Equilibrium & support reactions
About this app
Beam on pin and roller supports with movable loads: support reactions computed from ΣF=0 and ΣM=0 and shown as arrows, with a free-body diagram.
The basics
A rigid body is in equilibrium when all forces and all moments cancel. In the plane that means three conditions: ΣFₓ = 0, ΣF_y = 0 and ΣM = 0. For a beam on a pin support A and a roller support B, these conditions determine the unknown reactions uniquely. A convenient route: the moment equation about A does not contain the reaction at A and gives B directly. The vertical force balance then yields A = ΣF − B. Every such calculation starts with a free-body diagram in which the supports are replaced by their reaction forces.
What the app shows
A beam rests on a pin support A on the left and on a roller support B. Vertical point loads act as downward arrows, and the support reactions appear as upward arrows that update instantly. The panel lists A, B and the sum of the loads and confirms when ΣF = 0 and ΣM = 0 are satisfied. If a reaction turns negative because the support would have to pull, the app warns that the beam would lift off. A switch shows the free-body diagram, hiding the supports and keeping only the forces.
Things to try
Drag the loads along the beam or set position and magnitude with the sliders. Also change the beam length and the position of roller B. It gets interesting when a load moves past B onto an overhang: B grows, A shrinks and can become negative. Check a hand calculation by placing a single load exactly at mid-span, where each support carries half.
Frequently asked questions
- How do you calculate the support reactions of a beam?
- Draw the free-body diagram and write down the equilibrium conditions. Start with the moment equation about one support, because its reaction does not appear there. That gives the other reaction directly, and the remaining one follows from the force balance.
- What is the difference between a pin support and a roller support?
- A pin support prevents displacement in both directions of the plane and therefore transfers a horizontal and a vertical force. A roller can move along a track and only transfers a force perpendicular to it. Together they make a beam statically determinate.
- What does a negative support reaction mean?
- A negative reaction means it acts opposite to the assumed direction. For a support the beam simply rests on, this means the beam would lift off because the support would have to pull. This typically happens with loads on an overhang.
- What can I adjust in the equilibrium app?
- You can add, remove and move point loads and change their magnitude, and you can set the beam length and the position of roller B. The app computes the support reactions live, checks equilibrium and can switch to the free-body diagram.
Subject: Engineering mechanics
More from Engineering mechanics
- Bending stress & moment of inertia
- Düsenjet bis v_max (3D)
- Energy & oscillation
- Truss analysis (method of joints)
- Kinematics
- Kinetics & Newton's second law
- Forces & vector addition
- Force decomposition
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