Civil Engineering Reference
In-Depth Information
iii)
Assembly of the structure stiffness
iv)
Assembly of the load vector
v)
Solution for the nodal displacements
vi)
Determination of the element stresses
Additional steps in each load increment of the
non-linear analysis:
vii)
Check for cracking, yielding, and failure
viii)
Determination of the unbalanced nodal forces
ix)
Check for convergence
x)
If new crack appears: repeat steps ii)Îiii) and then followed by steps
x)
If yielding only occurs: repeat steps iv)Îix)
xi)
Stop when failure occurs or when full loading has been applied.
The linear solution procedure is well-known (Ghali, Neville and Cheung,
1971) and needs no further explanation.
Three different approaches are commonly used to solve a non-linear
problem, namely: incremental procedure, iterative procedure, and mixed
procedure.
9.15.1
Increment procedure
The total load is divided into a number of equal or unequal load increments.
At each step only one increment of load is added to the structure each time.
At each stage of loading the stiffness of the structure may have a different
value depending on the deformation reached and the constitutive law
adopted for the material as well as the method for estimating the stiffness at
that stage. After the application of the (
i
-l)th load increment
D
P
i-1
and the
s
i-1
, the elasticity matrix [
D
i-1
] can be determined
from the stress-strain relationship and hence the new stiffness [
K
i
] can be
estimated. The
i
th increment of displacement can then be determined from
determination of the stress
(9.50)
It is obvious that in the incremental procedure (
Figure 9.11
) a series of linear
solutions is used to yield the continuous non-linear solution. In fact the non-
linear curve is approximately represented by a number of short linear segments.
The total load and displacement at any stage is given by the sum of the
increments of all the loads and displacements of the previous stages.
(9.51)
This method has the advantage that it is simple to apply but the accuracy is
rather low unless the load increments are very small. However, the method has a
serious drawback that at each step the stiffness matrix has to be re-assembled
and the solution procedure for the linear equations has to be performed each
time. This is uneconomical in terms of computational efforts.
