Section Properties and Elastic Flexural Stress

Section Properties and Elastic Flexural Stress

Property first, stress second: The equation σ = My/I is simple only after y and I have been calculated for the correct geometry, material transformation, and cracked or uncracked state.

Centroid and Neutral Axis

For a homogeneous linear-elastic section in pure bending, the elastic neutral axis passes through the area centroid. Choose a datum and keep signed coordinates consistent:

ȳ = Σ(A_i y_i) / ΣA_i.

For a built-up homogeneous section, calculate centroidal inertia with the parallel-axis theorem:

I = Σ(I_i,c + A_i d_i²),

where I_i,c is each piece's inertia about its own centroidal axis and d_i is the distance to the composite centroid. Do not use an edge distance in place of d_i. Voids are handled as negative areas and negative inertias using the same datum.

These labels are not interchangeable. A concrete member may use gross stiffness for one prescribed check and cracked or effective stiffness for another. The problem statement and current reference control the choice. This 2026 guide uses the PE Civil Reference Handbook active for the test date; later reference editions do not change which property the question actually requests.

Transforming Materials

To express a composite section in reference material r, define the modular ratio

n_i = E_i/E_r

and replace material i with transformed area A'_i = n_i A_i. Use the transformed areas to find the neutral axis and transformed I. The transformed-section strain distribution is common to all bonded materials. Stress in the reference material is obtained from My/I; actual stress in material i is n_i times the stress that the reference material would have at that same fiber.

Always state the reference material. If the reciprocal modular ratio is used without relabeling the reference, both the transformed area and recovered stress will be wrong. Perfect composite action and linear elasticity are assumptions, not automatic facts for every built-up member.

Section Modulus and Flexural Stress

With a chosen moment sign, elastic normal stress varies linearly from zero at the neutral axis:

σ = -My/I.

The minus sign is one common convention: positive sagging moment compresses fibers at positive y. If only magnitude is requested, |σ| = M|y|/I. Section modulus is S = I/c, so the extreme-fiber magnitude is M/S. For an unsymmetric depth, use S_top = I/c_top and S_bot = I/c_bot; one section does not necessarily have one elastic section modulus.

Worked Cracked Transformed Section

A 12-in-wide, 20-in-deep reinforced concrete rectangle has A_s = 4.0 in² of tension steel at effective depth d = 17 in from the top. Use a problem-given modular ratio n = E_s/E_c = 8. After flexural cracking, neglect tensile concrete and treat the steel area as concentrated at d. Find the cracked neutral-axis depth c, transformed inertia, and stress magnitudes under M = 120 kip-ft.

The first moment of transformed area about the neutral axis must balance. The compression concrete is a 12 × c rectangle; transformed steel is nA_s = 32 in²:

12c(c/2) = 32(17 - c)

6c² + 32c - 544 = 0,

which gives the positive root c = 7.22 in below the top. The neutral axis is not at the 10-in gross-section centroid.

The cracked transformed inertia referenced to concrete is

I_cr = 12c³/3 + nA_s(d - c)²

I_cr = 12(7.2216³)/3 + 32(17 - 7.2216)² = 4,566 in⁴.

Convert moment before using inch-based properties: M = 120(12) = 1,440 kip-in. The top-concrete compression magnitude is

f_c = Mc/I_cr = 1,440(7.2216)/4,566 = 2.28 ksi.

At the steel level, the reference-concrete stress magnitude would be

f_ref = 1,440(17 - 7.2216)/4,566 = 3.08 ksi.

Recover actual steel stress:

f_s = n f_ref = 8(3.08) = 24.7 ksi.

For contrast, a gross-rectangle shortcut ignoring the bar transformation gives I_g = 12(20³)/12 = 8,000 in⁴ and top stress 1,440(10)/8,000 = 1.80 ksi. That smaller value is not an alternative answer to the cracked-section question; it comes from a different neutral axis and stiffness model.

Reliable Exam Workflow

1. Draw the section and axes; label every dimension and material.
2. Choose gross, uncracked transformed, or cracked transformed behavior from the question.
3. Select a reference material and calculate every transformed area.
4. Find the actual neutral axis; then use each piece's parallel-axis term to compute I.
5. Convert the applied moment into units consistent with I and fiber distance.
6. Evaluate stress at the requested fibers and convert transformed stress back to actual material stress.
7. Check that stress is zero at the neutral axis, linear across each elastic material, and largest in magnitude at a farthest relevant fiber.

Never combine y from one property model with I from another. That silent gross/cracked mix can look dimensionally correct while producing a physically unrelated stress.