Rigid end zone

Results.

Example	Reference	Records	Largest deviation
Beam fixed at both ends, rigid zone at both ends	Closed form, Published source	2	0%
Simply supported beam, rigid zone at both ends	Independent calculation	1	0%
Self weight, rigid zone at both ends	Closed form, Published source	2	0%

Problem. The beam-column joint zone is taken as rigid. The stiffness is built on the clear span and carried to the nodes by a rigid-body transformation.

Reference document. The closed-form solution is used. The end moment is the sum of the fixed-end moment over the flexible length, the shear transfer and the load falling on the rigid zone.

The flexible length of the element and the transformation across the rigid end zones are given by these relations. Here $H$ is the transformation that bridges the rigid zone.

L_{esnek} = L - r_i - r_j,\qquad k_{dugum} = H^{\mathsf T} k_{esnek} H

The end moment under self-weight is computed from this relation.

M = w\left(\dfrac{L_f^2}{12} + \dfrac{L_f\,a}{2} + \dfrac{a^2}{2}\right)

Numerical model. The stiffness comes from the clear span, and the mass and self-weight from the full length. End forces are reported at the node. The rigid zones are drawn thicker.

Beam fixed at both ends, rigid zone at both ends

Parameters. L = 6 · a = 1 · w = -12

Closed form. Flexible end fixity, shear transfer and the load on the rigid zone together

Quantity	Reference	Computed	Deviation
End moment $\lvert w\rvert (L_f^2/12 + L_f a/2 + a^2/2)$	46	46	0%
Mid-span deflection $\lvert w \rvert L_f^4 / 384EI$	0.0005	0.0005	0%

Note. The deflection arises in the flexible part only.

Simply supported beam, rigid zone at both ends

Parameters. L = 6 · a = 1 · w = 12

Independent calculation. Virtual work integral

\delta(a) = \int_0^L \dfrac{M_0(x)\,m_a(x)}{EI(x)}\,dx

Quantity	Reference	Computed	Deviation
Mid-span deflection	0.012	0.012	0%

Note. $EI$ is taken as infinite inside the rigid zone.

Self weight, rigid zone at both ends

Parameters. L = 6 · a = 1 · ρ = 2.5 · g = -9.81

Closed form. Flexible end fixity, shear transfer and the load on the rigid zone together

Quantity	Reference	Computed	Deviation
End moment $\lvert w\rvert (L_f^2/12 + L_f a/2 + a^2/2)$	1.12815	1.12815	0%
Mid-span deflection $\lvert w \rvert L_f^4 / 384EI$	0.0000122625	0.0000122625	0%

Note. The weight of the rigid zones goes straight to the node.

Beam fixed at both ends, rigid zone at both ends#

Simply supported beam, rigid zone at both ends#

Self weight, rigid zone at both ends#

Beam fixed at both ends, rigid zone at both ends

Simply supported beam, rigid zone at both ends

Self weight, rigid zone at both ends