OpenSpecEngineering

Solves two-dimensional statics problems and draws the free body diagram as you type. Students and engineers use it to find support reactions. It has three modes:

• Beam: the reactions of a beam on a pin and a roller (with overhangs) or a cantilever, under point, distributed and couple loads, with the shear and moment diagrams.
• Rigid body: the reactions of a pin and a roller, or a fixed support, for forces at any point and angle.
• Concurrent forces: the resultant of forces that meet at a point, or the forces in two cables or struts that hold them.

Equations

ΣF_x = 0     ΣF_y = 0     ΣM = 0

A body at rest has no net force and no net moment [1], [2]. These three equations give three unknown reactions; a joint of concurrent forces gives two. Along a beam, the shear V(x) and moment M(x) are the net force and moment of everything left of the section [1].

x points right and y up. Angles are in degrees, counterclockwise from +x, so 270 points down. Beam loads are positive downward, with their angle measured from the beam (90 is straight down). Couples are positive counterclockwise. Sagging moment (concave up) is positive. A member force is positive in tension.

In the free-body diagram, applied loads are amber and support reactions light blue. For a beam, the shear (V) and bending-moment (M) diagrams follow below it.

Limits and common mistakes

• The page handles statically determinate problems only: no beams on three supports, and no fixed-fixed beams.
• Load positions are measured from the beam’s left end, not from support A.
• A negative roller reaction means the support must pull down. A plain roller would lift off; the page says so.
• A cable can only pull. A negative member force means that member must be a strut.

Worked example

A 6 m beam sits on a pin at its left end and a roller 4.5 m along. It carries 4 kN/m over its length and 10 kN at 2 m. Find the reactions and moments.

1. Choose Problem Beam and Supports Pin (A) and roller (B). Enter Beam Length (L) 6000 mm, Left Overhang 0 and Right Overhang 1500 mm. Choose Number of Loads 2.
2. For Load 1, choose Point load: Point Load (P₁) 10000 N, Angle to the Beam 90, Position from the Left End (a₁) 2000 mm.
3. For Load 2, choose Distributed load: both intensities 4000 N/m, Start Position 0 and End Position 6000 mm. Press Solve. The Solution shows: Left Support Reaction (R_A, + up): 13,555.6 N
   Right Support Reaction (R_B, + up): 20,444.4 N
   Maximum Shear (|V|_max): 14,444.4 N, at x 4.5 m
   Maximum Sagging Moment (+M): 19,111.1 N·m, at x 2 m
   Maximum Hogging Moment (−M): −4,500 N·m, at x 4.5 m

Check by hand. Moments about A: 4.5 R_B = 24 × 3 + 10 × 2 = 92 kN·m, so R_B = 20.44 kN and R_A = 34 − 20.44 = 13.56 kN. Under the point load, M = 13.56 × 2 − 4 × 2²/2 = 19.11 kN·m.