**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:

- Fatigue of the longitudinal reinforcing steel,
- Failure of the concrete due to either a lack of confinement or a fracture of the transverse hoops,
- Compression buckling of the longitudinal reinforcement.

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 *e _{ap} *= plastic strain amplitude and

By assuming a linear strain profile across the critical section of a concrete column, plastic strains can be related to the
plastic curvature (* Phi_{p}*) 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 * Phi_{p}D* = a dimensionless plastic curvature amplitude.

Figure 1: Relationship Between DimensionlessPlastic 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:**

- Failure modes such as longitudinal bar buckling and transverse hoop fracture can be suppressed if sufficient
transverse reinforcement is used. The failure mode thus becomes the low cycle fatigue
*capacity*of the longitudinal reinforcement. - The fatigue failure
*capacity*of reinforced concrete bridge columns can be predicted by the theory presented herein without modification for low cycle fatigue failure mode. - The concept of a renewable plastic hinge has been introduced and validated experimentally. The fatigue life
*capacity*can be tuned to the fatigue*demand*by providing an appropriate length of fuse-bar and transverse confinement. - Fuse-bars can easily be replaced after the column hinge zone has been damaged. The repaired column performs as well as the undamaged virgin columns.
- The performance of renewable hinge columns is insensitive to changes in the axial load and the aspect ratio.

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
(*U _{sh}*) and concrete (

where *p _{s}* = volumetric ratio of the transverse reinforcement,

*External work done* results from the force actions of the neighboring concrete
(*U _{cc} *) and longitudinal reinforcement
(

and

in which *A _{st}* = total longitudinal steel area,

in which *a _{c}* = stress block factor is given by

where *f _{yh}* = yield strength of the lateral
reinforcement and p

where *f ^{'}_{cc}* = peak stress of the confined core concrete and

To simplify and reduce some nonlinearity in equation (7) simplifying assumptions are made. It is
assumed that * h _{c}* = 0.33 and

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
*p _{s}*. By using equation (3) to substitute for

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 (*p _{s} *) in terms of the axial load
intensity (

but

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.

**References**

*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.*

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