# Considering Elastic Slip Modulus of Timber Connection

## Technical Article on the Topic Structural Analysis Using Dlubal Software

### Technical Article

For a timber connection as shown in Figure 01, you can take into account the torsional spring rigidity (spring stiffness for rotation) of the connections. You can determine it by means of the slip modulus of the fastener and the polar moment of inertia of the connection.

#### Polar Moment of Inertia

The connection's polar moment of inertia 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 = 752 + 752 + 2252 +2252 = 112,500 mm2

#### Modulus of Displacement Determination for the Serviceability Limit State

The modulus of displacement for the serviceability limit state can be calculated according to [1], Table 7.1. For bolts with a diameter of 20 mm in C24 softwood, 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 density in kg/m³ d Diameter of the fastener

Kser = 4201.5 ⋅ 20/23 = 7,485 N/mm = 7,485 kN/m

This results in two shear planes for an internal steel plate. In addition, the modulus of displacement should be multiplied by a factor of 2.0 for steel plate-timber connections according to [1], Chapter 7.1 (3). You can determine the modulus of displacement for the bolt as follows:

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

#### Modulus of Displacement Determination 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

[2] and [3] require considering the design value of the modulus of displacement of a connection.

Design Value of 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

#### Torsional Spring Stiffness Determination

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

Torsional Spring Stiffness for 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 mm2 = 3,368 kNm/rad

Rotational Spring Stiffness for Ultimate Limit State

$$Cφ,ULS = Kd · Ip$$

 Cφ,ULS Rotational spring stiffness for 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 mm2= 1,727 kNm/rad

To take both rigidities into account, activate the "Modify Stiffness" sub-tab (select the corresponding check box in the Calculation Parameters sub-tab of the Load Combinations tab in the Edit Load Combinations and Calculations dialog box). As in this example, this allows you to multiply the torsional spring rigidity for all SLS combinations by Cφ,SLS / Cφ,ULS. The value of Cφ,SLS is entered in the support or hinge conditions. Thus, you can calculate with a torsional spring rigidity of 1.727 kNm/rad in all ULS combinations and with 3.368 kNm/rad in all SLS combinations. This approach is also shown in the video.

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

#### Torsional Spring Stiffness Determination Utilizing RF-/JOINTS Timber - Steel to the 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, you have to trandfer them 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: Design of timber structures - Part 1-1: General - Common rules and rules for buildings - Consolidated version with national specifications, national comments and national supplements for the implementation of OENORM EN 1995-1-1, ÖNORM B 1995-1-1:2015-06-15

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• Updated 21 March 2022

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