# Diaphragms

Consider the floor plan in Fig. 4.1. Imagine it to be a typical floor of a medium-rise building. For most of its design life the floor structure

Structural wall in y direction

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Section Inertia forces in one storey from y direction shaking

Inertia force from perimeter wall

Inertia force from floor slab

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Inertia forces acting in plan

Inertia forces acting in plan

Resistance from structural wall

▲ 4.1 Inertia forces within a multi-storey building shown in plan and section.

resists gravity forces; dead and imposed forces that act vertically. But during an earthquake, that perhaps lasts only between 10 to 100 seconds, the floor structure resists horizontal seismic forces. During this relative infinitesimally brief period of time, when the floor structure is called upon to resist not only gravity but also horizontal forces, it is described as a diaphragm. When the ground shakes in the y direction, inertia forces are induced in exterior and interior walls and the floor slab itself. Usually inertia forces acting upon a wall that is loaded

▲ 4.2 The weakness of unreinforced masonry walls and their connections prevented inertia forces being transferred safely to diaphragms at first floor and roof level. Santa Monica, 1994 Northridge earthquake.

(Reproduced with permission from A.B. King).

▲ 4.2 The weakness of unreinforced masonry walls and their connections prevented inertia forces being transferred safely to diaphragms at first floor and roof level. Santa Monica, 1994 Northridge earthquake.

(Reproduced with permission from A.B. King).

perpendicular to its length that are called face-loads or out-of-plane forces are transferred vertically, half up and half down to diaphragms, which then transfer them to vertical structure acting in the direction of shaking; in this case two structural walls. The walls then transfer the inertia forces to the foundations. Diaphragms play an identical role when wind forces act on a building.

Strong and ductile connections between walls and diaphragms are necessary. Where connections are lacking, or are brittle or weak as in many existing unreinforced masonry buildings, walls fall outwards from buildings (Fig. 4.2). For example, during the Northridge earthquake, inadequate connection details between walls and roof diaphragms of newer buildings also led to walls collapsing.

Inertia Structural wall providing force resistance in y direction

Inertia Structural wall providing force resistance in y direction x

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Plan of diaphragm

Diaphragm modelled as a simply supported beam

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Diaphragm modelled as a simply supported beam

Deflected shape

Shear force diagram

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Bending moment diagram

▲ 4.3 A diaphragm as a simply-supported beam.

When functioning as a diaphragm, a floor slab acts like a beam albeit resisting horizontal rather than vertical forces and possessing a span-to-depth ratio much smaller than that of a typical beam. Just like a simply-supported beam the diaphragm bends under the influence of the horizontal inertia forces, spanning not between piers or posts, but in this case between two structural walls. It experiences bending moments and shear forces whose distributions along its length are identical to that of a gravity-laden beam. So a diaphragm is modelled just the same except that we need to remember that the direction of force and bending is horizontal (Fig. 4.3).

A diaphragm is therefore a beam that acts horizontally. As such, it requires stiffness and strength. Its often squat geometry avoids excessive horizontal deflections. However, a structural engineer needs to check that a diaphragm slender in plan is not too flexible, particularly if called upon to resist and transfer inertia forces from heavy masonry walls orientated parallel to its length (see Fig. 6.10).

The maximum horizontal deflection of a typical reinforced concrete diaphragm resisting seismic forces is usually quite y x

Reinforced Concrete (reinforcing steel not shown)

▲ 4.4 A steel channel beam analogous to a diaphragm and cross-sections through diaphragms of different materials.

small relative to that of the vertical system it transfers force into. This is an example of a rigid diaphragm. Due to its in-plane stiffness it forces all vertical elements irrespective of their individual stiffness to deflect the same amount. The force each vertical element resists is therefore in proportion to its stiffness. A flexible diaphragm represents the other extreme. It is more flexible than the vertical structure beneath it. A common example of this is where relatively flexible timber diaphragms combine with stiff reinforced masonry or concrete walls. Since the diaphragm is too flexible to force all the walls to move together, each wall irrespective of its stiffness resists the inertia force only from the tributary area of floor connecting into it. So, depending upon the degree of diaphragm rigidity the forces resisted by individual vertical elements vary. This may require the structural engineer to fine-tune shear wall lengths or moment frame member dimensions, but the architectural implications of non-rigid diaphragms are usually minimal. Since small diaphragm deflections lead to less damage in a building, and it is always best to tie all building elements on any one level strongly to each other, diaphragm rigidity is definitely to be preferred.

A diaphragm can be considered analogous to that of a steel channel beam (Fig. 4.4). As its flanges provide bending strength, tension stress in one flange and compression in the other so chords - as they are called - do for diaphragms. Like the flanges of a beam, diaphragm chords should be continuous along the diaphragm length. Steel and wood diaphragms require the provision of specific chord members to carry the bending moment induced tensions and compressions. In concrete diaphragms these actions may be provided for by the simple addition of horizontal reinforcing steel along the diaphragm edges. However, a heavily laden concrete diaphragm might require specific chord members in the form of two longitudinal beams. Not only do they provide a location for accommodating the extra reinforcing steel but they prevent the compression edge of the diaphragm buckling.

The provision of diaphragm shear strength is also usually easy to provide. As the web of a steel beam withstands shear force, so does the horizontal plane of a diaphragm. Where a diaphragm is thin (constructed from plasterboard or plywood), the joists or rafters to which it is fixed provide out-of-plane stability to prevent it buckling under shear stress. Maximum shear force occurs at the diaphragm supports, namely adjacent to vertical elements such as structural walls into which the diaphragm transfers its shear. Strong connectors are required at these junctions. In reinforced concrete construction force

Plan of diaphragam

Diaphragm modelled as a continuous beam

Deflected shape forces

Deflected shape

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Shear force diagram

Shear force diagram

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Bending moment diagram ▲ 4.5 A continuous two-span diaphragm.

transfer occurs through the concrete and reinforcing bars that tie horizontal and vertical elements together.

Just as buildings include both simply supported and continuous gravity beams, continuous diaphragms are commonly encountered. Fig. 4.5 shows a continuous diaphragm that spans between three supports (moment frames) in one direction. In the other direction the diaphragm is simply supported between two lines of shear walls. As discussed previously, the diaphragm can be modelled as a continuous beam with typical bending moment and shear force diagrams. A single chord of a continuous diaphragm therefore experiences tension and compression simultaneously in different sections along its length.

Before considering the construction materials used for diaphragms, mention must be made about the structural design of diaphragms given the philosophy of Capacity Design (Chapter 3). If designers have chosen to absorb seismic energy in structural fuses within primary vertical structural elements like structural walls, then all other structural elements including diaphragms must be designed strong enough to avoid damage. Damage should occur only within the specially detailed fuse regions.