Chapter 7 Shear Resistance
Chapter 7 Shear Resistance
7.1 Shear Failure of RC Beams without Shear Reinforcement 7.1.1 Shear in homogeneous beams In mechanics of materials,the shear stress of homogeneous各向同性，elastic and uncracked beams can be calculated by S T二 Ib where -shear force S-first moment of area (S=4
7.1 Shear Failure of RC Beams without Shear Reinforcement 7.1.1 Shear in homogeneous beams • In mechanics of materials, the shear stress of homogeneous各向同性, elastic and uncracked beams can be calculated by VS I b τ where V - shear force S – first moment of area ( S A y )
Load My S O= t= I Ib 回萨， , +x2 2 4 (a) Reaction (b) 2 +x2 中 白 (c) Fig.7.1-1 Normal,shear and principal stresses in homogeneous uncracked beam.(a)Flexural and shear stresses acting on elements in shear span;(b)distribution of shear stress;(c) principal stresses on elements in shear span
(a) (b) (c) Fig. 7.1-1 Normal, shear and principal stresses in homogeneous uncracked beam. (a) Flexural and shear stresses acting on elements in shear span; (b) distribution of shear stress; (c) principal stresses on elements in shear span I My 2 2 1 2 4 2 2 2 2 4 - ) 2 arctan( 2 1 - VS I b τ
Crack ys flexural and shear stresses for a RC beam neutral axis (a (b) (c) 45°<0<90° 5 (d) (e) (D Fig.7.1-3 Orientation of principal stresses.(a)Geometry and loading;(b)axial stress at section x-x;(c)shear stress distribution at section x-x;(d)segment 4;(e)segment B;(f) segment C;(g)principal compressive stress paths
• Crack vs flexural and shear stresses for a RC beam Fig. 7.1-3 Orientation of principal stresses. (a) Geometry and loading; (b) axial stress at section x-x; (c) shear stress distribution at section x-x; (d) segment A; (e) segment B; (f) segment C; (g) principal compressive stress paths
7777777 Fig.7.1-2 Principal compressive stress trajectories in an uncracked beam
Fig. 7.1-2 Principal compressive stress trajectories in an uncracked beam
Crack patterns follow the principal compressive stress paths Stresses in different segments Segment 4 (at the bottom): =0 and =Omax Vertical flexural crack >Segment B(at the neutral axis): =t max and o=0 45-diagonal crack. >Segment C(between the N.A.and the bottom) Combination of t and o 45°-90°diagonal crack
ü Crack patterns follow the principal compressive stress paths ü Stresses in different segments Ø Segment A (at the bottom): τ = 0 and = max Vertical flexural crack. Ø Segment B (at the neutral axis): τ = τ max and = 0 45-diagonal crack. Ø Segment C (between the N.A. and the bottom): Combination of τ and 45-90 diagonal crack
7.1.2 Types of shear failure The type of shear failure of RC beams depends mainly on various factors: >Shear span-to-depth ratio(剪跨比) Quantity of longitudinal reinforcement ratio p=4./(bho) >Load configuration(荷载型式：集中、分布) Geometry of the beam,etc
7.1.2 Types of shear failure • The type of shear failure of RC beams depends mainly on various factors: Ø Shear span-to-depth ratio(剪跨比） Ø Quantity of longitudinal reinforcement ratio Ø Load configuration（荷载型式：集中、分布） Ø Geometry of the beam, etc. A /(bh0) s
One of the most significant factors is the shear-span/depth ratio(剪跨比)a,/ho ,Shear span a,is defined as the distance between points of zero and maximum moments. d moment moment zero zero L moment maximum (a) (b) Fig.7.1-4 Shear span. (a)Geometry and loading;(b)bending moment
• One of the most significant factors is the shear-span/depth ratio（剪跨比） , Shear span av is defined as the distance between points of zero and maximum moments. a / h0 v Fig. 7.1-4 Shear span. (a) Geometry and loading; (b) bending moment
·The effective shear-span/depth ratio(广义剪跨比)is defined as M/Th,where M is the bending moment and V the shear force.This is true for both distributed loading or concentrated loading. The failure mode is strongly dependent on the shear-span/depth ratio a./ho Why? 广义剪跨比实质上反映了截面上正应力与剪应力之间的相对关系， 近而决定了主应力的大小和方向，也就影响了斜截面受剪承载力和 破坏形态。 The shear-span/depth ratio can be divided into four general categories.For different members falling into the same category, the sequence of events and the nature of the failure are approximately the same
• The effective shear-span/depth ratio (广义剪跨比）is defined as , where M is the bending moment and V the shear force. This is true for both distributed loading or concentrated loading. M /Vh0 • The failure mode is strongly dependent on the shear-span/depth ratio a / h0 v The shear-span/depth ratio can be divided into four general categories. For different members falling into the same category, the sequence of events and the nature of the failure are approximately the same. Why? 广义剪跨比实质上反映了截面上正应力与剪应力之间的相对关系， 近而决定了主应力的大小和方向，也就影响了斜截面受剪承载力和 破坏形态
Category Shear span/depth ratio Mode of failure Category I a,<1 Diagonal compression failure ho (斜压破坏) Shear-compression failure Category II ≤3 ho (剪压破坏) 2<6 Diagonal tension failure) Category III 3< ho (斜拉破坏) Category IV Flexural failure ho
Category Shear span/depth ratio Mode of failure Category I Category II Category III Category IV 1 0 h a v 1 3 0 h a v 3 6 0 h av 0 6 h a v Diagonal compression failure Shear-compression failure (Diagonal tension failure) Flexural failure （斜压破坏） （剪压破坏） （斜拉破坏）