# Discrete Optimization with Fuzzy Constraints

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structural Optimization and Fuzzy Set Theory

**x**such that the objective function F(

**x**) is minimized subject to the equality and inequality constraints:

**x**) can be obtained from a fuzzy decision D, such that the membership function ${\mu}_{D}$ for the fuzzy decision D can be obtained from the intersection of the fuzzy membership functions for the objective function and constraints; see Equation (3):

**x**can be obtained by using the max-min procedure [13]; see Equation (4):

#### 2.1. ANFIS Architecture for the Development of Soft Constraint Functions

_{1}and y is B

_{1}, then:

_{i}and y is B

_{i}, then:

_{n}and y is B

_{n}, then:

_{i}, B

_{i}, C

_{i}, D

_{i}) is calculated with the following equations:

_{i}represents the fire strength of the rule i. The ratio of the i-th rule’s firing strength to the sum of all of the rule’s firing strengths is defined with:

## 3. Discrete Optimization

_{d}is the number of discrete variables and p

_{i}is the pre-established set of discrete values. If either n

_{d}or p

_{i}(or both) are large, it shows that much work will be required. It also shows an exponential growth in the calculations with the number of discrete variables. In a mixed optimization problem, this would involve the continuous optimum solution of a reduced model. It is not a serious problem if the mathematical model and its computer calculations are easy to implement. If the mathematical model requires extensive calculations, then some concerns may arise. Programming exhaustive enumeration is straightforward. The processing speed, large available desktop-memory and easy programming, through software like MATLAB [16], make exhaustive enumeration a very good idea today. This program is also ideal because the solution is a global optimum. The most important step is translating the mathematical model into a program code.

Step 1. ${s}^{*}=inf,\text{}K=\left[0,\text{}0,\text{}\dots ,0\right]$ |

For every allowable combination of $\left({y}_{1},\text{}{y}_{2},\dots ,{y}_{nd}\right)\Rightarrow \left({Y}_{b}\right)$ |

Solve optimization problem (solution K)^{*} |

$If\text{}h\left({K}^{*},{Y}_{b}\right)=\text{}\left[0\right]\text{}and$ |

$If\text{}g\left({K}^{*},{Y}_{b}\right)\le \text{}\left[0\right]\text{}and$ |

$If\text{}f\left({K}^{*},{Y}_{b}\right)\text{}{s}^{*}$ |

$Then\text{}{s}^{*}\leftarrow f\left({K}^{*},{Y}_{b}\right)$ |

$K\leftarrow {K}^{*}$ |

$Y\leftarrow {Y}_{b}$ |

$End\text{}If$ |

$End\text{}If$ |

$End\text{}If$ |

$End\text{}For$ |

## 4. Example Design of a Simply-Supported Laterally-Restrained Beam Application

- (1)
- resistance of the cross-section to bending (ULS),
- (2)
- resistance to shear buckling (ULS),
- (3)
- resistance to flange-induced buckling (ULS),
- (4)
- resistance of the web to transverse forces (ULS) and
- (5)
- deflection (SLS).

_{k}and a uniformly-distributed imposed load q

_{k}, as shown in Figure 1.

#### 4.1. Design Loads

#### 4.2. Resistance of Steel Cross-Sections

#### 4.2.1. Bending Moment

_{Ed}, at each section should satisfy the following:

_{c,Rd}is the design resistance for bending around one principal axis, taken as follows:

_{pl}is the plastic section modulus, for Class 1 and 2 sections only;

_{el,min}is the minimum elastic section modulus for Class 3 sections;

_{eff,min}is the minimum effective section modulus for Class 4 cross-sections only;

_{u,Rd}, if this is less than the appropriate values above. In calculating this value, fastener holes in the compression zone do not need to be considered; they would need to be if they were oversized, slotted or filled by fasteners. In the tension zone, holes do not need to be considered, provided that:

#### 4.2.2. Shear

_{Ed}at each cross-section should satisfy the following:

_{c,Rd}is the design shear resistance. For the plastic design, V

_{c,Rd}is taken as the design plastic shear resistance, V

_{pl,Rd}, given by:

_{v}is the shear area, which, for the rolled I and H sections, loaded parallel to the web, is:

- b overall breadth
- r root radius
- t
_{f}flange thickness - t
_{w}web thickness - h
_{w}depth of the web - η conservatively taken as 1.0
- A cross-sectional area

_{τ}is the buckling factor for shear and is given by:

#### 4.2.3. Resistance of Cross-Section-Bending and Shear

_{Ed}, exceeds 50 percent of the plastic shear design resistance, V

_{pl,Rd}, the design resistance moment of the section, M

_{v,Rd}, should be calculated using a reduced yield strength taken as:

_{y,v,Rd}, will be given by:

#### 4.2.4. Shear Buckling Resistance

_{w}/t

_{w}is usually less than 72ε, it was not discussed in this section.

#### 4.2.5. Flange-Induced Buckling

_{w}/t

_{w}of the web should satisfy the following criterion:

- A
_{w}is the area of the web = (h−2∙t_{f})∙t_{w} - A
_{fc}is the area of the compression flange = b∙t_{f} - f
_{yf}is the yield strength of the compression flange

#### 4.2.6. Resistance of the Web to Transverse Forces

- (a)
- forces resisted by shear in the web (loading Types (a) and (c)).
- (b)
- forces transferred through the web directly to the other flange (loading Type (b)).

- (i)
- crushing of the web close to the flange accompanied by yielding of the flange; the combined effect is sometimes referred to as web crushing
- (ii)
- localized buckling and crushing of the web beneath the flange; the combined effect is sometimes referred to as web crippling.

- (i)
- web crushing
- (ii)
- buckling of the web over most of the depth of the member.

- f
_{yw}is the yield strength of the web - t
_{w}is the thickness of the web - γ
_{M1}is the partial safety factor = 1.0 - L
_{eff}is the effective length of the web that resists transverse forces = χ_{F}l_{y}, in which χ_{F}is the reduction factor due to local buckling. - l
_{y}is the effective loaded length, appropriate to the length of the stiff bearing s_{s}. As stated in Clause 6.3 of Eurocode 3–5 [17], s_{s}should be taken as the distance over which the applied load is effectively distributed at a slope of 1:1, but s_{s}≤ h_{w}.

_{F}: The reduction factor χ

_{F}is given by:

_{y}: As stated in Clause 6.5 [17] for loading Types (a) and (b), the effective load length, l

_{y}, is given by:

_{y}is taken as the smallest value obtained from Equations (52) and (53), as follows:

#### 4.3. Deflections

#### 4.3.1. ANFIS for the Development of the Constraint Function

^{2}) and classification CLASS (-); and it has one output: LIMIT. The ANFIS-LIMIT model was proposed in order to calculate the deflection limits. MATLAB [16] and a Fuzzy Logic Toolbox were used as an interface for mathematical modeling and data handling.

_{1}and CLASS is B

_{1}, then:

_{2}and CLASS is B

_{2}, then:

- the membership grade of the fuzzy set (A
_{i}, B_{i}, C_{i}, D_{i}) is calculated; - the product of membership function for each rule is calculated;
- the ratio between the i-th rule’s firing strength and the sum of all rules’ firing strengths is calculated;
- the output of each rule is calculated; and
- the weighted average of each rule’s output is calculated.

_{i}, B

_{i}, C

_{i}, D

_{i}) is calculated with Equations (57) and (58):

_{Ai}, c

_{Bi}, σ

_{Ai}, σ

_{Bi}are premise parameters. In addition to this, the products between the membership functions for every rule are calculated; see Equations (59) and (60):

_{1}and w

_{2}represent the firing strength of the each rule. The weighted average of each rules’ output is defined as the ratio between the i-th rule’s firing strength and the sum of all of the rule’s firing strengths; see Equation (61):

^{2}) and the classification CLASS (-), as well as a single output deflection limit LIMIT (-).

## 5. Fuzzy Optimization Model: Beam Implementation

_{k}(kN/m) and imposed q

_{k}(kN/m) loads, the yield strength of the steel f

_{y}(MPa), the modulus of elasticity E (MPa), the density of the steel gam (kg/m

^{3}), bearings width ss (mm) and the allowable deflection of the beam lim (-). In addition to this, the data also include the coefficients involved in the design inequality constraints: safety factor for dead loads SFg (-), safety factor for imposed loads SFq (-), partial factor for resistance of cross-sections SFm0, partial factor for resistance of members to instability SFm1, modification factor k (-), non-dimensional slenderness lamflim (-) and the reduction factor for the relevant buckling curve ksiflim (-).

- Condition 1, resistance of the cross-section to bending (ULS): verified by Equation (28), by which the design bending moment M
_{Ed}(kNm) must not exceed the bending moment resistance M_{Rd}(kNm). - Condition 2, resistance of the cross-section to shear (ULS): verified by Equation (33), by which the design shear force V
_{Ed}(kN) must not exceed the shear resistance V_{Rd}(kNm). - Condition 3, deflection (SLS) is considered: the calculated vertical deflection of the steel beam must be less than specified by the ANFIS-LIMIT model.
- Condition 4, resistance to flange-induced buckling (ULS): to prevent the possibility of the compression flange buckling in the plane of the web.
- Condition 5, Condition 6, Condition 7 and Condition 8, resistance of the web to transverse forces (ULS): to prevent the possibility of the local buckling of webs.

_{k}= 5 kN/m and q

_{k}= 15 kN/m, as shown below. Steel was chosen from the material of the beam. The modulus of elasticity of the steel was E = 210 GPa. The weight of the steel was ρ = 7850 kg/m

^{3}. The yield strength in tension was f

_{y}= 335 MPa. The specified applied load LL was 3 kN/m

^{2}and used a classification of 1. A safety factor of 1.35 on the permanent load and 1.50 on the variable load were assumed. The results of an exhaustive enumeration computer code showed that the optimal section is HE 1000 × 393, with a weight of 9816.4 kg.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A

% Optimization of Fully Laterally Restrained Beams with soft constrain | |

% Simply supported steel beam | |

%-------------------------------------------------- | |

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |

% Discrete Optimization with soft constrain | |

% Dr. P. Jelusic | |

% See Text for Problem description | |

% The Beam Properties are loaded from the file | |

% BeamPropertiesEU.m | |

%*************************************************** | |

%%% | |

clear | |

clear global | |

clc | |

close | |

format compact | |

warning off | |

%%% Run File %%%%%%%%% | |

BeamPropertiesEU | |

%%%%%%%%%%%%%%%%%%%%%%%% | |

%%% monitor cpu time | |

starttime = cputime; | |

fprintf('\n********************************') | |

fprintf('\nSimply supported steel beam (Enumeration)') | |

fprintf | |

% %****************************** | |

% % Computer Code | |

% %******************************* | |

%Span, safety factors and loads | |

L = 25; | % span (m) |

SFg = 1.35; | % safety factor for dead load (-) |

SFq = 1.50; | % safety factor for imposed load (-) |

gk = 5; | % dead load(kN/m) |

qk = 15; | % imposed load(kN/m) |

ss = 100; | % bearings width (mm) |

eta = 1; | % shear factor eta (-) |

k = 0.3; | % factor k (-) |

lamflim = 0.5; | % factor lamflim (-) |

ksiflim = 1; | % factor ksiflim (-) |

CLASS = 1; | % use classification (-) |

LL = 3; | % Applied live load (kN/m^{2}) |

%ANFIS model coefficients | |

sigA1 = 6.61768494044089; | |

sigA2 = 7.47150990794045; | |

sigB1 = 0.962309787332703; | |

sigB2 = 0.979723951565027; | |

cA1 = 2.90117735987428; | |

cA2 = 2.08316049335067; | |

cB1 = 1.62121248482762; | |

cB2 = 1.27590144219536; | |

a01 = -771.211045670957; | |

a11 = 8.40911494744558; | |

a21 = 172.024353054372; | |

a02 = 1535.72815330126; | |

a12 = -29.7774593495827; | |

a22 = -158.884574516486; | |

%Material properties | |

fy = 355; | % yield strength (MPa) |

E = 210000; | % modulus of elasticity (MPa) |

SFmo = 1.00; | % safety factor for material bending (-) |

SFm1 = 1.05; | % safety factor for elastic resistance deflection (-) |

gam = 7850; | % density (kg/m^{3}) |

%%%Start Exhaustive Enumeration : | |

fstar = inf; | |

xstar = [inf inf inf inf]; | |

gstar = [inf inf inf]; | |

istar = 1; | |

% % | |

% | |

fprintf('\n----------------------------') | |

fprintf('\nFeasible Beams') | |

fprintf('\n-----------------------------\n\n') | |

for i = 1:length(RolledSteelBeamSI) | |

x1 = RolledSteelBeamSI(i).D; | |

x2 = RolledSteelBeamSI(i).B; | |

x3 = RolledSteelBeamSI(i).tw; | |

x4 = RolledSteelBeamSI(i).tf; | |

%******************************* | |

A = RolledSteelBeamSI(i).Area; | |

D = RolledSteelBeamSI(i).D; | |

B = RolledSteelBeamSI(i).B; | |

tw = RolledSteelBeamSI(i).tw; | |

tf = RolledSteelBeamSI(i).tf; | |

Rr = RolledSteelBeamSI(i).Rr; | |

dd = RolledSteelBeamSI(i).dd; | |

Ix = RolledSteelBeamSI(i).Ix; | |

Welx = RolledSteelBeamSI(i).Welx; | |

Wplx = RolledSteelBeamSI(i).Wplx; | |

%Design action. The reason for discrete optimization is to choose off-the-shelf I-beam which will keep the cost and production time down. Several mills provide information on standard rolling stock they manufacture. | |

Fed = (SFg*(gk+A*gam*9.81/10000000) + SFq*qk)*L; | %design action (kN) |

Med = Fed*L/8; | %design bending moment (kNm) |

Ved = Fed/2; | %design shear force (kN) |

%Section resistance | |

Mrd = Wplx*fy/(SFmo*1000); | %moment resistance (kNm) |

Av = A*100-2*B*tf + (tw + 2*Rr)*tf; | %shear area(mm^{2}) |

Vrd = Av*(fy/(3)^0.5)/(SFmo*1000); | %design shear resistance(kN) |

%Deflection | |

Mmax = (gk+qk)*L^2/8; | %maximum bending moment due to working load (kNm) |

Mcrd = Welx*fy/(SFm1*1000); | %elastic resistance (kNm) |

u = 5*qk*(L*1000)^4/(384*E*Ix*10000); | %deflection (mm) |

%ANFIS calculation procedure | |

A1ev = exp(-0.5*(((LL-cA1)/(sigA1))^2)); | |

A2ev = exp(-0.5*(((LL-cA2)/(sigA2))^2)); | |

B1ev = exp(-0.5*(((CLASS-cB1)/(sigB1))^2)); | |

B2ev = exp(-0.5*(((CLASS-cB2)/(sigB2))^2)); | |

w1 = A1ev*B1ev; | |

w2 = A2ev*B2ev; | |

w1n = w1/(w1+w2); | |

w2n = w2/(w1+w2); | |

fun1 = a01+a11*LL+a21*CLASS; | |

fun2 = a02+a12*LL+a22*CLASS; | |

nfun1 = w1n*fun1; | |

nfun2 = w2n*fun2; | |

lim = nfun1+nfun2; | |

uult = L*1000/lim; | %permissible deflection (mm) |

%Section classification | |

eps = (235/fy)^0.5; | %factor eps (-) |

c = (B-tw-2*Rr)/2; | %depth between fillets (mm) |

hw = D-2*tf; | %depth between flanges (mm) |

%Flange-induced buckling | |

Aw = (D-2*tf)*tw; | %area of the web (mm^{2}) |

Afc = B*tf; | %area of the compression flange (mm^{2}) |

Fib = hw/tw; | %criteria ratio of flange-induced buckling(-) |

Fibalw = k*(E/fy)*(Aw/Afc)^0.5; | %criteria ratio (-) |

%Web buckling | |

kf = 2+6*(ss/hw); | %buckling coefficient (-) |

kfalw = 6; | %limit of buckling coefficient(-) |

Fcr = (0.9*kf*E*tw^3)/hw; | %elastic critical buckling load(N) |

m1 = fy*B/(fy*tw); | %coefficient m1(-) |

m2 = 0.02*(hw/tf)^2; | %coefficient m2(-) |

le = min(kf*E*tw^2/(2*fy*hw),ss); | %effective loaded length(mm) |

ly = min(le+tf*(m1/2+(le/tf)^2+m2)^0.5,le + tf*(m1+m2)^0.5); | %(mm) |

lamf = (ly*tw*fy/Fcr)^0.5; | %reduction factor lamf (-) |

lamflim = 0.5; | %permissible reduction factor lamflim(-) |

ksif = 0.5/lamf; | %reduction factor ksif(-) |

leff = ksif*ly; | %effective length of web(mm) |

Frdweb = fy*leff*tw/1000; | %design resistance of web(kN) |

%Objective function | |

f = gam*L*A/10000; | %weight of steel beam (kg) |

%Constraints | |

g1 = Med - Mrd; | %bending (kNm) |

g2 = Ved - Vrd; | %shear (kN) |

g3 = u - uult; | %deflection(mm) |

g4 = Fib - Fibalw; | %flange-induced buckling (-) |

g5 = kf - kfalw; | %web buckling constraint 1 (-) |

g6 = lamflim - lamf; | %web buckling constraint 2 (-) |

g7 = ksif - 1; | %web buckling constraint 3(-) |

g8 = Ved - Frdweb; | %resistance of web constraint (kN) |

%%% total constraint vector | |

G = [g1 g2 g3 g4 g5 g6 g7 g8]; | |

if (g1 <= 0) & (g2 <= 0) & (g3 <= 0) | |

if (g4 <= 0) & (g5 <= 0) & (g6 <= 0) | |

if (g7 <= 0) & (g8 <= 0) | |

if (f <= fstar) | |

xstar = [x1 x2 x3 x4]; | |

fstar = f | |

Gstar = G; | |

istar = i | |

end | |

end | |

end | |

end | |

end | |

fprintf('\n******************************************') | |

fprintf('\nOptimum Fully Laterally Restrained Beam') | |

fprintf('\n******************************************\n\n') | |

fprintf('Rolled Beam Designation : '),disp(RolledSteelBeamSI(istar).Name) | |

fprintf('Depth(mm) Width(mm) Web Thickness(mm) Flange Thickness (mm)\n') | |

fprintf('%8.5f %8.5f %8.5f %8.3f\n',xstar) | |

fprintf('\nObjective Function(kg): '),disp(fstar) | |

fprintf('\nConstraints\n') | |

fprintf('---------------\n') | |

fprintf('Bending Stress Constraint - g1 (kNm): '),disp(Gstar(1)) | |

fprintf('Shear Stress Constraint - g2 (kN): '),disp(Gstar(2)) | |

fprintf('Deflection Constraint - g3 (mm): '),disp(Gstar(3)) | |

fprintf('flange-induced buckling - g4 (-): '),disp(Gstar(4)) | |

fprintf('web buckling constraint 1 - g5 (-): '),disp(Gstar(5)) | |

fprintf('web buckling constraint 2 - g6 (-): '),disp(Gstar(6)) | |

fprintf('web buckling constraint 3 - g7 (-): '),disp(Gstar(7)) | |

fprintf('resistance of web constraint - g8 (kN): '),disp(Gstar(8)) | |

%%% print time | |

totaltime = cputime - starttime; | |

fprintf('\n\nTotal cpu time (s)= %7.4f \n\n',totaltime) |

% EE - Exhaustive Enumeration | |

% For constrained optimization of fully laterally restrained beams | |

% Dr. P. Jelusic | |

% University of Maribor, Faculty of Civil Engineering | |

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |

%%% File UniversalbeamsEU.m | |

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |

% Discrete Variables | |

%-------------------------------------------------- | |

% See Text for Problem description | |

%********************************************** | |

%%% COMPANION FILE FOR PROBLEM Fully laterally restrained beams | |

%%% This file contains Beam Properties for universal beams | |

%%% beams in SI Units | |

%********************************************** | |

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |

%%% Define the section properties | |

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |

RolledSteelBeamSI(1).Name ='IPE AA 80'; | %beam identifier (-) |

RolledSteelBeamSI(1).Area = 6.31; | %area (cm2) |

RolledSteelBeamSI(1).D = 78; | %Depth of section (mm) |

RolledSteelBeamSI(1).B = 46; | %width of section (mm) |

RolledSteelBeamSI(1).tw = 3.2; | %web thickness (mm) |

RolledSteelBeamSI(1).tf = 4.2; | %flange thickness (mm) |

RolledSteelBeamSI(1).Rr = 5; | %root radius (mm) |

RolledSteelBeamSI(1).dd = 59.6; | %depth between fillets(mm) |

RolledSteelBeamSI(1).Ix = 64.1; | %second moment of area Ixx (cm4) |

RolledSteelBeamSI(1).Welx = 16.4; | %elastic modulus Welx (cm3) |

RolledSteelBeamSI(1).Wplx = 18.9; | %plastic modulus Wplx (cm3) |

RolledSteelBeamSI(2) = struct('Name','IPE A 80','Area',6.38, ... | |

'D',78,'B',46,'tw',3.3,'tf',4.2, ... | |

'Rr',5,'dd',59.6,'Ix',64.4, ... | |

'Welx',16.5,'Wplx',19); | |

RolledSteelBeamSI(3) = struct('Name','IPE 80','Area',7.64, ... | |

'D',80,'B',46,'tw',3.8,'tf',5.2, ... | |

'Rr',5,'dd',59.6,'Ix',80.1, ... | |

'Welx',20,'Wplx',23.2); | |

RolledSteelBeamSI(75) = struct('Name','HE 1000 X 584','Area',743.7, ... | |

'D',1056,'B',314,'tw',36,'tf',64, ... | |

'Rr',30,'dd',868,'Ix',1246100, ... | |

'Welx',23600,'Wplx',28039); | |

return; |

****************************************** | |||

Optimum Fully Laterally Restrained Beam | |||

****************************************** | |||

Rolled Beam Designation : HE 1000 X 393 | |||

Depth(mm) | Width(mm) | Web Thickness(mm) | Flange Thickness (mm) |

1016.00000 | 303.00000 | 24.40000 | 43.900 |

Objective Function(kg): | 9.8164e+003 | ||

Constraints | |||

--------------- | |||

Bending Stress Constraint | - g1 (kNm): | -3.8903e+003 | |

Shear Stress Constraint | - g2 (kN): | -5.1282e+003 | |

Deflection Constraint | - g3 (mm): | -12.9339 | |

flange-induced buckling | - g4 (-): | -193.5248 | |

web buckling constraint 1 | - g5 (-): | -3.3536 | |

web buckling constraint 2 | - g6 (-): | -0.0742 | |

web buckling constraint 3 | - g7 (-): | -0.1293 | |

resistance of web constraint | - g8 (kN): | -1.8169e+003 | |

Total cpu time (s)= | 0.2184 |

## Appendix B

Designation | A | h | b | t_{w} | t_{f} | r | d | I_{y} | W_{el.y} | W_{pl.y} | Designation | A | h | b | t_{w} | t_{f} | r | d | I_{y} | W_{el.y} | W_{pl.y} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Serial Size | cm^{2} | mm | mm | mm | mm | mm | mm | cm^{4} | cm^{3} | cm^{3} | Serial Size | cm^{2} | mm | mm | mm | mm | mm | mm | cm^{4} | cm^{3} | cm^{3} |

IPE AA 80 | 6.31 | 78 | 46 | 3.2 | 4.2 | 5 | 59.6 | 64.1 | 16.4 | 18.9 | IPE O 360 | 84.1 | 364 | 172 | 9.2 | 14.7 | 18 | 298.6 | 19,050 | 1047 | 1186 |

IPE A 80 | 6.38 | 78 | 46 | 3.3 | 4.2 | 5 | 59.6 | 64.4 | 16.5 | 19 | IPE A 400 | 73.1 | 397 | 180 | 7 | 12 | 21 | 331 | 20,290 | 1022 | 1144 |

IPE 80 | 7.64 | 80 | 46 | 3.8 | 5.2 | 5 | 59.6 | 80.1 | 20 | 23.2 | IPE 400 | 84.5 | 400 | 180 | 8.6 | 13.5 | 21 | 331 | 23,130 | 1160 | 1307 |

IPE AA 100 | 8.56 | 97.6 | 55 | 3.6 | 4.5 | 7 | 74.6 | 136 | 27.9 | 31.9 | IPE O 400 | 96.4 | 404 | 182 | 9.7 | 15.5 | 21 | 331 | 26,750 | 1324 | 1502 |

IPE A 100 | 8.8 | 98 | 55 | 3.6 | 4.7 | 7 | 74.6 | 141 | 28.8 | 33 | IPE A 450 | 85.6 | 447 | 190 | 7.6 | 13.1 | 21 | 378.8 | 29,760 | 1331 | 1494 |

IPE 100 | 10.3 | 100 | 55 | 4.1 | 5.7 | 7 | 74.6 | 171 | 34.2 | 39.4 | IPE 450 | 98.8 | 450 | 190 | 9.4 | 14.6 | 21 | 378.8 | 33,740 | 1500 | 1702 |

IPE AA 120 | 10.7 | 117 | 64 | 3.8 | 4.8 | 7 | 93.4 | 244 | 41.7 | 47.6 | IPE O 450 | 118 | 456 | 192 | 11 | 17.6 | 21 | 378.8 | 40,920 | 1795 | 2046 |

IPE A 120 | 11 | 117.6 | 64 | 3.8 | 5.1 | 7 | 93.4 | 257 | 43.8 | 49.9 | IPE A 500 | 101 | 497 | 200 | 8.4 | 14.5 | 21 | 426 | 42,930 | 1728 | 1946 |

IPE 120 | 13.2 | 120 | 64 | 4.4 | 6.3 | 7 | 93.4 | 318 | 53 | 60.7 | IPE 500 | 116 | 500 | 200 | 10.2 | 16 | 21 | 426 | 48,200 | 1930 | 2194 |

IPE AA 140 | 12.8 | 136.6 | 73 | 3.8 | 5.2 | 7 | 112.2 | 407 | 59.7 | 67.6 | IPE O 500 | 137 | 506 | 202 | 12 | 19 | 21 | 426 | 57,780 | 2284 | 2613 |

IPE A 140 | 13.4 | 137.4 | 73 | 3.8 | 5.6 | 7 | 112.2 | 435 | 63.3 | 71.6 | IPE A 550 | 117 | 547 | 210 | 9 | 15.7 | 24 | 467.6 | 59,980 | 2193 | 2475 |

IPE 140 | 16.4 | 140 | 73 | 4.7 | 6.9 | 7 | 112.2 | 541 | 77.3 | 88.3 | IPE 550 | 134 | 550 | 210 | 11.1 | 17.2 | 24 | 467.6 | 67,120 | 2440 | 2787 |

IPE AA 160 | 15.4 | 156.4 | 82 | 4 | 5.6 | 7 | 131.2 | 646 | 82.6 | 93.3 | IPE O 550 | 156 | 556 | 212 | 12.7 | 20.2 | 24 | 467.6 | 79,160 | 2847 | 3263 |

IPE A 160 | 16.2 | 157 | 82 | 4 | 5.9 | 9 | 127.2 | 689 | 87.8 | 99.1 | IPE A 600 | 137 | 597 | 220 | 9.8 | 17.5 | 24 | 514 | 82,920 | 2778 | 3141 |

IPE 160 | 20.1 | 160 | 82 | 5 | 7.4 | 9 | 127.2 | 869 | 109 | 124 | IPE 600 | 156 | 600 | 220 | 12 | 19 | 24 | 514 | 92,080 | 3070 | 3512 |

IPE AA 180 | 19 | 176.4 | 91 | 4.3 | 6.2 | 9 | 146 | 1020 | 116 | 131 | IPE O 600 | 197 | 610 | 224 | 15 | 24 | 24 | 514 | 118,300 | 3879 | 4471 |

IPE A 180 | 19.6 | 177 | 91 | 4.3 | 6.5 | 9 | 146 | 1063 | 120 | 135 | IPE 750 × 134 | 171 | 750 | 264 | 12 | 15.5 | 17 | 685 | 150,700 | 4018 | 4644 |

IPE 180 | 23.9 | 180 | 91 | 5.3 | 8 | 9 | 146 | 1317 | 146 | 166 | IPE 750 × 147 | 188 | 753 | 265 | 13.2 | 17 | 17 | 685 | 166,100 | 4411 | 5110 |

IPE O 180 | 27.1 | 182 | 92 | 6 | 9 | 9 | 146 | 1505 | 165 | 189 | IPE 750 × 173 | 221 | 762 | 267 | 14.4 | 21.6 | 17 | 685 | 205,800 | 5402 | 6218 |

IPE AA 200 | 22.9 | 196.4 | 100 | 4.5 | 6.7 | 12 | 159 | 1533 | 156 | 176 | IPE 750 × 196 | 251 | 770 | 268 | 15.6 | 25.4 | 17 | 685 | 240,300 | 6241 | 7174 |

IPE A 200 | 23.5 | 197 | 100 | 4.5 | 7 | 12 | 159 | 1591 | 162 | 182 | HE 100 A | 21.2 | 96 | 100 | 5 | 8 | 12 | 56 | 349.2 | 72.76 | 83.01 |

IPE 200 | 28.5 | 200 | 100 | 5.6 | 8.5 | 12 | 159 | 1943 | 194 | 221 | HE 100 B | 26 | 100 | 100 | 6 | 10 | 12 | 56 | 449.5 | 89.91 | 104.2 |

IPE O 200 | 32 | 202 | 102 | 6.2 | 9.5 | 12 | 159 | 2211 | 219 | 249 | HE 120 A | 25.3 | 114 | 120 | 5 | 8 | 12 | 74 | 606.2 | 106.3 | 119.5 |

IPE AA 220 | 27 | 216.4 | 110 | 4.7 | 7.4 | 12 | 177.6 | 2219 | 205 | 230 | HE 120 B | 34 | 120 | 120 | 6.5 | 11 | 12 | 74 | 864.4 | 144.1 | 165.2 |

IPE A 220 | 28.3 | 217 | 110 | 5 | 7.7 | 12 | 177.6 | 2317 | 214 | 240 | HE 140 A | 31.4 | 133 | 140 | 5.5 | 8.5 | 12 | 92 | 1033 | 155.4 | 173.5 |

IPE 220 | 33.4 | 220 | 110 | 5.9 | 9.2 | 12 | 177.6 | 2772 | 252 | 285 | HE 140 B | 43 | 140 | 140 | 7 | 12 | 12 | 92 | 1509 | 215.6 | 245.4 |

IPE O 220 | 37.4 | 222 | 112 | 6.6 | 10.2 | 12 | 177.6 | 3134 | 282 | 321 | HE 300 A | 112.5 | 290 | 300 | 8.5 | 14 | 27 | 208 | 18,260 | 1260 | 1383 |

IPE AA 240 | 31.7 | 236.4 | 120 | 4.8 | 8 | 15 | 190.4 | 3154 | 267 | 298 | HE 300 B | 149.1 | 300 | 300 | 11 | 19 | 27 | 208 | 25,170 | 1678 | 1869 |

IPE A 240 | 33.3 | 237 | 120 | 5.2 | 8.3 | 15 | 190.4 | 3290 | 278 | 312 | HE 300 M | 303.1 | 340 | 310 | 21 | 39 | 27 | 208 | 59,200 | 3482 | 4078 |

IPE 240 | 39.1 | 240 | 120 | 6.2 | 9.8 | 15 | 190.4 | 3892 | 324 | 367 | HE 700 A | 260.5 | 690 | 300 | 14.5 | 27 | 27 | 582 | 215,300 | 6241 | 7032 |

IPE O 240 | 43.7 | 242 | 122 | 7 | 10.8 | 15 | 190.4 | 4369 | 361 | 410 | HE 700 B | 306.4 | 700 | 300 | 17 | 32 | 27 | 582 | 256,900 | 7340 | 8327 |

IPE A 270 | 39.2 | 267 | 135 | 5.5 | 8.7 | 15 | 219.6 | 4917 | 368 | 413 | HE 800 AA | 218.5 | 770 | 300 | 14 | 18 | 30 | 674 | 208,900 | 5426 | 6225 |

IPE 270 | 45.9 | 270 | 135 | 6.6 | 10.2 | 15 | 219.6 | 5790 | 429 | 484 | HE 800 A | 285.8 | 790 | 300 | 15 | 28 | 30 | 674 | 303,400 | 7682 | 8699 |

IPE O 270 | 53.8 | 274 | 136 | 7.5 | 12.2 | 15 | 219.6 | 6947 | 507 | 575 | HE 900 AA | 252.2 | 870 | 300 | 15 | 20 | 30 | 770 | 301,100 | 6923 | 7999 |

IPE A 300 | 46.5 | 297 | 150 | 6.1 | 9.2 | 15 | 248.6 | 7173 | 483 | 542 | HE 900 A | 320.5 | 890 | 300 | 16 | 30 | 30 | 770 | 422,100 | 9485 | 10,810 |

IPE 300 | 53.8 | 300 | 150 | 7.1 | 10.7 | 15 | 248.6 | 8356 | 557 | 628 | HE 900 × 466 | 593.7 | 938 | 312 | 30 | 54 | 30 | 770 | 814,900 | 17,380 | 20,380 |

IPE O 300 | 62.8 | 304 | 152 | 8 | 12.7 | 15 | 248.6 | 9994 | 658 | 744 | HE 1000 AA | 282.2 | 970 | 300 | 16 | 21 | 30 | 868 | 406,500 | 8380 | 9777 |

IPE A 330 | 54.7 | 327 | 160 | 6.5 | 10 | 18 | 271 | 10,230 | 626 | 702 | HE 1000 A | 346.8 | 990 | 300 | 16.5 | 31 | 30 | 868 | 553,800 | 11,190 | 12,820 |

IPE 330 | 62.6 | 330 | 160 | 7.5 | 11.5 | 18 | 271 | 11,770 | 713 | 804 | HE 1000 × 393 | 500.2 | 1016 | 303 | 24.4 | 43.9 | 30 | 868 | 807,700 | 15,900 | 18,540 |

IPE O 330 | 72.6 | 334 | 162 | 8.5 | 13.5 | 18 | 271 | 13,910 | 833 | 943 | HE 1000 × 415 | 528.7 | 1020 | 304 | 26 | 46 | 30 | 868 | 853,100 | 16,728 | 19,571 |

IPE A 360 | 64 | 357.6 | 170 | 6.6 | 11.5 | 18 | 298.6 | 14,520 | 812 | 907 | HE 1000 × 438 | 556 | 1026 | 305 | 26.9 | 49 | 30 | 868 | 909,200 | 17,720 | 20,750 |

IPE 360 | 72.7 | 360 | 170 | 8 | 12.7 | 18 | 298.6 | 16,270 | 904 | 1019 | HE 1000 × 494 | 629.1 | 1036 | 309 | 31 | 54 | 30 | 868 | 1,028,000 | 19,845 | 23,413 |

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Use Classification | Deflection Limit |
---|---|

Roof beams (industrial) | L/180 |

Roof beams (commercial and institutional without plaster ceiling) | L/240 |

Roof beams (commercial and institutional with plaster ceiling) | L/360 |

Floor beams (ordinary usage) | L/360 |

Highway bridge stringers | L/200 to L/300 |

Railway bridge stringers | L/300 to L/400 |

LL < 2.5 kN/m^{2} | L/480 |

2.5 kN/m^{2} < LL < 4.0 kN/m^{2} | L/420 |

LL > 4.0 kN/m^{2} | L/360 |

Inputs | Output | |
---|---|---|

Applied Live Load LL | Classification | Deflection Limit |

(kN/m^{2}) | CLASS * | LIMIT |

20 | 1 | 360 |

10 | 1 | 360 |

4 | 1 | 360 |

3.5 | 1 | 420 |

3 | 1 | 420 |

2.5 | 1 | 480 |

2 | 1 | 480 |

1.5 | 1 | 480 |

1 | 1 | 480 |

0.5 | 1 | 480 |

0 | 1 | 480 |

20 | 2 | 240 |

10 | 2 | 240 |

4 | 2 | 240 |

3.5 | 2 | 280 |

3 | 2 | 280 |

2.5 | 2 | 320 |

2 | 2 | 320 |

1.5 | 2 | 320 |

1 | 2 | 320 |

0.5 | 2 | 320 |

0 | 2 | 320 |

20 | 3 | 180 |

10 | 3 | 180 |

4 | 3 | 180 |

3.5 | 3 | 210 |

3 | 3 | 210 |

2.5 | 3 | 240 |

2 | 3 | 240 |

1.5 | 3 | 240 |

1 | 3 | 240 |

0.5 | 3 | 240 |

0 | 3 | 240 |

Membership Function | Premise Parameters | Consequent Parameters | ||
---|---|---|---|---|

i | σ_{i} | c_{i} | - | |

A_{1} | 6.61768494044089 | 2.90117735987428 | a_{0}^{1} | −771.211045670957 |

A_{2} | 7.47150990794045 | 2.08316049335067 | a_{1}^{1} | 8.40911494744558 |

B_{1} | 0.962309787332703 | 1.62121248482762 | a_{2}^{1} | 172.024353054372 |

B_{2} | 0.979723951565027 | 1.27590144219536 | a_{0}^{2} | 1535.72815330126 |

- | - | - | a_{1}^{2} | −29.7774593495827 |

- | - | - | a_{2}^{2} | −158.884574516486 |

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**MDPI and ACS Style**

Jelušič, P.; Žlender, B.
Discrete Optimization with Fuzzy Constraints. *Symmetry* **2017**, *9*, 87.
https://doi.org/10.3390/sym9060087

**AMA Style**

Jelušič P, Žlender B.
Discrete Optimization with Fuzzy Constraints. *Symmetry*. 2017; 9(6):87.
https://doi.org/10.3390/sym9060087

**Chicago/Turabian Style**

Jelušič, Primož, and Bojan Žlender.
2017. "Discrete Optimization with Fuzzy Constraints" *Symmetry* 9, no. 6: 87.
https://doi.org/10.3390/sym9060087