Beam FEA with interactive preprocessor

3aaa6aeb-66d8-49f6-b5f7-a1463f90da18 125 Beam FEA with interactive preprocessor Martin Kraska 2D horizontal beam finite element analysis with exact representation of trapezoidal distributed loads. Features am interactive preprocessor and graphic and tabular output FHB 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

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 blue 2 blue 0.2 1 - 1 s * - 0011 s * - 1 - 1 s * - 42 mat (* blue 2 blue 24 mat 1104021 mat 14 mat :

support constraint w=0 1 - 01001 - 1 - 042 mat black 1 white 14 mat 1101021 mat 14 mat :

sym constraint φ=0 000122 mat black 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 blue 2 #00000000 s - 1 0.8 s * - 1.2 0.8 s * - 0.8 s - 142 mat blue 2 blue 1 - (s - 1 0.8 s * - 1.2 0.8 s * - 0.8 1 s * - 142 mat * blue 2 blue 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 :

t.cw 2 :

t.fneg 9 :

t.cφ 3 :

t.mpos 10 :

t.hinge 4 :

t.mneg 11 :

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 Identify required node positions t 1 time : 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 * x.1 x ≤ (x x.2 ≤ (& (* + : nodes nodes x.1 x.2 21 mat stack eval : 41 line for x EI EI : distributed load q 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 * 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 no dist trace if 21 line for x q q : nodal items (make sure that there are nodes at these positions) 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 0.05 : 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 node number for a given x value x n.N nn 0 : j 1 N# rows range x N# j 1 el - abs tol < nn j : 0 if for nn 31 line : create elements E# 05 matrix : j 1 N# rows 1 - range EI.e N# j 1 el N# j 1 + 1 el + 2 / EI eval : q.I N# j 1 el tol + q : q.J N# j 1 + 1 el tol - (q : EI.e 0 > E# E# j j 1 + EI.e q.I q.J 15 mat stack eval : 0 if 41 line for add nodal items: template nr, N column, value, position items t.cw 41 t.cφ 61 t.fpos 2 item 5 el t.fneg 2 item 5 el - t.mpos 3 item 4 el item 5 el (* t.mneg 3 item 4 el item 5 el - (* 63 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 : item trace n item 2 el n.N : pos items k 2 el : value items k 3 el : N# n pos el value eval : 61 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) n n.h EI 0015 mat trace E# E# n n.h EI 0015 mat stack eval : modify element to the right E# er 1 el n.h : 111 line for 0 if 1 time t - 331 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 : 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^* / : 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 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 reduce the system by adding 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 - eval : j 1 n.n range Nres# Nres# N# j 1 el u 2 j * 1 - el u 2 j * el F 2 j * 1 - el F 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 x.I - l / * q.I x.J 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 - 361 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 (: M.max M max M min abs 12 mat max : Q.max Q max Q min abs 12 mat max : w.max w max w min abs 12 mat max : P# 10055020015 mat : T# R x M M.max / augment x x rows el 0 x 1 el 022 mat stack red 1 #55ff0000 x Q Q.max / augment x x rows el 0 x 1 el 022 mat stack forestgreen 1 #5500ff00 x w w.max / augment black 2 #00000000 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 54 mat eval 110010021 mat 14 mat 11 sys : 1 time t - 111 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 17 mat :

out

EI 100000 :

q.0 0.1 :

Drag and drop objects from the palette to the canvas. Resize them with the mouse. Double-click them to set numerical values.

Function calls

out E N BeamPre 0.03 sec *

N E Res.N Res.E BeamSol 0.45 sec *

Res.E T P BeamPost 0.04 sec *

M.max Res.E 4 Absmax :

Q.max Res.E 3 Absmax :

w.max Res.E 6 Absmax :

out

M.max 704

Q.max 30

w.max 0.64

out w.max Q.max M.max P T Res.N N 'q.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 moment circle arc dist sym support force beam lastError out w.max Q.max M.max T Res.N N 'q.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 out w.max Q.max M.max 'q.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
-100,000	0,000	-0,015	0,000	0,000	-7,042	0,000
0,000	0,309	0,020	0,000	900,000	0,000	0,000
27,800	-0,208	0,018	0,000	0,000	0,000	0,000
27,800	-0,207	-0,010	0,000	0,000	0,000	0,000
50,000	-0,002	-0,008	0,200	-0,007	0,000	0,000
50,200	0,000	-0,008	34,900	-246,923	-23,241	0,000
100,000	0,000	0,006	14,900	164,757	-19,717	0,000
#
x	w	φ	F	M	S:F,R	S:M,R
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