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Analytical modeling of monolithic joints in concrete bridges

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Abstract

The behavior of bridge monolithic connections is modeled using a simplified mathematical model that accounts for stress equilibrium, compatibility of deformations, and the state of bond of longitudinal column bars anchored through the joint panel. In this regard, a stress gradient factor is introduced, to model the profile of bar stresses along the anchorage. To establish this factor, two independent mechanisms of stress transfer are considered: a bond mechanism between the anchored bars and the surrounding concrete and a friction mechanism between the anchored bars and the transverse bars that enclose and restrain the anchorages. The model is used for calculation of the shear stress–shear strain relationship of all tests found in the international literature on bridge monolithic connections that showed shear type of failure under simulated seismic loading. Joint strength values calculated with the proposed model are compared with the experimental results. Based on this comparison the proposed model is verified for use in interpretation of bridge monolithic connection behavior and design.

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Abbreviations

Ac :

Area of the column cross-section,

bb, bc :

Beam and column cross section width, in the plane of action considered,

bj :

Effective joint width according to EN 1998-2 (2005),

db, dbt :

Bar size of column longitudinal reinforcement and of additional joint horizontal reinforcement, respectively,

dc :

Column diameter in case of circular columns,

dfcontact, dfdistr :

Stress jump in the column longitudinal reinforcement occurring at points of contact with a transverse bar and in the segments between successive stirrups, respectively,

\({{\rm E}_{\rm c}^{0}}\) :

Initial (tangent) modulus of elasticity of concrete,

Es, Esh :

Initial modulus of elasticity of steel and modulus of elasticity at the strain hardening plastic phase of steel,

fcm, fctm :

Mean compressive and tensile strengths of concrete,

fb, fb,max and fb,ave :

Bond stress, bond strength (maximum bond stress) calculated according with EN 1992-1 (2005) and average bond stress along the elastic part of the anchorage,

fbs, fcs, fjx and fjz :

Average stresses of beam longitudinal, column longitudinal, additional joint horizontal and additional joint vertical reinforcement, defined at the joint center; superscript y identifies the yield stress of the associated reinforcement,

fo and fm :

Maximum attainable stress at the anchorage of the column longitudinal reinforcement, measured at the joint-column interface (beginning of the anchorage) and at mid-height of the joint, respectively,

fx, fz :

Average reinforcement stresses at the joint center in the x- and z-directions,

\({{\rm f}_{\rm x}^{\rm f}, {\rm f}_{z}^{\rm f}}\) :

Stresses of the effective joint reinforcement in the x- and z-directions, at joint failure,

\({{\rm f}_{\rm x}^{\rm y}, {\rm f}_{\rm z}^{\rm y}}\) :

Stresses of the effective joint reinforcement in the x- and z-directions, at first joint yielding,

fy, fty :

Yield strength of primary column and additional joint horizontal reinforcement, respectively,

Fy :

Estimated yield force of the test specimens,

hb, hc :

Beam and column cross section height, in the plane of action considered,

Ls, Ly and Le =  L s − L y :

Anchorage length, length of yielding penetration of the column longitudinal reinforcement inside the joint and length of the anchorage that remains elastic after yielding penetration,

n:

Number of the extreme longitudinal column bars that are anchored inside the joint,

N:

Number of the transverse bar legs that enclose and restrain the anchorage of the longitudinal column bars inside the joint at each horizontal joint reinforcement layer,

Nx :

The compressive beam axial load,

Nz =  0.5Nc :

Half the compressive column axial load for T- or knee-joints in bridge superstructures or for column-to-footing joints in bridge foundations,

nx =  Nx/(bjhb) and nz =  Nz/(bjhc):

Externally applied axial compressive stresses on the joint boundaries along the x and z directions, respectively,

s:

Distance between two subsequent layers of additional joint horizontal reinforcement,

s1, s2, s3 :

Characteristic slip values of the bond-slip tri-linear diagram,

v:

Average joint shear stress; subscripts cr, y, and f identify cracking, yielding and failure of the joint (shear strength); subscripts anal and exp identify analytical calculated and experimental recorded values,

\({{\rm v}_{\rm y}^{1}, {\rm v}_{\rm y}^{2}}\) :

The average joint shear stress at first joint yielding, related with yielding of horizontal or vertical joint reinforcement, respectively,

\({{\rm v}_{\rm f}^{1}, {\rm v}_{\rm f}^{2}, {\rm v}_{\rm f}^{3}, {\rm v}_{\rm f}^{4}, {\rm v}_{\rm f}^{5}}\) :

The average joint shear stress at joint failure, related with failure because of premature concrete crushing, yielding of vertical joint reinforcement after yielding of horizontal joint reinforcement, concrete crushing after yielding of horizontal joint reinforcement, yielding of horizontal joint reinforcement after yielding of vertical joint reinforcement, concrete crushing after yielding of vertical joint reinforcement, respectively,

X and Y:

Lengths between distinct points of the two joint diagonals used for the calculation of the joint distortion,

xpl :

Length at the elastic part of the anchorage along which bond is in the plastic range,

za, zy :

Required number of transverse bars in the yielded portion and in the elastic part of the anchorage, respectively, so that the anchorage can develop a maximum bar stress fo,

β x, β z :

Bond factors that relate the magnitude of maximum attainable stress in the longitudinal beam and column reinforcement, respectively, to their associated stress at the beam-joint interface and the joint-column interface, respectively. The maximum attainable stress is measured at the mid-height of the column and at the mid-height of the beam, respectively. Superscripts y and f (i.e. \({\beta_{\rm x}^{\rm y}}\) and \({\beta_{\rm x}^{\rm f}}\)) identify the values of β x or β z at the point of first joint yielding and at joint failure, respectively.

γ :

Shear distortion of the joint panel; subscripts cr, y, f and max identify joint cracking, yielding, failure and maximum recorded, respectively; subscripts anal and exp identify analytical calculated and experimental recorded values,

\({\gamma_{\rm y}^{1}, \gamma_{\rm y}^{2}}\) :

Shear distortion of joint panel at first joint yielding, related with yielding of horizontal or vertical joint reinforcement, respectively,

\({\gamma_{\rm f}^{1}, \gamma_{\rm f}^{2}, \gamma_{\rm f}^{3}, \gamma_{\rm f}^{4}, \gamma_{\rm f}^{5}}\) :

Shear distortion of joint panel at joint failure, related with failure because of premature concrete crushing, yielding of vertical joint reinforcement after yielding of horizontal joint reinforcement, concrete crushing after yielding of horizontal joint reinforcement, yielding of horizontal joint reinforcement after yielding of vertical joint reinforcement, concrete crushing after yielding of vertical joint reinforcement, respectively,

δ y :

Experimentally measured displacement of the test specimens at the point of estimated yield stress,

ΔX and ΔY:

Length change between distinct points of the two joint diagonals used for the calculation of joint distortion,

\({\varepsilon_{\rm o} = 2{\rm f}_{\rm cm}/{\rm E}_{\rm c}^{0}}\) :

Unconfined concrete crushing strain,

\({\varepsilon_{\rm y}, \varepsilon_{\rm plat}, \varepsilon_{\rm f}}\) :

Yield, maximum plateau and fracture strain of reinforcing steel of main column bars,

\({\varepsilon_{\rm x}^{\rm y} = {\rm f}_{\rm x}^{\rm y}/{\rm E}_{\rm s}}\) and \({\varepsilon_{\rm z}^{\rm y} = {\rm f}_{\rm z}^{\rm y}/{\rm E}_{\rm s}}\) :

Yield strain of the effective joint reinforcement in the x- and z- directions, respectively,

η = Es/Ec :

Ratio of material modulus of elasticity,

η 2 =  max{1.0; (132 − db)/100}:

Coefficient that takes into account the adverse effect on bond strength of large diameter bars (db >  32 mm), according with EN 1992-1 (2005),

θ :

Angle of the principal tensile stress σ 1 measured from the x-axis,

μ i,el, μ i,pl :

Friction coefficient at points of contact between transverse and main column bars on the elastic part of the anchorage and over the length of yield penetration, respectively,

ρ b, ρ c, ρ jx and ρ jz :

Area ratios of beam longitudinal, column longitudinal, additional joint horizontal and additional joint vertical reinforcement, respectively,

ρ x, ρ z :

The effective reinforcement ratios in the x- and z-directions,

σ x, σ z :

Normal concrete stresses in the joint, along the x- and z-directions,

\({\omega = [4 \cdot {\rm f}_{{\rm b},{\rm max}}/({\rm E}_{\rm s} \cdot {\rm d}_{\rm b} \cdot {\rm s}_{1})]^{0.5}}\) :

Bond constant that characterizes each anchorage.

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Authors and Affiliations

  1. Department of Civil Engineering, Demokritus University of Thrace, Xanthi, 67100, Greece

    D. Timosidis & S. J. Pantazopoulou

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  1. D. Timosidis
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  2. S. J. Pantazopoulou
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Correspondence to D. Timosidis.

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Timosidis, D., Pantazopoulou, S.J. Analytical modeling of monolithic joints in concrete bridges. Bull Earthquake Eng 7, 411–438 (2009). https://doi.org/10.1007/s10518-008-9102-5

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