4. Combined thrust front lightest possible braked train, and a wind-pressure on train and structure equal to one half of that specified.

Next determine by judgment the proper batter, and lay off the pier to scale; then divide it by horizontal planes from four to six feet apart, and compute the weights of all the portions of the masonry between these planes, making a proper reduction for weight of water for those parts which would be submerged by an average stage of river.

Next compute the wind-pressure on each vertical division of the pier, down to the assumed stage of water, in a direction parallel to the spans, using the same intensity and direction for the wind-pressure as were adopted in finding the longitudinal thrust from wind-pressure on the spans.

Next find graphically for all four cases the curves of pressure from the vertical and horizontal loads at top of pier, combined with the weights of the various divisions of the latter and the wind-pressures thereon, and see that none of the said curves at any plane of division pass outside of the middle third of the section at said plane. If any of them do, the batter will have to be increased, or, if all the curves fall much inside of the middle third points, it will have to be decreased; and in either case the graphical computations will have to be made again, and so on until a satisfactory batter is found.

The author is aware of the fact that this method of designing piers is not in general use, and it is quite possible that he is the sole engineer who adopts it; nevertheless he maintains that it is the only proper way to design masonry piers. The single concession which he would be willing to make on the score of economy would be to assume that a certain small portion of the thrust on a span is taken up at the roller end. But if the rollers are in good working order the amount of thrust that they will resist is very small indeed—so small, in fact, that the author prefers to neglect it entirely.

The ordinary method of proportioning piers is to make them as small as possible under coping and batter them all around, or at least on the sides, one-half inch to the foot. In some cases  this will suffice,  but in others it will not.  One of the