Course Experiment:Bridge Design J.J.Gao Systems Engineering.CLGO,SJTU 1 Requirements The experiment is team based.However,I suggest everyone of you do it yourself. In the last section,there are scores for each design task.The team who got the highest score win the contest.Please keep your design safely. Please send me your excel file including the model and the results by email after the class.For each team,only one result is required. Please follow my instruction. 2 Bridge design:problem formuation Your objective is to design a bridge to span a highway(Figure 1).The total span of the bridge(across both halves of the highway)must be L=30(m)meters,and it must support its own weight and a load g=33 x 104(N/m)along its span. Figure 1:Example of the simple bridge The bridge span will be supported by between one and four-I-beams(e.g.,see the Figure 2),In the figure,the I-beams would be parallel to each other going into the page, for example it could be one I-beam in the middle of the bridge,or two I-beam on both sides of the bridge,etc.(see the Figure 4).The number of the I-beam will be represented by design variable nrbeam.The shape of I-beams will be represented by three continuous design variables,the height,h,flange width,b,and thickness t.In the middle of the bridge,there is a support.The middle support is a rectangular(view from above)and will have two design variables,the width,w and depth,d.(See the Figure 3) (a)The designing detail:Let pream is the density of the material used for the I- beams,the mass of the I-beam can be computed as, MIbeam:=(2bt+(h-2t)t)LpIbeamnIbeam. 1
Course Experiment: Bridge Design J.J. Gao Systems Engineering, CLGO, SJTU 1 Requirements • The experiment is team based. However, I suggest everyone of you do it yourself. • In the last section, there are scores for each design task. The team who got the highest score win the contest. Please keep your design safely. • Please send me your excel file including the model and the results by email after the class. For each team, only one result is required. • Please follow my instruction. 2 Bridge design: problem formuation Your objective is to design a bridge to span a highway(Figure 1). The total span of the bridge(across both halves of the highway) must be L = 30(m) meters, and it must support its own weight and a load q = 33 × 104 (N/m) along its span. Figure 1: Example of the simple bridge The bridge span will be supported by between one and four-I-beams(e.g., see the Figure 2), In the figure, the I-beams would be parallel to each other going into the page, for example it could be one I-beam in the middle of the bridge, or two I-beam on both sides of the bridge, etc.(see the Figure 4). The number of the I-beam will be represented by design variable nIbeam. The shape of I-beams will be represented by three continuous design variables, the height, h, flange width, b, and thickness t. In the middle of the bridge, there is a support. The middle support is a rectangular(view from above) and will have two design variables, the width, w and depth, d.(See the Figure 3) (a) The designing detail: Let ρIbeam is the density of the material used for the Ibeams, the mass of the I-beam can be computed as, MIbeam := ( 2bt + (h − 2t)t ) LρIbeamnIbeam. 1
Front View Side View I-beams -Middle Support Figure 2:The bridge design problem Figure 3:Illustration of design variables Let ppor be the density of the material used for support,and H=5(m)is the height of the bridge above the ground,the mass of the middle support can be computed as, Msupport=wdHpsupport There is a constraint that the stress the I-beam is less than the material failure stress for the I-beam,or-ream Note that g=9.81m/s2 is the gravitational constant. q(L/2)2+MIeam(L/4)g OIbeam- 8IIbeamnIbeam ≤OFIbeam, where Iream,is the moment of interia for the I-beam given by: (h-2t)3t 12 (僧+ In addition that the shear stress in the I-beams is less than the material failure stress: TIbeam MIveamg+qL 4(2bt+(h-2t)t)nIbeam ≤OFIbeam: For teh middle-support,there are two constraints,the column can not bukle and the stress must be less than the material failure stress.Buckling is based on a requirement that the applied load is less than a critical load, 1, Policd=2 Mrcmg+gl）≤Prt, 2
Figure 2: The bridge design problem Figure 3: Illustration of design variables Let ρsupport be the density of the material used for support, and H = 5(m) is the height of the bridge above the ground, the mass of the middle support can be computed as, Msupport = wdHρsupport There is a constraint that the stress the I-beam is less than the material failure stress for the I-beam, σF −Ibeam. Note that g = 9.81m/s2 is the gravitational constant. σIbeam = ( q(L/2)2 + MIbeam(L/4)g 8IIbeamnIbeam ) h 2 ≤ σF Ibeam, where IIbeam, is the moment of interia for the I-beam given by: IIbeam = (h − 2t) 3 t 12 + 2 ( t 3 b 12 + tb(h − t 2 )2 ) . In addition that the shear stress in the I-beams is less than the material failure stress: τIbeam = MIbeamg + qL 4(2bt + (h − 2t)t)nIbeam ≤ σF Ibeam. For teh middle-support, there are two constraints, the column can not bukle and the stress must be less than the material failure stress. Buckling is based on a requirement that the applied load is less than a critical load, Papplied = 1 2 (MIbeamg + qL) ≤ Pcrit, 2
Figure 4:Illustration of how to put the I-beam Material Density Modulus of Elasticity Failure Stress Cost(③kg) kg/m） (GPa=10N/m2) (MPa=10N/m2) A16061 2700 70 270 2.05 A36 Steel 7850 210 250 0.62 A514 Steel 7900 210 700 0.90 Titanium 4500 120 760 16.00 Concrete 2400 31 70 0.04 Figure 5:The parameter of materials where the critical load,Perit,is a function of the lowest moment of inertia of the support and the modulus of elasticity of the support material,E 2Esupport min 11212 Perit- 4H2 The stress requirement is that the applied stress is less than the support material failure stress, Osupport Papplicd≤GFsupport wd (1) (b)The materials:The bridge span(I-beam)can be made of four materials,AL6061, A36 Steel,A514 Steel,or Titanium;however,the support can be made of AL6061, A36 Steel,A514 Steel,or Concrete.The reason for the difference is that concrete cannot be loaded in tension.The parameters of the materials are give in Figure 5. (c)Total cost:We Use Cream and Cpor to denote the cost per kilogram of the materials used for the I-beams and Support,the total bridge cost is: Ctotal =CIbeam MIbeam+Csupport Msupport. 3 Design Target (a)Please identify the design variable in this experiment,and what type of these de- sign variables(integer,continuous,binary)?Is there any more additional physical constraints of these design variables?(Discuss with your team member and list them) %
Figure 4: Illustration of how to put the I-beam Figure 5: The parameter of materials where the critical load, Pcrit, is a function of the lowest moment of inertia of the support and the modulus of elasticity of the support material, Esupport, Pcrit = π 2Esupport min{ w3d 12 , wd3 12 } 4H2 . The stress requirement is that the applied stress is less than the support material failure stress, σsupport = Papplied wd ≤ σF support (1) (b) The materials: The bridge span(I-beam) can be made of four materials, AL6061, A36 Steel, A514 Steel, or Titanium; however, the support can be made of AL6061, A36 Steel, A514 Steel, or Concrete. The reason for the difference is that concrete cannot be loaded in tension. The parameters of the materials are give in Figure 5. (c) Total cost: We Use CIbeam and Csupport to denote the cost per kilogram of the materials used for the I-beams and Support, the total bridge cost is: Ctotal = CIbeamMIbeam + CsupportMsupport. 3 Design Target (a) Please identify the design variable in this experiment, and what type of these design variables(integer, continuous, binary)? Is there any more additional physical constraints of these design variables?(Discuss with your team member and list them) 3
Variable(Ibeam) Value Variable(Support) Value b(m) 0.5 ) 0.9 t(m) 0.09 d 0.9 h(m) 1 Psupport(kg/m3) 2700 n Ibeam 2 Osupport(106 N/m2) 6.2718 PIbeam(kg/m3) 2700 OFsupport(106N/m2) 270 OIbeam(106N/m2) 104.3 Msupport(kg） 10935.5 OFIbeam(106 N/m2) 270 Csupport(S/kg） 2.05 MIbeam(kg） 26535.6 Papplied(10N) 5.08 CIbeam(S/kg） 2.05 Perit(10°N) 377.73 IIbeam(m） 0.0228 TIbeam(106N/m2) 7.7536 Table 1:Test data (b)This design problem can be formulated as a mixed-integer optimization problem. Discuss with your team member how to formulate the problem.(Hint:there are two options:(1)only use the continuous variables as decision variables,(2)use all design variable as the decision variable;What is your choice) (c)Formulate the problem in the EXCEL.Before you use the solver to solve the problem. Please use the data in Table 1 to verify whether your model is correct.In this test data,we use AL 6061 as the materia for support and I-beam. (d)Minimum cost design:(40 points) (d1)Your objective is to find the dimensions of the I-beams,number of the I-beams material type for I-beams,as well as the dimension of the support and material type of the support to minimize cost of the bridge. (d2)What is the applied load when the minimum cost is achieved?Is the minimum cost design a good design? (e)Pareto-optimal Design:(60 points) (el)(20 points)Find other two different design that the total cost Ctotat is at least 10%higher than the minimum cost generated in (d1). (e2)(20 points)Beside the total cost Cta,we regard the applied load Ppplied as our design target,i.e.,it is multi-criteria optimization problem.The specialist suggest that the applied load should be larger than 5.2 x 106(N).Find a designs satisfied this condition.If the applied load should be larger than 6.5 x 10(N), what is your design？ (e3)(20 points)Identify the Pareto-Optimal frontier of cost and applied load.(Plot the figure)
Variable(Ibeam) Value Variable(Support) Value b(m) 0.5 w 0.9 t(m) 0.09 d 0.9 h(m) 1 ρsupport(kg/m3 ) 2700 nIbeam 2 σsupport(106N/m2 ) 6.2718 ρIbeam(kg/m3 ) 2700 σF support(106N/m2 ) 270 σIbeam(106N/m2 ) 104.3 Msupport(kg) 10935.5 σF Ibeam(106N/m2 ) 270 Csupport($/kg) 2.05 MIbeam(kg) 26535.6 Papplied(106N) 5.08 CIbeam($/kg) 2.05 Pcrit(106N) 377.73 IIbeam(m4 ) 0.0228 τIbeam(106N/m2 ) 7.7536 Table 1: Test data (b) This design problem can be formulated as a mixed-integer optimization problem. Discuss with your team member how to formulate the problem. (Hint: there are two options: (1)only use the continuous variables as decision variables, (2) use all design variable as the decision variable; What is your choice) (c) Formulate the problem in the EXCEL. Before you use the solver to solve the problem. Please use the data in Table 1 to verify whether your model is correct. In this test data, we use AL 6061 as the materia for support and I-beam. (d) Minimum cost design:(40 points) (d1) Your objective is to find the dimensions of the I-beams, number of the I-beams, material type for I-beams, as well as the dimension of the support and material type of the support to minimize cost of the bridge. (d2) What is the applied load when the minimum cost is achieved? Is the minimum cost design a good design? (e) Pareto-optimal Design:(60 points) (e1) (20 points) Find other two different design that the total cost Ctotal is at least 10% higher than the minimum cost generated in (d1). (e2) (20 points) Beside the total cost Ctotal, we regard the applied load Papplied as our design target, i.e., it is multi-criteria optimization problem. The specialist suggest that the applied load should be larger than 5.2×106 (N). Find a designs satisfied this condition. If the applied load should be larger than 6.5 × 106 (N), what is your design? (e3) (20 points) Identify the Pareto-Optimal frontier of cost and applied load.(Plot the figure) 4