 # How is the rotational restraint stiffness calculated for a non-continuous rotational restraint (for example, purlins) in RF‑/STEEL EC3?

The total rotational spring comprises of several individual rotational springs, which are given in  as Equation 10.11.

In the case of a non-continuous rotational restraint due to purlins, RF-/STEEL EC3 considers the rotational restraint due to the connection stiffness CD, A , the rotational restraint CD, C from the bending stiffness of the applied purlins, and, if activated, also the rotational restraint CD, B from cross-section deformation .

Since the execution of the connection is unknown, the infinite value is set by default. The spring stiffnesses are considered as a reciprocal value 1/C, thus giving "infinitely" the result of spring stiffness = 0. If you know the rotational spring stiffness of the connection, you can specify this value manually.

The rotational stiffness CD, C from the bending stiffness is determined according to the following formula:

${\mathrm c}_{\mathrm D,\mathrm C}={\mathrm C}_{\mathrm D,\mathrm C}/\mathrm e\\{\mathrm C}_{\mathrm D,\mathrm C}=\frac{\mathrm k\cdot\mathrm E\cdot\mathrm I}{\mathrm s}\\$

where

E is the modulus of elasticity
k is the coefficient for position (inner span, outer span)
I is the moment of inertia Iy
s is the distance of the beams
e is the distance of the purlins

The rotational restraint CD, B from the section deformation is determined according to the following formula:

${\mathrm c}_{\mathrm D,\mathrm B}={\mathrm C}_{\mathrm D,\mathrm B}/\mathrm e\\{\mathrm C}_{\mathrm D,\mathrm B}=\sqrt{\mathrm E\cdot{\mathrm t}_{\mathrm w}^3\cdot\mathrm G\cdot{\mathrm I}_{\mathrm T,\mathrm G}/\left(\mathrm h-{\mathrm t}_{\mathrm f}\right)}\\{\mathrm I}_{\mathrm T,\mathrm S}=\left(\mathrm h-{\mathrm t}_{\mathrm f}\right)\cdot{\mathrm t}_{\mathrm w}^3/3\\{\mathrm I}_{\mathrm T,\mathrm G}=\left({\mathrm I}_{\mathrm T}-{\mathrm I}_{\mathrm T,\mathrm S}\right)/2$

where

E is the modulus of elasticity
tw = web thickness of truss or supported component
G is the G modulus
h is the height of the truss or the supported component
tf = flange thickness of the truss
b is the truss width
e is the distance of the purlins

The attached example includes two design cases.

Case 1 was designed without taking into account the cross-section deformation. The total rotational spring stiffness is
CD = CD, C = 4729 kNm/m

Case 2 was designed while taking into account the cross-section deformation. The total rotational spring stiffness is
CD = 72.02 kNm/m

Single spring CD, B = 73.14 kNm/m
Single spring CD, C = 4729 kNm/m

Total spring:

$\begin{array}{l}\frac1{{\mathrm C}_{\mathrm D}}=\frac1{{\mathrm C}_{\mathrm D,\mathrm B}}+\frac1{{\mathrm C}_{\mathrm D,\mathrm C}}\;=\;\frac1{73.14}+\frac1{4,729}\\{\mathrm C}_{\mathrm D}\;=72.02\;\mathrm{kNm}/\mathrm m\end{array}$

#### Reference

  Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; EN 1993‑1‑1:2010‑12  Eurocode 3: Design of steel structures - Part 1‑3: General rules - Supplementary rules for cold-formed members and sheeting; EN 1993‑1‑3:2010‑12 