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e4cdc970-3a6a-4b6e-9084-6b3eb7d45d34 27 4 5 decimal

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Element functions, stiffness matrix and load vector

Element field, expressed in terms of the DOFs and the element load parameters

x w 120 w.I * l 3^* EI * x x x 340 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^q.J * + (* l 2^7 l 2^* q.I * x x q.J * x - 5 l * + (q.I * + (* - (* + (* - 260 EI * 3 w.J - w.I + (* - l φ.J 2 φ.I * + (* + (* l 4^q.J * + (* 3 l 4^* q.I * + (l * + (* 120 φ.I * l 3^* EI * - (* + 120 l 3^* EI * / :

x φ 120 φ.I * EI * l 3^* x x 5 x * x q.J * x - 4 l * + (q.I * + (* l 2^* 3340 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^q.J * + (* 7 l 4^* q.I * + (* - (* 2260 EI * 3 w.J - w.I + (* - l φ.J 2 φ.I * + (* + (* l 4^q.J * + (* 3 l 4^* q.I * + (* l * + (* - 120 EI * l 3^* / :

x M 260 EI * 3 w.J - w.I + (* - l φ.J 2 φ.I * + (* + (* l 4^q.J * + (* 3 l 4^* q.I * + (l * x 10 x * x q.J * x - 3 l * + (q.I * + (* l 2^* 3340 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^q.J * + (* 7 l 4^* q.I * + (* - (* + 60 l 3^* / - :

x Q 340 EI * 2 w.J - w.I + (* - l φ.J φ.I + (* + (* l 4^q.J * + (* 7 l 4^* q.I * + 10 x * x q.J * x - 2 l * + (q.I * + (* l 2^* - 20 l 3^* / :

x q q.I l x - (* q.J x * + l / :

EI l K.e 12 EI * l 3^/ 6 EI * l 2^/ - 12 EI * l 3^/ - 6 EI * l 2^/ - 6 EI * l 2^/ - 4 EI * l / 6 EI * l 2^/ 2 EI * l / 12 EI * l 3^/ - 6 EI * l 2^/ 12 EI * l 3^/ 6 EI * l 2^/ 6 EI * l 2^/ - 2 EI * l / 6 EI * l 2^/ 4 EI * l / 44 mat :

q.I q.J l f.e l 3 q.J * 7 q.I * + (* 20 / l 2^2 q.J * 3 q.I * + (* 60 / - l 7 q.J * 3 q.I * + (* 20 / l 2^3 q.J * 2 q.I * + (* 60 / 41 mat :

Model description

Node input

Element input

x

node position

I left node

F

point force

J right node

M

point moment

EI bending stiffness

f.w

constraint flag for w (0 or 1)

q.I

left value of distributed load

w

prescribed value if constrained

q.J

right value of distributed load

f.φ

constraint flag for φ (0 or 1)

φ

prescribed value if constrained

x F M f.w w f.φ φ 17 mat

I J EI q.I q.J 15 mat

Beam with 3 supports and non-constant load

N 00010001 l * 10 kN * 010002 l * 10 kN * 000003 l * 001 0.1 m * 1047 mat :

E 12 EI 0 q.0 23 EI q.0 2 / q.0 34 EI q.0 035 mat :

l 2 m * :

φ.1 0 :

q.0 10 kN m / * :

u.0 1 mm * :

EI 100 kN * m 2^* :

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N 0001000 l.0 2 / F.0 00000 l.0 00100037 mat :

E 12 EI 0023 EI 0025 mat :

1

2

3

1

2

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N 0001010 l.0 00100027 mat :

E 12 EI q q 15 mat :

FEA procedure

Global stiffness matrix and load vector

n.e E rows :

n.n N rows :

K 2 n.n * 2 n.n * matrix :

F 2 n.n * 1 matrix :

u 2 n.n * 1 matrix :

i.e 1 E rows range I E i.e 1 el : J E i.e 2 el : EI E i.e 3 el : q.I E i.e 4 el : q.J E i.e 5 el : l N J 1 el N I 1 el - : j i.g 2 I 1 - (* j + j 2 ≤ 2 J 2 - (* j + cases : i 14 range j 14 range K i i.g j i.g el K i i.g j i.g el EI l K.e i j el + : for F i i.g el F i i.g el q.I q.J l f.e i el + : 21 line for 81 line for

Assemble element stiffness and element distributed loads into global stiffness and load vector

K 150000 kg * s 2^/ 150000 kg * m * s 2^/ - 150000 kg * s 2^/ - 150000 kg * m * s 2^/ - 0000150000 kg * m * s 2^/ - 200000 kg * m 2^* s 2^/ 150000 kg * m * s 2^/ 100000 kg * m 2^* s 2^/ 0000150000 kg * s 2^/ - 150000 kg * m * s 2^/ 300000 kg * s 2^/ 0150000 kg * s 2^/ - 150000 kg * m * s 2^/ - 00150000 kg * m * s 2^/ - 100000 kg * m 2^* s 2^/ 0400000 kg * m 2^* s 2^/ 150000 kg * m * s 2^/ 100000 kg * m 2^* s 2^/ 0000150000 kg * s 2^/ - 150000 kg * m * s 2^/ 300000 kg * s 2^/ 0150000 kg * s 2^/ - 150000 kg * m * s 2^/ - 00150000 kg * m * s 2^/ - 100000 kg * m 2^* s 2^/ 0400000 kg * m 2^* s 2^/ 150000 kg * m * s 2^/ 100000 kg * m 2^* s 2^/ 0000150000 kg * s 2^/ - 150000 kg * m * s 2^/ 150000 kg * s 2^/ 150000 kg * m * s 2^/ 0000150000 kg * m * s 2^/ - 100000 kg * m 2^* s 2^/ 150000 kg * m * s 2^/ 200000 kg * m 2^* s 2^/ 88 mat

F 3000 m * kg * s 2^/ 4000 kg * m 2^* 3 s 2^* / - 13500 m * kg * s 2^/ 1000 kg * m 2^* 3 s 2^* / - 15500 m * kg * s 2^/ 2000 kg * m 2^* 3 s 2^* / 3000 m * kg * s 2^/ 4000 kg * m 2^* 3 s 2^* / 81 mat

00 l f.e 000041 mat

in 1 N rows range F 2 in * 1 - el F 2 in * 1 - el N in 2 el + : F 2 in * el F 2 in * el N in 3 el + : u 2 in * 1 - el N in 4 el N in 5 el * : u 2 in * el N in 6 el N in 7 el * : c 2 in * 1 - el N in 4 el : c 2 in * el N in 6 el : 61 line for

add point forces and moments to global load vector

F 3000 m * kg * s 2^/ 4000 kg * m 2^* 3 s 2^* / - 23500 m * kg * s 2^/ 1000 kg * m 2^* 3 s 2^* / - 25500 m * kg * s 2^/ 2000 kg * m 2^* 3 s 2^* / 3000 m * kg * s 2^/ 4000 kg * m 2^* 3 s 2^* / 81 mat

set values of prescribed displacements and slopes in global displacement vector

vector of constraint flags

F.e F eval :

Store nodal element loads for use in postprocessing

System reduction by constraints

cc is an intermediate result to enable findrows to work

cc 12 n.n * range c augment :

constr cc 12 findrows 1 col (:

Indices of constrained degrees of freedom

free cc 02 findrows 1 col (:

Indices of unconstrained (free) degrees of freedom

F F K u * - (:

Take care of non-homogeneous constraints

u

i 1 free rows range F.R i el F free i el el : j 1 free rows range K.R i j el K free i el free j el el : f 21 line for 21 line for

Reduce system by removing all constrained dofs

Solution of the reduced system

u.R K.R 1 -^F.R * :

F.R

free rows 1 ≡ u.R u.R 11 mat : 0 if

Just solving the linear system. Single dof case gives scalar result, this has to be fixed for seamless postprocessing

Solution into full system

i 1 free rows range u free i el el u.R i el : for

Reactions

R K u * F - (:

R 2900 kg * m * s 2^/ 041825 kg * m * s 2^/ - 015000 kg * m * s 2^/ - 15000 kg * m 2^* s 2^/ 1075 kg * m * s 2^/ - 4700 kg * m 2^* 3 s 2^* / 81 mat

u 013500 / 049750 / - 503 m * 3000 / 392000 / - m 10 / 081 mat

Result extraction

w 011 sys :

φ 011 sys :

M 011 sys :

Q 011 sys :

q 011 sys :

i.e 1 E rows range I E i.e 1 el : J E i.e 2 el : EI E i.e 3 el : q.I E i.e 4 el : q.J E i.e 5 el : w.I u 2 I * 1 - el : φ.I u 2 I * el : w.J u 2 J * 1 - el : φ.J u 2 J * el : l N J 1 el N I 1 el - : w i.e el x x N I 1 el - w - x N I 1 el N J 1 el parametric : φ i.e el x x N I 1 el - φ x N I 1 el N J 1 el parametric : M i.e el x N I 1 el - M x N I 1 el N J 1 el explicit : Q i.e el x N I 1 el - Q x N I 1 el N J 1 el explicit : q i.e el x N I 1 el - q x N I 1 el N J 1 el explicit : 151 line for

M 100 x * kg * 25 x 2^* 87 m 2^* + (* 3 m * s 2^* / - x 02 m * explicit 25 kg * 1496 m 3^* x 2 m * - (50 x 2 m * - (* x - 2 x 2 m * + (* + (* 2271 m 2^* - (* + (* 3 m * s 2^* / - x 2 m * 4 m * explicit 25 kg * 1446 m 3^* - x 4 m * - (100 x 4 m * - (* x - 10 m * + (* 729 m 2^* + (* + (* 3 m * s 2^* / - x 4 m * 6 m * explicit 31 sys

Q 100 kg * 29 m 2^* 25 x 2^* + (* m s 2^* / - x 02 m * explicit 25 kg * 617 m 2^* 1014 m 2^* 5 x 2 m * - (* x - 2 x m + (* + (* - (* + (* m s 2^* / x 2 m * 4 m * explicit 25 kg * 523 m 2^* - 2014 m 2^* 5 x 4 m * - (* x - 8 m * + (* - (* + (* m s 2^* / x 4 m * 6 m * explicit 31 sys

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