Reinforced Concrete Design

4.

Flexural Analysis and Design of Beams

4.1.

Fundamental Assumptions

4.1.1.

Plane Sections Remain Plane

4.1.2.

Perfect Bond Between Materials

4.1.3.

Tensile Strength of Concrete Neglected

4.1.4.

Material Stress-Strain Relationships

4.2.

Beam Behavior Under Loading

4.2.1.

Stages of Loading

4.2.1.1.

Uncracked Stage

4.2.1.2.

Cracked Elastic Stage

4.2.1.3.

Inelastic Stage

4.2.1.4.

4.2.2.

Load-Deflection Response

4.2.3.

Moment-Curvature Relationship

4.3.

Singly Reinforced Rectangular Beams

4.3.1.

Elastic Analysis

4.3.1.1.

Transformed Section Method

4.3.1.2.

Cracked Section Analysis

4.3.1.3.

Moment of Inertia

4.3.1.4.

Cracking Moment

4.3.2.

Ultimate Strength Analysis

4.3.2.1.

Strain Distribution

4.3.2.2.

Stress Distribution

4.3.2.3.

Force Equilibrium

4.3.2.4.

Moment Equilibrium

4.3.3.

Whitney Stress Block

4.3.3.1.

Equivalent Rectangular Stress Block

4.3.3.2.

Stress Block Parameters

4.3.3.3.

Depth of Stress Block

4.3.4.

Design Equations

4.3.4.1.

Nominal Moment Capacity

4.3.4.2.

Required Steel Area

4.3.4.3.

Design Moment Capacity

4.4.

Reinforcement Ratios and Failure Modes

4.4.1.

Balanced Reinforcement Ratio

4.4.1.1.

Balanced Strain Condition

4.4.1.2.

Significance in Design

4.4.2.

Under-Reinforced Sections

4.4.2.1.

Tension-Controlled Failure

4.4.2.2.

Ductile Behavior

4.4.2.3.

Design Advantages

4.4.3.

Over-Reinforced Sections

4.4.3.1.

Compression-Controlled Failure

4.4.3.2.

Brittle Behavior

4.4.3.3.

Design Limitations

4.4.4.

Transition Zone

4.4.4.1.

Net Tensile Strain Limits

4.4.4.2.

Strength Reduction Factors

4.4.5.

Minimum Reinforcement

4.4.5.1.

Code Requirements

4.4.5.2.

Crack Control Considerations

4.4.6.

Maximum Reinforcement

4.4.6.1.

Practical Limits

4.4.6.2.

Constructability Issues

4.5.

Doubly Reinforced Beams

4.5.1.

Need for Compression Reinforcement

4.5.2.

Analysis Methods

4.5.2.1.

Strain Compatibility

4.5.2.2.

Stress in Compression Steel

4.5.2.3.

Force Equilibrium

4.5.3.

Design Procedure

4.5.3.1.

Contribution of Compression Steel

4.5.3.2.

Total Moment Capacity

4.5.4.

Practical Considerations

4.5.4.1.

Minimum Cover Requirements

4.5.4.2.

Lateral Support of Compression Bars

4.6.

T-Beams and L-Beams

4.6.1.

Effective Flange Width

4.6.1.1.

Code Provisions

4.6.1.2.

Geometric Limitations

4.6.2.

Neutral Axis in Flange

4.6.2.1.

Rectangular Beam Analysis

4.6.2.2.

Design Simplifications

4.6.3.

Neutral Axis in Web

4.6.3.1.

Flange and Web Contributions

4.6.3.2.

Design Equations

4.6.4.

Practical Design Considerations

4.6.4.1.

Construction Sequence Effects

4.6.4.2.

Shrinkage and Temperature Effects

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3. Design Philosophies and Building Codes

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5. Shear and Torsion in Beams