8788dd1b-9817-4243-9ee4-cb6850909ab3 49 4 5 decimal

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

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 :

out E N BeamPre #

N E Res.N Res.E BeamSol #

Res.E T P BeamPost #

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 / * :

config 125 - 10.25 150 10.5 250010102100 - 010 11.1 525 - 24.6 15018425 - 10.25 101040 10.25 1010240 - 010102100101085 mat :

2 beams

125 - 10.25 150 10.5 250010102100 - 0.35 - 10 11.1 525 - 24.6 15018125 - 10.25 100 10.5 55 mat

plf lbf ft / :

intomodeller config :

l.0 1 ft * :

Set length unit

E## 29000 ksi * :

F.0 1 lbf * :

Set force unit

w.0 1 in * :

I## 40000 in 4^* :

Set displacement unit

q.0 1 F.0 l.0 / * : plf 1

Set distributed load unit

Scale the moddel values

note that EI has been deleted

Eventually change sysmbols for deflection and length (update all symbols in calc)

L.0# ft :

F.0# 1 lbf * :

Δ.0 1 in * :

w.0# 0.1 F.0# L.0# / * :

outmodel

Input

supports# 0 ft * 10 ft * 20 ft * 40 ft * 110124 mat :

0=hinge, 1=pin support. Use hinge for cantilever enter from 0' to end of beam -> left to right

Support / Hinge

MoE

MoE# 29000 ksi * :

MoI

I# 100 in 4^* :

Uniform Load

w# 10 lbf ft / * 11 mat :

On all spans (use multiple rows for multiple input)

Partial Loads

w.p# 10 kip ft / * 10 kip ft / * 0 kip ft / * 5 kip ft / * 0 ft * 10 ft * 20 ft * 30 ft * 10 kip ft / * 0 kip ft / * 10 kip ft / * 10 kip ft / * 10 ft * 20 ft * 30 ft * 40 ft * 44 mat :

Enter wb in top row

Enter b in second row

Enter we in third row

Enter e in bottom row

Point Loads

Pt# 5 kip * 20 kip * 10 ft * 25 ft * 22 mat :

prebeaminput config :

Note that the input to the prebeam is the same as the input to the modeller

model prebeaminput :

Proposed inputs

model E# N# E## I## tol# fprebeam

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

Element matrix: [start node, end node, stiffness, dist load val start, dist. load value end]

Node matrix, N#: [x-position, Point Load value, Point Moment Value, support flag, support force, hinge flag

model 125 - 10.25 150 10.5 250010102100 - 010 11.1 525 - 24.6 15018425 - 10.25 101040 10.25 1010240 - 010102100101085 mat

tol# 1 in * :

Create input from beams:

Create Supports length# supports# 1 supports# cols el supports# 11 el - 2 / : Support# 05 matrix : n 1 supports# cols range type# supports# 2 n el : pos# supports# 1 n el : t# t.cw type# 1 ≡ t.hinge type# 0 ≡ unknown cases : t# unknown ≡ unknown support type type# num2str at column n num2str concyat trace continue 21 line 0 if Support# Support# t# pos# 0 ft * 0.1 length# * 0.1 length# * 15 mat stack eval : 51 line for 41 line Create beam Beam# 05 matrix : t# t.beam : posx# supports# 1 supports# cols el supports# 11 el - 2 / : posy# 0.1 length# * : length# supports# 1 supports# cols el supports# 11 el - : height_b# E## I## * l.0 * length# 3^F.0 * / : Beam# Beam# t# posx# posy# length# height_b# 15 mat stack eval : 81 line Create Distributed Loads UDistIP# 05 matrix : n 1 w# rows range t# t.qq : height# w# q.0 / : posx# supports# 1 supports# cols el supports# 11 el - 2 / : posy# height# 2 / height_b# + : length# supports# 1 supports# cols el supports# 11 el - : UDistIP# UDistIP# t# posx# posy# length# height# 15 mat stack eval : 61 line for 31 line Create Partial / Varying Loads Create new matrix which splits up the trapazoidal loads into new columns in a new w.p matrix w.p## 40 matrix : n 1 w.p# rows range q.1# w.p# 1 n el q.0 / : x1# w.p# 2 n el l.0 / : q.2# w.p# 3 n el q.0 / : x2# w.p# 4 n el l.0 / : Create a uniform load q.unp# q.1# q.2# Min : Create triangular (varying) load q.vp1# q.1# q.unp# - : q.vp2# q.2# q.unp# - : Create conditional cases matrices for uniform and varying loads. Return empy matrix if not in use w.punif# q.unp# x1# q.unp# x2# 41 mat q.1# q.2# ≡ 40 matrix q.1# 0 ≡ (q.2# 0 ≡ (| q.unp# x1# q.unp# x2# 41 mat cases : w.pvar# 40 matrix q.1# q.2# ≡ q.1# x1# q.2# x2# 41 mat q.1# 0 ≡ (q.2# 0 ≡ (| q.1# x1# q.2# x2# 41 mat cases : w.p## w.p## w.punif# w.pvar# augment eval : 131 line for Now use w.p## as the partial load matrix n 1 w.p## rows range t# t.qq w.p## 1 n el w.p## 3 n el ≡ t.q0 w.p## 1 n el w.p## 3 n el > t.0q w.p## 1 n el w.p## 3 n el < Unknown cases : t# unknown ≡ unknown partial load w.p## num2str at column n num2str concyat trace continue 21 line 0 if 21 line for 61 line 41 line

w.p# 1.4594 105^* kg s 2^/ * 1.4594 105^* kg s 2^/ * 0 72969.5147 kg s 2^/ * 0 3.048 m * 6.096 m * 9.144 m * 1.4594 105^* kg s 2^/ * 0 1.4594 105^* kg s 2^/ * 1.4594 105^* kg s 2^/ * 3.048 m * 6.096 m * 9.144 m * 12.192 m * 44 mat

q.1# plf 685.2177 s 2^kg / *

q.2# plf 0

w.punif# q.unp# x1# q.unp# x2# 41 mat q.1# q.2# ≡ 40 matrix q.1# 0 ≡ (q.2# 0 ≡ (| q.unp# x1# q.unp# x2# 41 mat cases :

w.punif# 40 mat 40 mat 40 mat 40 mat 41 mat

w.pvar# 40 matrix q.1# q.2# ≡ q.1# x1# q.2# x2# 41 mat q.1# 0 ≡ (q.2# 0 ≡ (| q.1# x1# q.2# x2# 41 mat cases :

w.pvar# 40 mat 40 mat 40 mat 40 mat 41 mat

q.1## q.1# : 10000

q.2## q.2# : 0

w.pvar## 40 matrix q.1## q.2## ≡ q.1## x1# q.2## x2# 41 mat q.1## q.1## > q.1## x1# q.2## x2# 41 mat cases :

ODD BEHAVIOUR WITH CASES

w.pvar## 40 mat 40 mat 40 mat 40 mat 41 mat

t 1 time : Initialize the variable for the element matrix E# 0 : Initalize the variable for the nodes matrix N# 0 : build EI(x) and create nodes at beam ends beams# model t.beam 1 findrows : Initilize the 'preliminary stiffness of beam segments matrix EI is given an initial value so that it can be used in a 'counter loop' for beams EI# 0 : A nodes matrix, containing nothing is created. We use '0' for the row becase we do not want anthing in the first element other wise the stack function will put a '0' as the first element, which is not correct Nodes will also be used in a 'counter' and therefore we are initilizing it the same as we do for the other variables 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 / + : Below EI is in the equation so that it can be added for each iteration For example if 2 beams are stacked on top of eachother in the modeller add EI EI# EI# beams# j 5 el E## I## * F.0# L.0# 2^* / (* x.1# x# ≤ (x# x.2# ≤ (& (* + : nodes nodes x.1# x.2# 21 mat stack eval : 61 line for x# EI# EI# : build q(x) and create nodes at distributed load ends initialize q# as 0 so that it can be use in a 'counter' for loads Stacked on top of each other 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 again are inculded in the equation as a 'counter' to add previous nodes and the new nodes for the distributed loads nodes nodes x.1# x.2# 21 mat stack eval : 61 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## tol# l.0 / : nsort nodes sort : Initialize k:1 k 1 : Create and initilize the nodal matrix N# 11 el nsort 1 el eval : N# 17 el 0 : j 2 nsort rows range We don't use for statement with 'k' becuase we don't know how many values need to be adjusted to meet the tolerance criteria. So we initilized k above N# k 1 el nsort j el - abs tol## > if the nodes are greater than tolerance then assign them to the N# (nodal matrix) k k 1 + : N# k 1 el nsort j el eval : 31 line If they are not, then do nothing. Note that the N# matrix will not be assigned zero Rather nothing happens. i.e. there is no node added to N# becasue we are saying that the values are so close that this is really the same node, so don't add another node 041 line if 31 line for Create a function which determines if the input x-value is at a node (wihin tolerance) then return that node number x# n.N# node number for a given x value initilize a variable for a counter (use nn) nn 0 : j 1 N# rows range Check if the input x is close enough (within tolerance) to node j x# N# j 1 el - abs tol## < if x is within tolerance then set it equal to that node number nn j : 21 line 0 if 21 line for nn 51 line : create elements. Mitigate zero stiffness to a low value Initilize an E# matrix E# 05 matrix : j 1 N# rows 1 - range EI of the jth element Send the midpoint of the element to the EI# function EI.e# N# j 1 el N# j 1 + 1 el + 2 / EI# 0.0000001 E## I## * F.0 l.0 2^* / * 12 mat max eval : Send element points to the q function to evaluate loads at each end of the element (noting the tolerance) q.I# N# j 1 el tol## + q# : q.J# N# j 1 + 1 el tol## - (q# : Create the element with its start number, end number, Stiffness (EI) and its distributed load value at each end E# E# j j 1 + EI.e# q.I# q.J# 15 mat stack eval : 101 line for add nodal items: template number, N column (position in N matirx), value note that t.cw is on twice to account for a displaced (vertically) support 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 : Fill in item (values) by defining below k 1 items rows range Build a matrix called 'tmp' which contains nodal items defined above tmp model items k 1 el 1 findrows : tmp 0 ≠ 1 continue if j 1 tmp rows range Set item equal to the 'j' row of the tmp matrix item tmp j row : Send the 'X' position of the item to the n.N# function, which determines which node (i.e. which row) of the nodal matrix that point load belongs to n# item 2 el n.N# : The 2nd column of the items matrix defines which column the 'value' (meaning the nodal force (i.e. point load) should be placed in the Nodal matrix N# pos# items k 2 el : Set value of nodal force which will be placed in the nodal matrix value# items k 3 el : Fill in the nodal matrix with location in row 1, and value of nodal forces in other rows N# n# pos# el value# eval : 131 line for 41 line for add hinges hinges model t.hinge 1 findrows : hinges 0 ≠ j 1 hinges rows range Send the x value for the hinge to the n.N# equation to find which node number the hinge is at. n# hinges j 2 el n.N# : Create a tmp matrix which the 1st colum is the number of the element and the following rows contain the element information tmp 1 E# rows range E# augment : define 'er' -> search for the node number on the left side of the element then return the element number er tmp n# 2 findrows 1 el eval : add new node at the same location note -> stack this node at teh bottom, must fill in with zeros to match vector size N# N# hinges j 2 el 00000017 mat stack : assign the right side of this zero length element Note that the rows of N# will increase through each loop so use rows(N#) n.h# N# rows : add hinge element (type is recognized by zero lenght in FEA) E# E# n# n.h# E## I## * F.0 l.0 2^* / 0015 mat stack eval : modify element to the right E# er 1 el n.h# : 191 line for 0 if 1 time t - 521 line

Konsistenz der Maßeinheiten (alle Quotienten müssen dimensionslos sein)

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

q.0 l.0 * F.0 / 1

w.0 l.0 / 0.0833

EI F.0 l.0 2^* / 2.4198 EI s 2^* kg m 3^* / *

E# 12 8.4583 1010^* 181823 8.4583 1010^* 181874 8.4583 1010^* 181885 8.4583 1010^* 181856 8.4583 1010^* 181837 8.0556 109^* 0048 8.0556 109^* 0075 mat

N# 100 - 00100040 - 00100025 - 0000000000000100010005000100025 - 000000000000087 mat