Civil Engineering Reference
In-Depth Information
V
V
Maximum (Ultimate)
Load
Limiting Compressive
V
max
Strain
V
δ
δ
δ
δ
O
O
u
u
Top Lateral Displacement
Top Lateral Displacement
(a) Based on a Limiting Compressive Strain
(b) Based on Peak Load
V
V
Maximum (Ultimate)
Load
First Fracture or Buckling
V
max
V
V
Small Reduction
in Load Capacity
O
δ
δ
O
δ
δ
u
u
Top Lateral Displacement
Top Lateral Displacement
(c) Based on Significant Load Capacity After Peak Load
(d) Based on Fracture and/or Buckling
Figure 2.38
D e fi nitions of ultimate deformations
Δ
e
Δ
tot
20.0
Δ
p
L
p/
L
= 0 05
L
p/
L
= 0 10
L
p/
L
= 0 15
F
G
m
L
p/
L
= 0 20
L
p/
L
= 0 25
L
p/
L
= 0 30
15.0
10.0
Ductile Response
θ
p
5.0
0.0
χ
p
χ
e
0.0
5.0
10.0
15.0
20.0
Member
Bending Moment
Curvatures
Displacements
Curvature ductility (
m
c
)
Figure 2.39
Relationship between local and global ductility for cantilever systems: free body and defl ection dia-
grams (
left
) and variation of displacement ductility as a function of geometric layout (
right
)
L
L
L
L
⎡
⎢
⎛
⎜
⎞
⎟
⎤
⎥
p
p
(
)
(2.16)
μ
=+
13
μ
−
1 10.5
−
δ
χ
where
L
p
and
L
are the plastic hinge length and the cantilever height, respectively. Thus, for ordinary
cross sections of columns and piers, to obtain global ductility factors
μ
δ
= 4 - 5 the required
μ
χ
- values
range between 12 and 16. The relationship in equation (2.16) accounts for total horizontal defl ections
δ
u
generated solely by fl exural deformations and assumes fi xed base for the cantilever; the ultimate
lateral displacement
δ
u
is given by:
p
(2.17)
δδδ
=+
y
u
where
δ
y
and
δ
p
are the yield and plastic displacements, respectively.
Shear deformations of the member, connection fl exibility (
see
for example Section 2.3.1.1 ) and soil -
structure interaction increase the yield displacement
δ
y
. Conversely, plastic displacements
δ
p
remain
