BGASCE7-10 Section 5.4
Arrangement of Live Loads
Last Revised: 11/04/2014
ASCE 7-10 Section 4.3.3 states "The full intensity of the appropriately reduced live load applied only to a portion of a structure or member shall be accounted for if it produces a more unfavorable effect than the same intensity applied over the full structure or member."
What this means is that you need to arrange the live load so as to cause maximum effect in your members. You must design your structural elements so that they have sufficient strength to support all possible arrangements of live load. Consequently your analysis needs to provide you with envelope diagrams for each member. Envelope diagrams are internal force diagrams that envelop all the possible values of force at each location along the member. So examples are used below to explain method for determining envelopes.
This can seem daunting task as you need to do multiple load cases to account for the various loadings on your structural system. For statically determinate structures, it is often easy to establish critical loading scenarios for shear, moment, reactions, and deflection. Unfortunately for continuous, statically indeterminate structures this is not so obvious and the use of influence lines becomes extremely useful.
In most structural analysis texts methods are presented for both detailed analytical and approximate influence lines. You will find that approximate methods will quickly identify where to place your loads. Let's look at a couple of general cases for shear, moment, and reactions.
Consider the continuous beam show in Figure 5.4.1.
Figure 5.4.1
Sample Beam #1
The likely critical locations for shear and moment are indicated as the points shown on the beam. To determine where to place live load so as to cause maximum effect at any of these points you need to decide which effect (shear or moment) you are interested in then draw the influence lines for that point. To get the upper bound of your internal force diagram analyze your structure under loading placed where the influence line is positive. To get the lower bound of your internal force diagram analyze your structure under loading placed where the influence line is negative.
Figure 5.4.2 shows the influence lines for shear and moment at point 5 and the associated critical load diagrams. Notice that span 1 is only partially loaded when considering shear. All the other spans are fully loaded. This will always be the case for shear, which results in an infinite number of load cases. In practice, however, shear is rarely the controlling load case for members subjected to distributed loads. Also, the critical shear is normally near the supports where the partial loading is close to a full span loading, so common practice is to consider only fully loaded spans and that the results will be close enough that the designer can feel assured that the shear limit states are satisfied. In cases where shear becomes a controlling limit state, then a more accurate partial span loading is in order.
Figure 5.4.2
Influence & Loading at Point 5
Figure 5.4.3 shows the influence lines for shear and moment at point 2 and the associated critical load diagrams.
Figure 5.4.3
Influence & Loading at Point 2
If we continue drawing influence lines we soon learn that there are four different full span load cases that need be analyzed in order to determine maximum shear and moment at every location.
Figure 5.3.4 shows the four load cases and the associated moment diagrams (the moment diagrams are approximate... will do more accurate ones at some future date, but this should be good enough to get the point across!). Finally, the moment envelope is found by superimposing the four moment diagrams. The envelope values are the maximum and minimum moments at each point. A similar diagram can be constructed for shear.
Figure 5.3.4
Load Cases & Moment Diagrams
This method can also be used on continuous two and three dimensional frames as well.
Figures 5.3.5 and 5.3.6 show the application for determining envelope moments at two different points. Additional loading cases need to be developed for other points in the structure if all necessary load arrangements are to be identified.
Figure 5.3.5
2D Frame Loadings for Moment at "A"
Figure 5.3.6
2D Frame Loadings for Moment at "B"
For three dimensional structures the load is applied to bays of the building in various patterns. One such pattern is depicted in Figure 5.3.7. The result is that you end up with a multitude of cases if live load to determine the maximum effect on the structure.
Figure 5.3.7
A 3D Loaded Frame
Another method, that I call influential superposition, has been developed that can eliminate the guess work regarding where to place the loads and is well suited to numerical analysis. A short paper on how influential superposition works can be downloaded here.