Lecture 3 – Moment Resistance of Rectangular Beams with Tension Steel Only PDF

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Lecture 3 – Moment Resistance of Rectangular Beams with Tension Steel Only

3.1 – Properly Reinforced Beams (Steel-Controlled Failure) • It fails in the steel-controlled mode: o Steel yields: the tensile strain is greater than or equal to the yield strain (𝜀s ≥ 𝜀y), while the stress in the steel is equal to the yield stress (fs = fy) o The maximum compressive strain in concrete (𝜀cmax) reaches the value of 0.0035 (𝜀cmax = 0.0035) • Consider the equilibrium of forces acting in figure c • The internal bending moment developed at any location within a beam is resisted by a force couple Tr and Cr • The resultant concrete compression force is equal to the product of the area of compression zone and the uniform stress of magnitude 𝛼 1$𝜑cf’c Cr = 𝛼 1$𝜑cf’c a b • The tension force in reinforcement is equal to the product of the factored stress in steel (𝜑 s fy) and the reinforcement area Tr = 𝜑 s fy As

• The equilibrium of forces in the horizontal direction gives Cr = T r 𝛼 1$𝜑cf’c a b = 𝜑 s fy As • By rearranging the above equation, the depth of the rectangular stress block (a) can be determined from 𝑎 =$

𝜑( 𝑓* 𝐴( 𝛼, 𝜑- 𝑓′- 𝑏

• The factored moment resistances (Mr) of the section can be obtained from either of the forces Tr or Cr by using the expression: Moment = force x distance 2

• In this case, the distance is 𝑑 − 3 2

∴ 𝑀5$ = $ 𝑇5 $(𝑑 − $ ) 3

2

𝑀5 = $𝜑s fy As$(𝑑 − $3 )

Example 3.1 : A typical cross-section of a reinforced concrete beam is shown in the figure below. The beam is reinforced with two 25M bars (2-25M) in the tension zone. The beam is properly reinforced. Concrete and steel material properties are given below. Find the factored moment resistance for the beam section. f’c = 25MPa fy = 400 MPa

3.2 – Overreinforced Beams (Concrete -Controlled Failure) • The concrete crushes before the steel yields: o The maximum compressive strain in the concrete (𝜀cmax) reaches the value of 0.0035 (𝜀cmax = 0.0035) o The tensile strain in the steel is less than the yield strain (𝜀s < 𝜀y), while the stress in the steel remains in the elastic range (fs < fy), so the tension reinforcement does not yield • The equation of equilibrium remains the same as in the case of properly reinforced beams: Cr = T r • Hook’s law applies and the stress in the steel is proportional to the strain: f s = Es 𝜀 s • The steel strain (𝜀s) can be determined using the similarity of triangles from the strain distribution diagram in the figure above:

where 𝜀cmax = 0.0035

𝜀-92: + 𝜀( 𝜀-92: =$ 𝑐 𝑑

also, rearranging the equation for a: 𝑐=

𝑎 𝛽,

where 𝛽 1 = 0.9

• In an overreinforced beam, the depth of the rectangular stress block is : 𝛼, 𝜑- 𝑓′- 𝑏 $𝑎3 + 𝑎 − 𝑑𝛽, = 0 𝜀-92: 𝜑( 𝐸( 𝐴(

Example 3.2: Consider the beam section shown below. The concrete and steel material properties are given below. Verify whether the beam is overreinforced and find the factored moment resistance. f’c = 25 MPa fy = 400 MPa Es = 200000 MPa

3.3 – Balanced Condition • The balanced condition is characterized by the simultaneous crushing of concrete and yielding of the tension reinforcement • The strain in the concrete reaches the maximum value 𝜀cmax = 0.0035 while the strain in the steel reaches the yield strain 𝜀s = $𝜀 y • The stress in the steel is equal to the yield stress; that is fs = fy • The balanced condition is shows below:

• Using the strain diagram, the following proportions can be obtained: 0.0035 0.0035 + $ 𝜀* = 𝑑 𝑐 Therefore, the following c/d ratio can be determined: 0.0035 𝑐 = 𝑑 0.0035 + $ 𝜀* • The area of tension reinforcement corresponding to the balanced condition is called the balanced reinforcement (Asb) and it can be found from the equation of equilibrium

Cr = Tr Tr = 𝜑 s fy Asb • Rearranging the equations will give Asb:

𝐴(C =

𝛼, 𝜑- 𝑓′- 𝑎$𝑏 𝐶5 = 𝜑( 𝑓* 𝜑( 𝑓*

• The corresponding reinforcement ratio, called the balanced reinforcement ratio (𝜌b), can be determined as : 𝜌b =

FGH CI

• The factored moment resistance (Mr) for a rectangular beam section in the balanced condition can be determined using the equation: 𝑎 𝑀5 = $ 𝜑( 𝑓* 𝐴(C (𝑑 − ) 2

Example 3.3: Consider the beam cross-section shown below. The concrete and steel material properties are given below. Find the following: a) The balanced area of reinforcement b) The balanced reinforcement ratio c) The factored moment resistance corresponding to the balanced condition f’c = 25 MPa fy = 400 MPa Es = 200000 MPa...