To fulfill the requirements of traffic on Indian railways it has become necessary to run the vehicle with high speed and heavier loads. With the increase in speed and load of trains the tractive and braking forces on rails will also increase. Consequently, it has become important to investigate the safety of sub-structures of existing bridges. Hence the share of force resisted by individual sub-structural elements should be estimated more accurately than given by the present specifications which are based on a few field trials, and are not realistic and reliable. The problem is studied in two parts. First is to investigate how the longitudinal forces developed over a bridge are shared by various supports and approaches. Secondly to ascertain if a particular support is capable of sustaining the loads determined above. This paper deals mainly with the first part of the problem, to develop a rational analysis for working out dispersion of longitudinal forces, and compare the values with the existing codal provisions which are based on the field tests. The constants and the factor affecting them are also given importance. Their ranges required for analysis purpose is also mentioned. 1.A fairly simple, reliable and universally applicable theoretical analysis using matrix method has been developed for working out dispersion of tractive and breaking forces of bridge substructure. 2. The constants required for the analysis such as approach constant, rail girder fixture constant and the support constants are given with their limiting range. 3. The analysis takes into account the effect of any number of axles exerting vertical loads and longitudinal forces. 4. The longitudinal forces on piers due to vertical loads are not as large as on abutments. These longitudinal forces are not being considered in design at present but the theory indicates that they should be given due consideration. 5. The maximum longitudinal force going to a support can be reduced by providing suitable rubber or neoprene bearings.