Templates

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Templates

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

force force 0001 - 22 mat blue 2 blue 0.2 1 - 1 - 0011 - 1 - 1 - 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 14 mat 112021 mat 14 mat :

dist distributed load 01 - 11 - 110101 - 52 mat blue 0 #550000ff 14 mat 1502021 mat 14 mat :

rdist distributed load 001011 - 0042 mat blue 0 #550000ff 14 mat 1502021 mat 14 mat :

ldist distributed load 01 - 10100042 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 15 ° * + range a a φ cos φ sin 12 mat stack : for a 31 line :

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

Vorlage beam force support sym dist rdist ldist circle 81 sys :

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 11 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 x.1 x.2 lele * + : nodes nodes x.1 x.2 21 mat stack eval : 41 line for distributed load q q 0 : items 51160171033 mat : k 1 items rows range dist model items k 1 el 1 findrows : dist GetType matrix ≡ 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 x.1 x.2 lele * + : nodes nodes x.1 x.2 21 mat stack eval : 41 line for no dist trace if 21 line for nodal items (make sure that there are nodes at these positions) k 234814 mat items model k 1 findrows : items GetType matrix ≡ 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 x n.N j 1 N# rows range x N# j 1 el - abs tol < nn j : 0 if for nn 21 line : create elements E# 05 matrix : j 1 N# rows 1 - range EI.e EI x N# j 1 el N# j 1 + 1 el + 2 / (≡ at eval : q.I q x N# j 1 el tol + ≡ at : q.J q x N# j 1 + 1 el tol - (≡ at : E# E# j j 1 + EI.e q.I q.J 15 mat stack eval : 41 line for add nodal items items 22 item 5 el 34146133 mat : k 1 items rows range tmp model items k 1 el 1 findrows : tmp GetType matrix ≡ 1 continue if k tmp rows tmp GetType trace 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 41 line for add hinges hinges model 81 findrows : 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 1 time t - 301 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 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 + 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# 10040020015 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 :

out

EI 100000 :

q.0 0.1 :

out E N BeamPre 0.031 sec *

N E Res.N Res.E BeamSol 0.702 sec *

Res.E T P BeamPost 0.047 sec *

M.max Res.E 4 Absmax :

Q.max Res.E 3 Absmax :

w.max Res.E 6 Absmax :

out

M.max 2014

Q.max 81

w.max 1.719