Chapter 3: FINITE ELEMENT METHOD AND BLAST RESISTANT DESIGN
3.2 Analysis Approach
Finite element method have been used extensively by structural designers and researchers worldwide because of its unique ability to transform real life problems into simple mathematical model and by solving the models using computer programs. Thus, this method saves a lot of time and helps avoid possible computational error made by human.
The main analysis method that have been used in this research is finite element method (FEM). The following subsections will discuss this method briefly.
3.2.1Finite element software
SAP is a general purpose finite element software which is used as a powerful tool to analyse and design any type of structure. SAP is abbreviated as Structural Analysis Program 2000 produced and developed by Computers and Structures, Incorporated (CSI).
It has the ability to perform static or dynamic, linear or nonlinear analysis and design of structural systems. To analyse a physical model, first a mathematical model is developed in the software. Next, material properties (linear or nonlinear) are defined and assigned to the model. Next, related loadings are applied followed by analysis approach i.e. static or dynamic is selected. After that the software divides the model into a number of finite
20 elements or pieces depending on the type of the model. The pieces are so selected that it represents the corresponding properties as close as possible to the physical model. These pieces are called meshes which plays an important role in the analysis. The accuracy of the results depends on this feature. Displacement, force and shape functions are assumed for each element. Analysis of each element, joint and nodes takes place according to their local axes taking into account the material properties and type of analysis selected. Then all the results are combined and expressed in terms of the global coordinate system.
Finally boundary conditions are applied and results are shown. Member forces are distributed among the elements using either weak or strong formulation.
3.2.2Mander concrete model
Mander concrete model is widely used in FEM software to model the behaviour of concrete because of its simplicity and acceptability among designers and researchers. The model is also used in SAP 2000 for concrete materials. The model is basically a stress- strain relation developed for both unconfined and confined concrete subjected to uniaxial compressive loading. The confinements may be any type of general confinement, i.e.
spiral or circular hoops or rectangular hoops with or without crossties. One of the basic feature of this model is that it considers the effect of strain rate (Mander et al., 1988). The model is available for both rectangular and circular section in SAP 2000. If sufficient data (i.e. section property and material property) is available, SAP 2000 automatically uses confined or unconfined stress-strain relationship curve for that section.
3.2.2.1 Mander unconfined concrete stress-strain model
The Mander unconfined concrete stress-strain curve is defined by the following equations-
For ε ≤ 2ε’c (curved portion),
f = f'cxr
r-1+xr (1)
Where,
x= ε
ε’c (2)
r = E
E-(f'c/ε’c) (3)
21 For 2ε’c < ε ≤ εu (linear portion),
f= ( 2f'cr
r-1+2r) ( εu-ε
εu-2ε'c) (4)
The tensile yield strength of Mander unconfined curve is taken as 7.5 √f'c in psi. The stress-strain curve is shown below in Figure 3.1.
Figure 3.1: Mander unconfined concrete stress-strain curve (Technical notes S-TN- MAT-001, Computer and Structures, Inc.).
3.2.3Rebar parametric stress-strain curve
Another widely used stress-strain curve for structural steel in FEM software is Rebar Parametric Stress-Strain Curve. In SAP 2000, this model is used by default unless otherwise the other available model “Park” being used. The simple model has four regions; elastic region, a perfect plastic region, a strain hardening region and a softening region which are defined by the following four equations-
For ε ≤ εy (elastic region),
f = Eε (5)
For εy < ε ≤ εsh (perfectly plastic region),
f=fy (6)
For εsh < ε ≤ εr (strain hardening region),
f = fy + (fu- fy)√ε-εsh
εu-εsh (7)
22 Where,
r = ε-εsh
εu-εsh (8)
Figure 3.2 shows a typical stress-strain curve of structural steel-
Figure 3.2: Typical simple structural steel parametric stress-strain curve (Technical Notes S-TN-MAT-001, Computer and Structures, Inc.).
3.2.4Nonlinear direct integration time-history analysis
Time-history analysis is a method used to determine the dynamic responses of a structure subjected to time dependent loading. This method solves the dynamic equilibrium equation which requires iterative solution. The dynamic equilibrium equation is given by-
Ku(t) +Cu̇(t)+Mü(t)=r(t) (9) Where K is the stiffness matrix, C is the damping matrix and M is the diagonal mass matrix. SAP 2000 has several type of time-history analysis methods like Modal or Direct Integration, linear or nonlinear and Transient or Periodic. Depending on the type of problem the analysis method is selected as direct integration time history analysis. The advantages of this method are as follows (CSI Reference Manual 2017, rev. 18)-
i. Full damping is considered coupling all the modes.
ii. Impact and wave propagation problems can be solved more efficiently.
iii. All types of nonlinearity can be considered.
23 3.2.4.1 Time steps
Direct integration time-history analysis method is extremely sensitive to the size of time steps. Selecting time step sizes for a particular problem should be investigated by trial and error. Always decreasing time steps should be selected as the accuracy of the analysis depends on it. Time steps used for the direct integration time-history analysis should be small enough so that the results are no longer affected by it. Newmark method is used for the time step integration. In this method the following equations are used-
u̇i+1 = u̇i + [(1 - γ) Δt]üi + (γΔt) üi+1 (10) ui+1 = ui + (Δt)ui+ [(0.5-β)(Δt)2]üi+[β(Δt)2]üi+1 (11)
For satisfactory results including accuracy of calculation, the values of β = 1/4 and γ = 1/2 is used (N.M. Newmark, 1959).