Beam Capacity Checking

Description
Core Engineering Principle: Concrete Beam Moment Resistance
This calculation follows the principle that in a reinforced concrete beam, concrete resists compression while steel reinforcement resists tension, and the beam's strength depends on achieving equilibrium between these forces.
1. Force Equilibrium Concept
- When a concrete beam bends, the top portion goes into compression (concrete handles this well)
- The bottom portion goes into tension (concrete is weak here, so steel rebar is needed)
- For the beam to be stable: Compression Force = Total Tension Force
- This is expressed as: C = T₁ + T₂ + T₃ (sum of all tension forces)
2. Stress Block Method
- The calculation uses the "equivalent rectangular stress block" - a simplified way to represent the actual curved stress distribution in concrete
- The concrete compression is calculated as: C = 0.85 × f'c × a × b
- Where 'a' is the depth of the equivalent compression block
3. Multi-Layer Reinforcement Analysis This beam has steel bars at different depths (layers), which affects capacity:
- Different grades of steel (Grade 40 vs Grade 60) have different yield strengths
- Different positions (closer to or farther from the compression zone) contribute differently to moment resistance
- Each layer contributes: T = As × fy (steel area × yield strength)
4. Moment Arm Principle
- The moment capacity depends not just on the forces, but on their lever arms (distances)
- Each tension force creates moment: M = T × (distance from compression center)
- Total moment = sum of all individual contributions
5. Tension-Controlled Verification
- The calculation checks that the steel yields before concrete crushes
- This ensures ductile failure (steel stretches and gives warning) rather than brittle concrete failure
- The fs > 1000 MPa check confirms the beam will fail safely
6. Design Comparison
- The "Actual Beam" represents an existing or proposed reinforcement layout
- The "Designed Beam" represents a simplified alternative arrangement
- Comparing their capacities determines if the actual beam is adequate
This is like comparing two different arrangements of support cables in a suspension system - you need to ensure the actual arrangement can handle at least as much load as your minimum design requirement.
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