Reinforced Concrete Analysis Types:

1. Axial Force and Bending Moment

   1. Pure flexure design example

2. Moment Curvature Analysis

Axial Force and Bending Moment:

Reinforced concrete analysis for axial force and bending moment is usually performed by assuming a given strain value at the extreme compression fiber with a linear strain distribution over the depth of the section.  The stress distribution typically assumes a rectangular stress block with a depth equal to some fraction of the neutral axis depth and a magnitude equal to some fraction of the concrete compressive strength.

Design Parameters:

1. Stress and strain

• Depth to neutral axis = c
• Maximum concrete strain = e
• Concrete compressive strength = f^'_c
• Reinforcing yield strength = f_y

2. Stress block

• Ratio of average concrete stress = a₁
• Ratio of stress block depth = b₁

3. Reduction factors (American, ACI 318)

• Reinf reduction factor for tension and flexure = f
• Reinf reduction factor for comp and flexure = f

Note: Strength reduction factors are used in the American codes, both ultimate strength design and load-resistance factor design. These factors are applied to the computed strength based on the mode of failure.

4. Resistance factors (Canadian, CSA A23.3)

• Concrete resistance factor = f_c
• Reinforcement resistance factor = f_r

Note: Resistance factors are used in the Canadian codes and are applied directly to the material strengths without regard to the mode of failure.

Pure Flexure Design Example:

Determine the bending moment resistance of a rectangular beam with tension reinforcement and no axial loads.

Depth to reinforcing = d = 21 inches
Beam width = b = 12 inches
Reinforcing steel area = A_s = 3.0 in²
Concrete compressive strength = f^'_c = 4000 psi
Reinforcing yield strength = f_y = 60000 psi

1. Determine moment capacity per ACI 318

Maximum concrete strain = e = 0.003
Reduction factor for flexure = f = 0.90
Ratio of average concrete stress = a₁ = 0.85
Ratio of stress block depth = b₁= 0.85

1. Solve for a using the SF = 0.

A_s f_y = a b b₁ f^'_c
3.0(60000) = a (12)(0.85)(4000)
a = 4.41 inches

2. Determine moment from force couple

f M_n = f A_sf_y (d - a / 2)
f M_n = 0.9(3.0)(60)(21 - 4.41 / 2) / 12
f M_n = 254 k-ft

2. Determine moment capacity per CSA A23.3

Maximum concrete strain = e = 0.0035
Concrete resistance factor = f_c = 0.60
Reinforcement resistance factor = f_r = 0.85
Ratio of average concrete stress = a₁ = 0.81
Ratio of stress block depth = b₁= 0.9

1. Solve for "a" using the SF = 0.

f_r A_s f_y = a b b₁ f_c f^'_c
0.85(3.0)(60000) = a (12)(0.81)(0.6)(4000)
a = 6.56 inches

2. Determine moment from force couple

M_r = f_r A_sf_y (d - a / 2)
M_r = 0.85(3.0)(60)(21 - 6.56 / 2) / 12
M_r = 226 k-ft

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