3. Design of Reinforced Concrete Beams by Limit State Method

3.1 Concept of Limit State, Stress Block Diagram, Introduction to Singly and Doubly Reinforced Sections, IS 456

Concept of Limit State: The Limit State Method is a design approach used in structural engineering that ensures a structure performs safely throughout its lifespan. It is based on the concept of the limit state, which refers to the condition beyond which a structure or part of a structure no longer fulfills the intended purpose.

There are two primary limit states in structural design:

Ultimate Limit State (ULS): This limit ensures the structure's safety from collapse or failure under extreme loading conditions.
Serviceability Limit State (SLS): This limit ensures that the structure remains functional and comfortable during regular use, without excessive deflection or cracking.

Stress Block Diagram: A stress block diagram is a graphical representation used to calculate the bending strength of reinforced concrete beams. The stress block shows the distribution of compressive and tensile stresses across the cross-section of the beam. It is based on the assumption that the concrete in compression behaves as a rectangular stress block and the reinforcement in tension carries the tensile stresses.

For a rectangular section, the stress block is typically assumed as follows:

Compressive Stress in Concrete: The stress block is assumed to be rectangular with a constant stress $f_{cd}$ (design compressive strength of concrete) up to a depth $a$ from the top of the beam.
Tensile Stress in Steel: The tensile stress in the reinforcement is assumed to be uniform up to the yield strength of the steel $f_{yd}$ .

Singly Reinforced Section: A singly reinforced beam is a beam that has only one layer of tension reinforcement (steel bars) placed at the bottom of the beam. The concrete in compression (at the top of the beam) and steel in tension resist the applied loads.

Doubly Reinforced Section: A doubly reinforced beam has both tension and compression reinforcement. This is typically used in cases where the beam has a large moment and the required depth for a singly reinforced beam would be excessively large.

IS 456: IS 456 is the Indian Standard code for the design and construction of reinforced concrete structures. It provides the guidelines for designing concrete beams, including the limit state method, materials, and safety factors.

3.2 Design of Singly Reinforced Beam, Concept of Under Reinforced, Over Reinforced and Balanced Section, Simple Numerical Problem on Ultimate Moment of Resistance and Design of Beam Section

Design of Singly Reinforced Beam: In a singly reinforced beam, the reinforcement is placed in tension. The steps for designing a singly reinforced beam are as follows:

Determine the design moment: The first step in the design process is to calculate the ultimate moment (M_u) that the beam needs to resist.
Calculate the depth of neutral axis ( $x$ ): The neutral axis is the point where the compressive stress in concrete and tensile stress in steel are equal. For singly reinforced beams, the neutral axis lies within the depth of the beam.
Calculate the lever arm (z): The lever arm is the distance between the centroid of the tensile reinforcement and the point of maximum compression in concrete.
Design the reinforcement area (Ast): Based on the moment and lever arm, calculate the required area of tensile reinforcement using the formula:
$M_u = f_{yd} \cdot A_s \cdot z$
where:
- $M_u$ = Ultimate moment
- $f_{yd}$ = Design yield strength of steel
- $A_s$ = Area of tensile reinforcement
- $z$ = Lever arm
Check the section for serviceability and crack width: Ensure the beam satisfies the deflection and crack width limits as per IS 456.

Under Reinforced, Over Reinforced, and Balanced Section:

Under Reinforced Section: This is a section where the area of reinforcement is less than the balanced condition. It is designed to fail in ductile mode, where the steel yields first, and then the concrete crushes. This is the most commonly designed type, as it provides safety with controlled failure.
Over Reinforced Section: In an over-reinforced section, the area of reinforcement is greater than required, resulting in the concrete crushing before the steel yields. This is an unsafe design as it leads to sudden failure.
Balanced Section: A balanced section is one where the steel and concrete are proportioned such that both the steel reaches its yield strength and the concrete reaches its crushing strength at the same time. It is the ideal condition, but it's difficult to achieve practically.

Numerical Example for Ultimate Moment of Resistance: Let’s assume a simply supported beam with a span of 4 meters, carrying a moment of 50 kNm. Assume the beam has the following parameters:

Concrete grade: M25 (f_ck = 25 MPa)
Steel grade: Fe415 (f_y = 415 MPa)
Effective depth of beam (d) = 500 mm
Width of beam (b) = 230 mm
Design moment (M_u) = 50 kNm

Step 1: Assume the value of neutral axis depth $x$ . Step 2: Calculate the design moment and area of steel reinforcement. Step 3: Calculate $A_s$ based on the moment.

3.3 Design of Doubly Reinforced Sections, Stress and Strain Diagrams, Depth of Neutral Axis, Simple Numerical Problems on Ultimate Moment of Resistance of Reinforced Beam, Calculation of Ast and Asc

Design of Doubly Reinforced Sections: In a doubly reinforced section, both the top and bottom of the beam are reinforced to resist large moments. This type of section is designed when the moment is too high for a singly reinforced section. The additional compression reinforcement helps prevent excessive depth and increases the beam's moment resistance.

Stress and Strain Diagrams: The stress-strain diagram for doubly reinforced sections involves:

Concrete in Compression: The stress is assumed to be uniformly distributed up to a depth $a$ (called the depth of the stress block) from the top of the beam.
Tensile Steel in Tension: The tensile steel reinforcement carries the tension in the lower part of the beam.
Compression Steel: The top reinforcement carries compression, which is needed in cases of very high moments.

The strain distribution in the beam is assumed to be linear, with the maximum strain occurring at the extreme fiber in compression.

Depth of Neutral Axis: In a doubly reinforced section, the neutral axis is located at a depth $x$ from the top of the beam. The position of the neutral axis depends on the ratio of the compression reinforcement to the tension reinforcement.

Calculation of Ast and Asc: The area of tension reinforcement $A_s$ and compression reinforcement $A_{sc}$ are calculated based on the design moment and the position of the neutral axis. The ultimate moment of resistance for a doubly reinforced section can be calculated using the formula:

M_u = f_{yd} \cdot A_s \cdot (d - a) + f_{cd} \cdot A_{sc} \cdot (d - x)

Where:

$A_s$ = Area of tension reinforcement
$A_{sc}$ = Area of compression reinforcement
$f_{yd}$ = Design yield strength of steel
$f_{cd}$ = Design compressive strength of concrete
$x$ = Depth of neutral axis

Hindi Notes:

3.1 लिमिट स्टेट का सिद्धांत, तनाव ब्लॉक आरेख, सिंगली और डबल्ली रीइंफोर्स्ड सेक्शन का परिचय, IS 456

लिमिट स्टेट का सिद्धांत: लिमिट स्टेट विधि संरचनाओं के डिजाइन का एक दृष्टिकोण है जो संरचना को उसके जीवनकाल के दौरान सुरक्षित रूप से कार्य करने की सुनिश्चितता प्रदान करता है। इसमें दो प्रमुख लिमिट स्टेट होते हैं:

अल्टीमेट लिमिट स्टेट (ULS): यह संरचना की सुरक्षा सुनिश्चित करता है ताकि वह अत्यधिक लोडिंग स्थितियों में भी विफल न हो।
सर्विसेबिलिटी लिमिट स्टेट (SLS): यह सुनिश्चित करता है कि संरचना सामान्य उपयोग के दौरान अत्यधिक विकृति या दरारों से मुक्त रहे।

तनाव ब्लॉक आरेख: तनाव ब्लॉक आरेख एक ग्राफिकल प्रतिनिधित्व है, जिसका उपयोग कंक्रीट बीम के मोड़ की ताकत को गणना करने के लिए किया जाता है। यह आरेख कंक्रीट और स्टील के तनाव वितरण को दर्शाता है।

सिंगली और डबल्ली रीइंफोर्स्ड सेक्शन:

सिंगली रीइंफोर्स्ड बीम: इस बीम में केवल एक परत (तल में) तनाव reinforcement होती है।
डबल्ली रीइंफोर्स्ड बीम: इसमें दोनों, तंग और संपीड़न reinforcement होती है, जब मोमेंट बहुत अधिक होता है।

3.2 सिंगली रीइंफोर्स्ड बीम का डिज़ाइन, अंडर रीइंफोर्स्ड, ओवर रीइंफोर्स्ड और बैलेंस्ड सेक्शन, अल्टीमेट मोमेंट ऑफ रेसिस्टेंस पर सरल अंकगणितीय समस्या

सिंगली रीइंफोर्स्ड बीम डिज़ाइन: इसमें बीम के नीचे तनाव reinforcement लगाई जाती है। बीम का डिज़ाइन मोमेंट, न्युट्रल अक्ष की गहराई, और लीवर आर्म से किया जाता है।

अंडर और ओवर रीइंफोर्स्ड सेक्शन:

अंडर रीइंफोर्स्ड: कम reinforcement क्षेत्र होता है, जिससे स्टील पहले टूटता है।
ओवर रीइंफोर्स्ड: अधिक reinforcement होती है, जिससे कंक्रीट जल्दी टूटता है।
बैलेंस्ड: इसमें कंक्रीट और स्टील समान समय पर क्रश होते हैं।

3.3 डबल्ली रीइंफोर्स्ड सेक्शन्स, तनाव और विकृति आरेख, न्युट्रल अक्ष की गहराई, अल्टीमेट मोमेंट ऑफ रेसिस्टेंस पर सरल अंकगणितीय समस्याएँ

डबल्ली रीइंफोर्स्ड सेक्शन डिज़ाइन: इसमें दोनों तंग और संपीड़न reinforcement होती है, जिसका उपयोग तब किया जाता है जब मोमेंट बहुत अधिक होता है।

तनाव और विकृति आरेख: यह आरेख दर्शाता है कि कंक्रीट और स्टील की तनाव वितरण कैसे होती है, जिसमें दोनों reinforcement की भूमिका है।

न्युट्रल अक्ष की गहराई और मोमेंट की गणना: न्युट्रल अक्ष की गहराई और दोनों reinforcement क्षेत्रों की गणना करके अंतिम मोमेंट का प्रतिरोध $A_s$ और $A_{sc}$ की गणना की जाती है।

3. Design of Reinforced Concrete Beams by Limit State Method