INDICE DE DESARROLLO HUMANO
EDUCACIÓN Y DESARROLLO
Use this as the new estimate of Ad.
6. Cycling once more through the sequence yields: [1 - 3.19, % — 0.147, and A ^ = 0.284
m (11.2 in.). The result has stabilized.
The reference effective stiffness is thus found, using T — Tc — 4.0 sec. in Eq.(3.1) as:
Gravity load or P-A requirements will govern the choice of the actual design strength. Fig.3.1(d). Second, (case (b)), the physical dimensions of the column cross section are
A415 5 = 0.533x0.35/0.7 = 0211m
% = 0.05 + 0.444—'—— — =0.150 (15%) 3.41;r
K e = 4 n 2 (5000/9.805)/4 2 = 1258 kN/m ,
and the maximum design base shear force, from Eq.(3.2) as
Chapter 3. Direct D isplacem ent-B ased D esign: Fundam ental C onsiderations 95
Case (b): Design Displacement: The yield curvature is unchanged, and the yield
displacement is calculated, from Eq.(3.7), as
A , = (pyH2/3 = 0.00264x252/3 = 0.55 m (21.7 in.).
Note that this is independent of the final strength. This displacement exceeds the elastic displacement at the corner period, A4 5 = 0.438 m. Hence there is no point in further
calculating the displacement capacity, as the pier will respond elastically to the design level of intensity, with a response displacement of 0.438 m.
Now, since the structure responds elastically, with a known displacement, the allocated strength is in fact arbitrary, as noted above. For example, if we allocated a yield strength of 500 kN, the stiffness would be K ei — 500/0.55 =909 kN/m. The calculated elastic period would be:
Td =
2^7(5000/(9.805x909))
= 4.71 seconds.The response displacement would be 0.438 m, and the maximum response force would be
VBose
= 909x0.438 = 398 kN (89.5 kips).However, if we arbitrarily allocated a yield strength of 350kN, the stiffness would be
Kei =350/0.55 = 636 kN/m, and the elastic period would be 5.63 seconds. The response
displacement, (see Fig.3.1 (d)), would still be 0.438 m, and the maximum response force would be 278 kN (62.5 kips).
P-A
moments for this case would be 36% of the base moment from the horizontal inertia force, and, in accordance with Section 3.6, would need to be carefully considered.Note that if the strength was arbitrarily set higher than 692 kN, (say 800 kN), then the elastic period would be found to be less than 4 sec. (in this case, Kei =1455 kN/m, and
Tei ~ 3 .72 sec, and the structure would respond with a displacement less than 0.438 m, (in
this case 0.407 m (16.0 in)), in accordance with the elastic 5% displacement spectrum for the calculated period.
In fact, the example is probably artificial. The base moment would be very high, in either case, and redesign with a larger column diameter (and hence smaller yield displacement) would be advisable. The pier would also be excessively flexible for gravity loads.
3.5 MULTI-DEGREE-OF-FREEDOM STRUCTURES
For multi-degree-of-freedom
(MDOF)
structures the initial part of the design process requires the determination of the characteristics of the equivalentSDOF
substitutestructure^. The required characteristics are the equivalentjnass, the design displacement,
and the effective damping. When these have been determined, the design base shear for the substitute structure can be determined. The base shear is then distributed between the mass elements of the real structure as inertia forces, and the structure analyzed under these forces to determine the design moments at locations of potential plastic hinges.
96 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures
3.5.1 Design Displacement
The characteristic design displacement of the substitute structure depends on the limit state displacement or drift of the most critical member of the real structure, and an assumed displacement shape for the structure. This displacement shape is that which corresponds to the inelastic first-mode at the design level of seismic excitation. Thus the changes to the elastic first-mode shape resulting from local changes to member stiffness caused by inelastic action in plastic hinges are taken into account at the beginning of the design. Representing the displacement by the inelastic rather than the elastic first-mode shape is consistent with characterizing the structure by its secant stiffness to maximum response. In fact, the inelastic and elastic first-mode shapes are often very similar.
The design displacement (generalized displacement coordinate) is thus given by
(3-26)
i= l i=1
where m; and A,- are the masses and displacements of the n significant mass locations respectively. For multi-storey buildings, these will normally be at the n floors of the building. For bridges, the mass locations will normally be at the centre of the mass of the superstructure above each column, but the superstructure mass may be discretized to more than one mass per span to improve validity of simulation (see Section 4.9.2(e)(iii)). ""~~~With tall columns, such as may occur in deep valley crossings, the column may also be
discretized into multiple elements and masses.
Where strain limits govern, the design displacement of the critical member can be determined using the approach outlined in Section 3.4.1. Similar conclusions apply when code drift limits apply. For example, the design displacement for frame buildings will normally be governed by drift limits in the lower storeys of the building. For a bridge, the design displacement will normally be governed by the plastic rotation capacity of the shortest column. With a knowledge of the displacement of the critical element and the design displacement shape (discussed further in the following section), the displacements of the individual masses are given by
A, = 8, ■
V
\ $ c J
(3.27)
where $ is the inelastic mode shape, and Ac is the design displacement at the critical mass, cy and 8C is the value of the mode shape at mass c.
Note that the influence of higher modes on the displacement and drift envelopes is generally small, and is not considered at this stage in the design. However, for buildings higher than (say) ten storeys, dynamic amplification of drift may be important, and the
Chapter 3. Direct D isplacem ent-B ased D esign: F undam ental C onsiderations 97
design drift limit may need to be reduced to account for this. This factor is considered in detail in the relevant structural design chapters.
3.5.2 Displacement Shapes
(a) Frame Budldings: For regular frame buildings, the following equations, though
approximate, have been shownl^17! to be adequate for design purposes:
Building frames: for n < 4: Sj = H [ / H n
cK‘ for n > 4: 8, — 1L 1- H :
4 H
(3.28a) (3.28b) n JIn Eq.(3.28) and H„ are the heights of level i} and the roof (level n) respectively. Displacement shapes resulting from Eq.(3.28b) provide improved agreement between predicted displacements and those resulting from inelastic time-history analysis for taller buildings, compared with the linear profile appropriate for shorter buildings
(b) Cantilever Wall Buildings: For^cantilever wall buildings the maximum drifts will
occur in the top storey. The value of this drift may be limited by the code maximum drift limit, or by the plastic rotation capacity of the base plastic hinge. Assuming a simple triangular distribution of first-mode curvature with height at yield, as shown in Fig. 3.18, to compensate for tension-shift and shear deformation, (see Section 6.2.1 for justification of this) the yield drift 6yn at the top of the wall will be
Qy„=<l>yHJ2
where (j)y — 2£y/lw from Eq.(3.6c), and lw is the wall length . Hence,
0yn=£yH J l w (3.29)
As a reasonable approximation (see Section 6.2.1(b)), the plastic rotation may be concentrated at the wall base. The critical drift at the top of the wall will thus be
= = i . 0 f , / / y / „ + ( ^ . - 2 . t e , / / .)£ , < e c (3.3 0)
where 6pn is the plasdc rotation at the top of the wall corresponding to the design limit state, <pm is the corresponding base curvature, and Lp is the plastic hinge length (see Section 4.2.7). The yield displacement at height Hj is given by:
98 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures
G'dn