENORM EN 1995‑1‑1, nationale Erläuterungen und nationale Ergänzungen; ÖNORM B 1995‑1‑1:2015‑06‑15

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• Updated 04/07/2021

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