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.
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Dam foundation uplift
Uplift pressure is one of the most dangerous — and historically most underestimated — factors in dam safety. It is the pressure of the water that seeps through the rock foundation and through the joints in the concrete and acts from below upward on the base of the dam, pushing it up and cutting the effective normal force that holds it to the ground. The uplift resultant per metre of dam, assuming a triangular distribution, is U = ½·γ_w·H·B, from the unit weight of water γ_w, the upstream hydraulic head H and the base width B. Its effect is insidious: by reducing the effective weight it lowers the available friction at the base and therefore the resistance to sliding, on top of adding an overturning moment. The failure of the St. Francis dam (California, 1928), which killed hundreds of people, is a tragic landmark that taught engineers to respect uplift. The real distribution depends on grout curtains (which seal the foundation) and on drains (which relieve the pressure downstream of the curtain), and both reduce it substantially; the triangular assumption (full head upstream, zero downstream) is conservative and used in preliminary sizing. Controlling uplift with a sound foundation, grouting and an efficient drainage system is absolutely essential to the safety of any gravity dam. Enter the unit weight of water, the hydraulic head and the base width.
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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.
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.
Pile Skin Resistance
Calculate a pile's side (friction) resistance, Q_l = f_s·A_s, from the average unit skin friction f_s (kPa) and the pile lateral surface area A_s (m², = π·D·L for a cylindrical pile). Side resistance is the share of pile capacity from FRICTION and ADHESION between the pile's lateral surface and the surrounding soil, along its whole buried length. As the pile tends to settle under load, the soil 'grips' its sides and resists — as a nail driven in wood resists pulling by face friction. The unit skin friction f_s depends on soil type (in clays, on undrained cohesion via the α method; in sands, on effective stress and friction via the β method), pile type (driven piles mobilize more friction than bored, displacing and compacting the soil) and surface roughness. Side resistance dominates in FLOATING (friction) piles, driven in soils without a firm bearing layer — they hang by friction. It is also the share mobilized FIRST under load (with small settlement), before the tip. This share adds to the tip resistance for the total capacity. Enter the unit skin friction and the side area.
Pile Allowable Load
Calculate a pile's allowable (working) load, Q_adm = Q_ult ÷ FS, from the ultimate bearing capacity Q_ult (kN) and the global safety factor FS. The allowable load is the maximum load that can be applied to the pile in service with adequate safety — obtained by dividing the ultimate capacity (the load that would cause FAILURE of the pile-soil system) by a safety factor covering uncertainties. The pile-foundation safety factor is typically HIGH (FS = 2.0-2.5 for ultimate capacity, higher if based only on theoretical formulas without a load test), reflecting the great uncertainty in determining soil capacity (unseen, heterogeneous and poorly known) and the severity of a foundation failure (which can collapse the whole structure). Codes often require different partial factors for tip and friction (which have different uncertainties), or limit-state methods. The allowable load sets how many piles are needed for the column loads: number of piles = column load ÷ allowable load. Load tests (measuring real field capacity) allow reducing the safety factor and optimizing design. Enter the ultimate capacity and the safety factor.
Reservoir Life (Sedimentation)
Estimate a reservoir's useful life from sedimentation, Vu = V ÷ V_s, from the reservoir's useful (or total) volume V (m³) and the sediment volume deposited per year V_s (m³/year). Every reservoir, by impounding a river, slows the flow and makes water lose its sediment-carrying capacity — sand, silt and clay from the watershed settle on the bottom, gradually reducing storage. The useful life is the number of years until sedimentation impairs the reservoir's function (power, supply, regulation). It is a crucial design parameter in hydrology and watershed management: reservoirs in basins with erodible soils, deforestation or intensive agriculture silt up fast (decades), while well-conserved basins last centuries. The sediment inflow V_s comes from the basin's sediment yield and the reservoir's trap efficiency (Brune curve). The simple constant-rate model gives the order of magnitude. Conserving the basin and flushing through bottom outlets extend the life. Enter the reservoir volume and the annual sediment inflow.
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.
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