Chapter 10
Dynamic Direct Stiffness Method
10.1 Introduction
The procedure presented in Chap. 8 for the determination of free vibration characteristics of the structural member, by considering it as a system having distributed mass,
requires the evaluation of constants which are dependent on the boundary conditions.
If the structure is made up of assemblage of structural members, then four constants
are to be determined for each member. For a three-span continuous beam, the number
of constants becomes 12, and with the increase in the number of members, it goes on
increasing. The whole calculation procedure becomes tedious in this straightforward
approach. A more methodical approach for carrying out the analysis for such cases
is performed by using dynamic direct stiffness approach. Once the free vibration
characteristics are evaluated, the method can be applied to forced vibration analysis,
in which case it becomes analogous to the displacement method used for statical
problems. We first look into the classical approach for solving what is termed as
statically indeterminate problems, and then we shall pass on to the dynamic direct
stiffness method [1–3].
10.2 Continuous Beam
Consider a two-span continuous beam having uniform flexural rigidity EI throughout
its length. Let us choose two coordinate systems for two spans—the origin of the
left span be at left-hand support and that of the right span be right-hand support and
positive directions of the axes are shown in Fig. 10.1.
The boundary conditions of the beam are
y 1 (0, t) = M 1 (0, t) = y 1 (L 1 , t) = y 2 (L 2 , t) = y 2 (0, t) = M 2 (0, t) = 0 (10.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_10
395
Dynamic Direct Stiffness Method
10.1 Introduction
The procedure presented in Chap. 8 for the determination of free vibration characteristics of the structural member, by considering it as a system having distributed mass,
requires the evaluation of constants which are dependent on the boundary conditions.
If the structure is made up of assemblage of structural members, then four constants
are to be determined for each member. For a three-span continuous beam, the number
of constants becomes 12, and with the increase in the number of members, it goes on
increasing. The whole calculation procedure becomes tedious in this straightforward
approach. A more methodical approach for carrying out the analysis for such cases
is performed by using dynamic direct stiffness approach. Once the free vibration
characteristics are evaluated, the method can be applied to forced vibration analysis,
in which case it becomes analogous to the displacement method used for statical
problems. We first look into the classical approach for solving what is termed as
statically indeterminate problems, and then we shall pass on to the dynamic direct
stiffness method [1–3].
10.2 Continuous Beam
Consider a two-span continuous beam having uniform flexural rigidity EI throughout
its length. Let us choose two coordinate systems for two spans—the origin of the
left span be at left-hand support and that of the right span be right-hand support and
positive directions of the axes are shown in Fig. 10.1.
The boundary conditions of the beam are
y 1 (0, t) = M 1 (0, t) = y 1 (L 1 , t) = y 2 (L 2 , t) = y 2 (0, t) = M 2 (0, t) = 0 (10.1)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_10
395