Structural Analysis Fundamentals / Composite beams
Reinforced concrete sections
Cracked-section analysis: where the neutral axis goes when the concrete cannot pull.
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Read the lesson register and outstanding review notesWhy this matters
Reinforced concrete is the most-used structural material on earth, and it only works because of one deliberate compromise: the concrete is allowed to crack. Understanding the elastic cracked section explains where the steel goes, why the neutral axis sits high, and what the lever arm actually is.
By the end of this lesson you should be able to
- State the cracked-section assumptions
- Find the neutral axis depth from first moments of area
- Calculate concrete and steel stresses
- Explain the lever arm and check by two routes
Concrete has a much lower tensile than compressive strength. In the ideal cracked-section model, neglect its tensile contribution and assign the tension force to reinforcement. Actual members retain some tension stiffening between cracks.
Cracking may be expected in reinforced-concrete bending, but crack width, durability, stiffness and reinforcement stress still need assessment. Reinforcement controls cracking; it does not guarantee that cracks stay closed or are harmless.
From first principles
Neutral axis of a cracked section
We want to show: the depth x of the neutral axis, from the condition that the section carries no net axial force.
The section is in pure bending, so the total compression must equal the total tension. Above the neutral axis we have a triangle of concrete compression. Below it we have steel in tension and nothing else, because the concrete there has cracked away. Balancing those two, after transforming the steel into equivalent concrete, gives one equation with one unknown — the neutral axis depth. It comes out as a quadratic.
Worked example
Cracked reinforced-concrete section
Given
- Rectangular section 300 mm wide, effective depth d = 450 mm
- Tension steel As = 1500 mm²
- Modular ratio n = 15
- Applied moment 150 kN·m
Find
The neutral axis depth, and the stresses in the concrete and the steel.
Assumptions
- Cracked section, concrete carries no tension
- Elastic materials
- Perfect bond
Practice
A composite beam is made of two materials with E₁ = 200 000 N/mm² and E₂ = 25 000 N/mm². What is the modular ratio n = E₁/E₂?
Practice
A cracked RC section is 250 mm wide with effective depth 400 mm, As = 1200 mm² and n = 15. What is the neutral axis depth x?
Check yourself
Why is the concrete term in Icr equal to bx³/3 rather than bx³/12?
Practice
A cracked rectangular section is 300 mm wide with effective depth 450 mm, reinforced with As = 1500 mm² and a modular ratio of 15. How deep is the neutral axis below the compression face, in mm?
Summary
- Cracked concrete carries no tension; the steel takes all of it
- bx²/2 = nAs(d − x) locates the neutral axis, independent of the applied moment
- Icr = bx³/3 + nAs(d − x)², with /3 because the axis is at the edge of the block
- σc = Mx/Icr and σs = nM(d − x)/Icr
- The lever-arm route z = d − x/3 gives an independent check on the steel stress