As suggested by SDC, the material properties considered in design are not the nominal strengths as generally adopted. In order to represent the actual behavior of existing bridges, actual material properties from test data should be used instead of the nominal design values. In the absence of actual material test data, the expected material properties based on SDC and other available technical research can be used for the evaluations.
4.1.1 Reinforcing Steels
The properties of reinforcing steels are modeled based on a stress-strain relationship that exhibits an initial linear elastic portion, a yield plateau, and a strain-hardening range in which the stress increases with strain. The yield point is defined by the expected yield stress of the steel, fye. The strain-hardening curve can be modeled as a parabola or some other non-linear relationship that terminates at the ultimate tensile strain . The ultimate strain should be set atue the point where the stress begins to drop with increased strain as the bar approaches fracture. The properties of reinforcing steels proposed by SDC are listed below:
Modulus of elasticity Es = 200000 MPa
Specified minimum yield strength fy = 420 MPa
Expected yield strength (typical) fye = 475 MPa
Specified minimum tensile strength fu
= 550 MPa
Expected tensile strength (typical) fue
= 655 MPa
Nominal yield strain ε = 0.0021
Expected yield strain (typical) ye
= 0.0023 Ultimate tensile strain (reduced by 33%)
Onset of strain hardening
It is possible that the expected yield stress of the steels in the ductile components may be less than 475 MPa (recommended by SDC); this will result in a reduced ratio of the actual plastic moment strength to the design strength, which will result in an underestimation of the strength requirement of the protected components. Therefore, a magnification factor of 1.1 has been proposed by NCHRP 12–49 [11] to define the value of fye; this definition is more conservative than that by SDC:
y
ye f
f 11.
(4.1)
This value will be adopted in the assessment herein. The stress-strain relationship of reinforcement is provided below:
4.1.2 Concrete
A stress-strain relationship model for confined and unconfined concrete is required in order to determine the capacity of the ductile concrete members.
The initial ascending curve may be represented by the same equation for both the confined and unconfined concrete since the confining steel has no effect in this range of deformation. As the curve approaches the nominal compressive strength f ' , the stress of the unconfined concrete begins to fall as the strainc increases and rapidly reduces to zero at the spalling strain . Typically, thesp value of spalling strain is 0.005. In the case of confined concrete, the curve continues to ascend until the confined compressive strength fccis reached.
This segment is followed by a descending curve that is dependent on the parameters of the confining steel. The ultimate strain should be the pointcu where strain energy equilibrium is reached between the concrete and confinement steel. Mander’s stress-strain model is a commonly used model for confined concrete [8]. The properties of concrete proposed by SDC are listed below:
28-day concrete strength (design/tested strength): f c
Expected concrete compressive strength: fce1.3fc
Modulus of Elasticity: Ec = 0.043w1.5 fcMPa Unconfined concrete compressive strain
at maximum compressive stress: c0 0.002 Ultimate unconfined compression (spalling strain): sp 0.005
The expected concrete compressive strength is recommended to be 1.3 times the design concrete strength according to SDC. However, in consideration of
the actual engineering environment in Taiwan, an overstrength factor of 1.1 is adopted in this study. That is,
c
ce f
f 11. (4.2)
According to ACI, the specified compressive strain of concrete is considered to be 0.003, as is the case with SDSHD2000. However, in accordance with a study by Blume [12], the unconfined compressive strain of concrete of 0.003 is too conservative and the value of 0.004 is considered adequate. Hence, an unconfined compressive strain of 0.004 is adopted in this study.cc
The ultimate concrete strain follows the model of Mander [8]. The value of
is calculated using the equation given belowcu
ce
ρs: steel ratio of confining reinforcement
fyh: yield strength of confining reinforcing steels
εhu: ultimate tensile strain of confining reinforcing steels; the value of 0.09 is recommended by Caltrans
4.1.3 Allowable Material Ultimate Strain
The displacement ductility capacity of a flexural component is calculated using a displacement-based approach based on a moment-curvature curve in which the ultimate curvature corresponds to the extreme structural response at the ultimate strain of steel or concrete, whichever is reached first. However, if the ductility capacity is considered at the ultimate strain, it is equivalent to allowing for structural collapse at the seismic intensity of design earthquakes.
In the absence of a safety margin, this would impose a high risk to life safety.
Therefore, it is favorable to design bridge structures to withstand small repairable cracks when the displacement ductile capacity is reached. To achieve this goal, a reduction of the ultimate material strain is considered in the estimation of the displacement ductility capacity [13]. The performances at various stages of seismic levels with their corresponding allowable displacement ductility ranges, as proposed by UCSD [14], are shown in Table 4-1. The suggested ductility range refers to the definition of repairable bridges proposed for the transit system in the San Francisco Bay Area, BART [15]. For BART, a reduction factor of 0.67 is suggested for the upper bound of the ultimate strain addressing the structural displacement ductility, and a reduction
factor of 0.5 is suggested as the lower bound to maintain an economical design.
Therefore, in this study, the ultimate strains of steel and concrete are respectively considered to be
R