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twenty four

concrete construction:

shear & deflection

FOUNDATIONS STRUCTURES:

FORM, BEHAVIOR, AND DESIGN

ARCH 331

DR. ANNE NICHOLS

SPRING 2014

Concrete Shear 1

Lecture 24

Foundations Structures

ARCH 331

Copyright © Kirk Martini

lecture

Concrete Shear 2

Lecture 24

Foundations Structures

ARCH 331

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Shear in Concrete Beams

• flexure combines with shear to form

diagonal cracks

• horizontal reinforcement doesn’t help

• stirrups = vertical reinforcement

Concrete Shear 3

Lecture 24

Foundations Structures

ARCH 331

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ACI Shear Values

• Vu is at distance d from face of support

• shear capacity:

– where bw meansthickness of web at n.a.

dbV wcc

Concrete Shear 4

Lecture 24

Foundations Structures

ARCH 331

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ACI Shear Values

• shear stress (beams)

– = 0.75 for shear

f’c is in psi

• shear strength:

– Vs is strength fromstirrup reinforcement

cc f 2

dbfV wcc 2

scu VVV

2

Concrete Shear 5

Lecture 24

Foundations Structures

ARCH 331

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Stirrup Reinforcement

• shear capacity:

– Av = area in all legs of stirrups

– s = spacing of stirrup

• may need stirrups when concrete has

enough strength!

s

dfAV

yv

s

Concrete Shear 6

Lecture 24

Foundations Structures

ARCH 331

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Required Stirrup Reinforcement

• spacing limits

0.75

Concrete Shear 7

Lecture 24

Foundations Structures

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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 8

Lecture 24

Foundations Structures

ARCH 331

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Shear Stress Distribution

• depend on the

deformation

• = angle of twist

– measure

• can prove planar

section doesn’t distort

3

Concrete Shear 9

Lecture 24

Foundations Structures

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Shearing Strain

• related to

• is the radial distance from

the centroid to the point

under strain

• shear strain varies linearly

along the radius: max is at

outer diameter

L

Concrete Shear 10

Lecture 24

Foundations Structures

ARCH 331

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Torsional Stress - Strain

• know and

• so

• where G is the Shear Modulus

Gfv

LG

L

Concrete Shear 11

Lecture 24

Foundations Structures

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Torsional Stress - Strain

• from

• can derive

– where J is the polar

moment of inertia

– elastic range

AT )(

JT

J

T

Concrete Shear 12

Lecture 24

Foundations Structures

ARCH 331

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Shear Stress

• max happens at outer diameter

• combined shear and

axial stresses

– maximum shear

stress at 45° “twisted”

plane

4

Concrete Shear 13

Lecture 24

Foundations Structures

ARCH 331

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Shear Strain

• knowing and

• solve:

• composite shafts:

LG

J

T

JG

TL

ii

ii

i GJ

LT

Concrete Shear 14

Lecture 24

Foundations Structures

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Noncircular Shapes

• torsion depends on J

• plane sections don’t

remain plane

• max is still at outer diameter

– where a is longer side (> b)

2

1

maxabc

T

Gabc

TL3

2

Concrete Shear 15

Lecture 24

Foundations Structures

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Open Thin-Walled Sections

• with very large a/b ratios:

23

1max

ab

T

Gab

TL3

31

Concrete Shear 16

Lecture 24

Foundations Structures

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Shear Flow in Closed Sections

• q is the internal shear force/unit length

• is the area bounded by the centerline

• si is the length segment, ti is the thickness

atT

2

i i

i

t

s

t

TL24 a

a

5

Concrete Shear 17

Lecture 24

Foundations Structures

ARCH 331

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Shear Flow in Open Sections

• each segment has proportion of T with respect to torsional rigidity,

• total angle of twist:

• I beams - web is thicker, so max is in web

33

1

maxmax

ii tb

Tt

33

1ii tbG

TL

Concrete Shear 18

Lecture 24

Foundations Structures

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Torsional Shear Stress

• twisting moment

• and beam shear

Concrete Shear 19

Lecture 24

Foundations Structures

ARCH 331

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Torsional Shear Reinforcement

• closed stirrups

• more longitudinal

reinforcement

• area enclosed by shear flow

Concrete Shear 20

Lecture 24

Foundations Structures

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Development Lengths

• required to allow steel to yield (fy)

• standard hooks

– moment at beam end

• splices

– lapped

– mechanical connectors

6

Concrete Shear 21

Lecture 24

Foundations Structures

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Development Lengths

• ld , embedment required both sides

• proper cover, spacing:

– No. 6 or smaller

or 12 in. minimum

– No. 7 or larger

or 12 in. minimum

c

yb

df

Fdl

25

c

yb

df

Fdl

20

Concrete Shear 22

Lecture 24

Foundations Structures

ARCH 331

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Development Lengths

• hooks

– bend and extension

• minimum

c

bdh

f

dl

1200

Concrete Shear 23

Lecture 24

Foundations Structures

ARCH 331

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Development Lengths

• bars in compression

• splices

– tension minimum is function of ld and

splice classification

– compression minimum

– is function of db and Fy

yb

c

yb

d Fd.f

Fd.l 00030

020

F2008abnConcrete Shear 24

Lecture 24

Foundations Structures

ARCH 331

Concrete Deflections

• elastic range

– I transformed

– Ec (with f’c in psi)

• normal weight concrete (~ 145 lb/ft3)

• concrete between 90 and 160 lb/ft3

• cracked

– I cracked

– E adjusted

7

Concrete Shear 25

Lecture 24

Foundations Structures

ARCH 331

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Deflection Limits

• relate to whether or not beam supports

or is attached to a damageable non-

structural element

• need to check service live load and long

term deflection against these

supporting masonry – live + long term

supporting plaster – live

floor systems (typical) – live + long term

roof systems (typical) – live

L/480

L/360

L/240

L/180