Dynamics of Structures
Structural Engineering Simplified
1.7 Examples
Figure 1.7.1 Example 1.7.1
Example 1.7.1
A W12x50 steel beam (E = 29000 ksi) is supporting a weight of 250 kips as illustrated in the figure below. Neglecting the weight of the beam itself, determine the natural frequency and the period of the motion.
From Example 1.3.1, the stiffness of the system was obtained as,
The mass of the system can be calculated as,
The angular frequency can be calculated as,
The frequency can be calculated as,
The period can be calculated as,
Figure 1.7.2 Example 1.7.2
Example 1.7.2
For the single story two bay steel frame shown in the figure, Assuming infinitely rigid horizontal girder, determine the system natural frequency and period of vibration (E = 29000 ksi).
The pinned-fixed columns has stiffness equal to,
The fixed-fixed middle column has stiffness equal to,
The total stiffness of the system,
The mass of the system is,
The natural frequency of the system is,
The period of the system is,
Figure 1.7.3 Example 1.7.3
Example 1.7.3
The concrete table-top platform is supporting a machine that weights w = 100 kips. The platform is 30'x30'x5' as shown in the figure and is supported by 4 concrete columns each has a 2'x2' cross section. Assuming that the columns are fixed to the base and rigidly connected to the platform, calculate the natural frequency of the platform (Ec = 3600 ksi, unit weight of concrete = 150 pcf).
The moment of inertia of each column can be calculated as,
The stiffness of each column can be calcuated as,
The total stiffness of the system,
The mass of the machine is,
The mass of the platform is,
The mass of the 4 columns is,
The total mass of the table-top structure is,
The natural frequency of the system is,
