Static Equilibrium — Force & Moment Balance Calculator
Set support conditions, loads, and moments to automatically compute unknown reactions. Real-time free body diagram drawing with numerical equilibrium verification.
Structure Settings
Beam Span L
m
Support Conditions
Left End A
Right End B
Concentrated Loads (max 5)
Magnitude [kN]Angle [°]Position x [m]Label
Concentrated Moment
Magnitude [kN·m]
Position x [m]
Positive: counter-clockwise (CCW)
Three equilibrium equations for a statically determinate 2D rigid body:
Determinacy condition: number of unknown reactions = 3 (equals number of equations)
Pin support → $A_x, A_y$ (2 reactions) Roller → $B_y$ (1 reaction) Fixed end → $A_x, A_y, M_A$ (3 reactions)
Degree of indeterminacy $= $ (no. of unknowns) $- 3$
Results
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Ax [kN]
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Ay [kN]
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By [kN]
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Equilibrium Error
Visualization
CAE Integration
Verify boundary conditions and hand-calculate reactions before FEM analysis / preliminary beam and frame design / check SPC setup errors in NASTRAN, ABAQUS, ANSYS. Early detection of modelling errors by comparing with statically determinate approximations.
What is Static Equilibrium?
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What exactly is "static equilibrium" for a beam? Is it just when nothing is moving?
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Basically, yes! It means the beam is perfectly balanced and stationary. In practice, this happens when all the forces pushing and pulling on it cancel out, AND all the twisting effects (moments) cancel out too. For instance, a simple bookshelf holding books is in static equilibrium—the shelf's supports push up exactly to counteract the weight pulling down. Try adding a point load in the simulator above; you'll see the support reactions instantly adjust to keep the beam balanced.
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Wait, really? So the three equations you mentioned ($\sum F_x=0$, etc.) are like a mathematical checklist for this balance?
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Exactly! They are the non-negotiable rules. The first two check force balance in the horizontal and vertical directions. The third, $\sum M_A=0$, checks rotational balance about any point A. A common mistake is forgetting the moment balance. For example, if you push on a wrench, you create a moment. In the simulator, try moving the load position along the beam span `L` using the slider. You'll see the reaction forces change, but the moment at the support changes dramatically—that's the moment balance in action.
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So if I get a solution here, does that mean my real-world beam design is safe?
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Great question! This calculator gives you the reactions—the forces the supports must provide for equilibrium. It tells you if the structure is statically determinate and solvable. However, "safe" depends on material strength, which comes later. In CAE, this is the crucial first step: verifying your boundary conditions. Before running a complex FEM analysis in ANSYS or NASTRAN, you use this hand calculation to spot errors. If the software reactions don't match this simple model, you likely made a support setup error.
Physical Model & Key Equations
The fundamental conditions for a 2D rigid body to be in static equilibrium. These equations ensure translational and rotational balance.
$\sum F_x$: Sum of all horizontal forces (e.g., horizontal reactions). $\sum F_y$: Sum of all vertical forces (e.g., loads, vertical reactions). $\sum M_A$: Sum of all moments about a conveniently chosen point A (often a support). A moment is a force multiplied by its perpendicular distance to point A.
For a simply supported beam with a point load $P$ at distance $a$ from support A, the vertical reactions are derived from these equilibrium equations.
$$R_{Ay}= P \frac{L-a}{L}\quad R_{By}= P \frac{a}{L}$$
$R_{Ay}, R_{By}$: Vertical reaction forces at supports A and B. $L$: Total beam span (the main parameter you control in the simulator). $a$: Distance of the point load from support A.
The equations show how reactions depend directly on the load position. When $a=L/2$, the load is central and reactions are equal.
Real-World Applications
Preliminary Structural Design: Engineers use these calculations for quick sizing of beams, columns, and supports in buildings or bridges. Before detailed analysis, they determine the maximum reactions to choose appropriate support types and estimate member sizes.
Verifying CAE/FEM Models: This is a critical step in CAE workflows. After setting up a complex finite element model in software like ABAQUS or ANSYS, the calculated support reactions are compared to hand-calculated equilibrium results. A mismatch often reveals incorrect boundary conditions or constraints in the digital model.
Design of Simple Machines & Components: From crane booms and forklifts to seesaws and shelf brackets, the principle of static equilibrium determines the forces at pins, hinges, and supports, ensuring the device doesn't collapse or tip over under load.
Educational & Code Compliance Checking: Used to check basic load paths and force distributions required by building codes. It also forms the foundational concept taught in engineering statics, which is applied in more advanced structural analysis.
Common Misconceptions and Points to Note
First, do you think "if the forces are balanced, it will absolutely never break"? This is a major misconception. The "equilibrium" calculated by this tool is merely a necessary condition for an object not to move (including rotation). Whether the member itself can withstand those forces (strength) or whether the deflection is within allowable limits (stiffness) are separate issues. For example, even if a reaction force is calculated as 100N, a wire supporting it that is too thin will snap immediately. Consider equilibrium calculations as the "first step" in safe design.
Next, the importance of load application points is often overlooked. Shifting the point of application of a concentrated load by just a little changes the moment and significantly alters the reaction forces. For instance, applying a 100N load at the center of a 10m beam versus at a point 1m from one end results in completely different reaction distributions. In design, it's a golden rule to calculate assuming the "worst-case load position".
Finally, don't forget the idealization that "roller supports have zero friction". In practice, rollers or movable supports have some friction. It's rare for the horizontal reaction to be perfectly zero as in a simulator, so a certain safety factor must be incorporated into the reaction forces obtained from simplified calculations. Understand that what you learn with this tool is an "ideal model", and reality adds various other factors on top of that.