Finite Elements in Civil Engineering Applications

Proceedings of the Third Diana World Conference, Tokyo, Japan, 9-11 October 2002

Max.A.N. Hendriks, J.A. Rots, Max.A.N. Hendriks, J.A. Rots

• Editore: CRC Press
• Anno: 2021
• ISBN:
  9781000446784
• Edizione: 1
• Argomento: Technology & Engineering
• Categoria: Civil Engineering

These proceedings present high-level research in structural engineering, concrete mechanics and quasi-brittle materials, including the prime concern of durability requirements and earthquake resistance of structures.

Reinforced concrete structures

Invited paper: Modal analysis versus time history analysis – concepts for the seismic design

Th. Baumann & J. Böhler

WALTER/DYWIDAG, Central Technics, Munich, Germany

ABSTRACT: The forces induced into a structure by earthquake ground motions are prescribed traditionally by elastic response spectra. They can be reduced by behaviour factors to account for the possibility of energy dissipation due to plastic deformations. Presupposition for the utilization of this favourable effect is a sufficient ductility. The consistent definition and verification of this ductility is an essential part of the structural design for earthquakes. In the present paper, the principal features of this task are discussed. In order not to persist in general terms, the governing effects are quantified within a case study of a 3-storey steel frame with composite beams, considering the following steps: Modal analysis, application of behaviour factors, usual assumptions for the required plastic deformations, push over-analysis with plastic rotation of hinges, connection of modal analysis and push over-analysis, nonlinear time history-analysis. Only the last step creates the basis for a reliable assessment of the relevant deformations of the plastifying frame, depending on the applied accelerogram and its peak value PGA.

1 BEHAVIOUR FACTOR, DUCTILITY AND MODAL ANALYSIS

The maximum seismic force of an elastic structure (F_el) is proportional to the peak value (PGA) of the ground acceleration a₀(t). The force which causes yielding of the structure is denominated F_y, the corresponding deformation Δ_y and the acceleration (PGA)_y. If the structure has a sufficient ductility allowing plastic deformations, it may sustain also stronger earthquakes. In Eurocode 8 (2001) and other codes this fact is taken into account by a behaviour factor q, which describes the possible increase of PGA up to q·(PGA)_y. This increase requires plastic deformations μ_Δ • Δ_y. For the definition of μ_Δ alternative assumptions have been proposed acc. to fig. 1 (Paulay et al. 1990): a) “same displacement” like an elastic structure with the same stiffness, and b) “same work”. As pointed out already by Baumann & Böhler (1997), these assumptions are questionable. Especially the “same work”–formula has no rational base, because the areas shadowed in fig. 1 do not designate “works” or energies at all, which are characteristic for the different behaviour of elastic and non-elastic structures.

On the other hand, the realistic valuation of the plastic deformations dependent on the value of PGA is an essential part of the seismic design. The following report shows, how the informations which are necessary in this respect can be found by various types of analyses, i.e. modal analysis (response spectrum analysis), push over-analysis and time history-analysis. In order not to remain in general terms, a steel frame with composite beams excited horizontally by earthquake is considered within a case study.

Figure 1. Plastic deformations as presupposition for the application of behaviour factors.

In areas with low seismicity, bracing of frames by diagonal rods acc. to fig. 2 is an economic solution (MGS 2001). However, this type of structure is not able to sustain plastic deformations and horizontal forces in reverse directions. Therefore a moment-resisting frame acc. to figs. 3 and 4 has been investigated. The frame distance of about 5 m is bridged by a concrete slab which is connected to the rolled steel girders by studs. In this way the stiffness of the composite beams is increased by the factor 3.5 compared to the steel girders only. The formation of plastic hinges adjacent to the columns is supported by omission of bond in this region.

Fig. 5 shows the first two eigenmodes for horizontal excitation. For ${\text{T}}_{\text{E}}=0.437\text{s}$ (mode 1), the response spectrum of fig. 6 (acc. to Eurocode 8 (2001) for Soil B/Type 1) defines an elastic response value of 2.5 for a viscous damping of $\text{ξ}=5\%$. For the lower damping of a steel or composite structure ($\text{ξ}=2\%$) this response is increased acc. EC 8 by the factor $\text{η}=1.2$. For an assumed value of $\text{PGA}=0.12\text{g}$ and a total weight of the structure of about $3\cdot 35\cdot 15=1575\text{kN}$ we get a total lateral force $\text{H}=0.12\cdot 2.5\cdot 1.2\cdot 1575=567\text{kN}$. From a more detailed modal analysis considering 10 horizontal modes results only $\text{H}=500\text{kN}$.

Figure 2. Bracing of steel frame with composite beams by diagonal rods.

Figure 3. Moment resisting frame with composite beams.

For time history-analyses we need natural or artificially generated accelerograms. Fig. 7 shows the two accelerograms, which have been applied for this case study. They are based on $\text{PGA}=\text{k}\cdot \text{S}\cdot {\text{a}}_{\text{g}}=1.0\cdot 1.2\cdot 0.10\text{g}=0.12\text{g}$. The value $\text{S}=\mathrm{1...}$