Geometric Nonlinearity - Theory of Nonlinear Structural Analysis - page 289

2500 5000

2000 4000

PHL #1 PHL #13

1500 3000

1000 2000

500 1000

0 0

-500 -1000

-1000 -2000

-1500 -3000

-2000 -4000

-2500 -5000

-0.04 -0.03 -0.02 -0.01 0 -0.04 -0.03 -0.02 -0.01 0

Plastic rotation at PHL #1 (rad) Plastic rotation at PHL #13 (rad)

4000 4000

3000 3000

2000 2000

1000 1000

0 0 PHL #56

-1000

-1000

-2000

-2000

-3000 -3000

PHL #49

-4000 -4000

0 0.01 0.02 0.03 0.04 -0.006 -0.004 -0.002 0 0.002 0.004

Plastic rotation at PHL #49 (rad) Plastic rotation at PHL #56 (rad)

Figure 7.33 Hysteresis loops of selected PHLs of the frame with geometric nonlinearity.

10 translational DOFs for the frame (i.e. d = 10). By applying the static condensation using
Eq. (7.147), the 10 × 10 K o matrix, 10 × 140 K 0 o matrix, and 140 × 140 K 0 o matrix are obtained,
where these condensed stiffness matrices are constant because updates of geometric nonline-
arity due to changes in axial force are ignored. Assume that a gravity load of Q = 5,978 kN is
applied on each floor of the leaning column, as shown in Figure 7.34. Then, the K f matrix
becomes

2
4

3
5

− 26600 12600 0 0 0 0 0 0 0 0

12600 − 13800 11200 0 0 0 0 0 0 0

0 11200 − 21000 9800 0 0 0 0 0 0

0 0 9800 −18200 8400 0 0 0 0 0

0 0 0 8400 − 15400 7000 0 0 0 0

K f = kN = m

0 0 0 0 7000 −12600 5600 0 0 0

0 0 0 0 0 5600 − 9800 4200 0 0

0 0 0 0 0 0 4200 − 7000 2800 0

0 0 0 0 0 0 0 2800 − 4200 1400

0 0 0 0 0 0 0 0 1400 − 1400

ð7 : 181Þ

K e can be computed accordingly using the third

and the condensed elastic stiffness matrix

equation of Eq. (7.148).