Environmental Engineering Reference
In-Depth Information
Characteristic Periods
Very rigid, 1 story structure: 0.1 sec (very high frequency, f
10 Hz)
Relatively stiff structures of 4 to 6 stories: 0.4-0.5 sec ( f
2 Hz)
Relatively flexible structures of 20 to 30 stories: 1.5 to 2.5 sec ( f
0.5 Hz)
Very flexible structures where deformation rather than strength governs design,
and wind loads become important: 3 to 4 sec or higher (very low, f
0.25 Hz)
The ground vibrates at its natural period which in the United States varies from about
0.4 to 2 sec depending generally on ground hardness. When the ground period approaches
the natural period of the structure, resonance may occur. This can result in a significant
increase in the acceleration of the structures.
Forces on Structural Members
Static loads result in stresses and deflections.
Dynamic loads result in time-varying deflections, which involve accelerations. These
engender inertia forces resisting the motion, which must be determined for the solution of
structural dynamics problems. The complete system of inertia forces acting in a structure
is determined by evaluating accelerations, and therefore displacements, acting at every
point in the structure.
Deflected Shape of Structure
The deflected shape of a structure may be described in terms of either a lumped-mass ide-
alization or generalized displacement coordinates.
In the lumped-mass idealization , it is assumed that the entire mass of the structure is con-
centrated at a number of discrete points, located judiciously to represent the characteris-
tics of the structure, at which accelerations are evaluated to define the internal forces
developing in the system.
Generalized displacement coordinates are provided by Fourier series representation.
In either case, the number of displacement components of coordinates required to spec-
ify the position of all significant mass particles is called the number of degrees of freedom
of the structure (Clough, 1970).
Single-Degree-of-Freedom System
Two types of single-degree-of-freedom systems are shown in Figure 11.41. In both cases,
the system consists of a single rigid (lumped) mass M so constrained that it can move with
only one component of simple translation. The dynamic forces acting on a simple build-
ing frame founded on the surface may be represented by a simple mass-spring-dashpot
system as shown in Figure 11.41a. (A dashpot is an energy adsorber.)
Translation motion is resisted by weightless elements having a total spring constant K (stiff-
ness) and a damping device which adsorbs energy from the system. The damping force C is
proportional to the velocity of the mass. The fundamental period T is expressed by
T
2
π
( M / K ) 1/2
(11.27)
During earthquakes, the motion is excited by an external load P ( t ) which is resisted by an
inertia force F I , a damping force F D , and an elastic force F s . The resisting forces are pro-
portional to the acceleration, velocity, and displacement of the mass given in terms of the
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