Se agregó esta notación al 497, según la nota del esquema de este proyecto

It was mentioned in Chapter 1 that, contrary to common assumptions made in force- based seismic design, the elastic stiffness of cracked concrete sections is essentially proportional to strength, and the concept of a constant yield curvature independent of strength is both valid, and important in terms of direct displacement-base design. A summary of the research leading to these statements, and to Fig. 1.4 is included below. A more complete presentation is available in [P3]. The research is based on moment- curvature analysis of different concrete sections, and the bilinear representation described in Section 4.2.7. It has been verified in numerous experiments^4!.

152 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

4.4.1 Circular Concrete Columns

Circular reinforced concrete columns are the most common lateral force-resisting elements for bridges in seismic regions^™]. In order to investigate the effective stiffness of circular columns, a parameter analysis was carried out varying the axial load ratio and flexural reinforcement ratio for a typical bridge column. The following basic data were assumed:

• Column diameter D — 2m (78.7 in)

• Cover to flexural reinforcement = 50 mm (2 in) • Concrete compression strength f*ce — 35 MPa (5.08 ksi) • Flexural reinforcement diameter d^i ~ 40 mm (1.575 in)

• Transverse reinforcement: spirals = 20mm (0.79in) at 100mm (4in) spacing

• Flexural reinforcement Ratio Pi!Ag — 0.005 to 0.04 (5 levels)

A selection of the moment-curvature curves resulting from analysis with the program CUMBIA provided on the attached CD is shown in Fig.4.10 for two levels of flexural reinforcement ratio, and a range of axial load ratios. Only the initial part of the moment- curvature curves has been included, to enable the region up to, and immediately after yield to be clearly differentiated. Also shown in Fig. 4.10 are the calculated bilinear approximations for each of the curves. Note that the apparent over-estimation by the bilinear representations of the actual curves is a function of the restricted range of curvature plotted, and is resolved when the full curve is plotted. It will be seen that the moment capacity is strongly influence by the axial load ratio, and also by the amount of reinforcement. However, the yield curvature of the equivalent bilinear representation of the moment-curvature curves does not appear to vary much between the curves.

Data from the full set of analyses for nominal moment capacity, and equivalent bilinear yield curvature are plotted in dimensionless form in Fig.4.11. The dimensionless nominal moment capacity and dimensionless yield curvature are respectively defined as

where £y = f yJE s is the flexural reinforcing steel yield strain.

The influence of both axial load ratio and reinforcement ratio on the nominal moment capacity is, as expected, substantial in Fig. 4.11(a), with an eight-fold range between maximum and minimum values. On the other hand, it is seen that the dimensionless yield curvature is comparatively insensitive to variations in axial load or reinforcement ratio.

• Steel yield strength • Axial load ratio

f ye = 4 5 0 MPa (65.3 ksi) Ni/TaAg - 0 to 0.4 (9 levels) 1V1 N r d 3 J ce (4.39) and <pDy=<l>yD l£ y (4.40)

D im en sio n le ss M o m en t (M N / f' cD 3)

Chapter 4. A nalysis Tools for Direct D isplacem ent-B ased D esign 153

Curvature (1/m) (a) Reinforcement Ratio = 1%

Curvature (1/m) (b) Reinforcement Ratio = 3% Fig.4.10 Selected Moment-Curvature Curves for Circular Bridge Columns

(D = 2m; Pce = 35 MPa; fye =450 MPa)l«l

Axial Load Ratio (Nu/f'cAg) (a) Nominal Moment

Axial Load Ratio (Nu/f'cAg) (b) Yield Curvature

Fig.4.11 Dimensionless Nominal Moment and Yield Curvature for Circular Bridge Columns (P3J

154 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

Thus the yield curvature is insensitive to the moment capacity. The average value of dimensionless curvature of (foy - 2.25 is plotted on Fig 4.11(b), together with lines at 10% above and 10% below the average. It is seen that all data except those for low reinforcement ratio coupled with very high axial load rado fall within the ±10% limits.

It should be noted that though the data were generated from a specific column size and material strengths, the dimensionless results can be expected to apply, with only insignificant errors, to other column sizes and material strengths within the normal range expected for standard design. The results would not, however apply to very high material strengths (say /*c>50MPa (7.25ksi), or ^>600M Pa (87ksi)) due to variadons in stress- strain characteristics.

The data in Figs. 4.10 and 4.11 can be used to determine the effective sdffness of the columns as a function of axial load rado and reinforcement rado, using Eq. (4.26). The rado of effecdve sdffness to initial uncracked section stiffness is thus given by

EL

EL

(4.41)

gross

Results are shown in Fig. 4.12 for the ranges of axial load and reinforcement ratio considered. It will be seen that the effective elastic stiffness ratio varies between 0.13 and 0.91. For the most common values of the variables, however, the ratio will be between 0.3 and 0.7.

Axial Load Ratio (Nu/ffcAg)

Chapter 4. A nalysis Tools for Direct D isplacem ent-B ased D esign 155

It should be noted that for convenience in computing the stiffness ratios of Fig 4.12, the gross stiffness of the uncracked section has been calculated without including the stiffening effect of the flexural reinforcing steel. That is

r 7UD4

gross ~ 64 (4'42)

Since the reinforcement increases the uncracked section moment of inertia by as much as 60% for the maximum steel ratio of 4%, the stiffness ratios related to true un-cracked sections would be lower, particularly for the higher reinforcement ratios. The value of the concrete modulus of elasticity used in computing Fig. 4.12 was

E = 5 0 0 0

E = 6 0 0 0 0

4.4.2 R ectangular Concrete Columns

Ductile rectangular columns can occur in bridge design, and at the base level of multi­ storey frame buildings. For the purposes of this study the special case of a square column with flexural reinforcement evenly distributed around the perimeter was investigated. The following basic data were assumed:

Column dimensions b — h — 1.6 m (63.5 in)

Cover to flexural reinforcement = 50mm (2 in) Concrete compression strength f*ce — 35 MPa (5.08 ksi) Flexural reinforcement diameter = 32 mm (1.26 in)

Transverse reinforcement: hoops “ 20mm dia. (0.^9in) /5 legs per layer Steel yield strength fye — 450 MPa (65.3 ksi)

Axial load ratio — 0 to 0.4 (9 levels) Flexural reinforcement ratio pJAg - 0.005 to 0.04 (5 levels)

Moment-curvature trends predicted by CUMBIA for the rectangular sections followed the same trends apparent for circular columns^3!.

Data from the full set of analyses for nominal moment capacity, and equivalent bilinear yield curvature are plotted in dimensionless form in Fig.4.13. The dimensionless nominal moment capacity, and dimensionless yield curvature are respectively defined as:

156 P riestley, Calvi and Kowalsky. D isplacem ent-B ased Seism ic D esign of Structures

Axial Load Ratio (Nu/ f ’cAg)