by John B. Mander, Anindya Dutta and Chin Tung Cheng
This article presents research conducted under NCEER's Highway Project to develop seismic design and capacity detailing recommendations for bridge substructures. This paper describes work performed during the last two years. Comments and questions should be directed to John Mander, University at Buffalo, at (716) 645-2114 ext. 2418.
The objective of this research task is to develop seismic design and capacity detailing recommendations for bridge substructures. The emphasis is on the ductile detailing of the primary energy dissipating zones - the plastic hinges of bridge piers. Design equations for confinement are developed that would ensure sufficient plastic rotational capacity to avoid undesirable failure mechanisms; the final and unavoidable failure mode being low cycle fatigue of the longitudinal reinforcement.
The importance of lateral hoop reinforcement in improving the ductility level of circular columns is well known. Failure in a circular bridge column can arise from:
Following the principles of capacity design, it is possible to avoid undesirable failure modes such as (b) and (c), leaving low cycle fatigue as the only unavoidable mode of failure. As a result, prediction of the amount of transverse reinforcement in the potential plastic hinge zones is of paramount importance to ensure "capacity protection" to the remainder of the structure. The first year of this two-year research effort therefore primarily focused on considering failure mode (a) fatigue of the longitudinal reinforcement. The second year has focused on failure modes (b) and (c) and thus methods of determining the appropriate quantity of transverse steel to "capacity protect" the structure. Early work by Mander et al. (1984, 1988a,b) led to an energy balance approach in which first hoop fracture of confined column sections could be predicted under concentric axial compression. This research extends that work by considering cyclic flexure in addition to axial compression. Thus design equations are developed for bridge columns for a range of cyclic demands.
Based on the experimental test results of recent studies by Mander et al. (1994) on the low cycle fatigue performance of reinforcing steels, it was shown that, regardless of the steel grade, a dependable plastic strain-life fatigue relationship is given by
where eap = plastic strain amplitude and Nf = number of cycles to the appearance of the first fatigue crack.
By assuming a linear strain profile across the critical section of a concrete column, plastic strains can be related to the plastic curvature (Phip) by
where D = overall column diameter (or depth) and d' = depth from the outermost concrete fiber to the center of reinforcement (Note: D-2d' = pitch circle diameter of the longitudinal steel in a circular column).
Substituting equation (2) into equation (1), one obtains a plastic curvature-life fatigue relationship for reinforced concrete columns
where PhipD = a dimensionless plastic curvature amplitude.
|Figure 1: Relationship Between Dimensionless|
Plastic Curvature and Cycles to Failure
The experimental results relating fatigue life and plastic curvature are plotted in figure 1. Equation (3) is also plotted and it is evident that there is good agreement between the theory and observed experimental performance.
The following conclusions are drawn based on this research:
Based on the energy balance theory proposed by Mander et al. (1988a), the gain in ductility of a confined concrete member can be attributed to the energy stored in the transverse reinforcement. Using a virtual work approach in which the external work done on the section (EWD) is equal to the internal energy absorption capacity (IWD) of the section it is possible to write
Internal work done on the critical section, that is the section's capacity to sustain plastic damage, is defined as the sum of the energy absorption capacity of the constituent materials: steel (Ush) and concrete (Uco), thus
where ps = volumetric ratio of the transverse reinforcement, Acc = area of the core concrete and Usf = area under the stress strain curve of steel reinforcement until fracture. (According to Mander et al. (1988a), this may be taken as Usf = 110 MJ / m3.) Also, in the absence of more rigorous analysis, the expression, 0.008 f'c Ag, is substituted as an approximation for the integral of the energy required to fail an equivalent unconfined column.
External work done results from the force actions of the neighboring concrete (Ucc ) and longitudinal reinforcement (Us ) on the critical section. Consider the circular column section in figure 2. It is assumed that the available strain energy is consumed during cyclic loading by the concrete and the steel doing plastic work in cyclic compression. The plastic work done by the steel and the concrete is obtained by multiplying the forces in the compression steel and concrete by the appropriate plastic strain. Further, assuming that in a circular section, one-half of the total steel is lumped at both the ends of the pitch circle diameter and the rest is distributed in a thin rectangular strip of depth (D" - d"), the external work done can be expressed in terms of the sum of the plastic work done on each load reversal
in which Ast = total longitudinal steel area, fy = yield strength of the longitudinal steel, Phip = plastic curvature, Bc = stress block depth factor and nc = an efficiency factor to account for the reduced area of the concrete stress-strain curve after the first reversal. Note 2Nc denotes the total number of load reversals. Also Ccc = core concrete compression force which for circular sections can be shown to be equal to
in which ac = stress block factor is given by
where fyh = yield strength of the lateral reinforcement and ps = volumetric ratio of that steel with respect to the core. This equation is based on a reanalysis of confined stress block parameters undertaken previously by Mander et al. (1984). The confined strength ratio K according to Mander et al. (1988a) is given by:
where f'cc = peak stress of the confined core concrete and f'l = effective confining stress provided by the transverse reinforcement at yield.
To simplify and reduce some nonlinearity in equation (7) simplifying assumptions are made. It is assumed that hc = 0.33 and Nc = 4, thus the terms in square brackets equal 0.5, and also Bc = 1.0. Note that the first term (0.25) in square brackets in equation (6) accounts for the fact that only half of the steel lumped at the extreme ends of the pitch circle diameter does work in compression.
Combining equations (6) and (7) and equating to equation (5), it is possible to obtain a fatigue-life equation in the form
where the fatigue-rotation coefficient
The neutral axis depth ratio (c"/D") can be obtained from force equilibrium across the section. For a circular section it can be shown
It is possible to obtain an expression for the required volumetric steel ratio ps. By using equation (3) to substitute for PhipD in equation (11), it is possible to express the volumetric steel ratio in the form as
where Psi = factor which depends on section, hoop type and effectiveness. For circular sections, as an example,
Similar expressions can be obtained for rectangular sections as well. The results of the analysis for determining the transverse steel requirements (ps ) in terms of the axial load intensity (Pe / fcAg ), longitudinal steel volume (pt ) and cyclic loading demand (Nc ) are presented in figure 3. Design curves are plotted for cyclic loading demands of 4, 10 and 20 cycles. These demands are based on recent work by Chang and Mander (1994) who found that for typical U.S. earthquakes, the equivalent number of constant amplitude cycles of loading is given by
where T = natural period of vibration of the structure. Equation (16) gives a cyclic demand spectra which is an implicit measure of earthquake duration effects.
The foregoing analysis provides a more rational basis for determining the transverse reinforcement required to prevent premature column failure. It considers the cyclic demand, the longitudinal steel volume and intensity of axial load - all factors that historically have been ignored in transverse steel design in the U.S.
Space does not permit the development of an energy-based analysis for determining the amount of transverse reinforcement necessary to prevent longitudinal bar buckling. This is another concern that is presently being investigated.
AASHTO (1994), AASHTO LRFD Bridge Design Specifications, 1st ed., American Association of State Highway and Transportation Officials, Washington, DC.
Chang, G.A. and Mander, J.B., (1994), "Seismic Energy Based Fatigue Analysis of Bridge Columns: Part I - Evaluation of Seismic Capacity," Technical Report NCEER-94-0006, University at Buffalo.
Mander, J.B. and Dutta, A. (1996), "A Practical Energy-Based Design Methodology for Performance Based Seismic Engineering," Proceedings, SEAOC Annual Convention, Maui, Hawaii.
Mander, J.B., Priestley, M.J.N. and Park, R., (1984), "Seismic Design of Bridge Piers," Research Report 84-2, Dept. of Civil Engineering, University of Canterbury, Christchurch, New Zealand.
Mander, J.B., Priestley, M.J.N. and Park, R., (1988a), "Theoretical Stress Strain Model of Confined Concrete," Journal of Structural Engineering, ASCE, Vol. 114, No. 8, pp. 1804-1826.
Mander, J.B., Priestley, M.J.N. and Park, R., (1988b), "Observed Stress Strain Behavior of Confined Concrete," Journal of Structural Engineering, ASCE, Vol. 114, No. 8, pp. 1827-1849.
Mander, J.B., Panthaki, F.D. and Kasalanti, A., (1994), "Low-cycle Fatigue Behavior of Reinforcing Steel," Journal of Materials in Civil Engineering, ASCE, Vol. 6, No. 4, pp. 453-468.
Click here to return to the Table of Contents
| Home | NCEER
| Information Assistance | Publications
| Databases, Software |
| Events | New! | Links | Data Resources |
Last modified May 16, 1997