[Illustration:  FIG. 5.]

[Illustration:  FIG. 6.]

The writer will take up next the question of pressures against the faces of sheeted trenches or retaining walls, in material of the same character as noted above.  Referring to Fig. 2, it is not reasonable to suppose that having passed the line, R F J, the character of the stresses due to the thrust of the material will change, if bracing should be substituted for the material in the area, W V J R, or if, as in Fig. 3, canvas is rolled down along the lines, E G and A O, and if, as this section is excavated between the canvas faces, temporary struts are erected, there is no reason to believe that with properly adjusted weights at W or W{2}_, an exact equilibrium of forces and conditions cannot be obtained.  Or, again, if, as in Fig. 5, the face, P Q, is sheeted and rodded back to the surface, keying the rods taut, there is undoubtedly a stable condition and one which could not fail in theory or practice, nor can anyone, looking at Fig. 5, doubt that the top timbers are stressed more heavily than those at the bottom.  The assumption is that the tendency of the material to slide toward the toe causes it to wedge itself between the face of the sheeting on the one hand and some plane between the sheeting and the plane of repose on the other, and that the resistance to this tendency will cause an arching thrust to be developed along or parallel to the lines, A N, B M, etc., Fig. 2, which are assumed to be the lines of repose, or curves approximating thereto.  As the thrust is greatest in that material directly at the face, A O, Fig. 6, and is nothing at the plane of repose, C O, it may be assumed arbitrarily that the line, B O, bisecting this angle divides this area into two, in one of which the weight resolves itself wholly into thrust, the other being an area of no thrust, or wholly of weight bearing on the plane of repose.  Calling this line, B O, the haunch line, the thrust in the area, A O B, is measured by its weight divided by the tangent of the angle, P Q R = [phi], which is the angle of repose; that is, the thrust at any given point, R = R Q / tan. [phi].

The writer suggests that, in those materials which have steeper angles of repose than 45 deg., the area of pressure may be calculated as above, the thrust being computed, however, as for an angle of 45 degrees.

In calculating the bending moment against a wall or bracing, there is the weight of the mass multiplied by the distance of its center of gravity vertically above the toe, or, approximately: 

2
Area, A O B x weight per unit x —–­ height,
3

where h = height,

W = weight per cubic foot of material = 90 lb.,

90 deg. — [phi]
and [beta] = -------------
2

P = pressure per linear foot (vertically),