JavaScript isn't enabled in your browser, so this file can't be opened. Enable and reload.

1 of 45

Lecture Goals

Doubly Reinforced beams
T Beams and L Beams

2 of 45

Analysis of Doubly Reinforced Sections

Effect of Compression Reinforcement on the Strength and Behavior

Less concrete is needed to resist the T and thereby moving the neutral axis (NA) up.

3 of 45

Analysis of Doubly Reinforced Sections

Effect of Compression Reinforcement on the Strength and Behavior

4 of 45

Reasons for Providing Compression Reinforcement

Reduced sustained load deflections.

Creep of concrete in compression zone
transfer load to compression steel
reduced stress in concrete
less creep
less sustained load deflection

5 of 45

Reasons for Providing Compression Reinforcement

Effective of compression reinforcement on sustained load deflections.

6 of 45

Reasons for Providing Compression Reinforcement

Increased Ductility

reduced stress block depth

increase in steel strain larger curvature are obtained.

7 of 45

Reasons for Providing Compression Reinforcement

Effect of compression reinforcement on strength and ductility of under reinforced beams.

ρ < ρ_b

8 of 45

Reasons for Providing Compression Reinforcement

Change failure mode from compression to tension. When ρ > ρ_bal addition of A_s strengthens.

Effective reinforcement ratio = (ρ − ρ’)

Compression zone

allows tension steel to yield before crushing of concrete.

9 of 45

Reasons for Providing Compression Reinforcement

Eases in Fabrication - Use corner bars to hold & anchor stirrups.

10 of 45

Effect of Compression Reinforcement

Compare the strain distribution in two beams with the same A_s

11 of 45

Effect of Compression Reinforcement

Section 1:

Section 2:

Addition of A’_s strengthens compression zone so that less concrete is needed to resist a given value of T. NA goes up (c₂ <c₁) and ε_s increases (ε_s2 >ε_s1).

12 of 45

Doubly Reinforced Beams

Under reinforced Failure

( Case 1 ) Compression and tension steel yields
( Case 2 ) Only tension steel yields

Over reinforced Failure

( Case 3 ) Only compression steel yields
( Case 4 ) No yielding Concrete crushes

Four Possible Modes of Failure

13 of 45

Analysis of Doubly Reinforced Rectangular Sections

Strain Compatibility Check Assume ε_s’ using similar triangles

14 of 45

Analysis of Doubly Reinforced Rectangular Sections

Strain Compatibility

Using equilibrium and find a

15 of 45

Analysis of Doubly Reinforced Rectangular Sections

Strain Compatibility The strain in the compression steel is

16 of 45

Analysis of Doubly Reinforced Rectangular Sections

Strain Compatibility Confirm

17 of 45

Analysis of Doubly Reinforced Rectangular Sections

Strain Compatibility Confirm

18 of 45

Analysis of Doubly Reinforced Rectangular Sections

Find c

confirm that the tension steel has yielded

19 of 45

Analysis of Doubly Reinforced Rectangular Sections

If the statement is true than

else the strain in the compression steel

20 of 45

Analysis of Doubly Reinforced Rectangular Sections

Return to the original equilibrium equation

21 of 45

Analysis of Doubly Reinforced Rectangular Sections

Rearrange the equation and find a quadratic equation

Solve the quadratic and find c.

22 of 45

Analysis of Doubly Reinforced Rectangular Sections

Find the f_s’

Check the tension steel.

23 of 45

Analysis of Doubly Reinforced Rectangular Sections

Another option is to compute the stress in the compression steel using an iterative method.

24 of 45

Analysis of Doubly Reinforced Rectangular Sections

Go back and calculate the equilibrium with f_s’

Iterate until the c value is adjusted for the f_s’ until the stress converges.

25 of 45

Analysis of Doubly Reinforced Rectangular Sections

Compute the moment capacity of the beam

26 of 45

Limitations on Reinforcement Ratio for Doubly Reinforced beams

Lower limit on ρ

same as for single reinforce beams.

(ACI 10.5)

27 of 45

Example: Doubly Reinforced Section

Given:

f’_c= 4000 psi f_y = 60 ksi

A’_s = 2 #5 A_s = 4 #7

d’= 2.5 in. d = 15.5 in

h=18 in. b =12 in.

Calculate M_n for the section for the given compression steel.

28 of 45

Example: Doubly Reinforced Section

Compute the reinforcement coefficients, the area of the bars #7 (0.6 in²) and #5 (0.31 in²)

29 of 45

Example: Doubly Reinforced Section

Compute the effective reinforcement ratio and minimum ρ

30 of 45

Example: Doubly Reinforced Section

Compute the effective reinforcement ratio and minimum ρ

Compression steel has not yielded.

31 of 45

Example: Doubly Reinforced Section

Instead of iterating the equation use the quadratic method

32 of 45

Example: Doubly Reinforced Section

Solve using the quadratic formula

33 of 45

Example: Doubly Reinforced Section

Find the f_s’

Check the tension steel.

34 of 45

Example: Doubly Reinforced Section

Check to see if c works

The problem worked

35 of 45

Example: Doubly Reinforced Section

Compute the moment capacity of the beam

36 of 45

Example: Doubly Reinforced Section

If you want to find the M_u for the problem

From ACI (figure R9.3.2)or figure (pg 100 in your text)

The resulting ultimate moment is

37 of 45

Analysis of Flanged Section

Floor systems with slabs and beams are placed in monolithic pour.
Slab acts as a top flange to the beam; T-beams, and Inverted L(Spandrel) Beams.

38 of 45

Analysis of Flanged Sections

Positive and Negative Moment Regions in a T-beam

39 of 45

Analysis of Flanged Sections

If the neutral axis falls within the slab depth analyze the beam as a rectangular beam, otherwise as a T-beam.

40 of 45

Analysis of Flanged Sections

Effective Flange Width

Portions near the webs are more highly stressed than areas away from the web.

41 of 45

Analysis of Flanged Sections

Effective width (b_eff)

b_eff is width that is stressed uniformly to give the same compression force actually developed in compression zone of width b_(actual)

42 of 45

ACI Code Provisions for Estimating b_eff

From ACI 318, Section 8.10.2

T Beam Flange:

43 of 45

ACI Code Provisions for Estimating b_eff

From ACI 318, Section 8.10.3

Inverted L Shape Flange

44 of 45

ACI Code Provisions for Estimating b_eff

From ACI 318, Section 8.10

Isolated T-Beams

45 of 45

Various Possible Geometries of T-Beams

Single Tee

Twin Tee

Box