The load on a beam due to a slab is typically calculated as an equivalent uniformly distributed load (UDL), especially for two-way slabs. This calculation is crucial for accurately designing the supporting beams and depends on the slab's geometry and how the load is distributed across its spans.
Understanding Slab Load Distribution to Beams
Slabs transfer the loads they carry (such as their own weight, furniture, occupants, etc.) to the supporting beams, which then transfer these loads to columns and foundations. The pattern of this load transfer depends primarily on the slab's aspect ratio (ratio of longer span to shorter span) and support conditions.
- One-way slabs primarily transfer loads in one direction, typically to two opposite beams. The load on these beams is generally a simple uniformly distributed load across their length.
- Two-way slabs transfer loads to all four supporting beams, distributing the load in two perpendicular directions. The load distribution to each beam is more complex, often idealized as triangular or trapezoidal areas of load, which are then converted into an equivalent UDL for beam design.
Formulas for Equivalent Uniformly Distributed Load (UDL) from a Two-Way Slab
For two-way slabs, the load transferred to a supporting beam is not uniform along its entire length. To simplify beam design, this non-uniform load is converted into an equivalent uniformly distributed load (w) per unit length of the beam. The specific formula used depends on whether the beam supports the shorter or the longer span of the slab panel.
Here are the key variables:
- W: The total design load per unit area of the slab. This includes the slab's self-weight, any floor finishes, and the applied live load (e.g., from people, furniture). It is typically expressed in kN/m² or psf.
- Lx: The shorter clear span of the slab panel, measured center-to-center of supporting beams.
- Ly: The longer clear span of the slab panel, measured center-to-center of supporting beams.
Formulas at a Glance
The formulas for calculating the equivalent UDL (w) on a beam from a two-way slab are as follows:
| Beam Supporting | Formula for Equivalent UDL (w) |
|---|---|
| Shorter Span | w = W * Lx / 3 |
| Longer Span | w = W * (1 - (1/3) * (Lx/Ly)^2) |
Detailed Explanation of Formulas
-
For Beams Supporting the Shorter Span (Lx):
- When a beam supports the shorter span of a two-way slab, it receives a load that is often approximated as a triangular distribution.
- The formula w = W * Lx / 3 calculates the equivalent UDL by considering that the beam effectively carries one-third of the total slab load distributed over its span from the shorter side.
-
For Beams Supporting the Longer Span (Ly):
- Beams supporting the longer span of a two-way slab receive a load that is typically approximated as a trapezoidal distribution.
- The formula w = W * (1 - (1/3) * (Lx/Ly)^2) accounts for this trapezoidal load pattern. The term
(Lx/Ly)^2reflects the aspect ratio of the slab, indicating how the load is proportioned between the longer and shorter spans. As the slab becomes more square (Lx approaching Ly), this term becomes more significant.
Practical Considerations and Example
Accurate calculation of these equivalent UDLs is fundamental for the safe and economical design of beams. Underestimating these loads can lead to structural failure, while overestimating them can result in an overly conservative and expensive design.
Determining 'W' (Total Design Load)
To calculate W, you need to sum up various load components:
- Slab Self-Weight: Calculated by multiplying the slab thickness by the density of the concrete (e.g., 24 kN/m³ or 150 lb/ft³) and any other materials in the slab assembly.
- Floor Finishes: Additional weight from tiles, screed, ceiling plaster, etc.
- Live Load: The variable load expected during the structure's use, specified by local building codes based on the occupancy type (e.g., residential, office, retail). Refer to standards like ACI 318 or Eurocode 2 for specific values.
Example Calculation
Consider a two-way slab panel with the following properties:
- Lx (shorter span) = 4 meters
- Ly (longer span) = 6 meters
- W (total design load on slab) = 8 kN/m² (This includes dead and live loads)
Let's calculate the equivalent UDL for the beams supporting this slab:
-
For a beam supporting the shorter span (Lx = 4m):
- w = W * Lx / 3
- w = 8 kN/m² * 4 m / 3
- w = 10.67 kN/m
-
For a beam supporting the longer span (Ly = 6m):
- w = W * (1 - (1/3) * (Lx/Ly)^2)
- w = 8 kN/m² * (1 - (1/3) * (4m / 6m)^2)
- w = 8 kN/m² * (1 - (1/3) * (0.6667)^2)
- w = 8 kN/m² * (1 - (1/3) * 0.4444)
- w = 8 kN/m² * (1 - 0.1481)
- w = 8 kN/m² * 0.8519
- w = 6.815 kN/m
These calculated UDLs (10.67 kN/m for the shorter span beam and 6.815 kN/m for the longer span beam) would then be used in the structural analysis and design of the respective beams.
It's important to remember that these formulas provide simplified approximations for design purposes. In complex structures or for more precise analysis, advanced methods considering continuity, torsional effects, and finite element analysis may be employed.
