Advanced Methods of Structural Analysis

Problems 305

Now assume that the number of spans is even. It means that additional vertical

member is placed on the axis of symmetry.In the case of symmetrical load, the

horizontal and angular displacements at the pointA, as in case of odd spans, are

zero, while the vertical displacement ofAis zero because the vertical member is

at axis of symmetry. Therefore, an equivalent half-frame must contain a support at

pointA, which would model corresponding displacements. Only clamped support

is related with displacements described above. In case of antisymmetrical loading,

the half frame contains the member at the axis of symmetry with bending stiffness

0.5EI.

Fundamental properties of internal force diagrams for symmetrical structures are

as follows:

1.In case of symmetrical loading, the internal force diagrams for symmetrical un-
knowns (M,N) are symmetrical and for antisymmetrical unknown.Q/they are
antisymmetrical.
2.In case of antisymmetrical loading, the internal force diagrams for symmetrical
unknowns (M,N) are antisymmetrical and for antisymmetrical unknown.Q/
they are symmetrical.
Ta b l e8.6also contains the number of unknowns for entire frame and for half-frame
by Force (FM) and Displacement (DM) methods. The rational method is shown by
bold. We can see that in case of symmetrical loading, the more effective is displace-
ment method, while in the case of antisymmetrical loading, the more effective is
the force method. From this table, we can see advantages of the combined method.
Bending moment diagram is constructed for each case and then summated (on the
basis of superposition principle) to obtain final bending moment diagram for origi-
nal frame.
The following procedure may be recommended for analysis of symmetrical
structures:

1.Resolve the entire load into symmetrical and antisymmetrical components
2.Construct the equivalent half-frame for both types of loading
3.Provide the analysis of each half-frameusing the most appropriate method
4.Find the final distribution of internal forces using superposition principle

Problems.......................................................................

In Problems 8.1 through 8.6 provide complete analysis by the displacement method,

including the following:

1.Determine the bending moment at the support and construct the internal force
diagrams
2.Calculate the reactions of the supports and provide static control
3.Provide kinematical control (check the slope and vertical displacement at the
intermediate support)
4.Compute the vertical displacement at the specific points and show the elastic
curve