Simplified Shear Design of Structural Concrete Members: Appendixes (2005) / Chapter Skim
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A-1 Appendix A: Models for Shear Behavior A.1 Introduction A.1.1 The Problem of Shear Transfer A flexural member supports loads by internal moments and shear forces. Classical beam theory, in which plane sections are assumed to remain plane, provides an accurate, simple, and effective model for designing a member to resist bending in combination with axial forces.
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A-2 Interface shear transfer: Local roughness in the crack plane provides resistance against slip and thus there is shear transfer across shear cracks. The contribution of interface shear transfer to shear strength is a function of the crack width and aggregate size.
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A-3 principal tensile stresses. Draping the prestressing tendons did not delay the formation of inclined cracks that developed out of flexural cracks.
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A-4 Size Effect: The shear strength of reinforced and prestressed beams without shear reinforcement decreases as the member depth increases; this is called the "size effect" in shear. Both the tests by Kani on size effect in 1967, and the tests by Shioya et al.
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A-5 height crack spacings are smaller and that improves shear strength significantly (Collins and Kuchma, 1999)
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A-6 The shear failure modes of beams without shear reinforcement were also discussed by ASCEACI Committee 426 (1973)
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A-7 compression zone, the shear transfer actions discussed previously for beams without shear reinforcement do not play a dominant role. For these reasons, the terms for the shear strength provided by the concrete are differentiated as follows: - Vct for members without transverse reinforcement, in order to indicate that the failure is governed by the concrete tensile strength, and - Vc for members with transverse reinforcement.
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A-8 available experimental results, the ACI-ASCE shear committee (1962) recommended the use of an empirical expression for the shear stress at inclined cracking as the shear failure load.
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A-9 A.2.2 45o Truss Model Truss models were widely used to understand the shear behavior of reinforced concrete beams in the early 1900's. Ritter (1899)
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A-10 ) cot(tan cossin 1 2 θθθθ +== jdb V jdb Vf ww (A-4a)
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A-11 From Fig.
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A-12 As shown in Fig.
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A-13 A.2.5 Modified Compression Field Theory The tensile stresses in cracked concrete provide significant shear resistance. The modified compression field theory (MCFT)
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A-14 width. A limit on civ was proposed by Vecchio and Collins (1986)
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A-15 ) (tan 63.0 243.0 16.2 tan ' 1 inandpsi a w f vf cci θθ ++ =≤ (A-27)
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A-16 Compatibility relationships can be derived from Mohr's strain circle. These relationships can simply be obtained from the Eqs.
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A-17 The principal tensile stress, dσ , predicted by Eq.
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A-18 tt ccc t fρφφτφσφσσ +−+= cossin2cossin 212122 (A-33b)
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A-19 If the crack angle coincides with the d -r coordinates, the concrete shear stress term, c21τ , vanishes and Eq.
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A-20 The "truss model with crack friction" for members with transverse reinforcement is such a failure mechanism approach and is a "discrete" approach with respect to cracking. It is based on the research of dei Poli et al.
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A-21 Therefore, the web must provide the following resisting shear force: VRd,web = Vswd + Vfd ≥ VSd,web (A-41) The shear force component Vswd carried by the vertical stirrups across the inclined crack at the ultimate capacity is given by: Vswd = (Asw /sw)
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A-22 the-art report on interface shear has been provided by Gambarova and di Prisco (1991)
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A-23 In the case of axial tension, the cracks may be steeper than 45° and the strain εx is positive. In the FIP Recommendations the following relations are given for members with axial tension: cotβr = 1.20 - 0.9 σxd /fctm ≥ 0 (A-45a)
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A-24 or zf s A V ywd w webSw webSd ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= , ,cotθ (A-46b) with VRd,web = Vswd + Vfd ≥ VSd,web This means that the angle θ varies with the magnitude of the applied shear force, i.e.
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A-25 or interface shear is modeled explicitly. The approach gives directly the shear capacity and the required amount of transverse reinforcement.
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A-26 However, the basic difference is that while the Vc term in the AASHTO Standard Specification and ACI 318 is assumed to be the shear force at diagonal cracking, the Vc term for the "truss-model with crack friction" is a function of the shear that can be transferred by friction across the inclined crack in a B-region. While the two Vc terms may have similar values the basic concepts associated with each are fundamentally different with the Vc term of the "truss-model with crack friction usually being less than the shear force at diagonal cracking.
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A-27 Figure A-1 Shear Transfer/Actions Contributing to Shear Resistance Figure A-2 Distribution of Internal Shear Resistance (ASCE-ACI Committee 426, 1973) d cot θ V : Shear in Compression Zonecc V : Dowel Action V : Aggregate Interlock V : Residual Tensile Stress in Concrete V : Steel Contribution V : Vertical Component of Prestressing Steel d ca cr s p Vsupport d V = A fs s v Vcr Vd Vca Vp θ Vd Vcc
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A-28 Figure A-3 Response of Concrete in Uniaxial Tension (Gopalaratnam and Shah, 1985) Figure A-4 Influence of Concrete Compressive Strength on Shear Strength (Kuchma and Kim, 2002)
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A-30 (a) Beam Deep Very Short Slender Very slender Short Flexural capacity Failure Inclined cracking and failure 1.0 2.5 6.5 a/d (b)
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A-31 Figure A-8 Diagonal Tension Failure (ASCE-ACI 426, 1973) Loss of bond due to splitting crack a)
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A-32 (a) Cross section V M M = 0 s 0.5V 0.5V jd / 2 A fv v f2 f2 s/ 2 f2 V 45o jd bw Av s (b)
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A-33 fsy fcy fsx fcxv Es fc| L |c 1 ρ fsxx f2 y v x ρ fsyv 12 2θ y x 0.5γm 12 2θ L 1 L 2 L x L y L x L 1 L 2 L y v γ/2 Initial cracks Later cracks fs f2 L s fy L 2 f2max (a) Free Body Diagram (b)
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A-34 v f2 y v x 12 2θcr v (c) Free Body Diagram (d)
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A-35 fcr 1 M Equilibrium Equations Es L s fs L Es s } (l) Average Stress-Strain Curve of Steel Bars in Concrete Reinforcement in Concrete: fs fy r (a)
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A-36 Figure A-19 The Approach by Mörsch (1909,1922) for Shear Design a)
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A-37 a) reinforced concrete members b)
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A-38 Figure A-23 Dimensioning Diagram for Vertical Stirrups for Prestressed Concrete Members and Members with Axial Compression a) truss action b)
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A-39 a) truss model with uniaxial compression field in B-region b)
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