# Considering End Releases Between Surfaces

### Technical Article

Figure 01 - Real Model and Structural System

#### End Releases Between Members

End releases between members are defined with member hinges when performing structural modeling and analysis. The definition is carried out comparable to the condition of statical indeterminacy to determine the statically indeterminacy of a structure:

n = r + 3 m - 3 n - h ≥ 0

with

r = support reactions

m = members

n = nodes

h = hinges

Therefore, it is necessary to assign always one hinge less than members with the same degree of freedom (h = m - 1) at a node. Figure 02 shows a valid definition (top) and an invalid definition (bottom).

Figure 02 - Member Hinge Members: Top Correct, Bottom Incorrect

#### End Releases Between Surfaces

The definition of end releases between surfaces is more complex, but is the same procedure as with members. Here too, two hinges generate a statically underdetermined structure with identical degree of freedom at a line. Contrary to members, the structures including surfaces are not instable so quickly. This is partly due to the fact that surfaces may warp in its plane and are therefore no more kinematic. Basically, when defining the hinges in Figure 03, the line will rotate around its own axis and the structure will therefore be kinematic.

Figure 03 - Kinematic Line Due to Two Hinges

#### Joint - Concrete Structures

The simplest case of line hinges is the joint between concrete surfaces mentioned above. It is used to model the assembly gap which is often necessary in concrete structures.

The linge hinges in u_{x}, u_{y} and u_{z} are released for this purpose (Figure 04). It is recommended to release the rotation of the line as well in this case. The degree of freedom which is released has to be selected for members and surfaces with hinges.

Figure 04 - Structural System Joint

#### Semi-Rigid Joint - Timber Structures

In timber structures, for example in cross-laminated timber or wood-based panel structures, the separation between surfaces is usually carried out flexibly. It is quite easy to consider a linear spring between two surfaces by using line hinges. However, the spring in timber structures is actually only available in tensile direction of the surface. In the contact area between the surfaces, wood-based panels or cross-laminated timber panels is an almost rigid pressure contact transmission. Thus, modeling such end releases is much more complex, since nonlinear properties have to be considered.

Nonlinear properties have disadvantages in terms of modeling, results evaluation, calculation time, number of unknowns, and so on. In the following, it is explained how it is possible to consider the nonlinearity pressure contact with linear line hinges. Figure 05 shows a structure consisting of four surfaces which are connected semi-rigidly. At the support node, the models have free supports in u_{x}. On the left, a surface is connected semi-rigidly with the fictive springs u_{x} = 100 kN/m² (longitudinal direction of the line) and u_{y} = 100 kN/m² (perpendicular to the line). On the right, the direction u_{x} = 100 kN/m² is connected identically. In u_{y}, the end release is rigid. At the head, the horizontal load is 15 kN/m.

Figure 05 - Comparison Stiffnesses

As you can see in Figure 05, the deformation of the left model is far too high. In addition, the upper surfaces intersect the lower surfaces. This deformation will not occur like this in practice. However, the deformation of the right model seems plausible. Figure 06 shows the shear strain n_{xy} between the surfaces. The design of the fasteners is carried out for this value. Regardless of the values, you can see that the shear strain of the left model has always a post-critical failure in both directions (positive and negative). This is due to the fact that the results of both surface sides are displayed and both sides consider the end release at the hinge. The shear strain reduces from the center towards the edge at the right model. This results from the overlapping of the stiffnesses inside the connected surfaces.

Figure 06 - Shear Strain nxy at Line Hinges

Figure 07 shows the force in n_{y}-direction. The forces displayed at the lines refer in each case to the orientation of the local surface axes.

Figure 07 - Forces in ny-direction

The direction of the force is displayed with the dashed red and violet arrows in Figure 07. The left model has a disturbed axial force distribution in vertical axis, which even results in a post-critical failure with tension component in the lower part.

#### Theory Line Release and Line Hinge

RFEM offers the option to define line releases to consider the nonlinearity of the above mentioned model, for example in the area of the pressure contact transmission. The theoretical basics are the same for line hinges and line releases. Both are subject to the so-called double node technology. By defining the release, virtually double nodes at the original nodes are generated. These nodes are then connected with each other by means of a spring. As soon as additional nonlinearities (for example pressure contact) are defined at this spring, a deformation alignment will be performed to check if the condition is met. The technical term for this method is Penalty method. Figure 08 shows a schematic view.

Figure 08 - Penalty Method [1]

It is also possible to carry out the alignment based on force. The nonlinearity shown in Figure 08 is controlled by the forces in the corresponding direction. Equation 1 shows a schematic view of the equation system for the penalty stiffness k in N/m. Additional derivation as well as explanation of the present structure are not included in this article.

Equation 1:

$\begin{bmatrix}2\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&-\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&0\\-\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&\frac{\mathrm E\;\mathrm A}{\mathrm l}\;+\;\mathrm k&-\;\mathrm k\\0&-\;\mathrm k&2\;\frac{\mathrm E\;\mathrm A}{\mathrm l}\;+\;\mathrm k\end{bmatrix}\;\begin{bmatrix}{\mathrm u}_1\\{\mathrm u}_2\\{\mathrm u}_3\end{bmatrix}\;=\;\begin{bmatrix}\mathrm F\\\mathrm k\;{\mathrm d}_0\\-\;\mathrm k\;{\mathrm d}_0\end{bmatrix}$

Equation 2 shows the identical equation system with Lagrange multipliers.

Equation 2:

$\begin{bmatrix}2\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&-\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&0\\-\;\frac{\mathrm E\;\mathrm A}{\mathrm l}&\frac{\mathrm E\;\mathrm A}{\mathrm l}\;+\;\mathrm k&-\;\mathrm k\\0&-\;\mathrm k&2\;\frac{\mathrm E\;\mathrm A}{\mathrm l}\;+\;\mathrm k\end{bmatrix}\;\begin{bmatrix}{\mathrm u}_1\\{\mathrm u}_2\\{\mathrm u}_3\end{bmatrix}\;=\;\begin{bmatrix}\mathrm F\\\mathrm k\;{\mathrm d}_0\;+\;\mathrm\lambda^\mathrm i\\-\;\mathrm k\;{\mathrm d}_0\;-\;\mathrm\lambda^\mathrm i\end{bmatrix}$

The equation systems only differ in the last part by the factor λ. It is clear now that the calculation with Penalty or Lagrange multipliers leads to identical results, at least in the first step. For more complex structures, it is better to use the Lagrange multipliers. After the start value zero, the iteration scheme is extended by the Langrange multipliers $\mathrm\lambda^{\mathrm{li}+1}\;=\;\mathrm\lambda^\mathrm i\;+\;\mathrm k\;\mathrm d^\mathrm i$.

#### Line Release

By defining a line release in RFEM, it is possible to fully consider a nonlinearity for the above mentioned example. As with the rigid model with end release in u_{x}, a comparable deformation occurs for the identical end release of the nonlinearity pressure contact (Figure 09).

Figure 09 - Deformation at Line Releases

The internal forces n_{xy }have an identical distribution of internal forces with regard to the vertical connection like the structure with only one end release (Figure 10). Only the horizontal line changes at the right side of the model because this surface is completely under pressure.

Figure 10 - Shear Strain at Line Releaes

#### Defining the Surface Side

Irrespective of selecting a line release or line hinge for defining the end release, it is important to display the model properly.

Figure 11 shows the nailing with cover panel (left) and notch (right). Figure 12 shows the corresponding structural model. When the structure is modeled, it is important to define the end release in u_{x}, thus in longitudinal direction of the connection, two times on the left and only one time on the right. Due to Hooke's law, the left model has the double end release.

#### Summary

Use the option line release or line hinge in RFEM to consider end release between surfaces.

#### Keywords

fasteners nonlinearity penalty lagrange