Design for Shear-Lecture 22

Lecture 22 – Shear Lecture 22 – Shear DesignDesign

March 17, 2003CVEN 444

Lecture GoalsLecture Goals

Shear Shear Design

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

Look at the shear and bending moment diagrams. The acting shear stress distribution on the beam.

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

The acting stresses distributed across the cross-section.

The shear stress acting on the rectangular beam.

Ib

VQ

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

The equation of the shear stress for a rectangular beam is given as:

Note: The maximum 1st moment occurs at the neutral axis (NA).

Ib

VQ

avemax

2

max

3

5.1*2

3

84*

2Q

Inertia ofMoment 12

bh

V

bhhbh

bhI

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

The ideal shear stress distribution can be described as:

Ib

VQ

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

A realistic description of the shear distribution is shown as:

Uncracked Elastic Beam Uncracked Elastic Beam BehaviorBehavior

The shear stress acting along the beam can be described with a stress block:

Using Mohr’s circle, the stress block can be manipulated to find the maximum shear and the crack formation.

Inclined Cracking in Inclined Cracking in Reinforced Concrete Reinforced Concrete BeamsBeams

Typical Crack Patterns for a deep beam

Inclined Cracking in Inclined Cracking in Reinforced Concrete Reinforced Concrete BeamsBeamsFlexural-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).

Inclined Cracking in Inclined Cracking in Reinforced Concrete Reinforced Concrete BeamsBeams

For deep beam the cracks are given as:

The shear cracks Inclined (diagonal) intercept crack with longitudinal bars plus vertical or inclined reinforcement.

Inclined Cracking in Inclined Cracking in Reinforced Concrete Reinforced Concrete BeamsBeams

For deep beam the cracks are given as:

The shear cracks fail due two modes:

- shear-tension failure - shear-compression failure

Shear Strength of RC Beams Shear Strength of RC Beams without Web Reinforcementwithout 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)

Strength of Concrete in Shear Strength of Concrete in Shear (No Shear Reinforcement)(No Shear Reinforcement)

(1) Tensile Strength of concrete affect inclined cracking load

Strength of Concrete in Strength of Concrete in Shear (No Shear Shear (No Shear Reinforcement)Reinforcement)

(2) Longitudinal Reinforcement Ratio, w

dbfV

db

A

wccw

w

sw

2:0025.00075.0for

cracks restrains

Strength of Concrete in Strength of Concrete in Shear (No Shear Shear (No Shear Reinforcement)Reinforcement)

(3) Shear span to depth ratio, a/d (M/(Vd))

2d

a

2

d

a Deep shear spans more detail design required

Ratio has little effect

Strength of Concrete in Strength of Concrete in Shear (No Shear Shear (No Shear Reinforcement)Reinforcement)

(4) Size of BeamIncrease Depth Reduced shear stress at

inclined cracking

Strength of Concrete in Strength of Concrete in Shear (No Shear Shear (No Shear Reinforcement)Reinforcement)

(5) Axial Forces - Axial tension Decreases inclined cracking load - Axial Compression Increases inclined cracking

load (Delays flexural cracking)

Function and Strength of Function and Strength of Web ReinforcementWeb 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:

Function and Strength of Function and Strength of Web ReinforcementWeb Reinforcement

Uncracked Beam Shear is resisted uncracked concrete.

Flexural Cracking Shear is resisted by vcz, vay, vd

bars. allongitudin fromAction Dowl

forceInterlock Aggregate ofcomponent Vertical

zonen compressioin Shear

d

ay

cz

V

V

V

Function and Strength of Function and Strength of Web ReinforcementWeb 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.

Designing to Resist Designing to Resist ShearShear

Shear Strength (ACI 318 Sec 11.1)

n u

capacity demand

V V

u

n

factored shear force at section

Nominal Shear Strength

0.75 shear strength reduction factor

V

V

Designing to Resist Designing to Resist ShearShear

Shear Strength (ACI 318 Sec 11.1)

n c sV V V

c

s

V

V

Nominal shear provided by the shear reinforcement

Nominal shear resistance provided by concrete

Shear Strength Provided by Shear Strength Provided by ConcreteConcrete

Bending onlyBending only

Simple formula

More detailed

Note:

Eqn [11.5]

Eqn [11.3]

dbf

dbfV

wc

wcc

3.5

2

dbf

dbM

dVfV

wc

w

u

uwcc

3.5

25001.9

1

u

u

M

dV

Shear Strength Provided by Shear Strength Provided by ConcreteConcrete

Bending and Axial CompressionBending and Axial Compression

Nu is positive for compression and Nu/Ag are in psi.

Simple formula

Eqn [11.4]

Eqn [11.7]

g

uwc

wc

g

uc

5001 3.5

2000

1 2

A

Ndbf

dbfA

NV

Typical Shear Typical Shear ReinforcementReinforcement

Stirrup - perpendicular to axis of members (minimum labor - more material)

ACI Eqn 11-15

s

dfAV

cossinyvs

s

dfAV yv

so90

Typical Shear Typical Shear ReinforcementReinforcement

Bent Bars (more labor - minimum material) see req’d in 11.5.6

ACI 11-5.6

s

dfAV

cossinyvs

s

dfAV yv

so 41.1

45

Stirrup Anchorage Stirrup Anchorage RequirementsRequirements

Vs based on assumption stirrups yield

Stirrups must be well anchored.

Stirrup Anchorage Stirrup Anchorage RequirementsRequirements

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 > 40 ksi ) standard hook plus a minimum embedment

Refer to Sec. 12.13 of ACI 318 for development of web reinforcement. Requirements:

Stirrup Anchorage Stirrup Anchorage RequirementsRequirements

Also sec. 7.10 requirement for minimum stirrups in beams with compression reinforcement, beams subject to stress reversals, or beams subject to torsion

Design Procedure for Design Procedure for ShearShear

(1) Calculate Vu

(2) Calculate Vc Eqn 11-3 or 11-5 (no axial force)

(3) Check

cu VV 2

1

If yes, add web reinforcement (go to 4)

If no, done.

Design Procedure for Design Procedure for ShearShear

(4) cuc VVV 2

1 If

v

w

ysv

y

wv A

b

fAs

f

sbA min for

50or 50 maxmin

Also:

(Done) 11.5.4 "24

2max d

s

Provide minimum shear reinforcement

Design Procedure for Design Procedure for ShearShear

(5)

cu

scus

scnu

scu

VV

VVVV

VVVV

VVV

d)(req' calulate , If

Check:

11.5.4 illegal otherwise, 8 dbfV wcs

Design Procedure for Design Procedure for ShearShear

Solve for required stirrup spacing(strength) Assume # 3, #4, or #5 stirrups

s

ysv

V

dfAs

(6)

from 11-15

Design Procedure for Design Procedure for ShearShear

(7) Check minimum steel requirement (eqn 11-13)

50max

w

ysv

b

fAs

Design Procedure for Design Procedure for ShearShear

(8) Check maximum spacing requirement (ACI 11.5.4)

illegal 8 If :Note

"124

4 If

"242

4 If

c

maxc

maxc

dbfV

dsdbfV

dsdbfV

ws

ws

ws

Design Procedure for Design Procedure for ShearShear

(9) Use smallest spacing from steps 6,7,8

Note: A practical limit to minimum stirrup spacing is 4 inches.

Location of Maximum Shear Location of Maximum Shear for Beam Designfor Beam Design

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.

Compression fan carries load directly into support.

Location of Maximum Shear for Beam Design

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.

When:

Location of Maximum Shear for Beam Design

Compression from support at bottom of beam tends to close crack at support

Homework – Design-(I)Homework – Design-(I)

Determine the lengths of top and bottom bars for the external span of the continuous beam. Concretebeam. Concrete is normal weight and bars are Grade 60. Total uniformly distribution factored gravity load on beam is wu = 6.0 k/ft (including weight of beam). The beam has fc = 4 ksi, fy= 60 ksi, b =16 in., h =22 in., and concrete cover =1.5 in.

Homework – Design(I)- Homework – Design(I)- due 3/24/03due 3/24/03

Determine the singly reinforced beam need for external beams for both the positive and negative moments. Draw the shear and bending moment diagrams for the beam and determine amount of steel needed for bending. Use the ACI code to find your moments. Hint: There is more than one bending moment diagram.