Solver catalogue

Resultant of Concurrent Forces

Adds concurrent coplanar forces (each as magnitude + direction angle CCW from +x, or as signed x/y components) into a single resultant: magnitude, direction, components — plus the equilibrant (the force that would balance the system).

Particle Equilibrium (Cable Tensions)

Equilibrium of concurrent forces at a point: known loads (e.g. a hung weight, direction 270°) plus one or two members/cables at known directions — solves ΣFx = 0 and ΣFy = 0 for the unknown forces. Positive result = tension (pulling away from the point); a negative result means compression, which a cable cannot carry (warned).

Moment of Forces About a Point

The moment (torque) of one or more 2D forces about any point, by the cross-product form M = rx·Fy − ry·Fx (equivalent to Varignon's theorem), plus applied couples — reporting each force's contribution, the total moment (CCW positive), and the resultant force of the system.

Friction on an Incline

The classic friction problem: a block of given weight (or mass) on a flat or inclined surface with coefficient of friction. Finds the normal force, the friction force available, whether the block slides on its own, the force parallel to the surface needed to START it moving up-slope (and to KEEP it moving with μk), and — when an applied force is given — whether that force suffices.

Normal, Shear & Bearing Stress

Chapter-one stress analysis: average normal stress σ = P/A in a member, average shear stress in bolts/pins (single or double shear, load shared over the bolt count), bearing stress on a plate σb = P/(d·t), and factors of safety against given allowable stresses.

Thermal Stress (Constrained Bar)

A bar restrained between rigid supports and subjected to a temperature change: free expansion δT = α·ΔT·L, the compressive (or tensile, on cooling) stress σ = E·α·ΔT when fully restrained, the reduced stress when an initial gap absorbs part of the expansion, and the restraint force when the area is given.

Thin-Walled Pressure Vessel

Thin-wall stresses in a pressurized cylinder (hoop σh = p·r/t, longitudinal σl = p·r/2t) or sphere (σ = p·r/2t) — or, given an allowable stress instead of a thickness, the minimum wall thickness. Warns when r/t < 10 (thin-wall theory becomes unreliable).

Beam Support Reactions

Support reactions for a determinate single-span beam (simply supported with optional overhangs, or a cantilever) under point loads, uniform/triangular/trapezoidal distributed loads, and applied moments. Reports the vertical reaction at each support and the fixed-end moment for a cantilever. Use only when reactions alone are asked; use beam-shear-moment for diagrams/max moment.

Beam Shear & Moment Diagrams

Support reactions plus complete shear V(x) and bending moment M(x) diagrams for a determinate single-span beam (simply supported with optional overhangs, or a cantilever) under point loads, uniform/triangular/trapezoidal distributed loads, and applied moments. Reports maximum shear and maximum/minimum bending moments with their locations.

Beam Deflection by Superposition

Deflection and slope of a simply supported beam (point loads anywhere, full-span UDL, end moments) or a cantilever (point loads anywhere, full-span UDL, free-end moment) by superposing the handbook closed forms (PL³/48EI, 5wL⁴/384EI, PL³/3EI, ...). Reports deflection/slope at requested points plus the maximum deflection and its location.

Beam Deflection by Double Integration

Deflection and slope anywhere on a determinate single-span beam (simply supported with optional overhangs, or a cantilever) under any combination of point loads, partial or full uniform loads, triangular/trapezoidal loads, and applied moments — by integrating the exact bending-moment polynomials twice (EI y'' = M). Reports the maximum deflection with its location, support/tip slopes, values at requested points, and the deflected-shape diagram.

Composite Section Properties

Area, centroid, centroidal moments of inertia Ix and Iy, and radii of gyration for a composite cross-section built from rectangles, circles, semicircles, and right triangles (holes supported via negative parts), using the parallel-axis theorem.

Axially Loaded Stepped Bar

Internal force, normal stress, and end deflection of a stepped or composite axially loaded bar fixed at one end, with concentrated axial loads at the section changes and an optional uniform temperature change (δ = Σ NL/AE + αΔT·L).

Torsion of Circular Shafts

Maximum shear stress and angle of twist for solid or hollow stepped circular shafts fixed at one end, under concentrated torques (τ = Tc/J, J = π(do⁴−di⁴)/32, φ = Σ TL/JG) or under transmitted power at a given rpm (T = P/ω).

Euler Column Buckling

Critical buckling load Pcr = π²EI/(KL)² for an ideal elastic column with pin-pin, fixed-free, fixed-fixed, or fixed-pin ends (theoretical K = 1, 2, 0.5, 0.7). Give Ix and Iy together to have the weak axis govern automatically; add the area for slenderness KL/r and critical stress, and the yield stress for the inelastic-range validity check.

Mohr's Circle / Stress Transformation

Principal stresses, maximum in-plane and absolute maximum shear stress, principal and shear-plane angles for a plane-stress state (σx, σy, τxy), plus the transformed stresses at an optional rotation angle. Hibbeler sign convention: tension positive; positive τxy acts in +y on the +x face.

Truss Analysis (Method of Joints)

Member forces (tension/compression/zero) and support reactions for a statically determinate plane truss, solved from joint equilibrium with a joint-by-joint walkthrough and zero-force member identification. Checks determinacy (m + r = 2j) and refuses unstable or indeterminate trusses.

Fluid Properties (ρ, γ, SG)

Converts between mass density ρ, unit weight γ = ρ·g, and specific gravity SG = ρ/ρ_water — give any one and get the others. Optionally: the mass and weight of a given volume, and the gage pressure p = γ·h at a given depth.

Buoyancy & Archimedes' Principle

Archimedes: the buoyant force equals the weight of displaced fluid. Handles the three classic setups — (1) does it float, and how much sits below the surface (submerged fraction = SG_object/SG_fluid); (2) apparent weight of a fully submerged object; (3) the crown problem: weight in air + weight in water gives the volume and the object's specific gravity.

Hydrostatic Pressure & Manometers

Pressure change through static fluid columns by the manometer traverse method: start at a known pressure and add γ·Δz moving down, subtract γ·Δz moving up, leg by leg (each leg its own fluid — water, mercury, oil by SG, unit weight, or density). Reports the end pressure in gage AND absolute terms plus each leg's contribution.

Hydrostatic Force on a Submerged Surface

Resultant hydrostatic force and center of pressure on a fully submerged plane surface (rectangle, circle, or triangle; vertical or inclined): F = γ·hc·A with ycp = yc + Ic/(yc·A) in slant coordinates, reported both along the plate and as vertical depth.

Continuity (Flow Between Pipe Sections)

Conservation of volumetric flow for incompressible fluid: Q = A1·V1 = A2·V2. Give each section's area OR diameter, the velocity where known (or the flowrate directly), and it finds the missing velocities, areas, or the flowrate — e.g. the velocity in a contracted pipe. Use ONLY when the question asks about velocity/area/flowrate alone. If the problem ALSO asks for a pressure (continuity + Bernoulli combined), use energy-equation instead — it handles the continuity part internally AND gives the pressure. Not for: friction losses (darcy-weisbach).

Energy Equation (Bernoulli with Machines & Losses)

The energy equation p₁/γ + V₁²/2g + z₁ + hp = p₂/γ + V₂²/2g + z₂ + ht + hL between two sections of one stream, rearranged for a single unknown: downstream pressure, downstream velocity, required pump head, head loss, downstream elevation, or pump power P = γQh/η. Velocities may come from flowrate + diameter via continuity. All pressures are gage.

Darcy–Weisbach Pipe Head Loss

Pressure-pipe friction by Darcy–Weisbach, hL = f·(L/D)·V²/2g plus ΣK·V²/2g minor losses, with f from 64/Re (laminar) or the Colebrook equation (Swamee–Jain seeded). Solves for head loss, flowrate, or required diameter; roughness by value or by material (steel/cast iron/concrete/PVC/glass); viscosity direct or from water temperature. Reports f, Reynolds number, velocity, and regime.

Hazen–Williams Pipe Flow

Hazen–Williams water-pipe friction, V = 0.849·C·R^0.63·S^0.54 (SI; 1.318 in US units), closed-form for head loss, flowrate, or required diameter of a full circular pressure pipe.

Manning Normal Depth

Normal (uniform-flow) depth from Manning's equation Q = (1/n)·A·R^(2/3)·√S for rectangular, trapezoidal, triangular, or circular (partly full) sections, with velocity, area, hydraulic radius, top width, and Froude number at that depth. Handles the circular-pipe capacity edge: flows above the open-channel maximum refuse as surcharge, and near-full flows with two normal depths report the lower with a warning listing both.

Critical Depth & Froude Number

Critical depth from the Fr = 1 condition Q²T/(gA³) = 1 for rectangular, trapezoidal, triangular, or circular sections (rectangular closed form (q²/g)^(1/3) shown as a check), plus — when an actual depth is given — the Froude number Fr = V/√(g·A/T) using the hydraulic depth, with sub/supercritical classification.

Rational Method Peak Flow

Peak stormwater runoff Q = C·i·A for a small catchment, with composite C over multiple subareas.

Runoff Depth & φ-Index

Storm water accounting for a uniform storm: converts a measured runoff volume to an equivalent depth over the watershed (or takes the depth directly), gives the runoff coefficient C = Q/P, the total abstractions, and the φ-index — the constant infiltration rate (P − Q)/t that explains the losses.

Horizontal Circular Curve Geometry

Complete simple circular curve geometry from the central angle Δ (decimal degrees or DMS) and either the radius R or the arc-definition degree of curve D: tangent length T, curve (arc) length L, long chord LC, external distance E, middle ordinate M, degree of curve, and PC/PT stationing from a PI station.

Minimum Curve Radius (Superelevation)

The minimum safe horizontal-curve radius for a design speed with maximum superelevation e and side-friction factor f: R = v²/(g(e + f)) — identical to the handbook forms R = V²/[127(e+f)] (km/h, meters) and R = V²/[15(e+f)] (mph, feet), shown for reconciliation.

Greenshields Traffic Flow

Greenshields' linear speed-density model: v = vf(1 − k/kj) and q = k·v. Give the free-flow speed and jam density plus ONE of current density, speed, or flow, and it returns the rest — with the capacity qmax = vf·kj/4 and (for a given flow) both the uncongested and congested density solutions. Units: densities in veh per the SAME distance unit as the speeds (km or miles).

Vertical Curve Elevations (Symmetric Parabola)

Symmetric parabolic vertical curve between grades g1 and g2: BVC/EVC stations and elevations, K value, elevation at any station, and the high/low (turning) point — reported only when it actually lies on the curve. Accepts curve length L or rate of vertical curvature K.

Stopping Sight Distance (AASHTO)

AASHTO stopping sight distance: brake-reaction distance plus braking distance, with grade adjustment, reporting the exact computed SSD and the design value rounded up per the Green Book tables (next 5 ft US / next 5 m metric). Defaults: 2.5 s reaction time, 11.2 ft/s² (US) / 3.4 m/s² (metric) deceleration.

Passing Sight Distance (AASHTO 2011+)

Passing sight distance for two-lane, two-way highways per the AASHTO 2011+ Green Book single-model design values (Table 3-4, from NCHRP Report 605): PSD and the assumed passing/passed vehicle speeds, with linear interpolation between tabulated design speeds.

Soil Phase Relationships

Complete soil phase (weight–volume) table — void ratio e, porosity n, water content w, degree of saturation S, moist/dry/saturated unit weights and buoyant unit weight γ′ — from any sufficient subset of those quantities plus specific gravity Gs.

USCS Soil Classification (ASTM D2487)

Unified Soil Classification System group symbol and name from gradation (percent gravel/sand/fines or sieve passing percents, D10/D30/D60 or Cu/Cc) and Atterberg limits (LL, PL or PI), following the ASTM D2487 flowchart: 50% fines split, A-line plasticity chart, dual symbols at 5–12% fines, CL-ML hatched zone.

AASHTO Soil Classification (M 145)

AASHTO group classification (A-1-a through A-7-6) with group index, from percent passing the No. 10, No. 40 and No. 200 sieves plus liquid limit and plasticity index, by left-to-right elimination per AASHTO M 145. Reports GI = (F−35)[0.2+0.005(LL−40)] + 0.01(F−15)(PI−10) rounded and clamped, with the partial (PI-term) GI for A-2-6/A-2-7.

Vertical Effective Stress Profile

Total stress σ, hydrostatic pore pressure u, and effective stress σ′ = σ − u at chosen depths in a layered soil profile with a water table (which may be at depth, at the surface, or ponded above it). Emits the σ/u/σ′ vs depth diagram.

Darcy Seepage Velocity & Discharge

Darcy's law through soil: discharge (Darcy) velocity v = k·i from hydraulic conductivity and gradient (or head loss over flow length), volumetric flow Q = v·A, and true seepage velocity v_s = v/n through the pores.

Triaxial Test — Effective Friction Angle

Interprets a triaxial compression test at failure: subtracts the pore pressure to get effective principal stresses, then finds the effective friction angle φ' from the Mohr-Coulomb criterion — sin φ' = (σ1'−σ3')/(σ1'+σ3') for c' = 0, or the general form with a given c'. Also reports deviator stress and the failure-plane angle.

Terzaghi Bearing Capacity

Terzaghi ultimate bearing capacity of a shallow strip, square, or circular footing: q_ult = sc·c·Nc + γ·Df·Nq + sγ·0.5·γ·B·Nγ, with the shape factors applied automatically and the N-factors taken from the problem (every textbook tabulates slightly different Terzaghi/Meyerhof values, so give the ones your problem provides). Divides by the factor of safety for the gross allowable capacity.

Stream Mixing & Dilution

Conservative steady-state mixing of two or more streams: mixed flow Q = ΣQi, flow-weighted concentration C = ΣQiCi/ΣQi, mass loading of each stream in kg/day, and an optional completely-mixed first-order decay C/(1 + kθ). Handles the mg/L ≡ g/m³ identity explicitly.

Tank Fill / Detention Time

The volume–flow–time triangle t = V/Q: time to fill or drain a tank, hydraulic detention (retention) time of a basin, required volume for a target time, or the flowrate implied by a fill time — with optional simultaneous outflow (net filling).

Sedimentation Tank (Overflow Rate)

Clarifier/sedimentation-basin performance numbers: surface overflow rate (surface loading) SOR = Q/A — the settling-velocity benchmark — plus detention time V/Q when the depth is given, weir loading rate Q/L_weir, and horizontal flow-through velocity.

BOD (Dilution Method)

Biochemical oxygen demand from the standard dilution bottle test: the oxygen depletion of the diluted mixture times the dilution factor, BOD = (DO_initial − DO_final)·(V_bottle/V_sample) — with the Standard Methods validity checks (at least 2 mg/L depletion and at least 1 mg/L residual DO).

Compound-Interest Factors (Time Value of Money)

Evaluates any of the eight standard discrete compound-interest factors — (F/P), (P/F), (A/P), (P/A), (F/A), (A/F), (P/G), (A/G) — at an effective rate per period, with optional nominal-to-effective rate conversion and an optional amount to apply.

Present / Annual / Future Worth Comparison

Compares mutually exclusive alternatives by present worth (PW), annual worth (AW), or future worth (FW) at the MARR, from signed cash flows (costs negative) with uniform-series shorthand and salvage values. Automatically switches to AW when lives are unequal and no common analysis period is given.

Internal Rate of Return (IRR)

Finds the internal rate of return — the interest rate at which the net present value of a signed cash-flow series equals zero — including ALL real roots when the flows change sign more than once (non-conventional series), with an accept/reject comparison against the MARR when given.

Particle Kinematics

Constant-acceleration motion. Rectilinear: give ANY 3 of {initial velocity, final velocity, acceleration, time, displacement} and it solves the rest, with a consistency check when over-specified. Projectile: launch speed + angle (+ optional launch height) gives time of flight, range, apex height, and impact velocity.

Work–Energy Theorem

Applies T1 + ΣU = T2 to a particle: kinetic energy plus the work of constant forces, gravity (height change), springs, and kinetic friction. Solves for the final speed, the initial speed, or an unknown shared distance (e.g. 'how far does it slide before stopping').

Impulse–Momentum & 1-D Collisions

Impulse mode: F·Δt = m·Δv for one body — solves the missing one of {final velocity, impulse, force, duration}. Collision mode: direct central impact of two bodies with a coefficient of restitution — post-impact velocities, impulse between the bodies, and kinetic energy lost.

Descriptive Statistics

Mean, median, mode, sample AND population standard deviation/variance (both reported — know which your course wants), coefficient of variation, range, quartiles and IQR for a list of values.

Linear Regression (Least Squares)

Fits y = a + b·x by least squares: slope, intercept, correlation coefficient r, r², standard error of the estimate, and predictions at requested x values (with extrapolation warnings).