GGU-RETAIN: 2nd order theory

Analysis of sheet pile walls under buckling loads is performed in GGU-RETAIN compliant with DIN EN 1993-1-1.

The differential equation for a normal flexural member is:

EI w''''(x) = q(x)

The normal force N is taken into consideration for a buckling member:

EI w''''(x) + N w''(x) = q(x)

For the sake of completeness, the differential equation for an additionally bedded system is shown here since the program also allows parallel processing of systems with 2^nd order theory and elastic bedding. The equation is:

EI w''''(x) + N w''(x) + k_s w(x) = q(x)

Analysis is performed on the deformed system. In

• DIN EN 1993-5
  Design of Steel Structures,
  Part 5: Piling

analysis using 2^nd order theory is recommended for analysis of sheet pile walls under buckling loads. DIN EN 1993-5 refers to

• DIN EN 1993-1-1
  Design of Steel Structures,
  Part 1-1: General Rules and Rules for Buildings

for these analyses. Analysis using 2^nd order theory produces more accurate results than the usual and simplified equivalent member method.

Analysis using 2^nd order theory requires a predeformation or pre-curvature of the underlying system. Pre-curvature values are given in Table 5.1 of DIN EN 1993-1-1.

Pre-curvatures are given as a function of the buckling line. In simplification, sheet pile walls can be analysed with a pre-curvature of e_0,d/L = 1/150.

In the case of embedded, non-anchored walls, the deformed system results from an inclined position of the wall as shown in the following figure.

In the case of embedded, simply back-anchored walls or a freely supported, simply back-anchored wall, the deformed system results from a linear pre-curvature from the support point to the wall head and a parabolic pre-curvature between the support points and the wall base (see following figure).

The length L is given per field. The procedure for two rows of anchors is shown in the following figure.

The settings are defined in GGU-RETAIN in the analysis menu under "System/Analyse".

The size and direction of the pre-curvature can be specified. Whether a pre-curvature towards the ground side or the atmosphere side provides the more unfavourable design values is system-dependent: the following message will therefore be displayed once analysis begins:

You must therefore deactivate the "Pre-curvature to ground side" check box following successful analysis and check, in a new analysis, whether pre-curvature to the atmosphere side delivers less favourable values.

In an analysis using 2^nd order theory the necessary iteration process in terms of displacement is carried out using the design normal force N_d. The analysis is based on a frame system, such that axial stiffnesses, and anchor and strut inclinations are taken into consideration correctly.

Subsequent design is based on a comparison of stresses

σ_d ≤ f_y,k / γ_M = f_y,k / 1,1 = f_y,d

The 'Example' folder contains 4 files, which deal with the classical Euler cases 1 to 4. If the vertical load V is increased slightly using the "Editor 2/Action boundary conditions" menu item and the system analysed, the following error message appears:

The normal force given in the data files thus corresponds to the buckling force determined after Euler.