Computational Engineering / Finite-element building blocks

Offsets and eccentricity

The rigid arm between a node and the member it actually connects to, and the moment it carries.

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Why this matters

Member centrelines and connection locations may differ. A rigid offset represents the assumed movement between a node and the member end. Check whether that region is sufficiently stiff for the assumption and include the force transfer across the eccentricity.

By the end of this lesson you should be able to

Say what a rigid offset represents physically
Compute the moment an offset introduces
Explain why offsets change the clear span as well as the moments
Decide when an offset is worth modelling

The rigid arm

An offset is a rigid connection between the node and the physical end of the member. It has two effects, and both are easy to overlook:

It shortens the member. The structural span is now between the offset ends, not between the nodes. A 6 m beam framing 250 mm into a column has a clear span of 5.75 m and is stiffer accordingly.

It carries force across a distance. The member's end shear reaches the node through the arm, and a force applied at a distance is a moment.

Exactly V·e

A beam framing into a column face at eccentricity e, carrying end shear V: the column receives the beam's end moment plus V·e. Not approximately — exactly, and it comes straight out of statics on the rigid arm.

For a 6 m beam at 20 kN/m into a 4 m column:

Eccentricity	Clear span	Beam end moment	Column top moment	Difference
0	6.00 m	59.68 kN·m	59.68 kN·m	0
0.10 m	5.90 m	54.71	61.53	6.83
0.20 m	5.80 m	49.81	63.13	13.32
0.30 m	5.70 m	44.99	64.46	19.47

At e = 0 the two moments are identical, as they must be — with no arm there is nothing to carry a moment across. At e = 0.30 m the difference is 19.47 kN·m, and the end shear at that eccentricity is 64.89 kN: 64.89 × 0.30 = 19.47. Exactly.

Read the two moment columns together and the effect is clear: the beam's end moment falls and the column's rises. Within a given offset-model solution, the moment difference across the arm is V·e with the chosen signs. Comparing with a separately solved centreline model also involves the changed member stiffness and force distribution, so the total modelling error is not generally V·e.

When it is worth modelling

Offsets add work and add places to make mistakes, so they should not be added everywhere. Three cases where they earn it:

Deep members into shallow ones. A 900 mm transfer beam into a 400 mm column has an eccentricity comparable with the member depth, and V·e is not a correction.

Bracing to gussets. A brace working to a gusset rather than to the theoretical work point puts moment into the connection that the centreline model has nowhere to show.

Composite floors. The slab's centroid sits above the beam's, which is the entire mechanism of composite action. Modelling it with an offset rather than a transformed section makes the eccentricity explicit.

And one case where it usually does not: ordinary beam-to-column connections in a regular frame. The eccentricity is small compared with the member sizes and the effect is within the other uncertainties. Recording that as a decision — in the exclusions field of the purpose statement — is better than either modelling it everywhere or forgetting it exists.

What a rigid offset carries

M_{\text{column}} = M_{\text{beam end}} + V\!\cdot\!e

What it calculates: The moment a supporting member receives when the beam frames into its face rather than its centreline

V: beam end shear (kN)
e: eccentricity from the node to the physical member end (m)
M: moment (kN·m)

This assumes

The offset is modelled as a rigid arm, with no flexibility of its own

In plain terms: The relation expresses signed moment equilibrium across the rigid arm. The change from a separate centreline-model solution also depends on how offsets alter stiffness and force distribution.

Try it

Offset and eccentricity explorer

A beam framing into a column face rather than its centreline. Watch the clear span, the two moments, and what the rigid arm carries.

Eccentricity (m)

Grid span (m)

Load (kN/m)

Clear span: 5.750 m
Beam end moment: 47.39 kN·m
Column top moment: 63.83 kN·m
Beam end shear: 65.74 kN
Carried by the rigid arm: 16.44 kN·m
Centreline model reports, for both: 59.68 kN·m
Column under-predicted by: 7.0 %

The beam's end moment falls and the column's rises. The difference between them is exactly the end shear times the eccentricity, and the centreline model has no mechanism by which it could show it.

What this shows: The arm carries the end shear across the eccentricity: exactly V·e, and a centreline model contains none of it.

Practice

A beam frames into a column face at an eccentricity of 0.25 m and its end shear is 80 kN. What moment does the rigid arm add to the column?

Practice

A 6 m beam frames 0.30 m into columns at both ends. What is its clear span?

Summary

An offset is a rigid arm between the node and the physical member end
It shortens the clear span, which stiffens the member
It carries the end shear across the eccentricity: exactly V·e
The beam's end moment falls and the supporting member's rises
The moment change across an offset arm is V·e; the error of a separate centreline model needs comparison of both complete solutions
Worth modelling for deep-into-shallow, bracing to gussets, and composite floors