Hydrostatic Thrust on Dam
Calculate the horizontal hydrostatic thrust per metre of length on a dam face, E = ½·γ·H², from the unit weight of water γ (≈ 9.81 kN/m³) and the water depth H (m) at the upstream face. Since hydrostatic pressure grows linearly with depth (p = γ·h), its diagram is triangular and the resultant is its area, ½·γ·H², applied at one third of the height from the base. This thrust is the main action tending to overturn and slide the dam, and the starting point of gravity dam stability analysis: it generates the overturning moment (about the downstream toe) and the horizontal force resisted by base friction. Dam stability depends on its self-weight (providing the stabilizing moment and normal friction force) exceeding these with adequate margin, also accounting for foundation uplift. Enter the unit weight of water and the depth.
Result
—
Hydrostatic thrust on a dam
The horizontal hydrostatic thrust per metre of length on the upstream face of a dam is E = ½·γ·H², from the specific weight of water γ (≈ 9.81 kN/m³) and the depth of the impounded water H. Since hydrostatic pressure grows linearly with depth (p = γ·h, zero at the surface and maximum at the bottom), its diagram is triangular, and the resultant force is the area of that triangle, ½·γ·H², applied at one third of the height above the base (at the centroid of the pressure diagram). This thrust is the governing action that tends to overturn the dam and to slide it downstream, and it is the starting point of the entire stability analysis of gravity dams: it generates the overturning moment about the downstream toe and the horizontal force that must be resisted by friction along the foundation base. The dam remains stable because its self-weight — supplied by the enormous mass of concrete or fill — produces a stabilising moment and a normal force that mobilises the friction required, exceeding those actions with an adequate margin. But there is a hidden enemy: uplift pressure in the foundation, which relieves the effective weight and has to be fought with drainage. Enter the specific weight of water and the depth.
Related Tools
Dam Sliding Safety Factor
Calculate the sliding safety factor of a gravity dam, FS = (μ·W) ÷ F_h, from the base friction coefficient μ (tan of the concrete-foundation friction angle, typically 0.6-0.75), the effective self-weight W (dam weight minus uplift, kN/m) and the destabilizing horizontal force F_h (hydrostatic thrust, kN/m). This factor compares the forces resisting the dam sliding on its foundation (mobilized base friction, proportional to the effective normal force) with those pushing it downstream (the reservoir thrust). It is one of the two fundamental gravity dam stability checks — the other being overturning. Codes typically require sliding FS ≥ 1.5 for normal loading. The simplified form uses friction only; fuller analyses add interface cohesion (c·B). Note how decisive uplift is: it reduces W and thus the numerator — hence the importance of foundation drainage. Enter the friction coefficient, effective weight and horizontal force.
Spillway Discharge (Creager/Ogee)
Calculate the discharge over a Creager/ogee dam spillway, Q = C·L·H^1.5, from the discharge coefficient C (typically 2.0-2.2 in SI for well-designed ogee profiles), the crest length L (m) and the head over the crest H (m). The spillway is a dam's most critical safety structure: it releases floods safely, preventing overtopping — the leading cause of dam failure. The ogee profile follows the shape of the underside of a free nappe, maximizing discharge while keeping crest pressure near atmospheric (avoiding cavitation). The coefficient C absorbs gravity and approach effects, exceeding that of a sharp-crested weir. Spillway design starts from the design flood (often the 10,000-year flood or the PMF) and sets the required crest length. Enter the discharge coefficient, crest length and head.
Dam Foundation Uplift
Calculate the uplift resultant per metre of length at a gravity dam base, assuming triangular distribution, U = ½·γ_w·H·B, from the unit weight of water γ_w, the head H (upstream water height) and the base width B. Uplift is the water pressure that percolates through the foundation and concrete joints acting upward on the dam base, reducing the effective normal force and thus the sliding friction resistance — one of the most dangerous and historically underestimated factors in dam stability (the 1928 St. Francis Dam failure is a landmark). The real distribution depends on grout curtains and drains, which reduce it; the triangular hypothesis (full upstream, zero downstream) is conservative and common in preliminary design. Uplift subtracts from self-weight in the sliding check and adds overturning moment. Enter the unit weight of water, the head and the base width.
Hydraulic Jump Length
Estimate a hydraulic jump's length, L ≈ 6.9·(y₂ − y₁), by the classic empirical formula, from the upstream y₁ and downstream y₂ sequent depths. Unlike the sequent depths (from momentum), jump length is empirical, from lab tests, since the jump has no mathematically sharp end — its length is the distance from the upstream face to where the surface stabilizes. Several formulas exist (Smetana ≈ 6(y₂−y₁), USBR vs Fr, Elevatorski ≈ 6.9(y₂−y₁)); all give the order of magnitude. Jump length sets the stilling basin size downstream of a spillway: the basin must be long enough to contain the whole jump so dissipation completes within the concrete-lined structure before water returns to the natural bed. Undersizing throws the still-erosive jump tail onto the unprotected bed. Enter the upstream and downstream sequent depths.
Metacentric Height (GM)
Calculates the metacentric height GM of a vessel by adding the centre of buoyancy height to the metacentric radius (waterplane moment of inertia divided by displaced volume) and subtracting the centre of gravity height. A positive GM means stable equilibrium; negative means a heeled hull will not right itself. Enter the moment of inertia, displaced volume, KB and KG.
Hydraulic Jump Energy Loss
Calculate the specific energy dissipated in a hydraulic jump, ΔE = (y₂ − y₁)³ ÷ (4·y₁·y₂), from the upstream y₁ (supercritical) and downstream y₂ (subcritical) sequent depths. The hydraulic jump is one of the most efficient energy dissipators in hydraulics: intense turbulence in the transition converts kinetic energy to heat and sound, removing excess flow energy. This head loss ΔE is exactly what is sought downstream of spillways, gates and bottom outlets — water arrives with very high energy (able to scour the riverbed and undermine the structure), and the stilling basin induces the jump to 'burn' that energy in a controlled way. The higher the incoming Froude number, the greater the dissipated fraction — jumps with Fr > 9 dissipate up to 85%. Enter the upstream and downstream sequent depths.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.