Qwestrum Engineering360 · Civil Engineering · Prestressed Concrete
Deflection and Cracking
Superpose the downward load deflection and the upward prestress camber to get the net deflection, apply a creep multiplier for long-term values, and check crack width for partially prestressed members.
Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.
Key formulas & points
Skim these first — then read the full notes below.
- Long-term deflection: multiply by 2–3 for creep effects
- Partial prestress: crack width limits stricter than RC
- SLS governs span/depth more often than ULS in PSC beams
Topic details
Introduction
Serviceability — deflection and cracking — frequently governs PSC design because prestressed members are slender. The net short-term deflection is the algebraic sum of the downward deflection from applied load and the upward camber produced by the eccentric prestress.
Scope in B.Tech and GATE syllabus
Because the prestress camber acts upward and the load deflection downward, a well-designed member can have very small or even zero net deflection at service. Long-term effects (creep and shrinkage) amplify the deflection, so a multiplier of 2–3 is applied to the sustained-load part.
Why this topic matters in practice
Cracking is controlled tightly: fully prestressed (Type 1/2) members are designed to remain uncracked or have minimal tension, while partially prestressed (Type 3) members permit limited, controlled cracking with crack widths checked against strict limits.
Key relations & formulas
(approximate)
Formulas (Indian textbook notation)
Formulas (Indian textbook notation)
Notation and sign conventions
Relation 1 —
(approximate)
Write this relation with symbols exactly as in Prestressed Concrete — N. Krishna Raju before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Formulas (Indian textbook notation)
Write this relation with symbols exactly as in Prestressed Concrete — N. Krishna Raju before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
Formulas (Indian textbook notation)
Write this relation with symbols exactly as in Prestressed Concrete — N. Krishna Raju before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Fundamentals and definitions
Deflection in PSC combines the sagging deflection from gravity loads with the hogging camber from the prestress. The camber from a parabolic tendon is proportional to Pe L²/EI, opposing the load deflection 5wL⁴/384EI, and the net value is what serviceability checks.
Governing relations in practice
Creep of concrete under the sustained prestress and dead load increases deflection over months and years, so the immediate elastic value is multiplied by a creep factor. Shrinkage adds further time-dependent movement. This is why long-term deflection, not the instantaneous value, usually governs.
Design and analysis considerations
Crack control depends on the class of prestressing. In fully prestressed members no flexural tension (or only a small limited value) is allowed at service, keeping the section uncracked and durable. Partially prestressed members trade some cracking for economy, and their crack width must be limited to protect the tendons from corrosion.
Advanced theory and extensions
Span-to-depth ratios provide a first-level deflection control without detailed calculation, and because PSC members are efficient in strength, this serviceability check often decides the required depth rather than the ultimate moment.
Assumptions and validity limits
State assumptions explicitly before using any relation for deflection and cracking — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Prestressed Concrete viva and GATE descriptive questions, listing valid assumptions often earns separate marks.
Step-by-step problem approach
1. Read the question and list given data with SI units (common in Prestressed Concrete papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to deflection and cracking.
4. Use equation 1:
5. Use equation 2:
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to deflection and cracking.
4. Use equation 1:
.
5. Use equation 2:
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.
Applications & exam relevance
Deflection and Cracking appears in long-span bridges and parking floors. In Indian civil curricula this topic is tested because it connects theory to PSC beams and loss calculations.
GATE and semester exams often combine deflection and cracking with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use deflection and cracking?" — answer with a lab, mini-project, or plant visit example if possible.
Common mistakes in exams
• Adding load deflection and camber instead of subtracting (they oppose).
• Reporting only the short-term deflection and ignoring the creep multiplier.
• Applying RC crack-width limits to a fully prestressed member where no cracking is allowed.
• Assuming the ULS governs depth when serviceability usually does.
• Reporting only the short-term deflection and ignoring the creep multiplier.
• Applying RC crack-width limits to a fully prestressed member where no cracking is allowed.
• Assuming the ULS governs depth when serviceability usually does.
Quick revision checklist
Before attempting deflection and cracking problems, confirm you can:
1. Long-term deflection: multiply by 2–3 for creep effects
2. Partial prestress: crack width limits stricter than RC
3. SLS governs span/depth more often than ULS in PSC beams
2. Partial prestress: crack width limits stricter than RC
3. SLS governs span/depth more often than ULS in PSC beams
Revise the solved examples in Prestressed Concrete — N. Krishna Raju and one previous-year GATE or university paper for this unit.
Worked examples
Try the problem first — open the solution when you are ready to check.
Net midspan deflection of a PSC beam
Problem
For a simply supported PSC beam, the downward deflection due to applied load is 18 mm and the upward camber due to prestress is 12 mm (both short-term). Estimate the net short-term deflection and the long-term deflection using a creep multiplier of 2.5 on the net value.
Solution
Net short-term deflection = 18 − 12 = 6 mm (downward). Applying the long-term multiplier: δ_long = 2.5 × 6 = 15 mm. This should be checked against the code limit (span/250 for total deflection), which for, say, a 10 m span is 40 mm — so the beam is satisfactory.
Conceptual check — Deflection and Cracking
Problem
In a Prestressed Concrete semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of deflection and cracking." What should a complete answer include?
Exams & GATE
Krishna Raju — state whether calculation is at transfer or service stage.
📖 Standard books (India)
Prestressed Concrete — N. Krishna Raju
Read: Syllabus unit
PSC systems, losses, and design
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