Chapter 6. Compression Reinforcement - Flexural Members

Chapter 6. Compression Reinforcement - Flexural Members

If a beam cross section is limited because of architectural or other considerations, it may happen that the concrete cannot develop the compression force required to resist the give bending moment. In this case, reinforcing is added in the compression zone, resulting in a so-called doubly reinforced beam, i.e., one with compression as well as tension reinforcement. Compression reinforced is also used to improve serviceability, improve long term deflections, and to provide support for stirrups throughout the beam.

6.1. Reading Assignment: Text Section 5.7; ACI 318, Sections: 10.3.4, 10.3.3, and 7.11.1

6.2. Strength Calculations

A's hd

d' cb

?u = 0.003

?s

ab = 1cb

0.85fc Cs

Cc

Abs

d-cb

h-cb

Tbs

b

?s = ?y

strains

stresses

forces

From geometry we can find the strain in compression steel at failure as:

?s

=

0.003

c

- c

d

(6.1)

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6.3. Nominal Resisting Moment When Compression Steel Yields

d'

A's

c

hd

As

d-c

b

?u = 0.003

?y

a

> ?y

0.85fc Cs

Cc =

Ts

0.85fc

Asfy a

Cc

+

Asfy

Ts = (As - As)fy

Case I

Case II

Doubly Reinforced Rectangular Beam

Total resisting moment can be considered as sum of: 1. Moment from corresponding areas of tension and compression steel 2. The moment of some portion of the tension steel acting with concrete.

Mn = (As - As) fy (d - 21c) + As fy (d - d) and from equilibrium:

0.85fc ab = (As - As)fy Solve for "a":

a=

As - As 0.85fc b

fy

(6.2) (6.3) (6.4)

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6.4. Compression Steel below Yield Stress (strain compatibility check). Whether or not the compression steel will have yielded at failure can be determined as fol-

lows:

A's

hd As lim

d' c

d-c

?u = 0.003 ?s = ?y a

0.85fc Cs

Cc

Ts

b

?y

From geometry:

?u ?s

=

c

c -

d

(6.5)

if compression steel yield ?s = ?y then:

??uy

=

c

c -

d

c

=

?u

?u -

?y d

Equilibrium for case II: (Alsim - As)fy = 0.85 ? (1c) b f c

(6.6) (6.7)

Substitute for "c" from Eq. (6.6) and (6.7) and divide both sides by "bd" gives:

(Alsim

- As)fy bd

=

0.85 ? 1 ? b ? f c

?u

?u -

?y

d

1 bd

or

Alsim

bd

=

A s bd

+

0.85

?

1

?

f c fy

?u

d

?u - ?y d

?lim

=

? s

+

0.85

?

1

?

f c fy

?

87, 000 87, 000 -

fy

d d

(6.8) (6.9) (6.10)

if

?actual > ?lim

this is common for shallow then compression steel will yield beams using high strength

steel

if

As

- A s bd

0.85

?

1

?

f c fy

?

87, 000 87, 000 -

fy

d d

then compression steel will yield

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6.5. Example of analysis of a reinforced concrete section having compression reinforcement.

Determine the nominal moment, Mn, and the ultimate moment capacity, Mu, of the reinforced concrete section shown below.

2.5"

fc = 5, 000 psi fy = 60, 000 psi

22.2

A's= 3.8 in2

As= 7.62 in2

12" Solution

Mn can be calculated if we assume some conditions for compression steel. Assume that compression steel yields:

Cc = 0.85f c 1 cb = 0.85 ? (5 ksi) ? (0.80) ? c ? (12) = 40.8c Cs = Asfy = 3.8 ? (60ksi) = 228 kips Ts = (7.62 in2) ? ( 60 ksi) = 457 kips

Equilibrium: Cs + Cc = Ts

solve for c:

c

=

457 - 228 40.8

=

5.6 in

check assumption

0.85fc

d' ?u = 0.003

c

?s

?s

=

c 0.003

- c

d

d

=

5.6 0.003

- 5.6

2.5 = 0.0017

Ts

d-c ?y

?s

=

0.0017

<

fy Es

=

60 29, 000

=

0.00207

wrong assumption

This means the compression steel does not yield. Therefore, our initial assumption was wrong. We need to make a new assumption.

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Assume f's < fy

Cs = Asf s = As ?s Es

=

(3.8 in2) ? (0.003 c

- c

2.5) ? (29, 000

ksi)

=

330 c

- c

2.5

Now for equilibrium: Cs + Cc = Ts

40.8c

+

330 ? c

- c

2.5

=

457 kips

solve for c c = 6.31 in

check assumption

f s

=

0.003

?

6.31 - 6.31

2.5 ? 29, 000

=

52.5 ksi

<

fy

=

60 ksi assumption o.k.

check ACI Code requirements for tension failure

c d

=

6.31 22.2

=

0.284

<

0.375

We are in the tension-controlled section and satisfy the ACI code requirements.

0.90

= 0.75 + (?t - 0.002)(50)

= 0.9

0.75

0.65

SPIRAL

OTHER

Compression Controlled

Transition

?t = 0.002

c dt

=

0.600

Tension Controlled

?t = 0.005

c dt

=

0.375

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