From what I know, for a nice rectangle slab with 4 edges bounded by beams, one can distribute the slab loading to adjacent beam via the Tributary Area method.

But what if the slab is irregular, or there are arbitrary line loads on the slab, or the slab is not of uniform method, or not all sides of the slabs are covered by beams? How should the slab's dead load and live load, and any load on slab be distributed to beams? Is there a general way?

• The most general method would be to use a numerical model consisting the slab as well, thus the load transferring mechanism is captured by the model. Within the finite element (FEM) paradigm plate elements would suffice. With today's computers and FEM applications it can be easily solved. Apr 29, 2016 at 8:24
• @Arpi, it's easy to say that FEM solves this problem, as if it's a panacea. But I would appreciate if you can elaborate on this. How does FEM solve it from first principle, and how does it compare with the tributary area method in the simplest case? Apr 29, 2016 at 9:45

Extension of my comment to an answer, feel free to edit and expand it!

The most general method would be to use a numerical model consisting the slab as well, thus the load transferring mechanism is captured by the model. Within the finite element (FEM) paradigm plate elements would suffice. With today's computers and FEM applications it can be easily solved.

I do not know about your background and experience but using plate elements with line elements to model slabs with beams is quite widespread. That is why I just mentioned FEM without any further details. If you could provide more details on your experience or formulate more specific questions, e.g. how to model the eccentricity of beams, beam slab connection, then we might be able to give equally specific answers.

Here are two practitioners' guides on modelling of reinforced concrete slabs:

Recommendations for finite element analysis for the design of reinforced concrete slabs

How to Design r.c. Flat Slabs Using Finite Element Analysis

Regarding how the tributary approach compares with FEM plate model for a simple case:

A Case Study Comparing Two Approaches for Applying Area Loads: Tributary Area Loads vs Shell Pressure Loads

Based on their simple example for a particular slab-to-beam stiffness ratio: The tributary area method is on the safe side and yields to about 20% overestimation of maximal bending moment and shear force.

These are based on a single, simple example thus the results are not conclusive, at best only indicative.

Since the structure type was not mentioned and maybe you have other than a building structure in mind, here are two relevant books on bridge decks:

O'Brien and Keogh: Bridge Deck Analysis

Hambly: Bridge Deck Behaviour