Consider a harbour breakwater constructed with massive concrete tanks filled with sand. It is necessary to evaluate the risk that the breakwater will slide under the pressure of a large wave during a major storm. Stability against sliding exists when the ratio of the resultant horizontal force R_h to the resultant of the vertical force R_v does not exceed the coefficient of friction, X₁.

The resultant of the vertical forces R_v is given by the algebraic sum of the weight of the tank reduced for buoyancy, X₂, and the vertical component of dynamic uplift pressure due to the breaking wave, F_v.

Rv = X₂ - F_v

In turn, F_v is proportional to the height of the wave, H_b, when the slope of the sea bottom deep-water harbour has a value, X₄.

F_v = a₁X₃X₄

where a₁ is the constant of proportionality and X₃is an additional variate to represent the uncertainties caused by the above simplification.

The resultant R_hof the horizontal forces depends on the balance between the static and dynamic pressure components and under a simplifying hypothesis, the depth of the breakwater can be taken to be a quadratic function of the wave height H_bwhich in turn is proportional to X₄

R_h = X₃(a₃X₄+a₂X²₄)

The constants a₁ to a₃ depend on the detailed geometry of the breakwater system.

The following data refers to conditions in La Spezia harbour, Italy. From the sea bottom profile and the geometry of the breakwater wall at La Spezia, engineers have estimated that

a₁ = 70 , a₂ = 17 kN/mand a₃ = 145.

The variables X₁ to X₄ are independent random variables (because for example the friction coefficient along a unit width of the vertical breakwater wall at La Spezia is not known precisely).

Question:

Suppose the mean value for X₁ is 0.70, the mean value for X₂ is 3,266 kN/m, the mean value for X₃ is 1.1 and the mean value for X₄ is 4.81m. What then is the mean value for R_h (answer to 2 decimal places).