Numerical Analyses

In this section, the presented modifications have been verified through some numerical examples including three, nine and twenty story braced frames. The geometric data of the models are presented in table 1 and fig. 5. The steel properties are, yield stress Oy = 245MPa, initial elastic modulus E = 2.10e5MPa with a bilinear nonlinear behavior with five percent strain hardening or second modulus Es = a. E,a = 0.05 . The dead load is assumed to be 3.9 kPa. and the reduced live load 1.4

kPa at floor levels. At roof level these values are assumed to be 3.2 kPa. and 1.0 kPa. respectively. The assumed data may be sufficient for DBD, but for the nonlinear push over and time history analyses the detail design of the members must also be available. This has been performed using the capacity design procedures and was performed using SAP2000 [21] commercial program. The modified strength reduction factors for MDOF structures taking into account the reductions due to structural over strength have been calculated based on the equations proposed in [23]. The effects of cumulative damage may also be considered using the idea presented in [5] that has been discussed subsequently. Nonlinear dynamic analyses have been performed using DRAIN2DX [22] program using three selected earthquake record which were compatible with the obtained response spectrum shown in fig. 1.

Table 1: Geometric data for numerical examples (Dimensions in meter)

n story

H1

(n-1)*Hi

LS1

No*Lspan1

LS2

No*Lspan2

l

3 story

3.5

2*3.5

2(4.0)

2*4

2(3.5)

2*4

1.5

9 story

4

8*3.5

2(4.2)

2*4.2

2(3.5)

2*4

1.5

20 story

4

19*3.5

2(4.2)

3*4.2

2(3.5)

3*4

1.5

Figure 6 shows the effect of the ductility demand distribution on story force and story drift using modal ductility distribution and equations 21 and 22 for a nine story building with eccentric bracing. The maximum ductility is assumed to be 2 according to fig. 8. It has been shown that the lateral story force has not been so sensitive to the ductility pattern but the displacement and drift directly change with the ductility in the DBD and equation 21 with a=-3 and equation 22 with b=3 may be acceptable comparing to the dynamic analysis results. In table 2 the effect of ductility pattern on DBD parameters (Effective parameters on the equivalent SDOF structure) has also been presented. As shown the effective damping, total base shear, mass and effective height ratio which may be assumed as the representative for lateral force distribution are not sensitive to the ductility distribution over height

and just depend on the maximum ductility value. In figure 7 the period of the first mode from eigen value analysis has been compared to the periods obtained from linear and nonlinear time history analyses showing the effects of vertical loads and nonlinear behavior and also the effective periods from DBD method taking into account the effect of column deformation for the last story based ductility model.

Figure 6: Effect of ductility demand distribution on story force and drift

Table 2: Effect of ductility demand distribution on DBD parameters

Effective

Parameters

Units

Modal

Pattern

Exponential (Equation 21)

Polynomial (Equation 22)

a=0.01

a=-3

a=-1

b=0.7

b=2

b=3

Effective Period

sec

0.85

1.30

0.92

1.14

1.35

1.25

1.01

Effective Damping

%

4.36

4.79

4.42

4.65

4.81

4.74

4.53

Mass Ratio

0.71

0.73

0.71

0.72

0.74

0.72

0.70

Effective Stiffness

N/mm

13601

5986

11783

7622

5658

6427

9550

Effective Displacement

mm

92

134

97

116

140

128

104

Effective Height Ratio

0.73

0.73

0.73

0.73

0.72

0.73

0.74

Total Base Shear

kN

798

674

756

683

685

674

705