In structural engineering, the terms rigid diaphragm and flexible diaphragm refer to the behavior of horizontal structural elements (like floors or roofs) in distributing lateral forces—such as those from wind, earthquakes, or other dynamic loads—to the vertical elements of a building (e.g., walls, columns, or braces). The classification of a diaphragm as rigid or flexible depends on its stiffness relative to the vertical resisting elements and affects how forces are analyzed and distributed in the structure.
1. Rigid Diaphragm
A rigid diaphragm is a horizontal structural element that is assumed to have infinite in-plane stiffness, meaning it does not deform significantly under lateral loads. Instead, it behaves like a rigid plate, distributing forces to the vertical elements based on their relative stiffness and maintaining compatibility of displacements across the structure.
Characteristics
- Infinite Stiffness: The diaphragm does not deform in its own plane (shear or bending deformations are negligible).
- Force Distribution: Lateral forces are distributed to vertical elements (e.g., shear walls or frames) proportional to their stiffness.
- Rotation: The diaphragm can rotate as a rigid body about a vertical axis, especially in asymmetric structures, leading to torsional effects.
- Displacement Compatibility: All points on the diaphragm move together, maintaining a uniform translation and rotation.
Examples
- Reinforced concrete slabs or thick steel decks in buildings.
- Floor systems with significant reinforcement or deep structural members.
Structural Behavior
When a lateral load (e.g., earthquake force) is applied, the rigid diaphragm:
- Transfers the load to the vertical elements based on their stiffness.
- Ensures that the horizontal displacement at all points on the diaphragm is the same (plus any rotational component).
- May induce torsion if the center of mass (where the load acts) and the center of rigidity (where resistance is provided) do not align.
Mathematical Representation
For a rigid diaphragm, the distribution of forces to vertical elements is calculated using:
The additional force due to torsion is proportional to the distance of each vertical element from the center of rigidity and its stiffness.
Advantages
- Simplifies analysis by assuming uniform displacement.
- Provides good resistance to lateral loads in stiff, robust structures.
Disadvantages
- May overestimate forces in some elements if the diaphragm has some flexibility.
- Torsion can complicate design in irregular structures.
2. Flexible Diaphragm
A flexible diaphragm is a horizontal structural element with relatively low in-plane stiffness, meaning it deforms significantly under lateral loads. Instead of behaving as a rigid plate, it acts more like a series of independent segments, distributing forces to vertical elements based on the tributary area or load path rather than their stiffness.
Characteristics
- Finite Stiffness: The diaphragm deforms (shear or bending) under lateral loads.
- Force Distribution: Lateral forces are distributed to vertical elements based on the tributary area or the load each element directly supports, not their stiffness.
- No Rigid Body Rotation: Torsional effects are minimal because the diaphragm does not enforce uniform displacement.
- Independent Behavior: Each portion of the diaphragm may deflect differently, depending on the local loading and support conditions.
Examples
- Plywood or light-gauge metal decking in wood-frame or steel buildings.
- Thin, unreinforced floors or roofs with large spans and minimal bracing.
Structural Behavior
When a lateral load is applied, the flexible diaphragm:
- Deforms in-plane, with deflections varying across its span.
- Transfers forces to vertical elements based on the area of the diaphragm they support (tributary area method).
- Does not enforce compatibility of displacements, so vertical elements may experience different deflections.
Mathematical Representation
For a flexible diaphragm, the force to each vertical element is typically:
Torsion is generally neglected because the diaphragm’s flexibility prevents it from acting as a rigid body.
Advantages
- Simpler force distribution based on area, avoiding complex stiffness calculations.
- Common in lightweight, low-rise structures where flexibility is acceptable.
Disadvantages
- Less resistance to lateral loads due to deformation.
- May lead to uneven load sharing if vertical elements have significantly different stiffnesses.
Key Differences
| Aspect | Rigid Diaphragm | Flexible Diaphragm |
|---|---|---|
| In-Plane Stiffness | Infinite (assumed) | Finite (deforms under load) |
| Force Distribution | Based on stiffness of vertical elements | Based on tributary area |
| Displacement | Uniform across diaphragm | Varies across diaphragm |
| Torsion | Significant if eccentricities exist | Negligible |
| Examples | Concrete slabs, stiff steel decks | Plywood, thin metal decking |
| Analysis Complexity | Requires stiffness and torsion calculations | Simpler tributary area approach |
Practical Classification
In real structures, diaphragms are rarely perfectly rigid or perfectly flexible—they often fall somewhere in between (semi-rigid diaphragms). Building codes (e.g., ASCE 7 in the U.S.) provide guidelines to classify diaphragms:
- Rigid: If the diaphragm’s in-plane deflection is less than or equal to twice the average deflection of the vertical elements under the same load.
- Flexible: If the diaphragm’s deflection is significantly greater than that of the vertical elements (e.g., more than twice).
For example:
- A concrete floor slab with a deflection of 0.1 inches compared to shear wall deflections of 0.05 inches is likely rigid.
- A plywood roof deflecting 1 inch while supporting walls deflect 0.2 inches is likely flexible.
Applications in Structural Dynamics
In the context of structural dynamics (e.g., seismic analysis):
- Rigid Diaphragms: Common in multi-story buildings where floors tie the structure together and distribute seismic forces to shear walls or frames. Torsional effects are critical in irregular buildings.
- Flexible Diaphragms: Typical in single-story buildings (e.g., warehouses with long-span roofs) where the roof deforms and each wall resists loads based on its tributary area. Dynamic amplification of diaphragm flexibility can increase deflections during earthquakes.
Example
Imagine a rectangular building with shear walls at both ends:
- Rigid Diaphragm: A concrete roof slab distributes a seismic force of 100 kN to the walls based on their stiffness (e.g., 60 kN to a stiff wall, 40 kN to a less stiff one).
- Flexible Diaphragm: A plywood roof distributes the 100 kN equally (50 kN each) if the tributary areas are equal, regardless of wall stiffness.