Structural Dynamics and Degree of Freedom
Structural dynamics and earthquake engineering
1. Define the concept of dynamic degree of freedom. Give some examples of single degree of freedom systems and multi degree of freedom systems.
Answer: Dynamic degrees of freedom are a set of independent displacements/rotations that completely define the displaced position of the mass with respect to its initial position.
• A vertical cantilever with the mass concentrated at its tip can be idealised as a single degree of freedom (SDOF) system. The degree of freedom is represented by the lateral displacement u of the mass.
• A multi-storey frame with the masses concentrated the storey levels can be idealised as a multi degree of freedom (MDOF) system. The degrees of freedom are the lateral displacements of the storey masses. The system in the next figure has 4 dynamic degrees of freedom – displacements u1 – u4.
2. Write the equation of motion of a single degree of freedom systems subjected to a dynamic force and explain its terms. Show using a sketch an example of a dynamic system like this.
Answer: The equation of motion of a single degree of freedom (SDOF) system subjected to an external force p(t) is the following:
3. Explain the lateral force method used to assess the seismic response of multi-storey structures. State the limitations of the method.
Answer: The lateral force method (LFM) is a simplified structural analysis method which can be used in the case of multi-storey structures whose response is governed by the fundamental mode of vibration. According to P100-1/2006, the LFM can be used only for structures which:
• are regular in elevation,
• have a period of vibration less than 1.5 sec. and
• are less than 30 m high.
The LFM is essentially a modal response spectrum analysis, which considers only the fundamental mode of vibration. The method consists in a static analysis of the structure subjected to the lateral forces Fiof the structure the base shear force Fb, which is determined according to the following
. These forces are determined by distributing along the height
4. Discuss the measures for conceptual seismic design of buildings from the point of view of torsional resistance and stiffness.
Answer: Lateral force resisting system should be as symmetrical as possible (see the figure below), in order to obtain as small difference as possible between the centre of mass (CM) and centre of rigidity (CR) of a structure. If an eccentricity between the CM and CR is present, the floor will be subjected to a torsional motion, in addition to the translational one. This results in larger displacements at the flexible side of the building in the direction of the seismic action. Additionally, displacements perpendicular to the direction of seismic action will be present.
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