ACI COncrete

Lecture 9 - Flexure

June 20, 2003CVEN 444

Lecture Goals

Load EnvelopesResistance Factors and LoadsDesign of Singly Reinforced Rectangular Beam Unknown section dimensions Known section dimensions

MomentEnvelopes

Fig. 10-10; MacGregor (1997)

The moment envelope curve defines the extreme boundary values of bending moment along the beam due to critical placements of design live loading.

MomentEnvelopes ExampleGiven following beam with a dead load of 1 k/ft and live load 2 k/ft obtain the shear and bending moment envelopes

MomentEnvelopes ExampleUse a series of shear and bending moment diagrams

Wu = 1.2wD + 1.6wL

0

1

2

3

4

5

0 5 10 15 20 25 30 35 40

(ft)

kips

-80

-60

-40

-20

0

20

40

60

80

0 5 10 15 20 25 30 35 40

ft

kips

-250

-200

-150

-100

-50

0

50

100

150

0 5 10 15 20 25 30 35 40

ft

k-ft

Shear Diagram Moment Diagram


Wu = 1.2wD + 1.6wL


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

ft

k/ft

-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30 35 40

ft

kips

-80

-60

-40

-20

0

20

40

0 5 10 15 20 25 30 35 40

ft

k-ft

(Dead Load Only)


Wu = 1.2wD + 1.6wL


00.5

11.5

22.5

33.5

44.5

5

0 5 10 15 20 25 30 35 40

ft

k/ft

-60-50-40-30-20-10

01020304050

0 5 10 15 20 25 30 35 40

ft

kips

-200

-150

-100

-50

0

50

100

150

200

0 5 10 15 20 25 30 35 40

ft

k-ft

MomentEnvelopes ExampleThe shear envelope

Shear Envelope

-80-60-40-20

020406080

0 10 20 30 40

ft

kips

Minimum ShearMaximum Shear

MomentEnvelopes ExampleThe moment envelope

Moment Envelope

-300

-200

-100

0

100

200

0 5 10 15 20 25 30 35 40

ft

k-ft

Minimum Moment Maximum Moment

Flexural Design of Reinforced Concrete Beams and Slab Sections

Analysis Versus Design:Analysis: Given a cross-section, fc , reinforcement

sizes, location, fy compute resistance or capacity

Design: Given factored load effect (such as Mu) select suitable section(dimensions, fc, fy, reinforcement, etc.)


ACI Code Requirements for Strength Design

Basic Equation: factored resistance factored load effect

Ex.

un M M

ACI Code Requirements for Strength Design

un M M Mu = Moment due to factored loads (required

ultimate moment)

Mn = Nominal moment capacity of the cross-section using nominal dimensions and specified material strengths.

= Strength reduction factor (Accounts for variability in dimensions, material strengths, approximations in strength equations.


Required Strength (ACI 318, sec 9.2)

U = Required Strength to resist factored loadsD = Dead LoadsL = Live loadsW = Wind Loads E = Earthquake Loads


Required Strength (ACI 318, sec 9.2)H = Pressure or Weight Loads due to soil,ground

water,etc. F = Pressure or weight Loads due to fluids with

well defined densities and controllable maximum heights.

T = Effect of temperature, creep, shrinkage, differential settlement, shrinkage compensating.

Factored Load Combinations

U = 1.2 D +1.6 L Always check even if other load types are present.

U = 1.2(D + F + T) + 1.6(L + H) + 0.5 (Lr or S or R)U = 1.2D + 1.6 (Lr or S or R) + (L or 0.8W)U = 1.2D + 1.6 W + 1.0L + 0.5(Lr or S or R) U = 0.9 D + 1.6W +1.6HU = 0.9 D + 1.0E +1.6H

Resistance Factors, ACI Sec 9.3.2 Strength Reduction Factors

[1] Flexure w/ or w/o axial tension

The strength reduction factor, , will come into the calculation of the strength of the beam.

Resistance Factors, ACI Sec 9.3.2 Strength Reduction Factors[2] Axial Tension = 0.90

[3] Axial Compression w or w/o flexure(a) Member w/ spiral reinforcement = 0.70(b) Other reinforcement members = 0.65

*(may increase for very small axial loads)

Resistance Factors, ACI Sec 9.3.2 Strength Reduction Factors

[4] Shear and Torsion = 0.75

[5] Bearing on Concrete = 0.65

ACI Sec 9.3.4 factors for regions of high seismic risk

Background Information for Designing Beam Sections

1. Location of Reinforcementlocate reinforcement where cracking occurs (tension region) Tensile stresses may be due to :

a ) Flexureb ) Axial Loadsc ) Shrinkage effects


2. Construction

formwork is expensive - try to reuse at several floors


3. Beam Depths

• ACI 318 - Table 9.5(a) min. h based on l (span) (slab & beams)

• Rule of thumb: hb (in) l (ft)

• Design for max. moment over a support to set depth of a continuous beam.


4. Concrete Cover

Cover = Dimension between the surface of the slab or beam and the reinforcement

Background Information for Designing Beam Sections4. Concrete Cover

Why is cover needed?[a] Bonds reinforcement to concrete[b] Protect reinforcement against corrosion[c] Protect reinforcement from fire (over

heating causes strength loss)[d] Additional cover used in garages, factories,

etc. to account for abrasion and wear.


Minimum Cover Dimensions (ACI 318 Sec 7.7)

Sample values for cast in-place concrete

• Concrete cast against & exposed to earth - 3 in.

• Concrete (formed) exposed to earth & weather No. 6 to No. 18 bars - 2 in.No. 5 and smaller - 1.5 in


Minimum Cover Dimensions (ACI 318 Sec 7.7)

•Concrete not exposed to earth or weather- Slab, walls, joists

No. 14 and No. 18 bars - 1.5 inNo. 11 bar and smaller - 0.75 in

- Beams, Columns - 1.5 in


5.Bar Spacing Limits (ACI 318 Sec. 7.6)

- Minimum spacing of bars

- Maximum spacing of flexural reinforcement in walls & slabs

Max. space = smaller of

.in 18 t3

Minimum Cover Dimension

Interior beam.

Minimum Cover DimensionReinforcement bar arrangement for two layers.

Minimum Cover Dimension

ACI 3.3.3

Nominal maximum aggregate size.

- 3/4 clear space - 1/3 slab depth - 1/5 narrowest dim.

Example - Singly Reinforced BeamDesign a singly reinforced beam, which has a moment capacity, Mu = 225 k-ft, fc = 3 ksi, fy = 40 ksi and c/d = 0.275

Use a b = 12 in. and determine whether or not it is sufficient space for the chosen tension steel.

Example - Singly Reinforced BeamFrom the calculation of Mn

u

n

c c

2c 1

2c

size

R

210.85 0.85 1

2 210.85 1 where, 2

10.85 12

aM C d

a a af ba d f bd dd d

a cf bd k k kd d

f k k bd

Example - Singly Reinforced BeamSelect c/d =0.275 so that =0.9. Compute k’ and determine Ru

1

u c

0.85 0.275

0.23375

0.85 12

0.233750.85 3 ksi 0.23375 12

0.5264 ksi

ckd

kR f k

Example - Singly Reinforced BeamCalculate the bd 2

U

2 N

u u

3

12 in225 k-ftft

0.9

5699 in0.5264 ksi

MMbdR R

Example - Singly Reinforced BeamCalculate d, if b = 12 in.

32 25699 in 440.67 in 21.79 in.

12 ind d

Use d =22.5 in., so that h = 25 in.

0.275 0.275 22.5 in 6.1875 in.c d

Example - Singly Reinforced BeamCalculate As for the beam

c 1s

y

2

0.85

0.85 3 ksi 12 in. 0.85 6.1875 in.40 ksi

4.02 in

f b cAf

Example - Singly Reinforced BeamChose one layer of 4 #9 bars

Compute

2 2s 4 1.0 in 4.00 inA

2

s 4.00 in12.0 in 22.5 in

0.014815

Abd

Example - Singly Reinforced BeamCalculate min for the beam

y

min minc

y

200 200 0.00540000

0.0053 3 3000 0.00411

40000

f

ff

0.014815 0.005 The beam is OK for the minimum

Example - Singly Reinforced BeamCheck whether or not the bars will fit into the beam. The diameter of the #9 = 1.128 in.

b stirrup4 3 2 cover

4 1.128 in. 3 1.128 in. 2 1.5 in. 0.375 in.11.65 in

b d s d

So b =12 in. works.

Example - Singly Reinforced BeamCheck the height of the beam.

Use h = 25 in.

bstirrupcover

21.128 in.

22.5 in. 1.5 in. 0.375 in.2

24.94 in

dh d d

Example - Singly Reinforced BeamFind a

Find c

2s y

c

4.0 in 40 ksi0.85 0.85 3 ksi 12.0 in.5.23 in.

A fa

f b

1

5.23 in.0.85

6.15 in.

ac

Example - Singly Reinforced BeamCheck the strain in the steel

Therefore, is 0.9

t cu22.5 in. 6.15 in. 0.003

c 6.15 in.0.00797 0.0056.15 in. 0.273322.5 in.

d c

cd

Example - Singly Reinforced BeamCompute the Mn for the beam

Calculate Mu

N s y

2

25.23 in.4.0 in 40 ksi 22.5 in.

23186.6 k-in

aM A f d

U N

0.9 3186.6 k-in 2863.4 k-inM M

Example - Singly Reinforced BeamCheck the beam Mu = 225 k-ft*12 in/ft =2700 k-in

Over-designed the beam by 6%

2863.4 2700 *100% 6.05%2700

6.15 in. 0.273322.5 in.

cd Use a smaller c/d

ratio

ACI COncrete

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