Sign in Start free trial

Frame element, shear deformation

Updated: 2026-09-21

On this page

Simply supported beam, distributed load, L/h = 8
Simply supported beam, distributed load, L/h = 12
Simply supported beam, distributed load, L/h = 15
Simply supported beam, distributed load, L/h = 20
Simply supported beam, distributed load, L/h = 30
Cantilever, point load at the free end
Cantilever, under uniformly distributed load
Beam fixed at both ends, under distributed load
Point load at a = 0.35 L
Point moment at a = 0.35 L
How the critical load converges with the mesh
Pin-ended column, with shear deformation
Simply supported beam, distributed load, 1-3 plane

Results.

Example	Reference	Records	Largest deviation
Simply supported beam, distributed load, L/h = 8	Published source	1	0%
Simply supported beam, distributed load, L/h = 12	Published source	1	0%
Simply supported beam, distributed load, L/h = 15	Published source	1	0%
Simply supported beam, distributed load, L/h = 20	Published source	1	0%
Simply supported beam, distributed load, L/h = 30	Published source	1	0%
Cantilever, point load at the free end	Published source	1	0%
Cantilever, under uniformly distributed load	Published source	1	0%
Beam fixed at both ends, under distributed load	Published source	1	0%
Point load at a = 0.35 L	Independent calculation	1	0%
Point moment at a = 0.35 L	Independent calculation	1	0%
How the critical load converges with the mesh	Published source	4	26.60%
Pin-ended column, with shear deformation	Published source	1	0.0239%
Simply supported beam, distributed load, 1-3 plane	Published source	1	0%

Problem. The Timoshenko element includes shear deformation. Deflections, fixed-end forces and the buckling load of a shear-flexible column are tested for span-to-depth ratios from eight to thirty.

Reference document. Timoshenko beam theory is used for the deflections, the force method for the fixed-end forces and the Engesser relation for buckling.

The deflection of a simply supported beam under a uniformly distributed load is the sum of the bending and shear contributions.

\delta = \dfrac{5wL^4}{384\,EI} + \dfrac{wL^2}{8\,GA_v}

The effect of shear deformation is measured by the Timoshenko shear parameter.

\Phi = \dfrac{12\,EI}{G A_v L^2}

Shear deformation reduces the critical load according to the Engesser relation.

\dfrac{1}{P_{kr}} = \dfrac{1}{P_E} + \dfrac{1}{G A_v}

Numerical model. The section is 200 by 600 mm and the ratio of elastic to shear modulus is 16.7. The shear area is five sixths of the section area. In the point load and point moment cases no system is solved, and the element's equivalent nodal loads are read.

Simply supported beam, distributed load, L/h = 8

Parameters. L = 4.8 · h = 0.6 · E/G = 16.7 · Φ = 0.3125 · element count = 1

Published source. Timoshenko beam theory

5wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.00250435	0.00250435	0%

Note. Shear deformation accounts for 20.00%.

Simply supported beam, distributed load, L/h = 12

Parameters. L = 7.2 · h = 0.6 · E/G = 16.7 · Φ = 0.13889 · element count = 1

Published source. Timoshenko beam theory

5wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.0112696	0.0112696	0%

Note. Shear deformation accounts for 10.00%.

Simply supported beam, distributed load, L/h = 15

Parameters. L = 9 · h = 0.6 · E/G = 16.7 · Φ = 0.08889 · element count = 1

Published source. Timoshenko beam theory

5wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.0265231	0.0265231	0%

Note. Shear deformation accounts for 6.64%.

Simply supported beam, distributed load, L/h = 20

Parameters. L = 12 · h = 0.6 · E/G = 16.7 · Φ = 0.05 · element count = 1

Published source. Timoshenko beam theory

5wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.0813913	0.0813913	0%

Note. Shear deformation accounts for 3.85%.

Simply supported beam, distributed load, L/h = 30

Parameters. L = 18 · h = 0.6 · E/G = 16.7 · Φ = 0.02222 · element count = 1

Published source. Timoshenko beam theory

5wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.403239	0.403239	0%

Note. Shear deformation accounts for 1.75%.

Cantilever, point load at the free end

Parameters. L = 4 · P = 30

Published source. Timoshenko beam theory

PL^3/3EI + PL/GA_v

Quantity	Reference	Computed	Deviation
Tip deflection	0.0171981	0.0171981	0%

Cantilever, under uniformly distributed load

Parameters. L = 4 · w = 15

Published source. Timoshenko beam theory

wL^4/8EI + wL^2/2GA_v

Quantity	Reference	Computed	Deviation
Tip deflection	0.0133333	0.0133333	0%

Beam fixed at both ends, under distributed load

Parameters. L = 6 · w = 20 · element count = 1

Published source. Timoshenko beam theory

wL^4/384EI + wL^2/8GA_v

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.00293478	0.00293478	0%

Point load at a = 0.35 L

Parameters. L = 6 · a = 2.1 · Φ = 0.2

Independent calculation. The force method, with a simply supported primary system

Quantity	Reference	Computed	Deviation
Fixed-end moment $m_i$	34.125	34.125	0%

Note. Shear deformation also changes the equivalent nodal load. The Euler element gives 35.4900 for the same load.

Point moment at a = 0.35 L

Parameters. L = 6 · a = 2.1 · M₀ = 50

Independent calculation. The force method, with a simply supported primary system

Quantity	Reference	Computed	Deviation
Fixed-end moment $m_i$	4.0625	4.0625	0%

Note. For a point moment the equivalent load comes from the slope in the Euler element and from the section rotation in the Timoshenko element. At this position the difference also changes the sign.

How the critical load converges with the mesh

Published source. The Engesser relation

1/P_{cr} = 1/P_E + 1/(GA_v)

Quantity	Condition	Reference	Computed	Deviation
Critical load	element count = 1 · L = 4	2,725.43	3,450	26.60%
Critical load	element count = 2 · L = 4	2,725.43	2,755.18	1.09%
Critical load	element count = 4 · L = 4	2,725.43	2,728.99	0.130%
Critical load	element count = 8 · L = 4	2,725.43	2,725.45	< 0.001%

Note. The tolerance is the convergence band measured on this mesh. The few parts per thousand left on a fine mesh come from the approximation in the Engesser relation.

Pin-ended column, with shear deformation

Parameters. L = 4 · mesh = 16 elements · $P_E$ = 2,837.5

Published source. The Engesser relation

1/P_{cr} = 1/P_E + 1/(GA_v)

Quantity	Reference	Computed	Deviation
Critical load	2,725.43	2,724.78	0.0239%

Note. The column is divided into sixteen elements, and the critical load converges as the mesh is refined.

Simply supported beam, distributed load, 1-3 plane

Parameters. L = 6 · w = 20 · element count = 1

Published source. Timoshenko beam theory, in the lateral plane

5wL^4/384EI_{22} + wL^2/8GA_{v3}

Quantity	Reference	Computed	Deviation
Mid-span deflection, from the solution along the member	-0.0746739	-0.0746739	0%