Simulation of steel truss design

Henki W. Ashadi, Universitas Indonesia

henki.ashadi@gmail.com,henki@eng.ui.ac.id

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8.4 μs

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5.7 μs

In order to design a structure, we need to analyse the structure and evaluating the internal forces at each structural members and its deflection produce by design loads.

Several technical decision should be made to fullfill the good practice of design are :

Selecting the efficient, economical structural form
Evaluating the strength, stiffness, and deflections to achieve a serviceable structure
Simulate many loading case, different systems, and steel section will develop a sense of how the structural behave, to prevent the eccessive deflection and impaired its function.

We use the stiffness method to solve the equation of equilibrium. We write the equilibrium equation in term of unknown joint displacement and stiffness coefficient (forces produced by unit displacement). Once the joint displacement is known, the internal forces of the structure can be calculated using the force-displacement relationship.

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12.4 μs

Equilibrium equations of the system : $k_{f f} \cdot Δ = R_{f}$

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6.7 μs

Constructing the element global stiffness matrix

The element stiffness matrix in global coordinate system can be define as

$K_{g l o b a l} = [\begin{matrix} \cos θ & 0 \\ \sin θ & 0 \\ 0 & \cos θ \\ 0 & \sin θ \end{matrix}] \frac{E A}{l} [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}] [\begin{matrix} \cos θ & \sin θ & 0 & 0 \\ 0 & 0 & \cos θ & \sin θ \end{matrix}]$

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7.6 μs

Example of stiffness matrix element 7 :

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3.5 μs

UndefVarError: kg4 not defined

top-level scope@Local: 1

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kg4

---

The stiffness matrix of the structure system is :

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3.5 μs

UndefVarError: ksa not defined

top-level scope@Local: 1

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ksa

---

After swapping the zeros displacement DOF we get the following stiffness matrix

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5.8 μs

UndefVarError: kff not defined

top-level scope@Local: 1

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kff

---

We define the nodal load $R_{f}$ according to the govern building code

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5.3 μs

Union{Missing, Float64}1

missing

0.0

missing

0.0

missing

0.0

missing

0.0

missing

0.0

0.0

missing

missing

missing

missing

missing

missing

missing

missing

missing

missing

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9.3 μs

Solving the equilibrium equation using $d = k_{f f} ∖ R_{f}$ , we get the displacement for each degree of freedom as follow :

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3.3 μs

UndefVarError: u not defined

top-level scope@Local: 1

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u

---

Internal Force can be define using the following formula

$N = \frac{E A}{l} [\begin{matrix} - \cos θ & - \sin θ & \cos θ & \sin θ \end{matrix}] {\begin{matrix} u_{1} \\ v_{1} \\ u_{2} \\ v_{2} \end{matrix}}$

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9.1 μs

Node coordinate of the truss :

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3.6 μs

TypeError: non-boolean (Missing) used in boolean context

fh@Local: 4[inlined]
_broadcast_getindex_evalf@broadcast.jl:648[inlined]
_broadcast_getindex@broadcast.jl:621[inlined]
getindex@broadcast.jl:575[inlined]
macro expansion@broadcast.jl:984[inlined]
macro expansion@simdloop.jl:77[inlined]
copyto!@broadcast.jl:983[inlined]
copyto!@broadcast.jl:936[inlined]
copy@broadcast.jl:908[inlined]
materialize(::Base.Broadcast.Broadcasted{Base.Broadcast.DefaultArrayStyle{1}, Nothing, typeof(Main.workspace48.fh), Tuple{Vector{Missing}, Base.RefValue{Missing}, Base.RefValue{Missing}}})@broadcast.jl:883
top-level scope@Local: 14

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begin
b=6;an=ang*pi/180
function fh(x,an,l)
    if x ≤ l/2
        h=x*tan(an)
    else
        h=(tan(an)*l/2)-((x-l/2)*tan(an))
    end
end
​
yb=zeros(b+1,1)
lx=[i*l/b  for i=0:b]
​
yu=fh.(lx,an,l)
​
xb=collect(1:b+1)
xu=[i for i in (b+2:2b)]
​
nb=[1:b+1 lx yb]
nu=[ b+2:2b lx[2:b] yu[2:b]]
nid=vcat(nb,nu)
end

---

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4.4 s

Setup maximum deflection of the structure $Δ / 240$
Using A36 steel : f_y = 240 N/mm2

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7.7 μs

Δmax

missing

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Δmax= l/240

100 ns

The simulation will be done using roof angle, length of the structure, area of the steel section members. Also the load at the bottom cords, top cords may applied to the truss systems.

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3.3 μs

UndefVarError: nid not defined

top-level scope@Local: 4

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# Another Plots
begin
    fac=1
    xn=nid[1:12,2].+fac.*uxy[1:12,1];
    yn=nid[1:12,3].+fac.*uxy[1:12,2];
    plot(size=(750,300),nid[1:7,2],nid[1:7,3],color=:purple,lw=2,legend=false,ylims=(-.5,5.1),xlims=(0,25))
    plot!(nid[[1 8 9 10 11 12 7],2]',nid[[1 8 9 10 11 12 7],3]',lw=2,color=:purple)
    plot!(nid[[2 8 3 9 4 10 4 11 5 12 6],2]',nid[[2 8 3 9 4 10 4 11 5 12 6],
            3]',color=:purple,lw=2)
    plot!(xn[1:7],yn[1:7],color=:green,lw=2)
    plot!(xn[[1 8 9 10 11 12 7]]',yn[[1 8 9 10 11 12 7]]',color=:green,lw=2)
    plot!(xn[[2 8 3 9 4 10 4 11 5 12 6]]',yn[[2 8 3 9 4 10 4 11 5 12 6]]',color=:green,lw=2)
    hline!([-Δmax],color=:blue,linestyle=:dash)
end

---

Angle = missing $^{\circ}$ , Length = missing m, Area = missing m2, Pa= missing kN, Pb= missing kN, Ph= missing kN

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12.2 μs

Angle : Length : ; Section area :

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12.8 ms

Pby : kN; Pty : kN ; Pth : kN

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80.7 μs

Node Displ-x, m Displ-y, m Internal Force, kN Stress F/a, N/mm2

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4.3 μs

UndefVarError: nid not defined

top-level scope@Local: 31

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begin
e=200e6
#a=700e-6
ea=e*a
    # nid, x, y
#nid=[1 0 0 ; 2 5 0 ; 3 10 0 ; 4 15 0 ; 5 20 0 ; 6 25 0 ; 7 30 0 ;
#     8 5 2.88 ; 9 10 5.77 ; 10 15 8.66 ; 11 20 5.77 ; 12 25 2.88 ]
​
# eid, i, j
#eid = [1 1 2 ;2 2 3 ;3 3 4 ;4 4 5 ;5 5 6 ; 6 6 7;
#      7 1 8 ;8 8 9 ;9 9 10 ;10 10 11 ;11 11 12 ;
#     12 7 12 ;13 2 8 ;14 3 8 ;15 3 9 ;16 4 9 ;
#     17 4 10 ;18 4 11 ;19 5 11 ;20 5 12 ;21 6 12 ]
​
# A plot
# using Plots
# eld=[0 0 ; 5 0 ;10 0 ;15 0;20 0;25 0;30 0;25 2.88;20 5.77;15 8.66;10 5.77;5 2.88;0 0 ; NaN NaN ; 5 0 ; 5 2.88; 10 0; 10 5.77; 15 0; 20 5.77;20 0;25 2.88;25 0 ; NaN NaN ; 15 0 ; 15 8.66]
# scatter(nid[:,2],nid[:,3], legend=false,size=(800,200))
# plot!(eld[:,1],eld[:,2])
​
# Another Plots
# plot(size=(600,200),nid[1:7,2],nid[1:7,3],color=:blue,legend=false)
# plot!(nid[[1 8 9 10 11 12 7],2]',nid[[1 8 9 10 11 12 7],3]',color=:blue)
# plot!(nid[[2 8 3 9 4 10 4 11 5 12 6],2]',nid[[2 8 3 9 4 10 4 11 5 12 6], 3]',color=:blue)
​
#fungsi 
​
​
​
# elemen 1 : node 1 to node 2
e1=fe(nid[1,2],nid[1,3],nid[2,2],nid[2,3])
# matrix elemen local
kg1 =fkg2(ea,e1)
# elemen 2 : node 2 to node 3
e2=fe(nid[2,2],nid[2,3],nid[3,2],nid[3,3])
kg2 =fkg2(ea,e2)
# elemen 3 : node 3 to node 4
e3=fe(nid[3,2],nid[3,3],nid[4,2],nid[4,3])
kg3 =fkg2(ea,e3)
# elemen 4 : node 4 to node 5
e4=fe(nid[4,2],nid[4,3],nid[5,2],nid[5,3])
kg4 =fkg2(ea,e4)
# elemen 5 : node 5 to node 6
e5=fe(nid[5,2],nid[5,3],nid[6,2],nid[6,3])
kg5 =fkg2(ea,e5)
# elemen 6 : node 6 to node 7
e6=fe(nid[6,2],nid[6,3],nid[7,2],nid[7,3])
kg6 =fkg2(ea,e6)
​
# elemen 7 : node 1 to node 8
e7=fe(nid[1,2],nid[1,3],nid[8,2],nid[8,3])
kg7 =fkg2(ea,e7)
# elemen 8 : node 8 to node 9
e8=fe(nid[8,2],nid[8,3],nid[9,2],nid[9,3])
kg8 =fkg2(ea,e8)
# elemen 9 : node 9 to node 10
e9=fe(nid[9,2],nid[9,3],nid[10,2],nid[10,3])
kg9 =fkg2(ea,e9)
# elemen 10 : node 10 to node 11
e10=fe(nid[10,2],nid[10,3],nid[11,2],nid[11,3])
kg10 =fkg2(ea,e10)
# elemen 11 : node 11 to node 12
e11= fe(nid[11,2],nid[11,3],nid[12,2],nid[12,3])
kg11 =fkg2(ea,e11)
# elemen 12 : node 7 to node 12
e12= fe(nid[7,2],nid[7,3],nid[12,2],nid[12,3])
kg12 =fkg2(ea,e12)
# Batang pengisi
​
# elemen 13 : node 2 to node 8
e13= fe(nid[2,2],nid[2,3],nid[8,2],nid[8,3])
kg13 =fkg2(ea,e13)
# elemen 14 : node 3 to node 8
e14= fe(nid[3,2],nid[3,3],nid[8,2],nid[8,3])
kg14 =fkg2(ea,e14)
# elemen 15 : node 3 to node 9
e15= fe(nid[3,2],nid[3,3],nid[9,2],nid[9,3])
kg15 =fkg2(ea,e15)
# elemen 16 : node 4 to node 9
e16= fe(nid[4,2],nid[4,3],nid[9,2],nid[9,3])
kg16 =fkg2(ea,e16)
# elemen 17 : node 4 to node 10
e17= fe(nid[4,2],nid[4,3],nid[10,2],nid[10,3])
kg17 =fkg2(ea,e17)
# elemen 18 : node 4 to node 11
e18= fe(nid[4,2],nid[4,3],nid[11,2],nid[11,3])
kg18 =fkg2(ea,e18)
# elemen 19 : node 5 to node 11
e19= fe(nid[5,2],nid[5,3],nid[11,2],nid[11,3])
kg19 =fkg2(ea,e19)
# elemen 20 : node 5 to node 12
e20=fe(nid[5,2],nid[5,3],nid[12,2],nid[12,3])
kg20 =fkg2(ea,e20)
# elemen 21 : node 6 to node 12
e21= fe(nid[6,2],nid[6,3],nid[12,2],nid[12,3])
kg21 =fkg2(ea,e21)
​
elid=[e1;e2;e3;e4;e5;e6;e7;e8;e9;e10;e11;e12;e13;e14;e15;e16;e17;e18;e19;e20;e21]
​
# matrix elemen global : u1,v1,u2,v2,u3,v3,u4,v4,u5,v5,u6,v6,u7,v7,u8,v8,u9,v9,u10,v10,u11,v11,u12,v12
ks = zeros(24,24)
​
fk(ks,kg1,1,2);fk(ks,kg2,2,3);fk(ks,kg3,3,4)
fk(ks,kg4,4,5);fk(ks,kg5,5,6);fk(ks,kg6,6,7)
fk(ks,kg7,1,8);fk(ks,kg8,8,9);fk(ks,kg9,9,10)
fk(ks,kg10,10,11);fk(ks,kg11,11,12);fk(ks,kg12,7,12)
fk(ks,kg13,2,8);fk(ks,kg14,3,8);fk(ks,kg15,3,9);
fk(ks,kg16,4,9);fk(ks,kg17,4,10);fk(ks,kg18,4,11);
fk(ks,kg19,5,11);fk(ks,kg20,5,12);fk(ks,kg21,6,12)
​
ksa=copy(ks)
​
swaprowcol(ksa,3,14)
​
kff=ksa[4:24,4:24]
​
#displacement
d=kff\Rf
​
u=[0;0;d[:11];d[:1];d[:2];d[:3];d[:4];d[:5];d[:6];d[:7];d[:8];d[:9];
 d[:10];0;d[:12];d[:13];d[:14];d[:15];d[:16];d[:17];d[:18];d[:19];d[:20];d[:21]]
​
# uxy=reshape(u,2,12)'
​
N1= fN1(ea,e1, u[:1] ,u[:2] ,u[:3] ,u[:4])
N2= fN1(ea,e2, u[:3] ,u[:4] ,u[:5] ,u[:6])
N3= fN1(ea,e3, u[:5] ,u[:6] ,u[:7] ,u[:8])
N4= fN1(ea,e4, u[:7] ,u[:8] ,u[:9] ,u[:10])
N5= fN1(ea,e5, u[:9] ,u[:10],u[:11],u[:12])
N6= fN1(ea,e6, u[:11],u[:12],u[:13],u[:14])
N7= fN1(ea,e7, u[:1] ,u[:2] ,u[:15],u[:16])
N8= fN1(ea,e8, u[:15],u[:16],u[:17],u[:18])
N9= fN1(ea,e9, u[:17],u[:18],u[:19],u[:20])
N10=fN1(ea,e10,u[:19],u[:20],u[:21],u[:22])
N11=fN1(ea,e11,u[:21],u[:22],u[:23],u[:24])
N12=fN1(ea,e12,u[:23],u[:24],u[:13],u[:14])
N13=fN1(ea,e13,u[ :3],u[:4] ,u[:15],u[:16])
N14=fN1(ea,e14,u[:15],u[:16],u[:5] ,u[:6])
N15=fN1(ea,e15,u[:5] ,u[:6] ,u[:17],u[:18])
N16=fN1(ea,e16,u[:17],u[:18],u[:7] ,u[:8])
N17=fN1(ea,e17,u[:7] ,u[:8] ,u[:19],u[:20])
N18=fN1(ea,e18,u[:7] ,u[:8] ,u[:21],u[:22])
N19=fN1(ea,e19,u[:9] ,u[:10],u[:21],u[:22])
N20=fN1(ea,e20,u[:9] ,u[:10],u[:23],u[:24])
N21=fN1(ea,e21,u[:11],u[:12],u[:23],u[:24])
​
#GAYA DALAM
N=[N1; N2; N3; N4; N5; N6; N7; N8; N9; N10; N11; N12; N13; N14; N15;N16;N17; N18; N19; N20; N21];
R=[N;0;0;0]
rxy= reshape(R,2,12)'
uxy= reshape(u,2,12)'
str= rxy ./(a .* 1000)
[1:12 uxy rxy str]
end

---

fN11 (generic function with 1 method)

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186 μs