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TABLE 11.5
Calculation Procedure
Define the system, cross-sectional properties, length.
Theory of First order
Basic load Cases
Linear calculations (Fundamental state)
Stress resultants of the fundamental state, e.g. N 0
Theory of Second Order
Choose load combinations with imperfections such as initial deformations and initial
curvatures, and factors of safety
Calculation steps
1. N 0 for each element
2. Stiffness matrix and load vector for each element
3. Assemble the system stiffness matrix and the system load vector
4. Solve the system equations
5. Compute the member stress resultants
6. Compare with the values of Step 1.
7. If necessary, proceed through Steps 1 to 6 using the improved values of N 0 . Thus,
implement an interation procedure.
8. Post processing calculations:
Max M , corresponding Q and N .
Max N , corresponding M,
Q, etc.
state, which have to be known in the sense of Eq. (11.19) to linearize the basic equations or
the principle of virtual work. The steps of the procedure are summarized in Table 11.5.
EXAMPLE 11.4 Influence of Geometric Imperfections on a Framework
In this example, the influence of imperfections on the behavior of the framework of
Fig. 11.16a is investigated. The structure consists of three beam column elements that are
rigid with respect to extension but flexible for bending.
In addition to the applied loading, it is assumed that element 1 is initially inclined and
has an initial parabolic curvature, both imperfections caused by the manufacturing process.
Element 3 experiences the same initial inclination, see Fig. 11.16b.
As shown in Fig. 11.17a, these imperfections can be replaced by equivalent loads or
reactions at the ends of the elements. For the case of the parabolic shape of the imperfection,
the equivalent end loads are obtained by setting t h e midspan moment M
=
N
w 0 of the
2
imperfection equal to the midspan mom e nt M
=
q
/
8 for an element with end reac ti on
forces and a uniformly distributed load q
.
This gives the equivalent load magnitude q
=
2
8 N
w
/
and end reactions 4 N
w
/
shown in Fig. 11.17a. For an initial inclination, the
0
0
resulting moment N
The corresponding loading terms for the
forces and moments at the element ends are given in Fig. 11.17b. These loading terms are
employed during the process of assembling the nodal forces in the system analysis.
The results of this analysis are provided in Table 11.6, in which various values of the
state variables are given for three cases: linear analysis and second-order analyses with
and without imperfections. The distribution of the stress resultants for the three cases are
shown in Fig. 11.18.
Comparison of the results shows that the linear theory underestimates the deflections and
stress resultants and could lead to an unsafe design. In addition, there are substantial differ-
ences between the results with and without the prescribed imperfections. From Fig. 11.18,
θ
leads to the reaction N
θ
.
0
0
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