The degrees-of-freedom in a structure can be regarded as the possible components of displacements of the nodes including those at which some support conditions are provided. In pin-jointed, plane-frames each node, unless restrained, can displace a small amount 5 which can be resolved in to horizontal and vertical components 5H and 5V as shown in Figure 1.29.
Figure 1.29
Each component of displacement can be regarded as a separate degree-of-freedom and in this frame there is a total of three degrees-of-freedom:
The vertical and horizontal displacement of node B and the horizontal displacement of node C as indicated.
In a pin-jointed frame there are effectively two possible components of displacement for each node which does not constitute a support. At each roller support there is an additional degree-of-freedom due to the release of one restraint. In a simple, i.e. statically determinate frame, the number of degrees-of-freedom is equal to the number of members. Consider the two frames indicated in Figures 1.20(a) and (b):
In Figure the number of members m=3
possible components of displacements at node B =2
possible components of displacements at node =1
support C
the number of members m=11
possible components of displacements at nodes =10
possible components of displacements at =1
support E
Total number of degrees-of-freedom (=m)=11
In the case of indeterminate frames, the number of degrees-of-freedom is equal to the (number of membersâ€”ID); consider the two frames indicated in Figures 1.20(c) and (d):
In Figure the number of members m=14 1.20(c):
possible components of displacements at =12 nodes possible components of displacements at =1 support G
degree-of-indeterminacy /d=1
Total number of degrees-of-freedom (m-ID)=13
In Figure the number of members m=15 1.20(d):
possible components of displacements at =10 nodes degree-of-indeterminacy Id=5
In rigid-jointed frames there are effectively three possible components of displacement for each node which does not constitute a support; they are rotation and two components of translation e.g. 0, 5H and 5V. At each pinned support there is an additional degree-of-freedom due to the release of the rotational restraint and in the case of a roller, two additional degrees-of-freedom due to the release of the rotational restraint and a translational restraint. Consider the frames shown in Figure 1.23.
In Figure 1.23(a): the number of nodes (excluding supports) =2
possible components of displacements at nodes =6
possible components of displacements at support D =1
Total number of degrees-of-freedom
In Figure the number of nodes (excluding supports)
possible components of displacements at nodes =12
possible components of displacements at support =1
possible components of displacements at support F =1
Total number of degrees-of-freedom
In Figure 1.23(c): the number of nodes (excluding supports)
possible components of displacements at nodes =9
possible components of displacements at support A =1
In Figure 1.23(d): the number of nodes (excluding supports) =1
possible components of displacements at nodes =3
possible components of displacements at support C =2
possible components of displacements at support D =1
Total number of degrees-of-freedom =6
The introduction of a pin in a member at a node produces an additional degree-of-freedom. Consider the typical node with four members as shown in Figure 1.30. In (a) the node is a rigid connection with no pins in any of the members and has the three degrees-of-freedom indicated. In (b) a pin is present in one member, this produces an additional degrees-of-freedom since the rotation of this member can be different from the remaining three, similarly with the other members as shown in (c) and (d).
Figure 1.30
Degrees-of-freedom:
(d)total^6 four of rotation -9
two of translation - ^ <?v two of translation - <?n. $v two of translation - Sv two of translation - <Shi, $v
In many cases the effects due to axial deformations is significantly smaller than those due to the bending effect and consequently an analysis assuming axial rigidity of members is acceptable. Assuming axial rigidity reduces the degrees-of-freedom which are considered; consider the frame shown in Figure 1.31.
Figure 1.31
1.6.1 Problems: Indeterminacy and Degrees-of-Freedom
Determine the degree of indeterminacy and the number of degrees-of-freedom for the pin-jointed and rigid-jointed frames indicated in Problems 1.1 to 1.3. and 1.4 to 1.6 respectively.
Problem 1.2
Problem 1.3
Problem 1.4
Problem 1.6
Structural analysis and design 27 1.6.2 Solutions: Indeterminacy and Degrees-of-freedom
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