Considering SecondOrder Theory in Dynamic Analysis
Technical Article
For the ultimate limit state design, EN 1998‑1 [1], Section 2.2.2 and 4.4.2.2, require the calculation considering the second‑order theory (P‑Δ effect). This effect need not be taken into account only if the interstorey drift sensitivity coefficient θ is less than 0.1.
The coefficient θ is defined as follows:
$$\mathrm\theta\;=\;\frac{\displaystyle{\mathrm P}_\mathrm{tot}\;\cdot\;{\mathrm d}_\mathrm r}{{\mathrm V}_\mathrm{tot}\;\cdot\;\mathrm h}\;\;\;\;\;\;\;\;\;\;\;(1)$$where
The secondorder effects may approximately be taken into account by a factor equal to 1 / (1 − θ), if 0.1 < θ ≤ 0.2. For θ > 0.2, it is necessary to consider the geometric stiffness matrix when calculating eigenvalues and multimodal response spectrum analysis.
Geometric Stiffness Matrix
For dynamic analyses, the iterative calculations for the nonlinear determination of second‑order theory are not suitable. The problem can be linearised, and it is sufficient to use the geometric stiffness matrix based on axial loads to consider the second‑order theory. For this, it is assumed that the vertical loads do not change due to horizontal effects and the deformations are small compared to the building dimensions [2]. The loads to be considered should correspond to those for the seismic design situations in accordance with EN 1990, Section 6.4.3.4 [3]:
$${\mathrm E}_\mathrm d=\sum_{\mathrm j\geq1}{\mathrm G}_{\mathrm k,\mathrm j}+\sum{\mathrm\Psi}_{2,\mathrm i}{\mathrm Q}_{\mathrm k,\mathrm i}\;\;\;\;\;\;\;\;\;\;(2)$$where
Axial tensile forces increase the stiffness, for example, in a prestressed cable. Compression forces reduce the stiffness and can lead to a singularity in the stiffness matrix. The geometric stiffness K_{g} is not dependent on mechanical properties of the structure, but only on the member length L and axial force N.
To illustrate the basic problem, there is a simple example of a cantilever displayed in Figure 01. The single mass points of the cantilever represent the individual storeys of a building. The building is subjected to a dynamic analysis considering the second‑order theory. The axial forces N_{i} on the individual storeys i = 1...n result from the vertical forces in the seismic design situation (see Expression 2). The storey height is defined by h_{i}.
The geometric stiffness matrix K_{g} can be derived from the static equilibrium conditions:
$$\begin{bmatrix}{\mathrm F}_\mathrm i\\{\mathrm F}_{\mathrm i+1}\end{bmatrix}\;=\;\underbrace{\frac{{\mathrm N}_\mathrm i}{{\mathrm h}_\mathrm i}\left[\begin{array}{rc}1.0&\;1.0\\1.0&\;1.0\end{array}\right]}_{{\mathbf K}_\mathbf g}\;\begin{bmatrix}{\mathrm u}_\mathrm i\\{\mathrm u}_{\mathrm i+1}\end{bmatrix}\;\;\;\;\;\;\;\;\;\;(3)$$For the purpose of simplification, only the degrees of freedom of the horizontal displacement are displayed here. The derivation shown is based on the overturning moment approach due to the linear displacement application. This is a simplification for the bending element, and an accurate assumption for the truss element.
More precise determination of the geometric stiffness matrix for bending beams can be obtained by using the cubic displacement approach or the analytical solution of the differential equation of the bending line. More information and derivations are provided by Werkle [4].
The geometric stiffness matrix K_{g} is added to the system stiffness matrix K, and thus the modified stiffness matrix K_{mod} is obtained:
$${\mathbf K}_\mathbf{mod}\;=\;\mathbf K\;+\;{\mathbf K}_\mathbf g\;\;\;\;\;\;\;\;\;\;(4)$$In the case of compression normal forces, this consequently leads to the stiffness reduction.
Example: Natural Frequencies and MultiModal Response Spectrum Analysis Considering SecondOrder Theory
The following shows how the geometric stiffness matrix can be considered in RFEM and the RF‑DYNAM Pro add‑on modules. As an example, the cantilever shown in Figure 01 is used. The cantilever consists of five concentrated mass points. Here, 4,000 kg act in the global X‑direction in each case.
The cross‑section is IPE 300 made of the material S 235 with:
${\mathrm l}_\mathrm y\;=\;8.356\;\cdot\;10^{5}\;\mathrm m^4\;$
$\mathrm E\;=\;2.1\;\cdot\;10^{11}\;\mathrm N/\mathrm m^2$
To be able to consider the geometric stiffness matrix in the dynamic analysis, a load combination is initially defined for the seismic design situation in the main program RFEM (see Equation 2).
The RF‑DYNAM Pro  Natural Vibrations add‑on module allows you to determine natural frequencies, mode shapes and effective modal masses of a structure, taking into account various stiffness modifications (see RF‑DYNAM Pro Manual [5], Chapter 2.4.7, and Technical Article [6]). Two natural vibration cases are defined. In NVC2, CO1 is imported in order to consider the geometric stiffness matrix and thus the second‑order theory. For comparison, the NVC1 is defined, which does not include any stiffness modifications.
Figure 04  Parameters for Eigenvalue Analysis in RF‑DYNAM Pro  Natural Vibrations
The following table includes the determined natural frequencies f [Hz], natural periods T [sec], and the acceleration values S_{a} [m/s²] based on the response spectrum, with and without the geometric stiffness matrix K_{g} resulting from the axial forces of CO1.
Figure 05  Natural Frequencies, Periods and Acceleration Values
The multimodal response spectrum analysis uses natural frequencies to determine the acceleration values from the defined response spectrum. Based on these acceleration values, the equivalent loads and the response spectrum internal forces are determined. The graphic display of a user‑defined response spectrum is shown in Figure 06, and the acceleration values S_{a} [m/s²] determined from the response spectrum for each eigenvalue are listed in the table above.
Figure 06  UserDefined Response Spectrum
In order to ensure the correct allocation of the modified frequencies, the right natural vibration case (NVC) must be assigned to the dynamic load case (DLC).
Figure 07  Assigning Natural Vibration Case to Dynamic Load Case to Determine Equivalent Loads
In the case of compression axial forces, the consideration of the geometric stiffness matrix leads to the reduction of the natural frequency, and can cause lower acceleration values S_{a}, such as in our example. Sole modification of the natural frequencies is not enough to consider the second‑order theory. In fact, this may actually lead to smaller results, which can be thus incorrect. It is very important to also use the modified stiffness matrix for the determination of internal forces and deformations.
In RFDYNAM Pro  Forced Vibrations, the modified stiffness is automatically used to determine the response spectrum results, because the calculation is performed within RF‑DYNAM Pro. InRF‑DYNAM Pro  Equivalent Loads, the equivalent loads are determined and exported as load cases into the main program RFEM. Therefore, the calculation is performed partially in RF‑DYNAM Pro and partially in RFEM.
Theoretical background for the equivalent load calculation is explained in the Manual RF‑DYNAM Pro [5]. The verification example [7] shows the calculation on a specific example. The determined equivalent loads, with and without the geometric stiffness matrix, are displayed in Figure 08.
The export of the equivalent loads has many advantages, but the most important is the correct transfer of stiffness modifications in the load cases. The calculation parameters of the exported load cases must be adjusted as shown in Figure 09.
The individual load cases are superimposed using the SRSS or CQC method. This is automatically set in RF‑DYNAM Pro and exported into result combinations. The results with and without the geometric stiffness matrix are displayed in Figure 10.
The consideration of the geometric stiffness matrix leads to larger deformations and internal forces. However, the equivalent loads and resulting support loads are slightly smaller when considering the geometric stiffness matrix.
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Figure 01  Reduction of Building on Cantilever Structure

Figure 02  SelfWeight and Imposed Loads

Figure 03  Definition of Load Combination for Seismic Design Situation and Resulting Axial Forces

Figure 04  Parameters for Eigenvalue Analysis in RFDYNAM Pro  Natural Vibrations

Figure 05  Natural Frequencies, Periods and Acceleration Values

Figure 06  UserDefined Response Spectrum

Figure 07  Assigning Natural Vibration Case to Dynamic Load Case to Determine Equivalent Loads

Figure 08  Equivalent Loads for Mode Shape 1

Figure 09  Calculation Parameters of Load Cases with Exported Equivalent Loads

Figure 10  Deformations uX, moment MY, and support reactions PX