Deflection Calculation for Girder and Box Beams

Calculating the deflection of girder beams and box girders (often used in cranes and bridges) requires adherence to engineering standards. Below is an overview of the methodology, key formulas, and standards to follow:

Deflection Calculation for Girder and Box Beams

1. Deflection Formula for Simply Supported Beams
   For a beam subjected to a uniformly distributed load or a point load:
   • For a Point Load at the Center:

• Where:
  • $P$: Applied load (N or kN)
  • $L$: Span length (m)
  • $E$: Modulus of elasticity (Pa or N/m²)
  • $I$: Moment of inertia of the beam’s cross-section (m⁴)
• For a Uniformly Distributed Load (UDL):

• • Where:
    • $w$: Load per unit length (N/m)
• Box Girder Deflection:
  The formula is similar, but you must calculate the moment of inertia $I$ for the box girder’s cross-section, which typically has a higher bending stiffness due to its closed geometry.
  • Moment of inertia $I$ for box girders:

1. • Where:
     • $b,h$: Outer width and height of the box girder
     • binner,hinnerb_{\text{inner}}, h_{\text{inner}}binner​,hinner​: Inner dimensions of the box girder
2. Combined Loads:
   If the girder experiences both point loads and distributed loads, deflections from each source are calculated separately and summed.

Crane Girder Deflection Limits

Cranes often have stricter deflection limits than bridges to ensure operational stability and safety.

Standards for Crane Girders:

1. CMAA 70 & 74 (USA):
   Crane Manufacturers Association of America specifies deflection limits for crane girders:
   • Maximum deflection under live load: L/600
     (L = span length)
2. FEM 1.001 (European Federation of Materials Handling):
   Specifies deflection limits for overhead cranes:
   • Serviceability deflection: L/750
3. BS 2853 (British Standards):
   Applies to supporting structures for cranes:
   • Maximum deflection: L/500
4. Indian Standards (IS 800):
   • Similar limits are applied, with crane beams often adhering to L/500L/500 or L/750 depending on application.

Dynamic Effects:

Crane girders may experience additional deflection due to dynamic loads (e.g., moving trolleys and hoist). FEM or dynamic analysis is often performed.

Steps to Calculate Girder Deflection

1. Determine Load and Geometry:
   • Identify the type of load (point load, UDL, or moving load).
   • Measure the span and cross-sectional dimensions.
2. Calculate Properties:
   • Find the modulus of elasticity (E) and moment of inertia (I).
3. Apply the Appropriate Formula:
   • Use the deflection formula relevant to the load type and girder configuration.
4. Verify Against Standards:
   • Check calculated deflection against applicable limits (e.g., $L/750$).