Equivalent Nodal Loads Calculator
Enter beam-element length and transverse line-load intensity at each end. The calculator returns local nodal shears and end moments using cubic Hermite beam shape functions.
- Node 1 Moment
- 30 force·length
- Node 2 Transverse Force
- 30 force
- Node 2 Moment
- -30 force·length
- Total Distributed Load
- 60 force
- Resultant Location from Node 1
- 3 length
- Force Equilibrium Residual
- 0 force
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The four primary outputs form the local consistent nodal-load vector [F1, M1, F2, M2]. Additional checks show the total distributed load, resultant location, and force-equilibrium residual.
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How to Use This Calculator
Use one consistent length unit and one force unit. If length is in meters and q is in kN/m, forces are returned in kN and moments in kN·m. Positive q acts in the chosen positive local transverse direction. Positive nodal moments follow the element's positive local rotational degree of freedom.
Formula & Methodology
The consistent load vector is the integral of Nᵀq(x) over the element. For a linearly varying q from q1 to q2, F1 = L(7q1+3q2)/20, M1 = L²(3q1+2q2)/60, F2 = L(3q1+7q2)/20, and M2 = −L²(2q1+3q2)/60.
Example: uniform load of 10 kN/m over 6 m
With q1 = q2 = 10 and L = 6, the consistent vector is F1 = 30 kN, M1 = 30 kN·m, F2 = 30 kN, and M2 = −30 kN·m.
This formulation is for a straight, prismatic, two-node Euler–Bernoulli beam element with transverse load varying linearly in the element's local coordinate. It is not a support-reaction calculator. Sign conventions, released degrees of freedom, Timoshenko elements, follower loads, geometric nonlinearity, partial-span loads, point loads, global transformations, and software-specific conventions require separate treatment.
Engineering calculation aid only. Verify the element formulation, interpolation functions, local axes, units, releases, and finite-element software conventions before design or analysis.
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Frequently Asked Questions
- What are equivalent nodal loads?
- They are nodal forces and moments that produce the same finite-element virtual work as the distributed element load under the chosen interpolation functions.
- Are these the same as support reactions?
- No. They are element load-vector components. Structural support reactions are obtained after assembly, constraints, and solution.
- Does the calculator handle a uniform load?
- Yes. Set q1 equal to q2. Each nodal shear becomes qL/2 and the end moments are equal and opposite with magnitude qL²/12.
- Does it handle a triangular load?
- Yes. Set one end intensity to zero and the other to the peak intensity.
- Why do the end moments have opposite signs for a uniform load?
- The local rotational shape functions and degree-of-freedom sign convention produce equal and opposite consistent nodal moment components.
Last updated 7/20/2026