Civil Engineering Reference
In-Depth Information
Problem 2.1
What are the differences between ' direct ' and ' capacity ' design? In a multi -storey reinforced con-
crete frame that is to be capacity-designed, state the sequence of dimensioning of each of the com-
ponents of the frame, from the foundations to the roof.
2.3 Structural Response Characteristics
2.3.1 Stiffness
Stiffness defi nes the relationship between actions and deformations of a structure and its components.
Whereas member stiffness is a function of section properties, length and boundary conditions, system
stiffness is primarily a function of the lateral resisting mechanisms utilized, e.g. moment-resisting
frames, braced frames, walls or dual systems, as illustrated in Appendix A. Relationships between
geometry, mechanical properties, actions and deformations can be established from principles of
mechanics. Their complexity depends on the construction material used. Cracking of concrete, yielding
of reinforcement bars and other sources of inelasticity in RC structures pose problems in defi ning a
fi xed value of stiffness. For RC and masonry structures, the stiffness can be taken as the secant to the
yield point or to any other selected point on the response curve. Slippage at connections, local buckling
and yielding in steel structures are the counterparts to the above discussion on RC structures.
Figure 2.9 shows a plot of the structural response of a system subjected to lateral loads; the response
curve is represented by base shear
V
versus top horizontal displacement
δ
. In the fi gure, the initial slope
K
0
is the elastic stiffness of the structure, while the secant stiffness is the slope
K
s
of the line corre-
sponding to a given level of load. The initial stiffness
K
0
is higher than the secant stiffness
K
s
for con-
ventional materials of construction. In the case of rubber and other special materials, used for example
in devices for structural vibration control, the stiffness may increase as loads increase. For the latter,
values of
V
-
δ
pairs are generally utilized to defi ne the secant stiffness. Variations in stiffness in the
inelastic range are often expressed by the tangent stiffness
K
t
, which is the slope of the tangent to the
response curve in Figure 2.9 for a given
V
-
δ
pair. A decrease in the values of
K
t
indicates that softening
of the structure is taking place. In analysis of inelastic structures, use is often made of secant stiffness
to avoid dealing with negative tangent stiffness beyond the peak action resistance. Since inelastic
response problems are solved by iterations, the solution will normally converge by using the secant
stiffness even before reaching the point peak action resistance, but the rate of convergence will be lower
than in the case of using tangent stiffness.
δ
δ
V
K
0
K
s
F
K
t
V
j
V
i
V
y
δ
y
δ
i
δ
j
δ
u
δ
O
Top Lateral Displacement
Figure 2.9
D e fi nition of initial and secant structural stiffness
