Calculators
Mechanical calculators
81 tools
Statics, sections, strength, and beam deflection.
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Family pages are registry-driven. Calculator URLs stay /calc/mechanical/… Tools may appear in more than one topic.
- Friction & Stability Incline, ladder, slip/tip, and pull problems. · 11 tools
- Section Properties Centroid, area moment, principal axes, and section modulus. · 9 tools
- Strength of Materials Bending stress, transverse shear, and plastic modulus. · 9 tools
- Beams & Deflection Euler–Bernoulli closed forms and station queries. Workspace: /workspace/beam. · 51 tools
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- Planar Equilibrium Calculator Sum 2-D point forces and concentrated moments, or solve two or three unknown reactions. Uses the vector and linear-algebra engines. Not a beam calculator.
- Incline Friction Calculator Force parallel to a rough incline at impending slip. Normal and friction come from the friction engine. Equilibrium is solved with the linear-algebra engine. Runs locally.
- Ladder Friction Calculator Minimum floor friction for a uniform ladder against a smooth wall. The three reactions are a linear system. μ_min comes from the friction engine. Runs locally.
- Slip or Tip Calculator Whether a block on a level surface slips or tips under a horizontal force. Slip uses the friction engine. Tip is the moment of the weight about the leading edge. Runs locally.
- Angled Pull Calculator Pull at an angle above the horizontal that puts a block on a level surface at impending slip. The normal is W − P sinα. Runs locally.
- Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally.
- Section Modulus Calculator Elastic section modulus of a composite plane area about its centroid. Sx = Ix/cy and Sy = Iy/cx. Ix and Iy come from the area-moment engine. Runs locally. Not bending stress and not beam deflection.
- Plane Centroid Calculator Centroid of composite plane areas from each part's area and centroid. Negative area is a hole. Runs locally. Not a moment of inertia and not a beam calculator.
Statics & Equilibrium
Resultants, reactions, and planar equilibrium.
Friction & Stability
Incline, ladder, slip/tip, and pull problems.
- Angled Pull Calculator Pull at an angle above the horizontal that puts a block on a level surface at impending slip. The normal is W − P sinα. Runs locally.
- Angled Pull on an Incline Calculator The pull at an angle above the horizontal that puts a block on an incline at impending slip. α = 0 is horizontal. α = θ is parallel to the plane. Runs locally.
- Horizontal Force on an Incline Calculator Horizontal force that puts a block on an incline at impending slip. P is positive toward the uphill side. The normal grows by P sinθ. Runs locally.
- Incline Friction Calculator Force parallel to a rough incline at impending slip. Normal and friction come from the friction engine. Equilibrium is solved with the linear-algebra engine. Runs locally.
- Incline Slip or Tip Calculator Whether a homogeneous rectangle on an incline slips or tips under its own weight. The slip angle is the friction angle. The tip angle is the moment about the downhill edge. Runs locally.
- Ladder Friction Calculator Minimum floor friction for a uniform ladder against a smooth wall. The three reactions are a linear system. μ_min comes from the friction engine. Runs locally.
- Least Pull Calculator The angle and magnitude of the smallest pull that puts a block on a level surface at impending slip. α = arctan μ. Runs locally.
- Least Pull on an Incline Calculator The smallest pull that moves a block up an incline. The pull is at the friction angle above the incline. Runs locally.
Section Properties
Centroid, area moment, principal axes, and section modulus.
- Area Moment Calculator Second moment of composite plane areas about the composite centroid. Supply each part's centroidal Ix and Iy. Runs locally. Not a section modulus and not a beam calculator.
- Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid circle from its radius. Runs locally.
- Hollow Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a hollow circle from the outer and inner radii. Runs locally.
- Inclined-Axis Moment Calculator Centroidal area moments about axes rotated by a given angle. Ix, Iy, and Ixy come from the principal-moment engine. Runs locally. Not the principal values, and not a Mohr circle.
- Plane Centroid Calculator Centroid of composite plane areas from each part's area and centroid. Negative area is a hole. Runs locally. Not a moment of inertia and not a beam calculator.
- Principal Moment Calculator Product of inertia and principal moments of a composite plane area. Ix and Iy reuse the area-moment engine. Runs locally. Not a section modulus and not a beam calculator.
- Principal Section Modulus Calculator Elastic section modulus about the principal axes. S1 = I1/c1 and S2 = I2/c2. I1, I2, and θ come from the principal-moment engine. Runs locally. Not Sx and Sy, and not bending stress.
- Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally.
Strength of Materials
Bending stress, transverse shear, and plastic modulus.
- Bending Stress Calculator Elastic bending stress from a moment and a section modulus. S can also be formed from Ix and c. Runs locally.
- Circular Plastic Modulus Calculator Plastic modulus and shape factor of a solid circle. The elastic modulus comes from the circular-section engine. Runs locally.
- Circular Shear Calculator Neutral-axis shear stress of one solid circle. The area moment comes from the circular-section engine. Runs locally.
- Hollow Circular Plastic Modulus Calculator Plastic modulus of a hollow circle from the outer and inner radii. The elastic modulus comes from the hollow-circular-section engine. Runs locally.
- Hollow Circular Shear Calculator Neutral-axis shear stress of one hollow circle. The area moment comes from the hollow-circular-section engine. Runs locally.
- Plastic Moment Calculator Fully plastic moment from a yield stress and a plastic modulus. One centroidal axis. Runs locally.
- Rectangular Plastic Modulus Calculator Plastic moduli and shape factor of a solid rectangle. Elastic Sx and Sy come from the rectangular-section engine. Runs locally.
- Rectangular Shear Calculator Neutral-axis shear stress of one solid rectangle. The area moment comes from the rectangular-section engine. Runs locally.
Beams & Deflection
Euler–Bernoulli closed forms and station queries. Workspace: /workspace/beam.
- Beam Deflection Calculator Midspan deflection of a simply supported span with one concentrated load. δ = P L³/(48 E I). I is an input. Runs locally.
- Beam Moment Deflection Calculator Midspan deflection of a simply supported span with one end moment. δ = M L²/(16 E I). I is an input. Runs locally.
- Beam Moment Maximum Deflection Calculator Largest deflection of a simply supported span with one end moment. δ = M L²/(9√3 E I) at x = L/√3. I is an input. Runs locally.
- Cantilever Deflection Calculator Free-end deflection of a cantilever with one concentrated load at the tip. δ = P L³/(3 E I). I is an input. Runs locally.
- Cantilever Intermediate Deflection Calculator Deflection at a concentrated load on a cantilever. δ = P a³ / (3 E I). a is measured from the fixed end. Runs locally.
- Cantilever Load at a Station Calculator Deflection at station x for a cantilever load at a. x = a is the load page. x = L is the free-end page. Runs locally.
- Cantilever Moment at the Couple Calculator Deflection at a concentrated moment on a cantilever. δ = M a² / (2 E I). a is measured from the fixed end. Runs locally.
- Cantilever Moment Deflection Calculator Free-end deflection of a cantilever with one concentrated moment. δ = M L²/(2 E I). I is an input. Runs locally.
All Mechanical calculators
Full list for search and crawl.
- Angled Pull Calculator Pull at an angle above the horizontal that puts a block on a level surface at impending slip. The normal is W − P sinα. Runs locally.
- Angled Pull on an Incline Calculator The pull at an angle above the horizontal that puts a block on an incline at impending slip. α = 0 is horizontal. α = θ is parallel to the plane. Runs locally.
- Area Moment Calculator Second moment of composite plane areas about the composite centroid. Supply each part's centroidal Ix and Iy. Runs locally. Not a section modulus and not a beam calculator.
- Beam Deflection Calculator Midspan deflection of a simply supported span with one concentrated load. δ = P L³/(48 E I). I is an input. Runs locally.
- Beam Moment Deflection Calculator Midspan deflection of a simply supported span with one end moment. δ = M L²/(16 E I). I is an input. Runs locally.
- Beam Moment Maximum Deflection Calculator Largest deflection of a simply supported span with one end moment. δ = M L²/(9√3 E I) at x = L/√3. I is an input. Runs locally.
- Bending Stress Calculator Elastic bending stress from a moment and a section modulus. S can also be formed from Ix and c. Runs locally.
- Cantilever Deflection Calculator Free-end deflection of a cantilever with one concentrated load at the tip. δ = P L³/(3 E I). I is an input. Runs locally.
- Cantilever Intermediate Deflection Calculator Deflection at a concentrated load on a cantilever. δ = P a³ / (3 E I). a is measured from the fixed end. Runs locally.
- Cantilever Load at a Station Calculator Deflection at station x for a cantilever load at a. x = a is the load page. x = L is the free-end page. Runs locally.
- Cantilever Moment at the Couple Calculator Deflection at a concentrated moment on a cantilever. δ = M a² / (2 E I). a is measured from the fixed end. Runs locally.
- Cantilever Moment Deflection Calculator Free-end deflection of a cantilever with one concentrated moment. δ = M L²/(2 E I). I is an input. Runs locally.
- Cantilever Outer Uniform Calculator Tip deflection when a uniform load starts away from the wall. δ = w [3 L⁴ − a³ (4L − a)] / (24 E I). Runs locally.
- Cantilever Partial Load Calculator Deflection where a uniform load that starts at the wall stops. δ = w a⁴ / (8 E I). Runs locally.
- Cantilever Tip Deflection Calculator Free-end deflection of a cantilever with an intermediate point load. δ = P a² (3L − a) / (6 E I). Runs locally.
- Cantilever Tip-Peak Deflection Calculator Tip deflection of a cantilever whose triangular load peaks at the free end. δ = 11 w0 L⁴ / (120 E I). Runs locally.
- Cantilever Triangular Deflection Calculator Tip deflection of a cantilever whose triangular load peaks at the fixed end. δ = w0 L⁴ / (30 E I). Runs locally.
- Cantilever Uniform Deflection Calculator Free-end deflection of a cantilever with a uniform load. δ = w L⁴/(8 E I). I is an input. Runs locally.
- Cantilever Uniform Load at a Station Calculator Deflection at distance x from the wall. δ = w x² (6 L² − 4 L x + x²) / (24 E I). Runs locally.
- Center-Peaked Triangle at a Station Calculator Deflection at station x when the triangle is 0 at both supports and w0 at midspan. x = L/2 is the midspan page. Runs locally.
- Center-Peaked Triangle Calculator Midspan deflection when a triangular load peaks at midspan. δ = w0 L⁴ / (120 E I). Runs locally.
- Center-Peaked Triangle plus End Moment Calculator Deflection at station x for a center-peaked triangle plus a sagging moment at the right end. x = L/2 is the sum of the two midspan pages. Runs locally.
- Circular Plastic Modulus Calculator Plastic modulus and shape factor of a solid circle. The elastic modulus comes from the circular-section engine. Runs locally.
- Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid circle from its radius. Runs locally.
- Circular Shear Calculator Neutral-axis shear stress of one solid circle. The area moment comes from the circular-section engine. Runs locally.
- End Moment at a Station Calculator Deflection at station x for one end moment. x is measured from the unloaded end. Runs locally.
- End-Peaked Triangle at a Station Calculator Deflection at station x when the triangle is w0 at both supports and 0 at midspan. x = L/2 is the midspan page. Runs locally.
- End-Peaked Triangle Calculator Midspan deflection when a triangular load peaks at both supports. δ = 3 w0 L⁴ / (640 E I). Runs locally.
- End-Peaked Triangle plus End Moment Calculator Deflection at station x for an end-peaked triangle plus a sagging moment at the right end. x = L/2 is the sum of the two midspan pages. Runs locally.
- Equal End Moments at a Station Calculator Deflection at station x for equal sagging end moments. x = L/2 is the midspan page. Runs locally.
- Equal End Moments Calculator Midspan deflection of a simply supported span with equal sagging end moments. δ = M L² / (8 E I). Runs locally.
- Hollow Circular Plastic Modulus Calculator Plastic modulus of a hollow circle from the outer and inner radii. The elastic modulus comes from the hollow-circular-section engine. Runs locally.
- Hollow Circular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a hollow circle from the outer and inner radii. Runs locally.
- Hollow Circular Shear Calculator Neutral-axis shear stress of one hollow circle. The area moment comes from the hollow-circular-section engine. Runs locally.
- Horizontal Force on an Incline Calculator Horizontal force that puts a block on an incline at impending slip. P is positive toward the uphill side. The normal grows by P sinθ. Runs locally.
- Incline Friction Calculator Force parallel to a rough incline at impending slip. Normal and friction come from the friction engine. Equilibrium is solved with the linear-algebra engine. Runs locally.
- Incline Slip or Tip Calculator Whether a homogeneous rectangle on an incline slips or tips under its own weight. The slip angle is the friction angle. The tip angle is the moment about the downhill edge. Runs locally.
- Inclined-Axis Moment Calculator Centroidal area moments about axes rotated by a given angle. Ix, Iy, and Ixy come from the principal-moment engine. Runs locally. Not the principal values, and not a Mohr circle.
- Ladder Friction Calculator Minimum floor friction for a uniform ladder against a smooth wall. The three reactions are a linear system. μ_min comes from the friction engine. Runs locally.
- Least Pull Calculator The angle and magnitude of the smallest pull that puts a block on a level surface at impending slip. α = arctan μ. Runs locally.
- Least Pull on an Incline Calculator The smallest pull that moves a block up an incline. The pull is at the friction angle above the incline. Runs locally.
- Midspan Load at a Station Calculator Deflection at station x when the load is at midspan. δ = P c (3 L² − 4 c²) / (48 E I). Runs locally.
- Offset Load at a Station Calculator Deflection at station x for one offset load. x = a is the under-load page. x = L/2 is the midspan page. Runs locally.
- Offset Load Deflection Calculator Deflection under a concentrated load that is not at midspan. δ = P a² b² / (3 E I L). I is an input. Runs locally.
- Offset Load Maximum Calculator Largest deflection of a simply supported span with an offset point load. δ = P c (L² − c²)^(3/2) / (9 √3 E I L). Runs locally.
- Offset Load Midspan Calculator Midspan deflection when one concentrated load is not at midspan. δ = P c (3 L² − 4 c²) / (48 E I). Runs locally.
- One-Sided Triangle plus End Moment Calculator Deflection at station x for a one-sided triangle plus a sagging moment at the right end. x = L/2 is the sum of the two midspan pages. Runs locally.
- One-Sided Triangle plus Equal End Moments Calculator Deflection at station x for a one-sided triangle plus equal sagging end moments. x = L/2 is the sum of the two midspan pages. Runs locally.
- Outer Uniform Load at a Station Calculator Deflection at distance x from the wall when the uniform load starts at a and runs to the tip. x = L is the tip page. a = 0 is the full uniform curve. Runs locally.
- Overhang Arm at a Station Calculator Deflection at station z on the overhang, measured from the right support out to the load. z = a is the load page. Runs locally.
- Overhang End Deflection Calculator Deflection at a load beyond a simply supported span. δ = P a² (L + a) / (3 E I). Runs locally.
- Overhang Span at a Station Calculator Deflection at station x on the span when a load sits at the end of an overhang. x = L/2 is the span-midspan page. Runs locally.
- Overhang Span Deflection Calculator Midspan of the span moves opposite a load on the overhang. δ = −P a L² / (16 E I). Runs locally.
- Paired Load at the Load Calculator Deflection under either of two equal loads. δ = P a² (3L − 4a) / (6 E I). Runs locally.
- Paired Load Deflection Calculator Midspan deflection of two equal loads, each at distance a from the nearer support. δ = P a (3 L² − 4 a²) / (24 E I). Runs locally.
- Paired Loads at a Station Calculator Deflection at station x for two equal loads, each at distance a from the nearer support. x = L/2 is the midspan page. x = a is the under-load page. Runs locally.
- Paired Point Loads plus End Moment Calculator Deflection at station x for two equal point loads plus a sagging moment at the right end. x = L/2 adds the paired midspan page. Runs locally.
- Partial Uniform Load at a Station Calculator Deflection at distance x from the wall when a uniform load runs from the wall to a. x = a is the loaded-end page. x = L is the tip page. Runs locally.
- Person on a Ladder Calculator How far a person can climb a uniform ladder before impending slip. s is a fraction of the length from the floor. μ_wall = 0 is a smooth wall. Runs locally.
- Planar Equilibrium Calculator Sum 2-D point forces and concentrated moments, or solve two or three unknown reactions. Uses the vector and linear-algebra engines. Not a beam calculator.
- Plane Centroid Calculator Centroid of composite plane areas from each part's area and centroid. Negative area is a hole. Runs locally. Not a moment of inertia and not a beam calculator.
- Plastic Moment Calculator Fully plastic moment from a yield stress and a plastic modulus. One centroidal axis. Runs locally.
- Principal Moment Calculator Product of inertia and principal moments of a composite plane area. Ix and Iy reuse the area-moment engine. Runs locally. Not a section modulus and not a beam calculator.
- Principal Section Modulus Calculator Elastic section modulus about the principal axes. S1 = I1/c1 and S2 = I2/c2. I1, I2, and θ come from the principal-moment engine. Runs locally. Not Sx and Sy, and not bending stress.
- Rectangular Plastic Modulus Calculator Plastic moduli and shape factor of a solid rectangle. Elastic Sx and Sy come from the rectangular-section engine. Runs locally.
- Rectangular Section Calculator Centroidal Ix, Iy, and elastic section moduli of a solid rectangle from its width and height. Runs locally.
- Rectangular Shear Calculator Neutral-axis shear stress of one solid rectangle. The area moment comes from the rectangular-section engine. Runs locally.
- Rough-Wall Ladder Calculator Critical angle of a uniform ladder when both the floor and the wall are at impending slip. μ_wall = 0 recovers the smooth-wall ladder. Runs locally.
- Section Modulus Calculator Elastic section modulus of a composite plane area about its centroid. Sx = Ix/cy and Sy = Iy/cx. Ix and Iy come from the area-moment engine. Runs locally. Not bending stress and not beam deflection.
- Shear and Moment Calculator Shear and bending moment at a cut in a simply supported span. Point forces and concentrated moments only. Runs locally. Not beam deflection.
- Slip or Tip Calculator Whether a block on a level surface slips or tips under a horizontal force. Slip uses the friction engine. Tip is the moment of the weight about the leading edge. Runs locally.
- Tip Load at a Station Calculator Deflection at distance x from the wall under a tip load. δ = P x² (3L − x) / (6 E I). Runs locally.
- Tip Moment at a Station Calculator Deflection at distance x from the wall for a moment at the free end. δ = M x² / (2 E I). Runs locally.
- Tip-Peak Triangle at a Station Calculator Deflection at distance x from the wall when the triangle peaks at the tip. x = L is the tip page. Runs locally.
- Transverse Shear Calculator Shear stress at one line from the shear force, the first moment, the area moment, and the width. Runs locally.
- Triangular Load at a Station Calculator Deflection at station x for a triangular load that is zero at the left support. x = L/2 is the midspan page. Runs locally.
- Triangular Load Deflection Calculator Midspan deflection of a simply supported span with a triangular load. δ = 5 w0 L⁴ / (768 E I). Intensity is 0 at the left support. Runs locally.
- Triangular Load Maximum Calculator Largest deflection of a simply supported span with a triangular load. The peak is not at midspan. Runs locally.
- Uniform Load at a Station Calculator Deflection at station x on a simply supported uniform load. δ = w x (L³ − 2 L x² + x³) / (24 E I). Runs locally.
- Uniform Load Deflection Calculator Midspan deflection of a simply supported span with a uniform load. δ = 5 w L⁴/(384 E I). I is an input. Runs locally.
- Wall-Peak Triangle at a Station Calculator Deflection at distance x from the wall when the triangle peaks at the wall. x = L is the tip page. Runs locally.