Dynamics of Structures
Structural Engineering Simplified
1.1 Degrees of Freedom Concept
FIgure 1.1.a A single story frame structure
FIgure 1.1.b A single degree of freedom mass spring system
Degrees of Freedom Concept
Degrees of freedom (DOF) in the dynamics of structures is a crucial concept that quantifies the independent ways in which a structure can move, vibrate, or deform in response to external forces. This concept is pivotal in understanding the behavior of various structural systems, ranging from simple beams and trusses to complex high-rise buildings and bridges. DOF helps engineers design and analyze structural systems by creating mathematical models that emulate real-world behavior. For instance, a simple beam supported at both ends has three degrees of freedom: one vertical translation and two rotational displacements at each support. In contrast, a complex high-rise building may exhibit numerous degrees of freedom, accounting for the three-dimensional movement of each floor and individual components. Practical applications include determining the natural frequencies and mode shapes of structures, assessing their dynamic response to external loads like earthquakes and wind, and optimizing structural designs for improved performance and resilience. In this section we will focus on structures that can be modeled as system with only one degree of freedom.
In this example
A single story frame can be modeled as a single degree of freedom assuming that the story mass is concentrated in the beam. The two columns of the frame represent the source of stiffness. Assuming that there is nothing that oppose the frame horizontal movement other than the columns resistance, the system can be modeled as a single degree of freedom mass-spring-system as modelled.