OUTLINE - Precast Concreteprecast.org/wp-content/uploads/2019/03/CP4-Reinforced-Concrete-Design-Kayla...BASICS OF REINFORCED CONCRETE DESIGN Kayla Hanson, Director of Technical Services

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BASICS OF REINFORCED CONCRETE DESIGN

Kayla Hanson, Director of Technical Services

National Precast Concrete Association


OUTLINE

• Loads, stresses, and concrete strength

• Stress, strain, and stress-strain curves

• Beam anatomy and failure modes

• Common beam cross sections

• Neutral axis

• Bending stress and rupture stress

• Summation of forces

• Moments

• Stress distributions

• Example problems

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POINT LOADS

• What kind of loads and stresses do we anticipate?• Point loads

• Distributed loads


DISTRIBUTED LOADS

• What kind of loads and stresses do we anticipate?• Point loads

• Distributed loads

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CONCRETE STRENGTH

• How do these forces act on the structure?• Compression

• Tension

• Flexure

• Torsion

• Axial


RELATING LOADS & STRESS TO CONCRETE STRENGTH

• Stress = load per unit area• Load = 36,000 lb

• Area = 9 in.2

• Stress = 36,000 lb/9 in.2 = 4,000 lb/in.2 = 4,000 psi

3 in.

3 in

.

Load = 36,000 lb

A = 9 in.2


CONCRETE STRENGTH


STRESS AND STRAIN


STRESS

• Stress (σ, sigma): force or load per area• σ = P / A

• Pounds per square inch, psi or lb/in.2

• Kips per square inch, ksi or kip/in.2

• Note: (1 kip = 1,000 lb)

ε = ΔL / L

σ = P / A


STRAIN

• Strain (ε, epsilon): change in length per original length

• ε = ΔL / L

• Unitless or dimensionless

ε = ΔL / L

σ = P / A

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STRESS-STRAIN CURVES

Ductile

Brittle

σ

ε

Ductile materials show warning signs before failure. They also

have a tell-tale “double kink” in their stress-strain curve.


CONCRETE STRESS-STRAIN CURVE


DUCTILE VS. BRITTLE MATERIALS


CONCRETE MIX DESIGN

• Refer to ACI 211, ACI 237, and ACI 318

• Covered in PQS II QA/QC

• We need certain pieces of information from the mix design to use in our analysis:

• Concrete compressive strength

• Steel tensile strength

• Cross sectional dimensions

• Area of steel


REINFORCED CONCRETE BEAMS

• Anatomy of a beam: beam, supports, applied loads

• Common failure modes: tension, compression, shear, or a combination

These supports produce upward

reactions as a result of the

applied load, P.


TENSION FAILURE

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COMPRESSION FAILURE


SHEAR FAILURE


REINFORCED CONCRETE BEAM IN FLEXURE

Flexure


COMMON STEEL BEAM CROSS SECTIONS

Steel unit weight ≈ 490 lb/ft3


COMMON REINFORCED CONCRETE BEAM CROSS SECTIONS

• Reinforcement usually appears at the top or bottom, sometimes in the middle, or a combination of 2 or 3 of those locations

With stirrups

Concrete unit weight ≈ 150 lb/ft3


CLASSIC BEAM THEORY

• Euler-Bernoulli Beam Theory: provides a way to calculate load-carrying capacity and deflection characteristics of beams

Bernoulli1700-1787

Euler1707-1783

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CAPACITY VS. DEMAND

• Capacity: maximum moment the element or beam is capable of withstanding without failure

• Demand: based on anticipated loads, the minimum required moment the element or beam must withstand without failure

Capacity Must Be > Demand


BENDING STRESS (PSI OR KSI)

• Beams will always go through bending

• Bending produces flexure, and introduces both compressive and tensile stresses


BENDING STRESS (PSI OR KSI)

• Bending stresses increase and decrease as the load on the beam increases and decreases

• Bending stress: σ = M*c / I, or: f = M*y / I• M: moment (lb*in. or kip*ft)

• c: distance from neutral axis

to top or bottom extreme fiber (in.)

• I: moment of inertia (in.4)• Note: ACI uses f to represent stress; we’ll use Φ


RUPTURE STRESS (PSI OR KSI)

• Rupture stress is the stress that causes the first crack to occur

• Rupture stress: fr = 7.5*√f’c• f’c: concrete compressive strength (psi)

• Note: must use f’c in psi for this equation, not ksi


SUMMATION OF FORCES

• x-direction (horizontal)

• y-direction (vertical)

• All forces in the x-direction add up to zero

• All forces in the y-direction add up to zero

• A force acting at an angle has an x-component and a y-component


MOMENTS (LB*FT OR KIP*FT)

• Nominal moment (Mn): service moment, the beam’s calculated moment capacity

• Ultimate moment (Mu = ΦMn): factored nominal moment, beam’s ultimate capacity

Moment = Effort x Lever Arm

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MOMENTS

• What are some everyday examples?


STRENGTH REDUCTION FACTOR (UNITLESS)

• Strength reduction factor (Φ, phi): accounts for uncertainties in material strength, dimensions, and workmanship

• Varies from Φ = 0.90 for beams to Φ = 0.65 for columns

Mn = 1000 lb*ft

Mu = Φ*Mn = 0.90 * 1000 lb*ft = 900 lb*ft


CRACKING MOMENT (LB*FT OR KIP*FT)

• Cracking moment (Mcr): moment at which the beam is expected to experience its first crack

• Cracking moment: Mcr = fr*I / y• fr: rupture stress (psi or ksi)

• I: moment of inertia (in.4)

• y: distance to middle of beam (in.)


REINFORCED CONCRETE BEAM CROSS SECTION

b

hAs


COMPRESSION AND TENSION

Never assume the neutral axis falls at

the beam’s horizontal centerline!

The distance to the NA will be used to find Mn.

The distance to the beam’s centerline will be used to find stresses before any

cracks occur.


REINFORCED CONCRETE BEAM CROSS SECTION

Dimensions:

• b: width (in.)

• h: height (in.)

• d: effective depth (in.)

Other Known Values:

• As: area of steel (in.2)

• fy: steel tensile strength (psi)

• f’c: concrete compressive

strength (psi)b

hd

As

NA

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Neutral Axis

c

ACTUAL COMPRESSION AND TENSION STRESS VARIATION

Compression

Tension

Everything above NA = Compression

Everything below NA = Tension


IDEALIZED COMPRESSION AND TENSION STRESS VARIATION

N.A.

c a Compression

Tension

0.85*f’c

b

h


RESULTANT COMPRESSION AND TENSION FORCES

N.A.

c a C = 0.85 * f’c * a * b

T = As * fy

d – a/2

C and T are “coupled forces” meaning they’re equal and opposite.C = T


NOMINAL MOMENT CALCULATION

C = 0.85 * f’c * a * b T = As * fy

Mn = C * (d – 0.5a) Mn = T * (d – 0.5a)

Diagram Reference

Sheet


NOMINAL MOMENT CALCULATION

C = 0.85 * f’c * a * b T = As * fy

Mn = C * (d – 0.5a) Mn = T * (d – 0.5a)

Mn = 0.85*f’c*a*b*(d-0.5a) Mn = As*fy*(d-0.5a)

C = T

0.85*f’c*a*b = As*fy

Then rearrange to solve for “a”:

a = As*fy / 0.85*f’c*b


REINFORCEMENT RATIO (UNITLESS)

• The amount of steel reinforcement should be limited

• A “balanced condition” occurs when the stress in the steel and the stress in the concrete are both at their yield points

• The amount of steel required to reach the balanced condition is given by the reinforcement ratio (ρ, rho): ρ = As / b*d

Max. and min. reinforcement ratios exist for

different conditions.

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EQUATION REFERENCE SHEET

• Bending stress:

σ = M * c / I

• Rupture stress:

fr = 7.5 * √f’c• Cracking moment:

Mcr = fr * y / I

• C and T Forces:

C = 0.85 * f’c * a * b

T = As* fyC = T

• Nominal moment:

Mn = C * (d – 0.5a)

Mn = 0.85 * f’c * a * b * (d-0.5a)

Mn = T * (d – 0.5a)

Mn = As* fy * (d-0.5a)

a = As * fy / 0.85 * f’c * b

• Ultimate moment:

Mu = Φ * Mn, where Φ = 0.9

• Reinforcement ratio:

ρ = As / b * d


REBAR REFERENCE SHEET


EXAMPLE 1


EXAMPLE 1

Calculate the bending stresses in the extreme fibers of the beam shown below if it is subjected to a moment of 50 kip*ft.

b

h

GIVEN:

Beam:I = 5,832 in.4

h = 18 in.


EXAMPLE 2


EXAMPLE 2

Calculate the cracking moment for the beam shown below if f’c = 5,000 psi.

b

h

GIVEN:

Beam:I = 5,832 in.4

h = 18 in.

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EXAMPLE 3 – 5


EXAMPLE 3

Find the ultimate moment capacity of a reinforced concrete beam with the following dimensions.

GIVEN:

Steel:Three #9 barsfy = 80,000 psi

Concrete:f’c = 10,000 psi


EXAMPLE 4

What happens to the moment capacity if the steel is accidentally placed 0.5” higher in the form?

GIVEN:

Steel:Three #9 barsfy = 80,000 psi

Concrete:f’c = 10,000 psi

d = 14.5”


EXAMPLE 5


EXAMPLE 5

• The equipment that was supposed to use for this installation was past due for inspection. The angle of the 4 chains was originally expected to be 70 degrees with respect to the product’s horizontal surface. With the new equipment, the actual lifting angle during installation is going to be 60 degrees with respect to the product’s horizontal surface. How does this change the maximum moment the product experiences during lifting? Will the moment demand exceed the capacity?


EXAMPLE 5

70° 70° 60° 60°

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EXAMPLE 6

Find the maximum moment acting on the beam from Example 3. Will the moment demand exceed the capacity?

R1R5R3

R4R2


RESOURCES

NPCA: www.precast.org

(800) 366-7731

Kayla Hanson: [email protected]

American Concrete Institute: www.concrete.org

Concrete Reinforcing Steel Institute: www.crsi.org


BASICS OF REINFORCED CONCRETE DESIGN

Kayla Hanson, Director of Technical Services

National Precast Concrete Association

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http://www.precast.org/

mailto:[email protected]

http://www.concrete.org/

http://www.crsi.org/

Documents

OUTLINE - Precast Concreteprecast.org/wp-content/uploads/2019/03/CP4-Reinforced-Concrete-Design-Kayla...BASICS OF REINFORCED CONCRETE DESIGN Kayla Hanson, Director of Technical Services