# Area Loads in One-way and Two-way Systems

When you’re designing a structure to support a slab, it’s important to consider how the slab transmits its load to the structural members.

In your model, you might consider adding a plate element to model the slab. However in the context of considering the adequacy of your beams, joists, and girders, this may not be appropriate. This is because loads are transferred from plate to nodes (no mid-member interaction) and unwanted stiffness may be introduced into your model.

A more appropriate approach would be to instead consider what is known as an “area load” or “tributary load”. This involves converting a pressure load (that acts along a surface) to distributed loads that act on the beams, joists, and girders (members) which support the surface.

In this article we’re going to explore two examples of how this can be done. Mathematically, the calculation for the distributed load is simply expressed as:

[math]$$w=qt_w$$[math]

Where w is distributed/area load magnitude, q is pressure load magnitude, and tw is tributary width. Tributary width is the width of an area that is divided up according to the area load type.

The consideration at this point is to determine whether a system is one-way or two-way. This is so that the tributary width and load distribution can be assigned to the member. In general, if the load from the slab is delivered to the beams in one direction, then the system is one-way. Conversely, if the load is delivered to the beams and the girders in two directions, then the system is considered two-way.

### Example: One-way System

In this example, the pressure load from the slab is considered to be transferred directly to the beams. Since the girders are not directly supporting the slab, the system is considered to be one-way. The area load is thus calculated as,

[math]$$w=qt_w=1.2 \times 3/2=1.8 kip / ft$$[math]

A one-way system will divide up the area formed by the two members selected. In this example, since it is a rectangle, the profile of the area load is also rectangular as shown below. Note that the profile is not always rectangular, but rather it is always quadrilateral and represents an equal division of area between the two members. In some cases, point loads are added in order to force balance the system if the slab overhangs the member.

### Two-way System

For the same structure as the previous example, if the slab were to instead be directly supported by members 2,8,5,9 the system would be considered two-way. The slab occupies the same area as the previous question, however the load profile will change. For two-way systems, bisecting lines are drawn from the corners to create a pair of triangles and a pair of quadrilaterals as shown in the diagram below.

Thus the calculation for area load for members 8 and 9 is given by,

[math]$$w=qt_w=1.2 \times 3/2=1.8 kip / ft$$[math]

And for members 2 and 5 (by simple trigonometry),

[math]$$w=qt_w=1.2 \times 1.5\tan(45^\circ)=1.8 kip / ft$$[math]

Applying these area loads to the structure is shown below,