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How to calculate concrete slab volume
Imagine covering a rectangular floor with a layer of concrete. First find how much floor is covered, then account for how thick that layer is. We are measuring volume, not designing the slab.
Steps + calculator + practiceHow to calculate percentage increase or decrease
A change of 20 means more when you started with 40 than when you started with 400. Percentage change compares the difference with the starting value.
Steps + calculator + practiceHow to find a percentage of a number
“20 percent of 150” asks for 20 of every 100 parts of the whole. Turn the percentage into a decimal share, then take that share of the whole value.
Steps + calculator + practiceTriangle area using base and perpendicular height
Two matching triangles can form a parallelogram with the same base and height. One triangle occupies half that area, which explains the factor of one half.
Steps + calculator + practiceHow to find the hypotenuse of a right triangle
The hypotenuse is opposite the right angle and is the longest side of a right triangle. You can calculate it when the two perpendicular side lengths are known.
Steps + calculator + practiceHow to calculate rectangle area
Area counts how many square units cover a flat surface. A rectangle 5 m long and 3 m wide can be thought of as five rows of three 1 m² squares.
Steps + calculator + practiceHow to calculate axial stress
A rod is pulled along its length. Stress tells you how much force is carried by each square metre of its cross-section. Think of spreading the same load over a narrow or a wide piece: the narrow one carries more force per unit area.
Steps + calculator + practiceHow to calculate strain from a change in length
Strain measures the change in length relative to the starting length. A 2 mm extension is a much bigger change for a short sample than for a very long one.
Steps + calculator + practiceHow to use Hooke’s law for stress
For a linearly elastic material, stress and strain grow in proportion. Young’s modulus E describes stiffness: a stiffer material needs more stress to reach the same strain.
Steps + calculator + practiceHow to calculate a rod’s axial extension
A uniform rod stretches when pulled. This formula combines its load, length, area and stiffness to estimate the small elastic change in length.
Steps + calculator + practiceHow to calculate thermal strain
An unconstrained material can change size as its temperature changes. Thermal strain measures the fractional length change, before you multiply by the actual length.
Steps + calculator + practiceHow to calculate thermal expansion in length
Two bars of the same material experience the same temperature rise. The longer bar changes length more because the same small fractional change acts over more material.
Steps + calculator + practiceHow to calculate a rectangle’s second moment of area
This geometric quantity measures how the area is distributed away from a chosen axis. For bending about the horizontal centroidal axis, depth matters strongly because it is cubed.
Steps + calculator + practiceHow to calculate elastic section modulus
The outermost fibres of a bending section are furthest from its neutral axis. Section modulus combines the area distribution I with this outer-fibre distance.
Steps + calculator + practiceHow to calculate maximum elastic bending stress
Bending stretches one side of a beam and compresses the other. For simple elastic bending, the largest stress magnitude is found at an extreme fibre.
Steps + calculator + practiceHow to calculate radius of gyration of an area
This radius is a geometric summary of how far an area is spread from an axis. It is useful when forming slenderness ratios, but is not necessarily a physical radius you can measure on the shape.
Steps + calculator + practiceBeam reactions: one load at the centre
A simply supported beam rests on a pin and a roller. With one downward load exactly in the middle, the two supports share the vertical load equally.
Steps + calculator + practiceBeam reactions under a uniform load
A uniformly distributed load applies the same force to every metre of a beam. Before sharing the load between supports, turn this force-per-metre into a total force.
Steps + calculator + practiceMaximum beam moment from a centre point load
For a simply supported beam with a central point load, the largest bending moment is at midspan. You can find it by considering only the left half of the beam.
Steps + calculator + practiceMaximum beam moment from a uniform load
Use this lesson for a simply supported beam carrying a uniform load over its entire span. The largest sagging moment is at the middle because the loading is symmetric.
Steps + calculator + practiceCantilever moment from a load at the free end
A cantilever is fixed at one end and free at the other. A downward tip load creates the largest bending-moment magnitude at the fixed end, where its lever arm is longest.
Steps + calculator + practiceCantilever moment from a uniform load
A uniform load covers a cantilever’s full length. Replace the many small forces with one equivalent force, then measure its distance from the fixed end.
Steps + calculator + practiceHow to calculate water pressure at a depth
Water lower down has more water above it. Hydrostatic pressure tells you the extra pressure caused by that vertical column of still liquid.
Steps + calculator + practiceHow to calculate hydrostatic force on a flat surface
Pressure acts across a surface, so a larger submerged gate can feel a greater total force. For a flat surface in still liquid, use pressure at its area centroid to find the total resultant.
Steps + calculator + practiceHow to calculate buoyant force
A submerged object displaces liquid. The liquid pushes upward with a force equal to the weight of the displaced liquid, not automatically the weight of the object.
Steps + calculator + practiceHow to calculate flow rate from area and velocity
Flow rate means how much fluid volume passes a section each second. Imagine a short moving slice of water: its cross-section is A and it travels v metres each second.
Steps + calculator + practiceHow to find velocity after a pipe area change
When the same incompressible flow passes from a wide section to a narrower section, it must move faster through the narrower area to carry the same volume each second.
Steps + calculator + practiceHow to calculate Reynolds number in a pipe
Reynolds number compares inertial effects with viscous effects in a flow. It has no unit and helps describe the flow regime; it is not itself a pressure loss.
Steps + calculator + practiceHow to calculate hydraulic radius
Hydraulic radius is flow area divided by wetted perimeter. In an open channel, the water surface touches air and is not part of the wetted perimeter.
Steps + calculator + practiceHow to calculate hydraulic pump power
This is the rate at which a pump adds useful hydraulic energy to the water. It is not the electrical input power printed on a motor.
Steps + calculator + practiceHow to calculate soil water content
Water content compares the mass of water with the mass of dry soil solids. The denominator is dry soil, not the total wet sample.
Steps + calculator + practiceHow to calculate soil void ratio
A soil sample contains solid particles and spaces between them. Void ratio compares the total space volume, whether occupied by water or air, with the volume of solids.
Steps + calculator + practiceHow to convert void ratio to porosity
Porosity asks what fraction of the total sample volume is voids. It uses a different denominator from void ratio, even though both describe the same sample.
Steps + calculator + practiceHow to calculate effective stress in saturated soil
Total stress is shared between pore-water pressure and the soil skeleton. Effective stress represents the part carried through the soil skeleton in this saturated-soil model.
Steps + calculator + practiceHow to find horizontal distance from coordinate differences
East and north coordinate differences form two perpendicular sides of a right triangle. The straight horizontal distance between the points is its hypotenuse.
Steps + calculator + practiceHow to calculate gradient from rise and run
Gradient compares vertical change with horizontal travel. A rise of 1 m over 20 m means the height changes by 0.05 m for every horizontal metre.
Steps + calculator + practiceHow to find a level from backsight and foresight
A level instrument creates a horizontal line of sight. A staff reading tells you how far a point lies below that line. Start from a point with a known elevation.
Steps + calculator + practiceEarthwork volume by the average-end-area method
Two cross-sections bound a length of earthwork. This method estimates the volume by using the average of the two end areas across the distance between them.
Steps + calculator + practiceEarthwork volume using a middle cross-section
A middle cross-section tells you more about the shape than two ends alone. The prismoidal rule gives the middle area a weight of four before forming the volume.
Steps + calculator + practiceHow to calculate density from mass and volume
Density tells you how much mass is packed into a unit of volume. Two blocks can have the same size but different masses because their densities differ.
Steps + calculator + practiceHow to convert density to unit weight
Density uses mass; unit weight uses force. Gravity converts the kilograms in each cubic metre into the force that volume exerts.
Steps + calculator + practiceHow to calculate kinetic energy
A moving object carries kinetic energy. Speed matters strongly: doubling speed makes this energy four times larger when mass is unchanged.
Steps + calculator + practiceHow to calculate gravitational potential energy
Lifting an object raises its gravitational potential energy relative to a reference height. The height reference must be stated because a height is never meaningful by itself.
Steps + calculator + practiceHow to calculate linear momentum
Momentum combines mass with velocity and has direction. In one-dimensional problems, choose one direction as positive before using the signs.
Steps + calculator + practiceHow to calculate impulse from force and time
Impulse measures the effect of a force acting over time. A smaller force acting for longer can give the same momentum change as a larger force acting briefly.
Steps + calculator + practiceHow to estimate runoff volume from rainfall
Rain depth spread over an area represents a volume of water. A specified runoff fraction C estimates the part that becomes runoff for this simple event-volume exercise.
Steps + calculator + practiceLearn: Poisson's ratio
Pull a bar gently: it usually becomes longer and narrower. Poisson’s ratio compares that sideways strain with the lengthwise strain; it compares relative changes, not the two changes in millimetres.
Steps + calculator + practiceLearn: Shear modulus
Young’s modulus describes resistance to stretching; shear modulus describes resistance to a change of shape. This relationship lets you find the shear stiffness of an isotropic material from two familiar elastic properties.
Steps + calculator + practiceLearn: Fully restrained thermal stress
A free bar expands when it is heated. If rigid supports prevent all axial movement, the prevented expansion becomes elastic stress instead; the restraint, not temperature alone, creates this stress.
Steps + calculator + practiceLearn: Circle second moment of area
This geometric property describes how area is spread away from an axis for bending about a centroidal diameter. It is not the area of the circle and it is not a mass moment of inertia.
Steps + calculator + practiceLearn: Polar second moment — solid circle
This geometric property describes how area is spread away from an axis for torsion about the centre of a solid circular section. It is not the area of the circle and it is not a mass moment of inertia.
Steps + calculator + practiceLearn: Parallel-axis theorem
You may know the second moment of an area about its centre but need it about another parallel axis. The parallel-axis theorem adds the effect of moving the entire area away from its centroidal axis.
Steps + calculator + practiceLearn: Beam shear stress
Shear force does not normally produce a uniform stress over a beam cross-section. This formula finds the local shear stress at a chosen level by using the area on one side of that level.
Steps + calculator + practiceLearn: Circular-shaft torsional shear
Twisting a circular shaft creates shear stress that increases with distance from its centre. The formula gives the stress at the particular radius you choose, not necessarily the maximum surface stress.
Steps + calculator + practiceLearn: Angle of twist
A shaft can remain safe in stress yet twist too much for its function. Angle of twist measures the relative rotation between the two ends of a shaft segment under torque.
Steps + calculator + practiceLearn: Principal stresses — plane stress
The stresses on a tiny element change when you rotate the axes, even though its physical loading stays the same. Mohr’s circle turns the three plane-stress components into a centre and a radius so you can identify the important extremes.
Steps + calculator + practiceLearn: Maximum in-plane shear stress
The stresses on a tiny element change when you rotate the axes, even though its physical loading stays the same. Mohr’s circle turns the three plane-stress components into a centre and a radius so you can identify the important extremes.
Steps + calculator + practiceLearn: Force equilibrium check — three forces
Equilibrium means there is no unbalanced force in the component being checked. This lesson adds three signed contributions and reports the residual; it does not silently assume that the data already balance.
Steps + calculator + practiceLearn: Moment equilibrium check — three moments
Equilibrium means there is no unbalanced moment in the component being checked. This lesson adds three signed contributions and reports the residual; it does not silently assume that the data already balance.
Steps + calculator + practiceLearn: Simply supported — centre-load deflection
Find the maximum transverse deflection of a simply supported beam carrying one central point load. The maximum is at midspan; this is a displacement calculation, not a bending-strength check.
Steps + calculator + practiceLearn: Simply supported — UDL deflection
Find the maximum transverse deflection of a simply supported beam carrying a uniform load over its entire span. The maximum is at midspan; this is a displacement calculation, not a bending-strength check.
Steps + calculator + practiceLearn: Cantilever — end-load deflection
Find the maximum transverse deflection of a cantilever with one point load at its free end. The maximum is at the free end; this is a displacement calculation, not a bending-strength check.
Steps + calculator + practiceLearn: Cantilever — UDL deflection
Find the maximum transverse deflection of a cantilever carrying a uniform load over its entire length. The maximum is at the free end; this is a displacement calculation, not a bending-strength check.
Steps + calculator + practiceLearn: Euler critical buckling load
A slender compressed column may become laterally unstable before its material yields. Euler’s load predicts the ideal elastic bifurcation load for a column represented by an effective buckling length.
Steps + calculator + practiceLearn: Euler critical stress
Euler buckling can also be expressed as an average axial stress instead of a total load. This form uses the geometric slenderness L/i to express how vulnerable a column is to elastic buckling.
Steps + calculator + practiceLearn: Member slenderness
Slenderness compares a column’s effective length with the spread of its cross-sectional area. A long member with a small radius of gyration is more slender and generally more sensitive to buckling.
Steps + calculator + practiceLearn: Limit-state utilization Ed/Rd
A resistance check compares what the structure must carry with what it can resist in a particular failure mode. The utilization ratio places demand and resistance on a common scale: 1 is the boundary for this one comparison.
Steps + calculator + practiceLearn: Generic material design value
A characteristic material property is a statistical reference value, not automatically the value used in a design equation. This calculation applies a conversion factor and a material partial factor to obtain the supplied design-property model.
Steps + calculator + practiceLearn: Simple ULS combination — study form
An ultimate-limit-state load combination applies factors to permanent and variable actions before adding their contributions. This lesson contains one permanent action and one leading variable action so the basic bookkeeping is easy to see.
Steps + calculator + practiceLearn: Self-weight from unit weight
Self-weight is a permanent action caused by the material itself. When a component has uniform unit weight, multiplying that unit weight by the actual material volume gives its total weight force.
Steps + calculator + practiceLearn: Roof snow load — coefficient form
Ground snow load is not automatically the load on a roof. This teaching equation adjusts a supplied ground value for roof shape, exposure and thermal conditions to obtain one roof-load case.
Steps + calculator + practiceLearn: Basic wind velocity pressure
Basic velocity pressure converts a supplied basic wind speed into an energy-per-volume scale. It is a starting quantity for wind action calculations, not yet the pressure on a particular wall or roof.
Steps + calculator + practiceLearn: Wind surface pressure
A surface coefficient translates peak velocity pressure into a signed pressure on one surface. A positive coefficient represents pressure in the chosen convention; a negative coefficient represents suction.
Steps + calculator + practiceLearn: SLS characteristic combination — study form
The characteristic serviceability combination represents a particular level of actions for checks such as movement, cracking or long-term response. It is not interchangeable with an ultimate-limit-state combination or the other two serviceability combinations.
Steps + calculator + practiceLearn: SLS frequent combination — study form
The frequent serviceability combination represents a particular level of actions for checks such as movement, cracking or long-term response. It is not interchangeable with an ultimate-limit-state combination or the other two serviceability combinations.
Steps + calculator + practiceLearn: SLS quasi-permanent combination — study form
The quasi-permanent serviceability combination represents a particular level of actions for checks such as movement, cracking or long-term response. It is not interchangeable with an ultimate-limit-state combination or the other two serviceability combinations.
Steps + calculator + practiceLearn: Concrete design strength
Concrete’s characteristic cylinder strength is not the stress value normally inserted directly into a design resistance equation. The design compressive strength accounts for a specified strength coefficient and a concrete partial factor.
Steps + calculator + practiceLearn: Reinforcement design yield strength
Reinforcing bars have a characteristic yield strength, but a resistance calculation generally needs a design strength. This lesson converts the supplied characteristic yield value using the reinforcing-steel partial factor.
Steps + calculator + practiceLearn: Mean concrete compressive strength
Characteristic and mean concrete compressive strengths describe different statistical values. This first-generation normal-weight concrete relationship estimates the mean strength from the characteristic cylinder strength.
Steps + calculator + practiceLearn: Concrete secant modulus — study relation
Concrete stiffness controls elastic deformation and is not numerically equal to its compressive strength. This empirical expression estimates a mean secant modulus from mean cylinder strength for the stated concrete model.
Steps + calculator + practiceLearn: RC flexural resistance — tensile steel form
A singly reinforced concrete section resists bending through a compressive force in concrete and a tensile force in steel. Their separation is the lever arm z, which turns those balancing forces into a resisting moment.
Steps + calculator + practiceLearn: Required tensile steel area — study estimate
A singly reinforced concrete section resists bending through a compressive force in concrete and a tensile force in steel. Their separation is the lever arm z, which turns those balancing forces into a resisting moment.
Steps + calculator + practiceLearn: Minimum longitudinal tension steel — classic EC2 form
A beam needs a minimum amount of tensile reinforcement even when a simple bending calculation suggests very little steel. This teaching expression compares a strength-dependent minimum with a geometric lower bound and selects the larger.
Steps + calculator + practiceLearn: Shear reinforcement resistance
Vertical stirrups resist shear as part of a truss-like mechanism with inclined concrete compression struts. This equation estimates the stirrup contribution using the amount of steel per unit length and a specified strut angle.
Steps + calculator + practiceLearn: Punching shear design stress
A concentrated column load can punch through a slab around a control perimeter. This calculation spreads the design shear, adjusted by a supplied eccentricity factor, over the idealized vertical area u d.
Steps + calculator + practiceLearn: Crack width
A reinforced-concrete crack opens because reinforcement and concrete do not have the same average tensile strain between cracks. Multiplying that strain difference by the characteristic crack spacing estimates the crack width.
Steps + calculator + practiceLearn: Basic required anchorage length
A reinforcing bar transfers its tensile force to surrounding concrete through bond. Basic required anchorage length estimates how much straight bonded length is needed before additional design modifiers and minimum-length rules are considered.
Steps + calculator + practiceLearn: Design bond strength — classic EC2 form
Bond strength describes the design stress that transfers force between reinforcement and concrete along the bar surface. This teaching relationship derives it from concrete design tensile strength and two supplied bond-condition factors.
Steps + calculator + practiceLearn: Prestressed concrete fibre stress
Prestress produces a uniform axial stress and, when eccentric, a bending stress. An external bending moment adds another contribution, so the stress at one chosen fibre is found by adding three signed terms.
Steps + calculator + practiceLearn: RC compression resultant — rectangular block
A rectangular stress block replaces a nonlinear concrete compression distribution with a simpler uniform block. Its resultant force is stress times the area of that block, not the area of the whole section.
Steps + calculator + practiceLearn: RC lever arm — rectangular block
The lever arm is the distance between the tensile-steel force and the concrete compression resultant. For a rectangular uniform stress block, the compression force acts at the middle of that block.
Steps + calculator + practiceLearn: Concrete shear resistance — coefficient form
Concrete without calculated shear reinforcement still has a shear-resistance model based on concrete strength, longitudinal reinforcement and section size. This lesson evaluates the explicitly supplied main expression, including a compressive-stress contribution.
Steps + calculator + practiceLearn: Maximum shear compression-strut resistance — study form
Adding more stirrups cannot increase shear resistance indefinitely: the inclined concrete compression strut can crush. This lesson evaluates the stated simplified strut-crushing ceiling for a supplied angle and concrete-strength reduction factor.
Steps + calculator + practiceLearn: Maximum crack spacing — coefficient form
Crack spacing is influenced by cover, bar size, bond and effective reinforcement ratio. This teaching expression adds a cover-related term to a reinforcement-related term to estimate the maximum crack spacing for its stated model.
Steps + calculator + practiceLearn: Design anchorage length from basic length
The basic required anchorage length is adjusted for specific detailing and bond-related effects through a product of design modifiers. This lesson evaluates that product only, keeping every modifier visible so none is accidentally omitted.
Steps + calculator + practiceLearn: Prestress friction loss along tendon
A tendon loses force along its length as friction develops against its duct. This exponential model combines intended curvature and unintended wobble to estimate the remaining force at distance x from the stressing end.
Steps + calculator + practiceLearn: Steel gross-section yielding resistance
A steel tension member can yield across its gross cross-section. This equation estimates that gross-section design resistance by multiplying the area by yield strength and applying the supplied material partial factor.
Steps + calculator + practiceLearn: Non-dimensional buckling slenderness
Nondimensional member slenderness compares a steel section’s yielding force with its ideal elastic buckling load. It places material strength and buckling sensitivity into one ratio used by a selected buckling curve.
Steps + calculator + practiceLearn: Buckling Φ parameter
The buckling-curve parameter Φ is an intermediate quantity used to calculate the member reduction factor χ. It combines nondimensional slenderness with an imperfection factor chosen for the relevant buckling curve.
Steps + calculator + practiceLearn: Buckling reduction factor
The reduction factor χ lowers a steel member’s reference compression resistance to account for buckling in the selected model. This lesson uses an already calculated Φ and nondimensional slenderness, and caps the factor at 1.
Steps + calculator + practiceLearn: Steel buckling resistance
A compressed steel member may buckle before reaching gross-section yield. This equation applies the supplied buckling reduction factor to the reference yielding force and then introduces the member partial factor.
Steps + calculator + practiceLearn: Plastic bending resistance
A plastic section modulus represents the force couple when an admissible section develops its plastic stress distribution. Multiplying it by yield strength gives the plastic bending resistance before the supplied section partial factor.
Steps + calculator + practiceLearn: Elastic bending resistance
Elastic bending resistance corresponds to the extreme fibre reaching the design yield stress in an admissible section model. It uses elastic section modulus, which is not the same geometric property as plastic section modulus.
Steps + calculator + practiceLearn: LTB bending resistance
Lateral-torsional buckling can reduce a beam’s bending resistance before its cross-section reaches the reference moment capacity. This equation applies an already established χLT reduction to the appropriate section-modulus resistance.
Steps + calculator + practiceLearn: Plastic shear resistance
A steel cross-section has a shear area that carries the shear force in the chosen direction. This plastic shear expression combines that effective shear area with the yield-based shear stress and the section partial factor.
Steps + calculator + practiceLearn: LTB non-dimensional slenderness
A beam can deflect sideways and twist under bending. Lateral-torsional nondimensional slenderness compares its reference section moment with the elastic critical moment for that lateral-torsional mode.
Steps + calculator + practiceLearn: Composite modular ratio
Steel and concrete deform together differently because their elastic moduli differ. A modular ratio compares those stiffnesses and is used when transforming one material’s area into an equivalent area of another.
Steps + calculator + practiceLearn: Required shear connectors — study estimate
A composite member transfers longitudinal shear between its materials through discrete connectors. Dividing the total shear to be transferred by the design resistance of one connector gives a theoretical count, which must be rounded upward.
Steps + calculator + practiceLearn: Timber design strength
Timber strength depends on load duration and moisture conditions as well as the characteristic grade. This teaching equation adjusts the relevant characteristic strength by kmod and divides by the timber material partial factor.
Steps + calculator + practiceLearn: Characteristic masonry strength
Masonry is an assembly of units and mortar, so its characteristic compressive strength is not simply the strength of either ingredient. This empirical relationship combines normalized unit strength and mortar strength through specified powers.
Steps + calculator + practiceLearn: Masonry design strength
Characteristic masonry strength describes the assembled masonry under a specified reference model. Its design compressive strength is obtained by applying the appropriate masonry partial factor, which depends on the adopted design provisions.
Steps + calculator + practiceLearn: Steel net-section fracture resistance — coefficient form
Bolt holes reduce the area available to carry tension, and a member can fracture across that reduced section. This expression evaluates a net-section fracture resistance component using ultimate tensile strength rather than yield strength.
Steps + calculator + practiceLearn: Bolt shear resistance — coefficient form
A bolt can fail in shear, so its resistance must be checked for that mode independently. This equation evaluates the stated resistance of one bolt on one shear plane.
Steps + calculator + practiceLearn: Bolt tension resistance — coefficient form
A bolt can fail in tension, so its resistance must be checked for that mode independently. This equation evaluates the stated resistance of one bolt in tension.
Steps + calculator + practiceLearn: Bolt bearing resistance — coefficient form
A bolt can press against the side of its hole and damage the connected plate. This bearing-resistance component uses the plate’s ultimate strength, bolt diameter, plate thickness and supplied geometry-dependent coefficients.
Steps + calculator + practiceLearn: Fillet weld effective throat area
A fillet weld transfers force through its effective throat rather than through its visible leg size alone. The effective throat area is the throat thickness multiplied by the effective weld length.
Steps + calculator + practiceLearn: Composite/internal-force moment sum
A composite-section resistance model may contain several force couples or resultants acting at different lever arms. This lesson adds two signed force-times-distance contributions about one common reference.
Steps + calculator + practiceLearn: Timber bending utilization
A simple timber bending check compares the calculated bending stress with the applicable design bending strength. It is a stress utilization for one axis, not a complete check of every way the member can fail.
Steps + calculator + practiceLearn: Masonry compression resistance
A masonry wall’s design compressive strength must be applied to an effective area and reduced for the specified wall behaviour. This equation evaluates the supplied compression-resistance model using a reduction factor Φ.
Steps + calculator + practiceLearn: Void ratio from porosity
Porosity and void ratio both describe empty space in soil, but they use different denominators. Porosity divides void volume by total volume; void ratio divides it by solid volume.
Steps + calculator + practiceLearn: Degree of saturation
Degree of saturation tells you how much of a soil’s void space is filled with water. The remaining void space contains air; the solid grains do not belong in the denominator.
Steps + calculator + practiceLearn: Dry density from bulk density
Bulk density includes both solids and water, but dry density counts only the solids mass within the same total sample volume. Water content by mass lets you remove the water-mass contribution without changing the volume denominator.
Steps + calculator + practiceLearn: Relative density
Relative density locates a granular soil between its loosest and densest reference states. It compares void ratios, not the soil’s mass density directly.
Steps + calculator + practiceLearn: Hydrostatic pore pressure
Below a static water table, water pressure increases with depth because of the weight of water above the point. Hydrostatic pore pressure is found from water unit weight and pressure head.
Steps + calculator + practiceLearn: Rankine active earth pressure coefficient
Rankine’s active coefficient relates limiting horizontal effective stress to vertical effective stress in a simplified soil state. Active pressure develops when a wall moves sufficiently away from the soil.
Steps + calculator + practiceLearn: Rankine passive earth pressure coefficient
Rankine’s passive coefficient relates limiting horizontal effective stress to vertical effective stress in a simplified soil state. Passive resistance develops when a wall moves into the soil.
Steps + calculator + practiceLearn: At-rest earth pressure — Jaky approximation
Soil restrained from lateral movement can retain an at-rest horizontal stress different from the active and passive limits. Jaky’s empirical expression estimates the effective at-rest coefficient for normally consolidated soil.
Steps + calculator + practiceLearn: Triangular active earth thrust
For a dry cohesionless backfill, the ideal active pressure grows linearly from zero at the top to a maximum at the base. Its total horizontal thrust is the area of that triangular pressure diagram.
Steps + calculator + practiceLearn: Uniform surcharge earth thrust
A uniform surface surcharge adds horizontal pressure to a retaining wall in addition to the soil’s own weight. Under the stated active-pressure model, that added pressure is constant with depth.
Steps + calculator + practiceLearn: Classical bearing capacity — basic form
A simple shallow-foundation bearing model separates resistance into cohesion, surcharge and soil-weight contributions. This lesson evaluates that three-term expression using supplied bearing-capacity factors.
Steps + calculator + practiceLearn: Footing load eccentricity
A vertical force and a moment can be represented by an equivalent force acting away from the footing centre. Eccentricity is the signed distance needed for that force to reproduce the supplied moment.
Steps + calculator + practiceLearn: Rectangular footing contact pressure
A moment makes footing contact pressure larger at one edge and smaller at the opposite edge. The linear full-contact model starts with average pressure and adds or subtracts an eccentricity-related variation.
Steps + calculator + practiceLearn: Immediate elastic settlement
A loaded foundation compresses the supporting ground even when bearing failure is not reached. This elastic estimate combines contact pressure, foundation width, soil stiffness, Poisson’s ratio and an influence factor.
Steps + calculator + practiceLearn: 1D consolidation settlement — NC soil
A normally consolidated saturated clay layer can settle as increased effective stress compresses its soil skeleton. This one-dimensional estimate uses the compression index and a logarithmic change in effective stress.
Steps + calculator + practiceLearn: Consolidation time
Settlement magnitude and the time needed to develop it are different questions. The consolidation time factor relates elapsed time to the drainage path and coefficient of consolidation for a chosen degree of primary consolidation.
Steps + calculator + practiceLearn: Pile total compression resistance
An axially compressed pile can transfer load through its base and along its shaft. This calculation adds those two supplied resistance components to obtain their combined value under a compatible pile model.
Steps + calculator + practiceLearn: Infinite-slope factor of safety
A shallow slip surface parallel to a long uniform slope can be studied with an infinite-slope model. The factor of safety compares available shear strength on that plane with the downslope driving shear stress.
Steps + calculator + practiceLearn: Darcy's law — seepage
Groundwater flow through a porous soil is often proportional to hydraulic gradient in the laminar regime. Darcy’s law gives total discharge through the gross cross-sectional area normal to flow.
Steps + calculator + practiceLearn: Saturation relationship Sr·e = w·Gs
When direct phase volumes are unavailable, degree of saturation can be found from water content, solids specific gravity and void ratio. The identity links a mass-based measurement to how full the void space is with water.
Steps + calculator + practiceLearn: Seepage velocity
Darcy velocity spreads discharge over the whole soil cross-section, including solids. Actual average pore-water velocity is larger because water moves only through the connected pore space.
Steps + calculator + practiceLearn: Pile base resistance
A pile base transfers compressive load to the ground beneath it. When a representative unit base resistance is supplied, multiplying it by the effective base area gives the total base-resistance component.
Steps + calculator + practiceLearn: Classical retaining-wall sliding FS
A retaining wall needs a consistent balance against sliding. This traditional overall factor of safety compares the supplied resisting total with the supplied driving total; it does not derive those totals from geometry.
Steps + calculator + practiceLearn: Classical retaining-wall overturning FS
A retaining wall needs a consistent balance against overturning. This traditional overall factor of safety compares the supplied resisting total with the supplied driving total; it does not derive those totals from geometry.
Steps + calculator + practiceLearn: Centre of pressure — vertical plane
Water pressure increases with depth. On a submerged vertical plate, the lower part therefore pushes harder than the upper part, so the resultant force acts below the area centroid.
Steps + calculator + practiceLearn: Bernoulli total head at a section
Total hydraulic head expresses mechanical energy per unit weight as an equivalent height of liquid. A flowing liquid can carry that energy as elevation, pressure, or motion.
Steps + calculator + practiceLearn: Darcy–Weisbach friction loss
Flow along a pipe loses mechanical energy through wall friction. The Darcy–Weisbach relationship converts pipe length, diameter, speed and a supplied friction factor into an equivalent lost head.
Steps + calculator + practiceLearn: Laminar Darcy friction factor
At sufficiently low Reynolds number, liquid in a circular pipe moves in orderly layers. For fully developed laminar flow, the Darcy friction factor has a simple inverse relationship with Reynolds number.
Steps + calculator + practiceLearn: Colebrook–White friction factor
In turbulent pipe flow, the friction factor depends on both Reynolds number and pipe-wall roughness. The Colebrook equation contains the unknown factor on both sides, so ordinary one-step substitution cannot isolate it.
Steps + calculator + practiceLearn: Minor/local head loss
A fitting, inlet, bend or valve disturbs pipe flow and dissipates energy. A local loss coefficient K scales that loss to the velocity head in a specifically chosen reference pipe section.
Steps + calculator + practiceLearn: Pump input power
A pump must supply energy to raise liquid through a specified total head. Hydraulic output power is less than required input power because the pump or combined drive has losses.
Steps + calculator + practiceLearn: Simple system curve
A simple pump-system curve adds a static head to flow-dependent losses. The static part remains even when flow stops; the quadratic part grows as liquid moves faster through the system.
Steps + calculator + practiceLearn: Manning discharge
Manning’s equation estimates the discharge carried by an open channel under uniform-flow conditions. Roughness slows flow, while hydraulic radius and energy slope increase its carrying ability.
Steps + calculator + practiceLearn: Manning mean velocity
Manning’s velocity equation estimates the section-average speed in an open channel under uniform-flow conditions. Roughness slows flow, while hydraulic radius and energy slope increase its carrying ability.
Steps + calculator + practiceLearn: Froude number
The Froude number compares the mean speed of open-channel flow with the speed scale of shallow gravity waves. It helps distinguish subcritical flow, which can respond upstream, from supercritical flow.
Steps + calculator + practiceLearn: Open-channel specific energy
Specific energy measures open-channel energy relative to the local channel bed. It includes water depth and velocity head, but not the bed elevation above an external survey datum.
Steps + calculator + practiceLearn: Critical depth — rectangular channel
For a rectangular channel carrying a given discharge per unit width, critical depth is the depth at which specific energy is minimum. It separates the two ideal depth branches of open-channel flow.
Steps + calculator + practiceLearn: Hydraulic-jump sequent depth
A hydraulic jump changes shallow supercritical flow into deeper subcritical flow. In a horizontal rectangular channel, momentum balance relates the upstream depth and Froude number to the downstream conjugate depth.
Steps + calculator + practiceLearn: Hydraulic-jump energy loss
A hydraulic jump dissipates mechanical energy even though a momentum balance can relate the depths. For a rectangular horizontal channel, the loss can be written directly using the conjugate depths.
Steps + calculator + practiceLearn: Orifice discharge
An orifice is a small opening through which a pressure or water-level difference drives flow. The ideal jet speed follows from converting head into kinetic energy; a discharge coefficient corrects the ideal estimate.
Steps + calculator + practiceLearn: Rectangular sharp-crested weir
A rectangular sharp-crested weir estimates discharge from upstream head above its crest. The width of flowing water and jet speed both contribute to the head exponent.
Steps + calculator + practiceLearn: Triangular V-notch weir
A triangular V-notch weir estimates discharge from upstream head above its notch vertex. The width of flowing water and jet speed both contribute to the head exponent.
Steps + calculator + practiceLearn: Hydraulic diameter
Hydraulic diameter turns flow area and wetted perimeter into one equivalent length. If hydraulic radius R = A/P is already known, the diameter is simply four times that radius.
Steps + calculator + practiceLearn: Total pipe-system head loss
A pipe route can lose energy along straight lengths and at several localized fittings. Once each component is expressed as head loss, the compatible components can be added.
Steps + calculator + practiceLearn: Rational Method runoff
The Rational Method estimates a peak runoff rate for a catchment using rainfall intensity, catchment area and a runoff coefficient. It is a peak-flow estimate, not the full time history of a storm.
Steps + calculator + practiceLearn: Storage continuity rate
A storage volume rises when water enters faster than it leaves. The continuity equation compares inflow and outflow rates to find the instantaneous rate of storage change.
Steps + calculator + practiceLearn: Simple water balance
A water balance accounts for everything crossing a selected boundary during a selected period. Storage change is total water entering minus total water leaving during that same period.
Steps + calculator + practiceLearn: Return period from annual probability
Return period is a probability-based way to describe the rarity of exceeding a specified event magnitude. A “100-year” event has a 1% annual exceedance probability in the stated model, not a scheduled occurrence once per century.
Steps + calculator + practiceLearn: Probability of ≥1 exceedance in n years
A small annual risk can accumulate over many years. This calculation finds the probability of at least one exceedance during n years, rather than the expected number of exceedances.
Steps + calculator + practiceLearn: Steady radial flow — confined aquifer
The confined Thiem-type well relation connects steady radial flow with heads measured at two distances from a well. The radius order and head difference determine the sign of the result.
Steps + calculator + practiceLearn: Steady radial flow — unconfined aquifer
The unconfined Thiem-type well relation connects steady radial flow with heads measured at two distances from a well. The radius order and head difference determine the sign of the result.
Steps + calculator + practiceLearn: Natural circular frequency
A mass on an elastic support can vibrate after being disturbed. Its undamped natural angular frequency describes how fast the vibration phase advances, not the number of full cycles per second.
Steps + calculator + practiceLearn: Natural frequency
Angular frequency counts radians of phase per second, whereas ordinary frequency counts complete cycles per second. One complete cycle contains 2π radians.
Steps + calculator + practiceLearn: Natural period
Natural period is the time a linear mass–spring system needs to complete one undamped vibration cycle. A more flexible system takes longer, while a stiffer one vibrates more quickly.
Steps + calculator + practiceLearn: Critical damping coefficient
Critical viscous damping is the dividing level between an oscillatory and non-oscillatory free response in a linear single-degree system. It is a damping coefficient, not a percentage.
Steps + calculator + practiceLearn: Damping ratio
Damping ratio compares actual viscous damping with the critical value for the same dynamic system. The calculator reports the ratio as a percentage to make values such as 5% easier to interpret.
Steps + calculator + practiceLearn: EC8 base shear — study form
A simplified lateral-force seismic method estimates total horizontal base shear from design spectral acceleration, participating mass and an applicable correction factor. It is an equivalent-force model, not a time-history response.
Steps + calculator + practiceLearn: EC8 floor force distribution
Equivalent seismic base shear is distributed among floors using their masses and assumed modal displacement shape. This calculation finds one floor’s share of a supplied total base shear.
Steps + calculator + practiceLearn: Easting increment
A survey line can be split into east and north coordinate changes. This lesson finds its east component from horizontal length and azimuth measured clockwise from north.
Steps + calculator + practiceLearn: Northing increment
A survey line can be split into east and north coordinate changes. This lesson finds its north component from horizontal length and azimuth measured clockwise from north.
Steps + calculator + practiceLearn: Polygon area — shoelace
A boundary described by ordered survey coordinates encloses an area. The shoelace method adds signed cross-products of neighboring vertices, closes the final edge, and takes half the absolute total.
Steps + calculator + practiceLearn: Horizontal-curve radius
A vehicle following a horizontal curve needs inward acceleration. Superelevation and available side friction contribute to that demand in a simplified road-curve balance.
Steps + calculator + practiceLearn: Circular-curve tangent length
The tangent length runs from the tangent-intersection point PI to the start or end of a simple circular curve. It is a straight distance along a tangent, not a distance along the curved road.
Steps + calculator + practiceLearn: Circular-curve arc length
Arc length measures distance along the curved alignment between the beginning and end of a circular curve. It is longer than the straight chord joining those endpoints.
Steps + calculator + practiceLearn: Circular-curve long chord
The long chord is the straight line between the start and end of a circular curve. It is useful for geometric checks but does not equal distance traveled along the arc.
Steps + calculator + practiceLearn: Circular-curve external distance
External distance is the shortest distance from the tangent-intersection point PI to the midpoint of a simple circular arc, measured along the angle bisector. It is not the mid-ordinate measured from the long chord.
Steps + calculator + practiceLearn: Stopping sight distance — basic model
Stopping sight distance combines distance traveled while the driver perceives and reacts with distance traveled during braking. Road grade changes the simplified braking-distance term.
Steps + calculator + practiceLearn: Traffic flow relationship
Traffic flow rate counts vehicles passing a point per hour. Density describes how many vehicles occupy a kilometre, and space-mean speed links these spatial and time-based descriptions.
Steps + calculator + practiceLearn: Average time headway
Average time headway is the mean time between successive vehicles passing a fixed point in a stream. It is the time counterpart of a flow rate, not the physical gap between vehicle bumpers.
Steps + calculator + practiceLearn: Vertical-curve K value
The K value of a parabolic vertical curve expresses how much curve length is provided per one percentage-point change in grade. A larger K means a more gradual change for the same grade difference.
Steps + calculator + practiceLearn: Specific gravity
Specific gravity compares a material’s density with a reference water density. It tells you how dense the material is relative to water without carrying a physical unit.
Steps + calculator + practiceLearn: Moisture content from wet/dry mass
Dry-basis moisture content compares the mass of water removed from a sample with the dry mass left behind. It does not divide by the original wet mass.
Steps + calculator + practiceLearn: Approximate creep strain
A material under sustained stress can continue to deform with time. In a simple linear creep-coefficient model, creep strain is the additional time-dependent strain relative to an elastic reference strain.
Steps + calculator + practiceLearn: Total shrinkage strain
A simplified concrete shrinkage model separates drying shrinkage from autogenous shrinkage. Their compatible strain contributions add to give the total shrinkage strain at the specified age.
Steps + calculator + practiceLearn: Fatigue stress range
A fatigue stress cycle moves between a minimum and maximum stress. Stress range is the full difference between those extremes, not half the difference.
Steps + calculator + practiceLearn: Fatigue stress ratio
Fatigue stress ratio R compares the minimum stress in a cycle with its maximum stress. It describes the cycle’s mean-stress character and is not a stress range or a resistance utilization ratio.
Steps + calculator + practiceLearn: Miner's cumulative damage — three blocks
Miner’s rule estimates cumulative fatigue damage by adding the fractions of life consumed at several stress levels. This lesson uses three blocks of cycles and supplied fatigue lives for those blocks.
Steps + calculator + practiceHow to reorder PDF pages without uploading the document
Think of a PDF as a stack of numbered sheets. Reordering changes where a sheet sits in that stack; rotating turns a sheet; deleting removes it. These are different actions.
Guide + try itHow to compress a PDF and choose the right quality
A smaller PDF is easier to send, but smaller is not automatically better. Start by deciding whether you need selectable text, clear small lettering or simply a smaller scanned document.
Guide + try itHow to add page numbers to a PDF
Page numbering places a visible label on each sheet. It helps a reader follow a document, but it does not reorder sheets or change the original contents into a new sequence.
Guide + try itYour learning, your pace
Only optional understood checkmarks are saved locally. Lesson answers and documents are not sent to a progress server. Clearing browser data removes these checkmarks.