Reinforced Concrete II Hashemite University The Hashem ite University Departm ent of Civil Engineering Lecture 6 Lecture 6 – – Analysis and Analysis and Design for Torsion Design for Torsion Dr. Hazim Dwairi Dr Hazim Dwairi Dr Hazim Dwairi Dr. Hazim Dwairi Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Torsion in Pla in Concrete Torsion in Pla in Concrete Mem bers Mem bers • Torsion in circular members Torsion in circular members ρ T τ = I p T : Applied Torque ρ ρ : : Radial Radial Distance Distance I : Polar Moment of Inertia p Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 1
Reinforced Concrete II Hashemite University Torsion in Pla in Concrete Torsion in Pla in Concrete Mem bers Mem bers • Torsion in rectangular members Torsion in rectangular members – The largest stress occurs at the middle of the wide face a . The largest stress occurs at the middle of the wide face “a” The largest stress occurs at the middle of the wide face a . The largest stress occurs at the middle of the wide face “a” – The stress at the corners is zero. The stress at the corners is zero. – Stress distribution at any other location is less than that at the Stress distribution at any other location is less than that at the middle and middle and – greater than zero. greater than zero. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Torsion in Pla in Concrete Torsion in Pla in Concrete Mem bers Mem bers • Torsion in rectangular members Torsion in rectangular members T τ = α max 2 b a ∞ a/b 1.0 1.5 2.0 3.0 5.0 α 0.208 0.219 0.246 0.267 0.290 1/3 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 2
Reinforced Concrete II Hashemite University Cracking Strength Cracking Strength Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Thin Walled Tube Analogy Thin Walled Tube Analogy • The design for torsion is based on a thin walled The design for torsion is based on a thin walled tube, space truss analogy. A beam subjected to tube, space truss analogy. A beam subjected to tube space truss analogy A beam subjected to tube space truss analogy A beam subjected to torsion is idealized as a thin torsion is idealized as a thin- -walled tube with the walled tube with the core concrete cross section in a solid beam is core concrete cross section in a solid beam is neglected. neglected. Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 3
Reinforced Concrete II Hashemite University Thin Walled Tube Analogy Thin Walled Tube Analogy V 1 =V =V 3 , V , V 2 2 =V =V 4 4 According to thin walled theory: According to thin walled theory: q = V q = V 1 /x /x o =V =V 2 2 /y y o o =V =V 2 /x /x o =V =V 4 /y y o q = shear force/unit length q = shear force/unit length q = shear flow = constant q = shear flow = constant q = τ. τ. t � q = τ = shear stress = shear stress Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Thin Walled Tube Analogy Thin Walled Tube Analogy Take moment about centroid Take moment about centroid: : T T = (V 1 +V T T = (V (V (V +V 3 V ) 3 )y o o /2 + (V /2 /2 + (V 2 +V /2 (V (V +V 4 )x V ) )x o /2 /2 /2 /2 T = 2V T = 2V 1 y o o /2 + 2V /2 + 2V 2 x o /2 /2 Recall: V Recall: V 1 1 = = q.x q.x o & V & V 2 2 = = q.y q.y o o T = 2V T = 2V 1 x o o y o o /2 + 2V /2 + 2V 2 y o x o /2 /2 T = 2qA o T = 2qA q q o � τ = T/(2A = T/(2A o o t) t) Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 4
Reinforced Concrete II Hashemite University Threshold Torsion Threshold Torsion • Torques that do not exceed approximately one Torques that do not exceed approximately one- - quarter of the cracking torque quarter of the cracking torque T cr quarter of the cracking torque quarter of the cracking torque T cr will not cause a will not cause a will not cause a will not cause a structurally significant reduction in either the structurally significant reduction in either the flexural or shear strength and can be ignored. flexural or shear strength and can be ignored. • Cracking is assumed to occur when the principal Cracking is assumed to occur when the principal tensile stress reaches . In a tensile stress reaches . In a nonprestressed nonprestressed ' 0 . 33 f c beam loaded with torsion alone, the principal beam loaded with torsion alone the principal beam loaded with torsion alone the principal beam loaded with torsion alone, the principal tensile stress is equal to the tensile stress is equal to the torsional torsional shear stress, shear stress, τ τ . Recall that: . Recall that: T τ = 2 A t o Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Threshold Torsion Threshold Torsion According to ACI - R11.6.1 A A 2 3 = = cp A A ; t o cp 3 4 p cp = = + where : A xy ; p 2 ( x y ) cp cp Substituti ng values of A and t o ⎛ ⎛ ⎞ ⎞ 2 A A ⎜ ⎜ ⎟ ⎟ ∴ = cp ' T 0 . 33 f ⎜ ⎟ cr c p ⎝ ⎠ cp φ T ≤ cr Neglect Torsion wh en T u 4 Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 5
Reinforced Concrete II Hashemite University Torsion in Torsion in Reinforced Reinforced Concrete Concrete Mem bers Mem bers Mem bers Mem bers x & y distance from o o CL to CL of stirrup Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Reinforcem ent Requirem ent for Reinforcem ent Requirem ent for RC Mem bers in Torsion RC Mem bers in Torsion • Reinforcement is determined using space truss Reinforcement is determined using space truss analogy analogy analogy. analogy. • In space truss analogy, the concrete compression In space truss analogy, the concrete compression diagonals (struts), vertical stirrups in tension (ties), diagonals (struts), vertical stirrups in tension (ties), and longitudinal reinforcement (tension chords) and longitudinal reinforcement (tension chords) act together as shown in figure on the next slide. act together as shown in figure on the next slide. • The analogy derives that • The analogy derives that The analogy derives that torsional The analogy derives that torsional torsional shear stress will torsional shear stress will shear stress will shear stress will be resisted by the vertical stirrups as well as by be resisted by the vertical stirrups as well as by the longitudinal steel the longitudinal steel Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 6
Reinforced Concrete II Hashemite University Reinforcem ent Requirem ent for Reinforcem ent Requirem ent for RC Mem bers in Torsion RC Mem bers in Torsion Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Vertical Stirrup Reinforcem ent Vertical Stirrup Reinforcem ent T = τ = Recall : q t 2 A o θ T y cot = = = = o V qy y nA f A f = 4 o o t yv t yv n number of stirrups p crossing g 2 2 A A s s o θ diagonal crack y cot 2 A = o o T A f = A Area of one leg stirrup t yv s y t o = f Yield strength of shear yv A f A f = θ = θ t yv t yv T 2 x y cot 2 A cot reinforcem ent n o o oh s s Dr. Hazim Dwairi Dr. Hazim Dwairi The Hashemite University The Hashemite University Reinforced Concrete II Reinforced Concrete II Dr. Hazim Dwairi 7
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