Mode Shapes (2D Frames)
This page last updated August 18, 1999
Free vibration of a multiple degree of freedom (mdof) system is discussed by Chopra starting on page 365.
The Equations of Motion for a mdof system can be written in a form similar to the traditional Eigenvalue problem (Equation 10.2.6). The eigenvectors define the displacement configurations of the various modes of the system. Each mode has a natural frequency associated with it (the eigenvalue). For a mdof system with N degrees of freedom, N mode shapes and frequencies will exist.
The primary mathematical advantage of determining mode shapes is that they will be orthogonal to each other, in other words they allow for the Equations of Motion to be written in an uncoupled form (see Chopra, Section 10.4).
For we design engineers, mode shapes are useful because they represent the shape that the building will vibrate in free motion. These same shapes tend to dominate the motion during an earthquake (or windstorm). By understanding the modes of vibration, we can better design the building to withstand earthquakes.
A model of a four-story one-bay frame can be evaluated to determine the mode shapes. This model of a typical bay of the SJSU Engineering Building illustrates the various mode shapes in a typical building. One principle of modal decomposition (the calculation of the mode shapes and natural frequencies) is that the scale of the mode shape is unknown. In other words, the ratio of the maximum displacement of the fourth mode to the first mode is completely unknown, but we usually expect it to be very small. (To animate the mode shapes, right-click on the frame elevation and select PLAY).
For the mathematical solution of a simpler problem (2-story 1-bay) see Chopra's Example 10.5 on page 378.
The significance of a mode is indicated by the Mass Participation. This factor indicates the amount of the total structural mass that is activated by a single mode. If all modes of a structure are considered, the cumulative Mass Participation will be 100%. Structures with 80% of the mass in the first mode will be dominated by that mode shape.
Vibrational nodes of a mode shape are the points of the system that do not have any motion. In the first mode, the only node that exists is at the base, and some people would not even consider this a node. (Note we are talking about something different that the nodes and elements used in the analysis model). In the second mode, a distinct point can be seen (just below the third floor) that does not move laterally during the modal vibration. The third mode has two of these distinct points, the fourth has four and so on. For the Jth mode shape, there will be J vibrational nodes, if you consider the base to be one of the vibrational nodes.
Mode Shapes (3D Frames)
A structure is built up from several 2D frames into a 3D system. Each of these 2D frames has modes associated with it. When we perform 3D analysis, the mode shapes for the entire structure are based upon the interaction of the frames in their own 2D configuration. Suddenly the problem becomes much larger (Chopra wisely chooses not to do anything larger than a one-story building, see Example 10.6), but in reality the solution follows the same procedure.
Now we talk about translational building modes, and torsional building modes. These modes may be coupled (a mode that has motion contributed by two or more orthogonal directions), or completely uncoupled. In Chopra's Example 10.6, he has the x-direction translational building mode uncoupled (Third Mode), but the y-direction translational direction and the torsional direction are coupled in the first two modes.
For most buildings with NS number of stories, the building will have 2xNS translational modes and NS torsional modes. Design guidelines usually require enough modes be calculated, that at least 90% of the mass participation be considered. If a mode shape shows a Mass Participation in two different direction (X,Y,Z), then these directions are coupled in the mode shape.
Ideally, a structure is designed where the modes are clearly uncoupled, and the torsional modes occur at shorter periods than the relative translational modes. Thus the first two building modes should be translational, the third is torsional, the fourth and fifth are translational, the sixth is torsional, and so on. Also it is nice if 80% of the Mass Participation occurs in the first three building modes.
Unfortunately, it is easy to define what is nice, the more difficult question is what is too bad. As modes begin to couple, as Mass Participation moves into higher modes, and as torsional building modes occur at longer periods, we generally expect the overall performance of the structure to be decreased. Unfortunately, we do not have good understanding of what level of these values would be unacceptable.