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Numerical and Analytical Behavior of Beams Prestressed with Unbonded Internal or External Steel Tendons: A State-of-the-Art Review

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Abstract

This paper presents a detailed state-of-the-art chronological account of past studies dating back to the late 1980s consisting of an overview of the vast majority of prediction equations for the stress at ultimate in prestressed unbonded internal or external steel tendons of simply supported concrete beams. The paper also details past studies carried out to investigate the complete nonlinear behavior of such beams throughout loading. Past and relevant studies were carefully reviewed in detail and then synthesized in the most reduced, yet accurately depicted, format in an effort to capture the pertinent highlights. For each review, the most relevant factors considered by the authors were highlighted, including span-to-depth ratio, second-order effects, length of the plastic hinge, friction and/or slippage at deviators, and experimental verification. An evaluation was carried out to compare and contrast the various nonlinear analysis studies using a simple tabulated format. Gaps were identified, and recommendations for future research were further proposed by the authors.

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Courtesy of Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

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Abbreviations

A ps :

Area of prestressed unbonded reinforcement

A s :

Area of non-prestressed tensile reinforcement

b :

Beam width of a rectangular section

c :

Depth of neutral axis at ultimate nominal strength

d :

Distance from extreme compressive fiber to the centroid of the non-prestressed reinforcement

d ps :

Depth from the extreme to compressed fiber to the centroid of the pretsressing steel

e :

Eccentricity of the unbonded prestressed reinforcement at any point along the beam

e m :

Eccentricity of the prestressing steel at midspan

e s :

Eccentricity of the prestressing steel at the support

E ps :

Modulus of elasticity of the prestressing steel

f c :

Stress in the concrete top fiber of a simply supported beam

f pe :

Effective prestress in unbonded tendons

f ps :

Stress in the unbonded prestressed reinforcement at nominal strength

f pu :

Characteristic (ultimate) strength of the prestressed reinforcement

f py :

Yield stress of unbonded tendons

f s :

Stress in the non-prestressed tensile reinforcement

f y :

Yield stress in the non-prestressed reinforcement

f′ c :

Concrete compressive strength

FDM:

Finite difference method

FEM:

Finite element method

FRP:

Fiber-reinforced polymers

GIA:

Generalized iterative analysis

I cr :

Cracked moment of inertia

I t :

Transformed moment of inertia

L :

Length of the simply supported beam

L a :

Length of the region of constant moment [17]

L o :

Length of the equivalent plastic region [42]

L p :

Length of the equivalent plastic hinge [54, 64]

L t :

Length of the undeformed external tendon between end anchorages [64]

L 1 :

Length of the undeformed inclined external tendon between end anchorage and deviator [64]

M ext :

External moment at any point x along the span

M int :

Internal moment at x along the span calculated from force and moment equilibrium

M n :

Moment at ultimate nominal strength

M pl :

Plastic moment calculated excluding unbonded prestressing force at ultimate

P :

Concentrated load at midspan

P/2:

Concentrated load at the third point

R s :

Stress increment reduction factor responsible for second-order effects [58]

S d :

Spacing of the deviators [43, 64]

S e :

Elastic stiffness matrix

S g :

Geometric stiffness matrix

S t :

Tangent stiffness matrix

T P :

A ps x f ps

x :

Distance measured from simply supported end up to deviator at the third point [64]

Z :

Shear span or the distance between the point of maximum moment and the point of contra flexure

Δ:

Midspan deflection

Δf ps :

Stress increase in the unbonded tendons beyond effective prestress

ΔM :

Midspan deflection [54]

Δε p :

Average elongation in the unbonded external tendon at ultimate [64]

Δε ps :

Average elongation in the unbonded tendon at a specific loading stage

ε cu :

Strain at the concrete top fiber of a simply supported beam at ultimate

ε ps :

Strain in the prestressed reinforcement obtained from the stress–strain curve

β 1 :

A fraction between 0.65 and 0.85 used by the ACI Code to compute the width of the Whitney stress block

Φcr :

Beam curvature assuming an elastic cracked section

Φpy :

Beam curvature assuming yielding of the prestressed reinforcement

Φu :

Beam curvature assuming ultimate nominal strength

Φy :

Beam curvature assuming yielding of the non-prestressed reinforcement

Ω:

Strain reduction coefficient of unbonded to bonded tendons

Ωcr :

Strain reduction coefficient assuming elastic cracked analysis

Ωu :

Strain reduction coefficient assuming ultimate nominal strength

ζ :

Shape function that varies between 0 and 1 derived from experimental tests [64]

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Alqam, M., Alkhairi, F. Numerical and Analytical Behavior of Beams Prestressed with Unbonded Internal or External Steel Tendons: A State-of-the-Art Review. Arab J Sci Eng 44, 8149–8170 (2019). https://doi.org/10.1007/s13369-019-03934-3

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