Non-linear analysis of plane frames - Concrete Structures: Stresses and Deformations - page 449

0
0
938.0

10 −6

{ D } cycle 1 = { D *} cycle 1 = (b)

Treat AB as a cantilever, xed at B and subjected to a bending moment

diagram composed of two straight lines joining the above three M -
values. The magnitudes of these moments indicate that cracking occurs
at several sections in the negative and in the positive moment zones.
Calculate

fi

ε O and ψ at the eleven sections and apply Equation (13.21) to

obtain:

− 1119

6922
1530

{ D s } cycle 1 = 10 −6 (c)

For the member considered, end B is xed; thus, the displacements of

end A with respect to end B is the same as { D *}. From Equation (13.32),
determine { D *} error :

fi

1119
− { D *} error cycle 1 = {{ D *} − { D * s }} = 10 −6 6922 (d)

− 592.2

The exibility matrix of the cracked cantilever is (Equation

(13.8) ):

fl

5.569
−

− 16.790

4839
−

− 1.685

10 −9 { f *} cycle 1 = 16.790 − 569.2

82.07

(e)

− 1.685 569.2

Inversion of [ f *] gives the sti ff ness matrix of the cracked member:

216.2
6.908
52.35

6.908
1.343
9.454

52.35
9.454
78.83

[ S *] cycle 1 = [ f *] −1 cycle 1 = 10 6 (f )

The residual member end forces (Equation (13.34) ):