ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture twenty one concrete construction: Copyright Kirk Martini shear & deflection Concrete Shear 1
Shear in Concrete Beams flexure combines with shear to form diagonal cracks horizontal reinforcement doesn t help stirrups = vertical reinforcement Concrete Shear 2
ACI Shear Values V u is at distance d from face of support shear capacity: where b w means thickness of web at n.a. V c = c b w d Concrete Shear 3
ACI Shear Values shear stress (beams) = 0.75 for shear c 2 f c f c is in psi V c 2 b shear strength: V V V u d V s is strength from stirrup reinforcement f c c w s Concrete Shear 4
Stirrup Reinforcement shear capacity: V s = A vf y d s A v = area in all legs of stirrups s = spacing of stirrup may need stirrups when concrete has enough strength! Concrete Shear 5
Required Stirrup Reinforcement spacing limits 0.75 Concrete Shear 6
Torsional Stress & Strain can see torsional stresses & twisting of axi-symmetrical cross sections torque remain plane undistorted rotates not true for square sections... T T Concrete Shear 7
Shear Stress Distribution depend on the deformation = angle of twist measure can prove planar section doesn t distort Concrete Shear 8
Shearing Strain related to L is the radial distance from the centroid to the point under strain shear strain varies linearly along the radius: max is at outer diameter Concrete Shear 9
Torsional Stress - Strain know so f v G G L and L where G is the Shear Modulus Concrete Shear 10
Torsional Stress - Strain from T ( ) A can derive T J where J is the polar moment of inertia elastic range T J Concrete Shear 11
Shear Stress max happens at outer diameter combined shear and axial stresses maximum shear stress at 45 twisted plane Concrete Shear 12
Shear Strain knowing G L and T J solve: TL JG composite shafts: i T L J i i i G i Concrete Shear 13
Noncircular Shapes torsion depends on J plane sections don t remain plane max is still at outer diameter max c 1 T ab 2 c 2 TL ab 3 G where a is longer side (> b) Concrete Shear 14
Open Thin-Walled Sections with very large a/b ratios: max 1 3 T ab 2 1 3 TL ab 3 G Concrete Shear 15
Shear Flow in Closed Sections q is the internal shear force/unit length a TL a 4t T ta 2 2 i s t i i is the area bounded by the centerline s i is the length segment, t i is the thickness Concrete Shear 16
Shear Flow in Open Sections each segment has proportion of T with respect to torsional rigidity, Concrete Shear 17 max total angle of twist: 1 3 G 1 3 Tt TL max b i t i b i t i 3 I beams - web is thicker, so max is in web 3
Torsional Shear Stress twisting moment and beam shear Concrete Shear 18
Torsional Shear Reinforcement closed stirrups more longitudinal reinforcement area enclosed by shear flow Concrete Shear 19
Development Lengths required to allow steel to yield (f y ) standard hooks moment at beam end splices lapped mechanical connectors Concrete Shear 20
Development Lengths l d, embedment required both sides proper cover, spacing: No. 6 or smaller Concrete Shear 21 l No. 7 or larger l d d d 25 d 20 b b F F y f y f c c or 12 in. minimum or 12 in. minimum
Development Lengths hooks bend and extension minimum l dh d 1200 f c b Concrete Shear 22
Development Lengths bars in compression 0. 02d F l b y 0. 0003d d f splices tension minimum is function of l d and splice classification compression minimum is function of d b and F y c b F y Concrete Shear 23
Concrete Deflections elastic range I transformed E c (with f c in psi) normal weight concrete (~ 145 lb/ft 3 ) concrete between 90 and 160 lb/ft 3 cracked I cracked E adjusted E c = 57,000 f c E c = w c 1.5 33 f c Concrete Shear 24
Deflection Limits relate to whether or not beam supports or is attached to a damageable nonstructural element need to check service live load and long term deflection against these L/180 roof systems (typical) live L/240 floor systems (typical) live + long term L/360 supporting plaster live L/480 supporting masonry live + long term Concrete Shear 25