Result integration, frame element

Results.

Example	Reference	Records	Largest deviation
Cantilever beam, tip load, on a plane cut	Closed form	6	0%
Simply supported beam, distributed load	Closed form	2	0%

Problem. The section integration is tested on a cut through frame elements. On a cantilever the shear force and the moment are compared with the values that follow from statics.

Reference document. Static equilibrium is used.

On a cantilever the point load beyond the section produces this shear force and moment.

V = P,\qquad M = P\,(L-x)

Numerical model. The method is free body equilibrium. The end forces at the nodes on the cut are summed and reduced to a reference point.

Cantilever beam, tip load, on a plane cut

Closed form. Static equilibrium

V = P,\qquad M = P(L-x)

Quantity	Condition	Reference	Computed	Deviation
Shear force $V$	x = 1.5 · L = 6 · P = -30	30	30	0%
Bending moment $M$	x = 1.5 · L = 6 · P = -30	135	135	0%
Shear force $V$	x = 3 · L = 6 · P = -30	30	30	0%
Bending moment $M$	x = 3 · L = 6 · P = -30	90	90	0%
Shear force $V$	x = 4.5 · L = 6 · P = -30	30	30	0%
Bending moment $M$	x = 4.5 · L = 6 · P = -30	45	45	0%

Simply supported beam, distributed load

Closed form. The moment relation for a simply supported beam

M(x) = wx(L-x)/2

Quantity	Condition	Reference	Computed	Deviation
Bending moment at $x = 1.5$	x = 1.5 · L = 6 · w = -20	67.5	67.5	0%
Bending moment at $x = 3.0$	x = 3 · L = 6 · w = -20	90	90	0%

Cantilever beam, tip load, on a plane cut#

Simply supported beam, distributed load#

Cantilever beam, tip load, on a plane cut

Simply supported beam, distributed load