Equivalent Nodal Loads for a Beam Element
Derive and check the consistent local load vector for a two-node Euler–Bernoulli beam with a linearly varying transverse load.
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Consistent, Not Merely Static
Equivalent nodal loads in a finite element are obtained by integrating the distributed load against the element shape functions. This preserves virtual work for the element interpolation.
Use the Equivalent Nodal Loads Calculator for a straight two-node Euler–Bernoulli beam.
Linear Load Definition
Let q vary linearly from q1 at node 1 to q2 at node 2 over length L.
The consistent local vector is:
`F1 = L(7q1 + 3q2)/20`
`M1 = L²(3q1 + 2q2)/60`
`F2 = L(3q1 + 7q2)/20`
`M2 = -L²(2q1 + 3q2)/60`
Uniform-Load Check
Setting q1 = q2 = q gives:
- F1 = qL/2,
- F2 = qL/2,
- M1 = qL²/12,
- M2 = −qL²/12.
OpenSees documentation likewise describes equal nodal forces and equal-and-opposite nodal moments for a uniform gravity line load on an elastic beam.
Force Equilibrium
The integrated load is:
`W = L(q1 + q2)/2`
The calculator verifies that F1 + F2 − W is approximately zero.
Not Support Reactions
The vector belongs to the element before global assembly and solution. Supports, releases, transformations, and the structure's stiffness determine final reactions and displacements.
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Frequently Asked Questions
- Why not place half the total load at each node?
- That reproduces total force for a uniform load but omits the consistent nodal moments required by cubic beam interpolation.
- What sign convention is used?
- Loads are expressed in the element's chosen positive local transverse and rotational degrees of freedom.
- Can I use the formulas for a partial-span load?
- Not directly. A partial-span load requires integrating over its actual loaded interval or subdividing the element.
Last updated 7/20/2026