Advanced Methods of Structural Analysis

14.1 Fundamental Concepts 517

stiffnessEAD1), then the mass can move only in vertical direction and the struc-
ture has one degree of freedom. Structure 14.3c has three degrees of freedom; they
are displacement of the lumped mass inx-y-zdirections. Introducing additional de-
formable but massless members does not change the number of degrees of freedom.

c

b

a d

A B

Fig. 14.3 (a–d) Design diagrams of bar structures and truss

The truss (Fig.14.3d) contains five concentrated masses. The mass at the joint
Ais fixed, the mass at the jointBcan move only in horizontal direction, and the
rest masses can move in horizontal and vertical directions. So, this structure has
seven degrees of freedom. If we assume thathorizontal displacements of the joints
may be negligible in comparison with vertical displacements, then this truss may
be considered as a statically determinate structure with three degrees of freedom. If
additional members will be introduced in the truss (shown by dotted lines), then this
truss should be considered as two times statically indeterminate structure with three
degrees of freedom.
Figure14.4presents plane frames and arches. In all cases, we assume that all
members of a structure do not have distributed masses. The lumped massM in
Fig.14.4a, b can move in vertical and horizontaldirections, so these structures have
two degrees of freedom. Figure14.4c shows the two-story frame containing abso-
lutely rigid cross bars (the total mass of each cross bar isM). This frame may be
presented as shown in Fig.14.4d.

e

f

c d

M

M

M

M

EI=∞

EI=∞

b M

a M

Fig. 14.4 (a–f) Design diagrams of frames and arches

Arches with one and three lumped masses are shown in Fig.14.4e, f. Taking into
account their vertical and horizontal displacements, the number of their degrees