Frame3D

8/8

3D Coordinate Transformation

Transform element stiffness from local to global coordinates using rotation matrix

K_{g l o ba l} = T^{T} \cdot K_{l oc a l} \cdot T, T = diag (R, R, R, R)_{12 \times 12}

Referencia

Matrix Structural Analysis, McGuire, Gallagher & Ziemian — Ch. 5

Variables

Simbolo	Nombre	Unidad
R	3x3 rotation matrix (local-to-global)	-
T	12x12 transformation matrix	-

Trapezoidal Distributed Load — Fixed-End Forces

Fixed-end forces for a beam element with linearly varying (trapezoidal) distributed load from w_i to w_j

V_{i} = \frac{L ( 7 w _{i} + 3 w _{j} )}{20}, V_{j} = \frac{L ( 3 w _{i} + 7 w _{j} )}{20}, M_{i} = \frac{L ^{2} ( 3 w _{i} + 2 w _{j} )}{60}, M_{j} = - \frac{L ^{2} ( 2 w _{i} + 3 w _{j} )}{60}

Referencia

Matrix Structural Analysis, McGuire, Gallagher & Ziemian — Table 4.2

Variables

Simbolo	Nombre	Unidad
w_i	Load intensity at node i	N/m
w_j	Load intensity at node j	N/m
L	Element length	m

Casos de prueba (2/2)

Cantilever with uniform distributed load in Y

Fuente: δ = wL⁴/(8EIz); R = wL; M = wL²/2

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
tip_displacement_y_m	0.00607524	0.00607524	6.0752e-13	0.00%	m
reaction_y_n	-3000.00	-3000.00	3.0000e-7	0.00%	N

Fixed-fixed beam with uniform distributed load

Fuente: δ = wL⁴/(384EIz); M = wL²/12

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
mid_displacement_y_m	-0.00200008	-0.00200008	2.0001e-13	0.00%	m
moment_z_end_nm	6666.67	6666.67	6.6667e-7	0.00%	N·m

Internal Force Recovery

Recover element end forces in local coordinates from global displacements, including fixed-end forces from distributed loads

f_{l oc a l} = K_{l oc a l} \cdot T \cdot u_{e l e m} + Q_{f}

Referencia

Direct stiffness method — element force recovery

Variables

Simbolo	Nombre	Unidad
f_{local}	Element end forces in local coords (12x1)	N, N·m
u_{elem}	Element nodal displacements in global coords (12x1)	m, rad
Q_f	Fixed-end forces from distributed loads (12x1)	N, N·m

Static Condensation for Element Releases

Condensed stiffness matrix after releasing selected DOFs at element ends

K^{*} = K_{r r} - K_{r c} \cdot K_{cc}^{- 1} \cdot K_{cr}

Referencia

Matrix Structural Analysis, McGuire, Gallagher & Ziemian — Ch. 10

Variables

Simbolo	Nombre	Unidad
K_{rr}	Retained DOF stiffness block	-
K_{cc}	Condensed DOF stiffness block	-
K_{rc}	Coupling stiffness block	-

Casos de prueba (2/2)

Propped cantilever via moment release at end

Fuente: R₁ = 5wL/8; R₂ = 3wL/8; M₁ = wL²/8

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
reaction_y_fixed_n	25000.0	25000.0	2.5000e-6	0.00%	N
reaction_y_pin_n	15000.0	15000.0	1.5000e-6	0.00%	N
moment_z_fixed_nm	20000.0	20000.0	2.0000e-6	0.00%	N·m

Truss element via rotation releases — axial only

Fuente: δ = FL/(AE); all moments = 0

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
displacement_x_m	7.5000e-5	7.5000e-5	7.5000e-15	0.00%	m

Euler-Bernoulli 12x12 Element Stiffness

Local stiffness matrix for a 3D beam-column element with 6 DOFs per node: axial, biaxial bending, biaxial shear, and torsion

k_{l oc a l} = [\frac{E A}{L} ⋮ \dots ⋱]_{12 \times 12}, with \frac{12 E I _{z}}{L ^{3}}, \frac{6 E I _{z}}{L ^{2}}, \frac{4 E I _{z}}{L}, \frac{G J}{L}

Referencia

Matrix Structural Analysis, McGuire, Gallagher & Ziemian — Ch. 5

Variables

Simbolo	Nombre	Unidad
E	Elastic modulus	Pa
A	Cross-sectional area	m²
G	Shear modulus	Pa
I_y	Moment of inertia about local y	m⁴
I_z	Moment of inertia about local z	m⁴
J	Torsional constant	m⁴
L	Element length	m

Casos de prueba (4/4)

Cantilever beam with point load in Y

Fuente: δ = PL³/(3EIz); θ = PL²/(2EIz)

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
tip_displacement_y_m	0.00160006	0.00160006	1.6001e-13	0.00%	m
tip_rotation_z_rad	0.00120005	0.00120005	1.2000e-13	0.00%	rad

Simply supported beam with central point load

Fuente: δ = PL³/(48EIz); R = P/2

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
mid_displacement_y_m	-0.00800032	-0.00800032	8.0003e-13	0.00%	m
reaction_y_n	5000.00	5000.00	5.0000e-7	0.00%	N

Pure axial load — regression against truss solver

Fuente: δ = FL/(AE) = 10000×1/(0.01×200×10⁹) = 5×10⁻⁶ m

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
displacement_x_m	5.0000e-6	5.0000e-6	5.0000e-16	0.00%	m

Pure torsion — angle of twist

Fuente: φ = ML/(GJ) = 500/(76.92×10⁹×10⁻⁵)

PASS

Check	Esperado	Resultado	Tolerancia	Error rel.	Unidad
twist_rad	0.000650026	0.000650026	6.5003e-14	0.00%	rad