To solve this problem, we'll first calculate the various forces acting on the masonry dam and then determine the resultant pressure and its location on the base.
Given data:
Thickness at the top (t₁) = 0.80 m
Thickness at the bottom (t₂) = 3.6 m
Height (h) = 6 m
Water pressure (P) = 4.2 m
Weight of the masonry (W) = 2350 kg/m
To find the resultant pressure and its location on the base, we'll follow these steps:
Calculate the area and centroid of the trapezoidal cross-section:
Area (A) = (t₁ + t₂) * h
Centroid from the top (C) = (t₁ * h/2 + t₂ * h + t₁ * h/2) / (2 * (t₁ + t₂))
Calculate the total vertical force acting on the masonry:
Vertical force due to masonry weight (F_w) = W * A
Calculate the vertical force due to water pressure:
Vertical force due to water pressure (F_p) = P * A
Calculate the resultant vertical force:
Resultant vertical force (F_r) = F_w + F_p
Calculate the location of the resultant pressure on the base:
Distance from the top to the resultant (D) = C - (F_w / F_r) * (t₁ + t₂)
Calculate the intensity of pressure at the heel and toe:
Pressure at the heel (P_heel) = F_r / (t₁ + t₂)
Pressure at the toe (P_toe) = F_r / (t₁ + t₂)
Now let's calculate the values:
Calculate the area and centroid of the trapezoidal cross-section:
A = (0.80 + 3.6) * 6 = 26.4 m²
C = (0.80 * 6/2 + 3.6 * 6 + 0.80 * 6/2) / (2 * (0.80 + 3.6)) = 3.7714 m
Calculate the total vertical force acting on the masonry:
F_w = 2350 * 26.4 = 61940 N
Calculate the vertical force due to water pressure:
F_p = 4.2 * 26.4 = 110.88 kN = 110880 N
Calculate the resultant vertical force:
F_r = F_w + F_p = 61940 + 110880 = 172820 N
Calculate the location of the resultant pressure on the base:
D = 3.7714 - (61940 / 172820) * (0.80 + 3.6) = 2.1813 m
Calculate the intensity of pressure at the heel and toe:
P_heel = F_r / (0.80 + 3.6) = 172820 / 4.4 = 39368 N/m²
P_toe = F_r / (0.80 + 3.6) = 172820 / 4.4 = 39368 N/m²
A masonry dam of trapezoidal cross section, with one face vertical has a thickness of 0.80 m at the top, 3.6 m at the bottom, and a height of 6 m. The vertical face is subjected to a water pressure, the water standing 4.2 m above the base. The weight of the masonry is 2350 kg/m? Where will the resultant pressure intersect the base and what will be the intensity of pressure at the heel and at the toe (a) assuming no hydrostatic uplift and (b) considering hydrostatic uplift.