Balken-FEM mit grafischer Ein- und Ausgabe

3aaa6aeb-66d8-49f6-b5f7-a1463f90da18 187 Balken-FEM mit grafischer Ein- und Ausgabe Martin Kraska Finite-Elemente-Programm für horizontale 2D-Balken mit exakter Darstellung linearer Streckenlasten. Grafische Systemdefinition, Querkraft-, Momenten- und Biegelinie. Fachhochschule Brandenburg Beam FEA with interactive preprocessor Martin Kraska 2D horizontal beam finite element analysis with exact representation of trapezoidal distributed loads. Features an interactive preprocessor and graphic and tabular output FHB 4 5 decimal

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Templates

beam beam 01 - 11 - 110101 - 52 mat black 1 lightgray 14 mat 11001021 mat 14 mat :

These are the objects that can be placed in the modeller region to define the model.

s force force 0001 s * - 22 mat green 2 green 0.2 1 - 1 s * - 0011 s * - 1 - 1 s * - 42 mat (* green 2 green 24 mat 1104021 mat 14 mat :

support constraint w=w.0 1 - 01001 - 1 - 042 mat forestgreen 1 #5500ff00 14 mat 1101021 mat 14 mat :

sym constraint φ=0 000122 mat red 2 white 1 - 01022 mat #00000000 00000000024 mat 1103021 mat 14 mat :

I J dist distributed load 000 I - 1 J - 1042 mat blue 0 #550000ff 14 mat 1502021 mat 14 mat :

φ.1 φ.2 arc a 02 matrix : φ φ.1 φ.2 φ.1 30 ° * + range a a φ cos φ sin 12 mat stack : for a 31 line :

circle hinge 0 ° * 360 ° * arc eval black 0 white 14 mat 1101021 mat 14 mat :

s moment moment s - 10101 - s 1 - 42 mat red 2 #00000000 s - 1 0.8 s * - 1.2 0.8 s * - 0.8 s - 142 mat red 2 red 1 - (s - 1 0.8 s * - 1.2 0.8 s * - 0.8 1 s * - 142 mat * red 2 red 34 mat 1303021 mat 14 mat :

Vorlage beam support sym circle 11 dist 10 dist 01 dist 1 force 1 - force 1 - moment 1 moment 111 sys :

t.beam 1 :

t.fpos 8 :

Set "Vorlage" as template in the context menu of the modeler region.

t.cw 2 :

t.fneg 9 :

t.cφ 3 :

t.mpos 10 :

t.hinge 4 :

t.mneg 11 :

The items in the model are identified by template number.

t.qq 5 :

t.q0 6 :

t.0q 7 :

preprocessing

BeamPre(Mout,E,N) Beam FEA preprocessing. 
Mout: Modeller output, E,N: element  and node tables for the solver BeamSol() model E# N# BeamPre t 1 time : E# 0 : N# 0 : build EI(x) and create nodes at beam ends beams model t.beam 1 findrows : EI 0 : nodes 01 matrix : j 1 beams rows range x.1 beams j 2 el beams j 4 el 2 / - : x.2 beams j 2 el beams j 4 el 2 / + : EI EI beams j 5 el EI F.0 l.0 2^* / * x.1 x ≤ (x x.2 ≤ (& (* + : nodes nodes x.1 x.2 21 mat stack eval : 41 line for x EI EI : build q(x) and create nodes at distributed load ends q 0 : items t.qq 11 t.q0 10 t.0q 0133 mat : k 1 items rows range dist model items k 1 el 1 findrows : dist 0 ≠ j 1 dist rows range x.1 dist j 2 el dist j 4 el 2 / - : x.2 dist j 2 el dist j 4 el 2 / + : q q dist j 5 el q.0 l.0 * F.0 / * items k 3 el x x.1 - x.2 x.1 - / (* items k 2 el x.2 x - x.2 x.1 - / * + (* x.1 x ≤ (x x.2 ≤ (& (* + : nodes nodes x.1 x.2 21 mat stack eval : 41 line for 0 if 21 line for x q q : create nodes at hinges, constraints and point loads k t.cw t.cφ t.hinge t.fpos t.fneg t.mpos t.mneg 17 mat items model k 1 findrows : items 0 ≠ j 1 items rows range nodes nodes items j 2 el 11 mat stack eval : 11 line for 0 if 21 line for merge identical nodes tol 0.0005 : nsort nodes sort : k 1 : N# 11 el nsort 1 el eval : N# 17 el 0 : j 2 nsort rows range N# k 1 el nsort j el - abs tol > k k 1 + : N# k 1 el nsort j el eval : 21 line 0 if for x n.N node number for a given x value nn 0 : j 1 N# rows range x N# j 1 el - abs tol < nn j : 0 if for nn 41 line : create elements. Mitigate zero stiffness to a low value E# 05 matrix : j 1 N# rows 1 - range EI.e N# j 1 el N# j 1 + 1 el + 2 / EI 0.0000001 EI F.0 l.0 2^* / * 12 mat max eval : q.I N# j 1 el tol + q : q.J N# j 1 + 1 el tol - (q : E# E# j j 1 + EI.e q.I q.J 15 mat stack eval : 41 line for add nodal items: template nr, N column, value items t.cw 41 w constraint flag t.cw 5 item 3 el w.0 l.0 / * - w constraint value t.cφ 61 φ constraint flag t.fpos 2 item 5 el F value t.fneg 2 item 5 el - F value t.mpos 3 item 4 el item 5 el (* M value t.mneg 3 item 4 el item 5 el - (* M value 74 mat : k 1 items rows range tmp model items k 1 el 1 findrows : tmp 0 ≠ 1 continue if j 1 tmp rows range item tmp j row : n item 2 el n.N : pos items k 2 el : value items k 3 el : N# n pos el value eval : 51 line for 31 line for add hinges hinges model t.hinge 1 findrows : hinges 0 ≠ j 1 hinges rows range n hinges j 2 el n.N : tmp 1 E# rows range E# augment : er tmp n 2 findrows 1 el eval : add new node at the same location N# N# hinges j 2 el 00000017 mat stack : n.h N# rows : add hinge element (type is recognized by zero lenght in FEA) E# E# n n.h EI F.0 l.0 2^* / 0015 mat stack eval : modify element to the right E# er 1 el n.h : 101 line for 0 if 1 time t - 341 line :

Solution

BeamSol(N,E,Nres,Eres) Beam FEA solver. N,E node and element matrices. Each line in N has the entries
(x, F, M, w-flag, w-value, φ-flag, φ-value). Flags=1 means constraint to given value. 
Each line in E has the entries (I, J, EI, qI, qJ). Zero length elements are hinges with w-stiffnes EI.
Nres are nodal results (x, w, φ, F, M, RF, RM), Eres are element results for plotting with the columns 
(x, q, Q, M, φ, w) N# E# Nres# Eres# BeamSol t 1 time : 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 : provide functions for result computation 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^* / : Element stiffness (beam or hinge) and element load vector EI l K.e l 0 > 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 EI 0 EI - 00000 EI - 0 EI 0000044 mat if : 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 : Assemble global stiffness and element loads 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 j 2 ≤ (0 ≠ (j 21 - I + (* + (* j 2 ≤ (0 ≡ (j 22 - J + (* + (* + (: 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 + eval : for F i i.g el F i i.g el q.I q.J l f.e i el + eval : 21 line for 81 line for add concentrated loads and constraints 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 F.e F eval : reduce the system using constraints cc 12 n.n * range c augment : constr cc 12 findrows 1 col eval : free cc 02 findrows 1 col eval : F F K u * - eval : 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 : 11 line for 21 line for solve the system u.R K.R 1 -^F.R * eval : free rows 1 ≡ u.R u.R 11 mat : 0 if expand solution to full system i 1 free rows range u free i el el u.R i el : for Nodal (DOF) results: x, w, φ, F, M, FR, MR Nres# 07 matrix : R K u * F.e - eval : j 1 n.n range Nres# Nres# N# j 1 el u 2 j * 1 - el u 2 j * el ° / F.e 2 j * 1 - el F.e 2 j * el R 2 j * 1 - el R 2 j * el 17 mat stack eval : for Element results: x, q, Q, M, φ, w Eres# 06 matrix : n 8 : i.e 1 n.e range I E# i.e 1 el : J E# i.e 2 el : 12 mat x.I N# I 1 el : x.J N# J 1 el : q.I E# i.e 4 el : q.J E# i.e 5 el : w.I u 2 I * 1 - el : w.J u 2 J * 1 - el : φ.I u 2 I * el : φ.J u 2 J * el : 42 mat l x.J x.I - : EI E# i.e 3 el eval : l 0 > j 0 n range x l j n / * : q q.J x l / * q.I l x - l / * + : Eres# Eres# x x.I + q x Q x M x φ ° / x w 16 mat stack eval : 31 line for 0 if 51 line for 1 time t - 391 line :

Postprocessing

BeamPost[Res,T,P) Beam FEA postprocessing. Res: element results from BeamSol(), 
T: Modeller template, P: Modeller predefined layoutE Res# T# P# BeamPost t 1 time : M Res# 4 col - : Q Res# 3 col - : w Res# 6 col (: x Res# 1 col (: q Res# 2 col - : M.max M max M min abs 0.001 13 mat max : Q.max Q max Q min abs 0.001 13 mat max : w.max w max w min abs 0.001 13 mat max : q.max q max q min abs 0.001 13 mat max : P# 101405005020555001003055 - 50010040140 - 5005045 mat : Axes x 1 el 0 x x rows el 022 mat black 1 black x 1 el 1 - x 1 el 122 mat black 1 black 24 mat : T# q Axes x q q.max / augment x x rows el 0 x 1 el 022 mat stack blue 1 #550000ff 14 mat stack eval 15005021 mat 14 mat Q Axes x Q Q.max / augment x x rows el 0 x 1 el 022 mat stack forestgreen 1 #5500ff00 14 mat stack eval 150010021 mat 14 mat Q Axes x M M.max / augment x x rows el 0 x 1 el 022 mat stack red 1 #55ff0000 14 mat stack eval 150010021 mat 14 mat w Axes x w w.max / augment black 2 #00000000 14 mat stack eval 15005021 mat 14 mat 41 sys : 1 time t - 141 line :

M# c# Absmax D M# c# col : D min abs D max 12 mat max 21 line :

Nhead x w φ F M S:F.R S:M.R S:'l.0 S:'w.0 S:'° S:'F.0 S:'F.0*'l.0 S:'F.0 S:'F.0*'l.0 27 mat :

Instructions

Beam segment. Thickness scales EI. Segments can overlap, then EI adds up.

Vertical support. Place it at y=0 for zero displacement. Shift vertically for displaced supports (non-homogeneous constraints)

Constraint φ=0 (symmetry line)

Hinge, place anywhere. To a certain extent the solver can handle mechanisms (gives huge displacement values)

Distributed load, individual segments can be stacked or overlap to create arbitrary piecewise linear q(x). Scale the values with and with the height of the segments.

q.0

x Q

Point loads. Scale with the height.

x M

Moment load. Scale with the distance and the length of the arrows.

Drag and drop objects from the palette to the canvas. x and y values are taken from the center of the object. Resize them with the mouse. Double-click them to set numerical values (does not work in SMath cloud).

Re-calculation is triggered by any changes in the model.

Robustness features:

- Nodes with less than tol=0,01 distance are merged. - All nodes are connected with elements. Elements with zero stiffness (resulting from loads or constraints outside beams)are set to a very low stiffness in order to prevent solver crash - zero max result values are set to a small value in order to keep the diagram scaling alive.

out

EI 100000 kN * m 2^* :

l.0 1 m * :

F.0 1 N * :

w.0 1 mm * :

q.0 0.1 F.0 l.0 / * :

Konsistenz der Maßeinheiten (alle Quotienten müssen dimensionslos sein)

Unit consistency check (all results should be unit-free):

q.0 l.0 * F.0 / 0.1

w.0 l.0 / 0.001

EI F.0 l.0 2^* / 1108^*

Function calls

out E N BeamPre 0.067 sec *

N E Res.N Res.E BeamSol 1.945 sec *

Res.E T P BeamPost 0.151 sec *

M.max Res.E 4 Absmax F.0 * l.0 * :

Q.max Res.E 3 Absmax F.0 * :

w.max Res.E 6 Absmax l.0 * :

q.max Res.E 2 Absmax F.0 l.0 / * :

post

q.max q.0 20

Q.max F.0 135

M.max F.0 l.0 * 3236

w.max w.0 2.37

post q.max w.max Q.max M.max Res.N N 'q.0 'w.0 'F.0 'l.0 'EI Absmax BeamPost BeamSol BeamPre t.0q t.q0 t.qq t.mneg t.hinge t.mpos t.cφ t.fneg t.cw t.fpos t.beam moment circle arc dist sym support force beam lastError post q.max w.max Q.max M.max T Res.N N 'q.0 'w.0 'F.0 'l.0 'EI Nhead Absmax BeamPost BeamSol BeamPre t.0q t.q0 t.qq t.mneg t.hinge t.mpos t.cφ t.fneg t.cw t.fpos t.beam Vorlage arc lastError post q.max w.max Q.max M.max 'q.0 'w.0 'F.0 'l.0 'EI t.0q t.q0 t.qq t.mneg t.hinge t.mpos t.cφ t.fneg t.cw t.fpos t.beam

Res.N 1 csort eval
-101,000	0,000	0,001	0,900	-0,150	0,000	0,000
-100,000	0,000	0,001	36,900	-239,850	43,073	0,000
-60,000	0,000	-0,002	54,000	180,000	-252,335	0,000
-40,000	0,001	-0,004	58,180	59,994	0,000	0,000
-39,800	0,001	-0,004	0,180	0,006	0,000	0,000
-25,000	0,002	-0,005	0,000	0,000	0,000	0,000
-25,000	0,002	0,000	0,000	0,000	0,000	0,000
-23,400	0,002	0,000	33,570	-227,000	0,000	0,000
23,200	0,001	0,003	16,360	1 755,459	0,000	0,000
26,600	0,001	0,003	0,069	0,052	0,000	0,000
50,000	0,000	0,003	0,000	0,000	9,102	0,000
51,000	0,000	0,003	0,000	0,000	0,000	0,000
#
x	w	φ	F	M	S:F,R	S:M,R
S:'l,0	S:'w,0	S:'°	S:'F,0	S:'F,0*'l,0	S:'F,0	S:'F,0*'l,0
#
#
#

position

w

displacement

φ

slope in °

F, M

external loads, including contri- butions of distri- buted loads

F.R

Support reactions

M.R