Method of Analysis

The computer program “Plastique ” which is described herein, is capable to perform the different types of analysis, namely; Push-over analysis, Quasi-static analysis, Non-linear dynamic analysis and Eigenvalue analysis

The first two analysis types, although significantly simplified, can lead to valuable conclusions concerning the behavior of the structure and the possible collapse mechanism. The applied procedure can be described in brief as follows. In the case of 2-D analysis the structure is assumed to consist of a finite number of nodes interconnected by a finite number of elements. The types of elements have been described in section 3. In the case of 3-D analysis the structure is assumed to consist of the aforementioned 2-D frames, assuming a rigid diaphragm assemblage of their horizontal dof’s per floor slab. Loads may be applied at the nodes or along the elements. In both cases though, they are transformed to nodal loads.

Figure 5. Modified Newton Raphson Method

After the formation of the stiffness matrix the equilibrium equations are solved by an efficient algorithm based on the Gaussian elimination method. The structure stiffness is stored in a banded form

to optimize the use of core storage and during arithmetic operations are avoided. An incremental method is applied for all types of analysis. The specified loads are divided to sufficiently smaller sub-loads, in order to simulate more efficiently the stress redistribution which occurs due to the non linear behaviour of the structure. An iterative process (modified Newton Raphson Method) is incorporated in each load step so that a higher level of accuracy can be achieved, as shown in Figure 5.

The member forces are computed for each load increment and the tangent stiffness matrix is updated to account for changes in any stiffness coefficient of the element. A spread plasticity model is used, as described in section 4, in order to simulate the changes in the flexibility in each element. In the case of the dynamic analysis the Newmark Method is used for the direct integration of the equations of motion.

The equation of motion to be solved at any stage of the analysis is written as:

[M ] {U} + [C ] {U} + P} = P} = – M ] {S}{Ug} (10)

where [M] is the mass matrix; [C] is the damping matrix; [Pint] is the internal load vector of the structure; [Pext] is the external load vector of the structure; [S] is a modal influence vector; [U] is the structure displacement vector and {Ug} is the ground acceleration vector.

The above system of equations is solved using the constant acceleration method, according to which equation (10) can be rewritten for time t+At and iteration k as:

[M ]t+4t iU}ik) +[C ]+4t-{u (k)}+ *+4t {Pnt}(k-1) + * [ K, ] *+4t-{AU }(k) = *+4t{Pxt,} (11)

The hysteretic model applied is the Bouc Wen – Baber Noori model which, as mentioned previously, is able to simulate R/C behaviour such as stiffness degradation, strength deterioration and pinching. At every step of the analysis, the Damage Indices of the elements and the structure are calculated providing an evaluation not only of the inflicted damage, but also of the structure’s residual strength and capacity to withstand further loading. The Damage Index for each section is given by relation (7). The Damage Index for each element is computed as the maximum Damage Index of its sections’ and finally the Damage Index of the whole structure can be obtained from the following expressions:

where (DI)i is the Damage Index for each element and Etoti is the total amount of absorbed energy per element.