Login
OR
Don't worry. We'll fix this for you!
Concrete is a widely used construction material known for its excellent compressive strength but relatively poor tensile strength. In conventional reinforced concrete, steel reinforcement bars (rebars) are embedded to resist tensile stresses. However, under heavy loads or long spans, concrete members may develop cracks due to tensile stresses, affecting durability and serviceability.
Prestressed concrete is an advanced form of concrete construction where internal stresses are introduced deliberately before the application of external loads. These internal stresses are designed to counteract the tensile stresses that occur during service, effectively keeping the concrete mostly in compression. This technique enhances the load-carrying capacity, reduces cracking, and allows for longer spans and thinner sections.
Prestressing is essential in many civil engineering applications such as bridges, industrial floors, water tanks, and high-rise buildings, where structural efficiency and durability are critical. It offers several advantages over conventional reinforced concrete, including reduced material usage, improved structural performance, and better control over deflections and cracks.
There are two main types of prestressing methods:
In this chapter, we will explore the principles, analysis, and design of prestressed concrete members, focusing on practical problem-solving approaches relevant for competitive exams.
The fundamental idea behind prestressing is to apply a compressive force to a concrete member so that the tensile stresses induced by external loads are neutralized or minimized. This is achieved by tensioning high-strength steel tendons within the concrete.
Consider a simply supported beam subjected to a downward load. The bottom fibers experience tension, which can cause cracking. By applying a prestressing force with an eccentricity (offset from the neutral axis), a bending moment is created that induces compression in the tensile zone, counteracting the external tensile stresses.
There are two common methods of applying prestress:
Once prestressing force is applied, it does not remain constant throughout the life of the structure. Various factors cause a reduction in the effective prestressing force, known as prestress losses. Understanding and accurately calculating these losses is crucial for safe and economical design.
The main types of prestress losses are:
| Loss Type | Typical Percentage Loss Range | Influencing Factors |
|---|---|---|
| Elastic Shortening | 2% - 5% | Concrete strength, tendon stress, member size |
| Creep | 3% - 7% | Concrete age, humidity, sustained load duration |
| Shrinkage | 2% - 5% | Concrete mix, curing conditions, member size |
| Relaxation of Steel | 2% - 4% | Type of steel, initial stress level, temperature |
| Anchorage Slip | 0.5% - 1% | Quality of anchorage, tensioning method |
A post-tensioned concrete beam is prestressed with an initial force of 1500 kN. Calculate the total prestress loss if the losses due to elastic shortening, creep, shrinkage, relaxation, and anchorage slip are 3%, 5%, 3%, 2%, and 0.5% respectively of the initial prestress force.
Step 1: Identify the initial prestressing force, \( P_i = 1500 \, \text{kN} \).
Step 2: Calculate individual losses:
Step 3: Sum all losses to find total loss:
\[ \Delta P = 45 + 75 + 45 + 30 + 7.5 = 202.5 \, \text{kN} \]
Step 4: Calculate effective prestressing force:
\[ P_{eff} = P_i - \Delta P = 1500 - 202.5 = 1297.5 \, \text{kN} \]
Answer: Total prestress loss is 202.5 kN, and the effective prestressing force is 1297.5 kN.
To ensure safety and serviceability, it is essential to calculate stresses at critical points in a prestressed concrete beam under combined prestressing and external loads. The stresses at the top and bottom fibers depend on the prestressing force, its eccentricity, and applied bending moments.
A rectangular prestressed concrete beam has a cross-section 300 mm wide and 500 mm deep. The initial prestressing force is 1200 kN applied with an eccentricity of 150 mm from the centroidal axis. The beam is subjected to an external bending moment of 60 kNm. Calculate the stresses at the top and bottom fibers of the beam. Take modulus of elasticity of concrete \(E_c = 30,000 \, \text{MPa}\).
Step 1: Calculate cross-sectional properties.
Area, \( A = 300 \times 500 = 150,000 \, \text{mm}^2 \)
Moment of inertia, \( I = \frac{b d^3}{12} = \frac{300 \times 500^3}{12} = 3.125 \times 10^9 \, \text{mm}^4 \)
Step 2: Convert external moment to N·mm:
\( M = 60 \, \text{kNm} = 60 \times 10^6 \, \text{Nmm} \)
Step 3: Calculate stress due to prestressing force \(P = 1200 \, \text{kN} = 1.2 \times 10^6 \, \text{N}\).
Distance from neutral axis to top fiber, \( y_{top} = -250 \, \text{mm} \) (negative for compression)
Distance to bottom fiber, \( y_{bottom} = +250 \, \text{mm} \)
Stress at fiber due to prestress:
\[ \sigma_p = \frac{P}{A} \pm \frac{P e y}{I} \]
Calculate direct stress:
\[ \sigma_{p, direct} = \frac{1.2 \times 10^6}{150,000} = 8 \, \text{MPa} \]
Calculate bending stress due to prestress:
\[ \sigma_{p, bending} = \frac{1.2 \times 10^6 \times 150 \times y}{3.125 \times 10^9} \]
At top fiber (\(y = -250\)):
\[ \sigma_{p, bending} = \frac{1.2 \times 10^6 \times 150 \times (-250)}{3.125 \times 10^9} = -14.4 \, \text{MPa} \]
Total prestress at top fiber:
\[ \sigma_{p, top} = 8 - 14.4 = -6.4 \, \text{MPa} \quad (\text{compression}) \]
At bottom fiber (\(y = +250\)):
\[ \sigma_{p, bending} = \frac{1.2 \times 10^6 \times 150 \times 250}{3.125 \times 10^9} = +14.4 \, \text{MPa} \]
Total prestress at bottom fiber:
\[ \sigma_{p, bottom} = 8 + 14.4 = 22.4 \, \text{MPa} \quad (\text{compression}) \]
Step 4: Calculate stress due to external bending moment:
\[ \sigma_{ext} = \frac{M y}{I} \]
At top fiber:
\[ \sigma_{ext, top} = \frac{60 \times 10^6 \times (-250)}{3.125 \times 10^9} = -4.8 \, \text{MPa} \]
At bottom fiber:
\[ \sigma_{ext, bottom} = \frac{60 \times 10^6 \times 250}{3.125 \times 10^9} = +4.8 \, \text{MPa} \]
Step 5: Calculate total stresses by adding prestress and external stresses:
Top fiber:
\[ \sigma_{top} = \sigma_{p, top} + \sigma_{ext, top} = -6.4 - 4.8 = -11.2 \, \text{MPa} \quad (\text{compression}) \]
Bottom fiber:
\[ \sigma_{bottom} = \sigma_{p, bottom} + \sigma_{ext, bottom} = 22.4 + 4.8 = 27.2 \, \text{MPa} \quad (\text{compression}) \]
Answer: The top fiber is under 11.2 MPa compression, and the bottom fiber is under 27.2 MPa compression.
Design a rectangular prestressed concrete beam section to resist a factored bending moment of 200 kNm. Use high strength concrete with characteristic compressive strength \( f_{ck} = 40 \, \text{MPa} \) and prestressing steel with yield strength \( f_{py} = 1860 \, \text{MPa} \). Assume effective prestressing force \( P = 1000 \, \text{kN} \) with eccentricity \( e = 150 \, \text{mm} \). Use limit state design principles.
Step 1: Assume beam width \( b = 300 \, \text{mm} \) and effective depth \( d = 500 \, \text{mm} \).
Step 2: Calculate resisting moment due to prestressing force:
\[ M_p = P \times e = 1000 \times 10^3 \times 0.15 = 150 \, \text{kNm} \]
Step 3: Required additional moment to be resisted by concrete compression and steel tension:
\[ M_r = 200 - 150 = 50 \, \text{kNm} \]
Step 4: Calculate limiting moment capacity of concrete section using IS code formulas (simplified here):
Assuming maximum compressive stress in concrete \( f_{cd} = \frac{f_{ck}}{\gamma_c} = \frac{40}{1.5} = 26.7 \, \text{MPa} \).
Step 5: Calculate neutral axis depth \( x_u \) using equilibrium of forces:
Let \( C = 0.36 f_{ck} b x_u \) be compressive force, and \( T = P \) be tensile force from prestressing steel.
Equating \( C = T \):
\[ 0.36 \times 40 \times 300 \times x_u = 1000 \times 10^3 \]
\[ x_u = \frac{1000 \times 10^3}{0.36 \times 40 \times 300} = 231.5 \, \text{mm} \]
Step 6: Calculate moment capacity \( M_u \):
\[ M_u = C \times (d - 0.42 x_u) = 1000 \times 10^3 \times (0.5 - 0.42 \times 0.2315) = 1000 \times 10^3 \times 0.403 = 403 \, \text{kNm} \]
Step 7: Since \( M_u = 403 \, \text{kNm} > M_r = 50 \, \text{kNm} \), the section is safe.
Answer: A beam section of 300 mm width and 500 mm effective depth with given prestressing force and eccentricity is adequate to resist the factored bending moment of 200 kNm.
List the advantages and typical applications of pre-tensioning and post-tensioning methods of prestressing with examples.
Pre-tensioning:
Post-tensioning:
Answer: Pre-tensioning is preferred for factory-made elements with repetitive shapes, while post-tensioning suits cast-in-place structures requiring flexibility and longer spans.
Calculate the modular ratio for prestressed concrete if the modulus of elasticity of steel is 200 GPa and that of concrete is 30 GPa. Also, find the transformed area of steel with cross-sectional area 1000 mm².
Step 1: Calculate modular ratio:
\[ m = \frac{E_s}{E_c} = \frac{200,000}{30,000} = 6.67 \]
Step 2: Calculate transformed area of steel:
\[ A_{steel, transformed} = m \times A_s = 6.67 \times 1000 = 6670 \, \text{mm}^2 \]
Answer: Modular ratio is 6.67, and the transformed steel area is 6670 mm².
When to use: During quick calculations in entrance exams to save time.
When to use: While solving stress analysis problems involving prestressing force.
When to use: When solving complex problems involving combined loads.
When to use: While analyzing prestressed concrete sections under service loads.
When to use: When preparing for conceptual questions and design problems.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|
