Lecture 17 Design of Reinforced Concrete Beams for Shear

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Lecture 17 - Design of Reinforced Concrete Beams
for Shear

• November 1, 2001
• CVEN 444

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Lecture Goals

• Stirrup Design

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Uncracked Elastic Beam Behavior
Look at the shear and bending moment diagrams.
The acting shear stress distribution on the beam.
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Uncracked Elastic Beam Behavior
The acting stresses distributed across the
cross-section.
The shear stress acting on the rectangular beam.
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Uncracked Elastic Beam Behavior
The equation of the shear stress for a
rectangular beam is given as
Note The maximum 1st moment occurs at the
neutral axis (NA).
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Uncracked Elastic Beam Behavior
The ideal shear stress distribution can be
described as
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Uncracked Elastic Beam Behavior
A realistic description of the shear distribution
is shown as
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Uncracked Elastic Beam Behavior
The shear stress acting along the beam can be
described with a stress block
Using Mohrs circle, the stress block can be
manipulated to find the maximum shear and the
crack formation.
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Inclined Cracking in Reinforced Concrete Beams
Typical Crack Patterns for a deep beam.
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Inclined Cracking in Reinforced Concrete Beams
Flexural-shear crack - Starts out as a flexural
crack and propagates due to shear
stress. Flexural cracks in beams are vertical
(perpendicular to the tension face).
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Inclined Cracking in Reinforced Concrete Beams
For deep beam the cracks are given as The shear
cracks Inclined (diagonal) intercept crack
with longitudinal bars plus vertical or inclined
reinforcement.
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Inclined Cracking in Reinforced Concrete Beams
For deep beam the cracks are given as The shear
cracks fail due two modes - shear-tension
failure - shear-compression failure

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Shear Strength of RC Beams without Web
Reinforcement
vcz - shear in compression zone va - Aggregate
Interlock forces vd Dowel action from
longitudinal bars Note vcz increases from (V/bd)
to (V/by) as crack forms.
Total Resistance vcz vay vd (when no
stirrups are used)
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Strength of Concrete in Shear (No Shear
Reinforcement)
(1) Tensile Strength of concrete affect
inclined cracking load
(2) Longitudinal Reinforcement Ratio, rw
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Strength of Concrete in Shear (No Shear
Reinforcement)
(3) Shear span to depth ratio, a/d (M/(Vd))

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Strength of Concrete in Shear (No Shear
Reinforcement)
(4) Size of Beam Increase Depth
Reduced shear stress at inclined cracking
(5) Axial Forces - Axial tension
Decreases inclined cracking load - Axial
Compression Increases inclined cracking
load (Delays flexural cracking)
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Function and Strength of Web Reinforcement
Web Reinforcement is provided to ensure that the
full flexural capacity can be developed.
(desired a flexural failure mode - shear failure
is brittle) - Acts as clamps to keep shear
cracks from widening
Function
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Function and Strength of Web Reinforcement

• Uncracked Beam Shear is resisted
  uncracked concrete.
• Flexural Cracking Shear is resisted by
  vcz, vay, vd

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Function and Strength of Web Reinforcement

• Flexural Cracking Shear is resisted by
  vcz, vay, vd and vs

Vs increases as cracks widen until yielding of
stirrups then stirrups provide constant
resistance.
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Designing to Resist Shear
Shear Strength (ACI 318 Sec 11.1)
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Shear Strength Provided by Concrete
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Lightweight ConcreteShear Strength Provided by
Shear Reinforcement
Minimum Shear Reinforcement (11.5.5)
Except
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Lightweight ConcreteShear Strength Provided by
Shear Reinforcement
(provides additional 50 psi of shear strength)
Note
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Typical Shear Reinforcement
Stirrup - perpendicular to axis of members
(minimum labor - more material)
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Typical Shear Reinforcement
Bent Bars (more labor - minimum material) see
reqd in 11.5.6
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Stirrup Anchorage Requirements
Vs based on assumption stirrups yield
Stirrups must be well anchored.
Refer to Sec. 12.12 of ACI 318 for development of
web reinforcement. Requirements - each bend
must enclose a long bar - 5 and smaller can use
standard hooks 90o,135o, 180o - 6, 7,8(fy
40 ksi) - 6, 7,8(fy gt 40 ksi) standard hook
plus a min embedment
Also sec. 7.11 requirement for min. stirrups in
beams with compression reinforcement, beams
subject to stress reversals, or beams subject to
torsion
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Design Procedure for Shear
(1) Calculate Vu (2) Calculate fVc Eqn 11-3 or
11-5 (no axial force) (3) Check
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Design Procedure for Shear
(4)
Also (Done)
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Design Procedure for Shear
(5)
Check
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Design Procedure for Shear
(6) Solve for required stirrup spacing(strength)
Assume 3, 4, or 5 stirrups (7) Check
minimum steel requirement (eqn 11-13)
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Design Procedure for Shear
(8) Check maximum spacing requirement (ACI
11.5.4) (9) Use smallest spacing from steps
6,7,8
Note A practical limit to minimum stirrup
spacing is 4 inches.
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Location of Maximum Shear for Beam Design
Compression fan carries load directly into
support.
Non-pre-stressed members
Sections located less than a distance d from face
of support may be designed for same shear, Vu, as
the computed at a distance d.
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Location of Maximum Shear for Beam Design
When
The support reaction introduces compression into
the end regions of the member No concentrated
load occurs with in d from face of support .
1.
2.
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Location of Maximum Shear for Beam Design
Compression from support at bottom of beam tends
to close crack at support
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Example Design of Stirrups to Resist Shear
fc 4000 psi fy 60 ksi wsdl
1.2 k/ft wll 1.8 k/ft fys 40 ksi
wb 0.5 k/ft
From flexural design will use either a 3 or 4
stirrup