OPTIMIZATION AND EXPERIMENT OF COMPOSITE SQUARE BEAM

Transcription

1 THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS OPTIMIZATION AND EXPERIMENT OF COMPOSITE SQUARE BEAM T.Lili 1*, Y. Mingsen 1 1 College o Aerospace and Civil Engineering, Harbin Engineering Universit, Harbin, China * Corresponding author (tonglili@hrbeu.edu.cn) Kewords: composite; bo beam; optimal design; eperiment Abstract The aim o this article is to design a composite bo beam in three-point bending with a high ratio o load-bearing/weight. First o all, a three-point bending eperiments was conducted to the manuactured composite bo beam and it was ound the top o the bo broken or eceeding the compressive stress limitation. Then, a inite element model the same as the eperimental (original) parameter was established; comparing the analsis results with eperimental data, the were in good agreement. Then, a basic shape o the square beam was got b using topological optimiation o ANSYS. ANSYS APDL was used to model a parametric composite square beam, which can perorm the geometric optimiation o the square beam. The result shows that increase the height o the square beam s arch and the thickness o the top o the square beam within a certain range helps to reduce the maimum stress and increase the load appl on the square beam. Finall, the second composite square beam, which is almost the same as optimied result, was manuactured. An eperiment was conducted to the second beam. The eperiment results show: the square beam with optimied parameter has a better mechanical perormance (the ratio o load-bearing/weight is 55.99N/g) than the square beam with initial parameter (the ratio o loadbearing/weight is 49.01N/g). 1 Introduction The composite material is composed b two or more materials with diering properties [1]. The perormance o composite is oten better than their component, that s wh we preer composite. In this paper, we ocus on iber-reinorced composites, which composed o ibers embedded in a matri [2]. Furthermore, laminated composite made o iber cloth is what this paper concern about. Composite material is a new kind o material, which has a lot o advantage, such as lightweight, high-strength, antiatigue and designable. Bo beam structure has high bending and torsional stiness; additional, its overall perormance is prominent and has reasonable orce distribution. Bo beams are widel used in civilian and militar or its prominent structure perormance. It can overcome the weakness o low stiness o composites in some degree b using bo beam structure and make ull use o its advantage o high strength [3]. In 1957, Goldberg [4] proposes the theor o prismatic olded plate structures, in which discrete bo beam into a number o rectangular plates, then, establishing equations and calculating b the basement o elastic plane stress theor and plate bending theor and utiliing the deormation and statics conditions o each combination. Vlasov [5] is a pioneer o the theoretical research o thin-walled closed cross section structure composed o isotropic material; he established the theor o thin walled beams in The theoretical analsis is the basis o the actual structural stress analsis, but or man comple components, purel theoretical analsis is ver diicult or even impossible to carr out. With the development o computer technolog, numerical analsis methods can eicientl and accuratel anale the practical engineering structures o comple shape. In 2000, X.Xiangdong [6] utilied sotware he written based on plastic node method and general nonlinear inite element simulation method perorms a simulation o bo-beam hull model, and a result consistent with test was got; analtical calculation method to estimate the ultimate strength o bo beam hull structure was proposed based on eperiment and theoretical analsis. In 2007, S.Chang [7] analsis the reliabilit

2 o the bo beam b utiliing Newman series epansion Monte Carlo simulation, and established a calculation model o thin-walled bo beam considered material parameters with uncertaint. In 2013, Giovanni Belingardi perormed a geometr optimiation numerical simulation; a composite geometr sie was got [8], which has the same impact eect with steel. 2 Theoretical Foundations 2.1 Stress-Strain Relationships The stress is related to the strains b [9]: el = D (1) T stress vector= [D] = elasticit or elastic stiness matri or stressstrain matri. {ε el } = {ε} - {ε th } = elastic strain vector. {ε th } = thermal strain vector. {ε el } are the strains that cause stresses. The shear strains (ε, ε, and ε ) are the engineering shear strains, which are twice the tensor shear strains. The ε notation is commonl used or tensor shear strains, but is used here as engineering shear strains or simplicit o output. The stress vector is shown in the igure below. The sign convention or direct stresses and strains used throughout the ANSYS program is that tension is positive and compression is negative. For shears, positive is when the two applicable positive aes rotate toward each other. Fig.1. Stress Vector Deinition Equation (1) ma also be inverted to: th 1 D = (2) For the 3-D case, the thermal strain vector is: se th se se se T = T (3) = secant coeicient o thermal epansion in the direction. ΔT = T - T re T = current temperature at the point in question; T re = reerence (strain-ree) temperature. The leibilit or compliance matri, [D] -1 is: E E E D 1/ - / - / / E 1/ E - / E / E - / E 1/ E / G / G / G 1 where tpical terms are: E = Young's modulus in the direction ; ν = maor Poisson's ratio; ν = minor Poisson's ratio; G = shear modulus in the plane. (4) Also, the [D] -1 matri is presumed to be smmetric, so that: = (5) E E E E = (6) = (7) E E Because o the above three relationships, ν, ν, ν, ν, ν, and ν are not independent quantities and thereore the user should input either ν, ν, and ν (input as PRXY, PRYZ, and PRXZ), or ν, ν, and ν (input as NUXY, NUYZ, and NUXZ). The use o Poisson's ratios or orthotropic materials sometimes causes conusion, so that care should be taken in their use. Assuming that E is larger than E, ν (PRXY) is larger than ν (NUXY). Hence, ν is commonl reerred to as the maor Poisson's ratio, because it is larger than ν, which is commonl reerred to as the minor Poisson's ratio. For orthotropic materials, the user needs to inquire o the source o the material propert data as to which tpe

3 o input is appropriate. In practice, orthotropic material data are most oten supplied in the maor (PR-notation) orm. For isotropic materials (E = E = E and ν = ν = ν ), so it makes no dierence which tpe o input is used. Epanding equation (2) with equation (3) thru equation (7) and writing out the si equations eplicitl, = T - - (8) E E E = T- - (9) E E E = T- - (10) E E E = (11) G = (12) G = (13) G where tpical terms are: ε = direct strain in the direction; σ = direct stress in the direction; ε = shear strain in the - plane; σ = shear stress on the - plane. Alternativel, Equation (1) ma be epanded b irst inverting equation (4) and then combining that result with equation (3) and equation (5) thru equation (7) to give si eplicit equations: E 2 E E E = 1- - T h E h E E - T - T (14) h E E E 2 E = - T 1- h E h E E - T - T (15) h E E E = - T h h E E 2 E - T 1- - T (16) h E = G (17) = G (18) = G (19) h=1- E - E - E -2 E (20) E E E E The [D] matri must be positive deinite. The program checks each material propert as used b each active element tpe to ensure that [D] is indeed positive deinite. In the case o temperature dependent material properties, the evaluation is done at the uniorm temperature (or the irst load step. The material is alwas positive deinite i the material is isotropic or i ν, ν, and ν are all ero. When using the maor Poisson's ratios (PRXY, PRYZ, PRXZ), h as deined in equation (20) must be positive or the material to be positive deinite. 2.2 The Theor o Topological Optimiation The theor o topological optimiation seeks to minimie or maimie the obective unction () subect to the constraints (g) deined. The design variables (ηi) are internal, pseudodensities that are assigned to each inite element (i) in the topological problem. The pseudodensit or each element varies rom 0 to 1; where ηi 0 represents material to be removed; and ηi 1 represents material that should be kept. Stated in simple mathematical terms, the optimiation problem is as ollows: =a min imum/maimum w.r.t. (21) subect to 0 1 ( i1,2,3,..., N) (22) g g g ( i 1,2,3,..., M) (23) N = number o elements; M = number o constraints; g = computed th constraint value; g = lower bound or th constraint; g = upper bound or th constraint. 2.3 Theor o Design Optimiation The irst order optimiation method was used to anale the composite square beam. This method o

4 optimiation calculates and makes use o derivative inormation. Derivatives are ormed or the obective unction and the state variable penalt unctions, leading to a search direction in design space. Various steepest descent and conugate direction searches are perormed during each iteration until convergence is reached. Each iteration is composed o subiterations that include search direction and gradient (i.e., derivatives) computations. In other words, one irst order design optimiation iteration will perorm several analsis loops. Compared to the subproblem approimation method, this method is usuall seen to be more computationall demanding and more accurate. An unconstrained version o the problem is ormulated as ollows. n m1 m2 m 3 Q(,q)= P i q Pg gi Ph gi Pw g i (24) 0 i=1 i=1 i=1 i=1 Q = dimensionless, unconstrained obective unction; P, P g, P h, and P w = penalties applied to the constrained design and state variables; 0 = reerence obective unction value that is selected rom the current group o design sets. Constraint satisaction is controlled b a response surace parameter, q. Eterior penalt unctions (P ) are applied to the design variables. State variable constraints are represented b etended-interior penalt unctions (P g, P h, P w ). The unctions used or the remaining penalties are o a similar orm. As search directions are devised (see below), a certain computational advantage can be gained i the unction Q is rewritten as the sum o two unctions. Deining and Q ()= (25) 0 n m1 m2 m 3 Qp (,q)= P i q Pg gi Ph hi Pw w i (26) i=1 i=1 i=1 i=1 then equation (24) takes the orm Q(,q)= Q () Q (,q) (27) The unctions Q and Q p relate to the obective unction and the penalt constraints, respectivel. For each optimiation iteration () a search direction vector, d (), is devised. The net iteration (+1) is obtained rom the ollowing equation. 1 p = s d (28) Measured rom (), the line search parameter, s, corresponds to the minimum value o Q in the direction d (). The solution or s uses a combination o a golden-section algorithm and a local quadratic itting technique. The range o s is limited to S 100 ma * 0 s s (29) * s = largest possible step sie or the line search o the current iteration (internall computed) ; S ma = maimum (percent) line search step sie. The ke to the solution o the global minimiation o equation (27) relies on the sequential generation o the search directions and on internal adustments o the response surace parameter (q). For the initial iteration ( = 0), the search direction is assumed to be the negative o the gradient o the unconstrained obective unction. in which q = 1, and Q p d =, q =d d (30) Q p Qp d = and d = (31) Clearl or the initial iteration the search method is that o steepest descent. For subsequent iterations ( > 0), conugate directions are ormed according to the Polak-Ribiere recursion ormula.

5 -1 Q k -1 d =, q r d (32) T -1 Q,q -Q,q Q,q r -1= (33) 2-1 Q,q Notice that when all design variable constraints are satisied P ( i ) = 0. This means that q can be actored out o Q p, and can be written as p,q =q p i i i i (i=1,2,3,...,n) (34) Q Q I suitable corrections are made, q can be changed rom iteration to iteration without destroing the conugate nature o equation (32). Adusting q provides internal control o state variable constraints, to push constraints to their limit values as necessar, as convergence is achieved. The ustiication or this becomes more evident once equation (32) is separated into two direction vectors: p d =d d (35) where each direction has a separate recursion relationship, -1 Q -1 d = r d (36) -1 p Qp -1 p d = q r d (37) The algorithm is occasionall restarted b setting r -1 = 0, orcing a steepest decent iteration. Restarting is emploed whenever ill-conditioning is detected, convergence is nearl achieved, or constraint satisaction o critical state variables is too conservative. So ar it has been assumed that the gradient vector is available. The gradient vector is computed using an approimation as ollows: ie- Q Q Q i i (38) e = vector with 1 in its ith component and 0 or all other components. D i = i-i 100 ΔD = orward dierence (in percent) step sie. First order iterations continue until either convergence is achieved or termination occurs. These two events are checked at the end o each optimiation iteration. Convergence is assumed when comparing the current iteration design set () to the previous (-1) set and the best (b) set. and -1 τ = obective unction tolerance (39) b (40) It is also a requirement that the inal iteration used a steepest descent search. Otherwise, additional iterations are perormed. In other words, a steepest descent iteration is orced and convergence rechecked. Termination will occur when n=n 1 1 n i = number o iterations; N 1 = allowed number o iterations. 3 Eperiments beore Optimiation (41) An eperiment was conducted beore the optimiation. We can get some important data o the material used in the beam rom the eperiment, such as stiness and strength. The test load and the bridge s basic shape is shown below. It s a tpical shape o I beam in the igure below, but the composite beam we made is an arch square beam.

6 Load/N The thickness o bottom The thickness o webs The thickness o top The thickness o reinorcement 1.6mm 2.5mm 2.2mm 2mm Fig. 2. The test orm o the square beam The perspective view o the square beam, which we are going to design, is shown below in ig. 3. The initial date was made into an actual composite square beam, which is tested in the load instrument and the broken beam is shown below. Fig. 5. The broken composite square beam Fig. 3. Perspective view o the square beam From the ig. 4, it s clearl that there is a reinorcement o each side o the beam and the middle o it. The reinorcement can guarantee much more propert distribution o the weight. The geometric sie is shown in below igure Fig. 6. The broken place Load-displacement curve Fig. 4. Cross-sectional sie The length o the beam is 620mm (a little longer than 23 inch), the reinorcement length o the middle o the beam is 160mm and the reinorcement o the beam s sides is 40mm. There are much more parameters, which is design varies, are shown in below table. Table. 1. The initial date o the square beam Displacement/mm Fig. 7. The load-position curve o square beam The weight o the square beam is 561g and the load curve o three-point bending test is shown above. Table. 2. The ratio o load/weight o the initial beam The weight o the beam 561g The height o each side 25mm The maimum load o the beam 27493N The height o arch 35mm The ratio o load/weight 49.01N/g

7 Thickness/mm length/mm The composite o the beam is treated as isotropic in it s stiness, and the Young modulus o this material (carbon iber) is given 46Gpa. The square beam s simulation was perormed in the FEA sotware ANSYS, the displacement o which is close to the test above. Fig. 8. The displacement o the beam in simulation The maimun compress stress o the beam is 395MPa and the maimun tensile stress o the beam is 385MPa. From ig. 6, we can get the cause o the beam s broken is the compress o up o the beam eceed its limitation. However, the tensile strength o the carbon composite is much bigger than the compress strength; there is some kind o unresonable or the tensile stress is smaller than the compress stress, which means there is long wa to go to optimie the composite square beam. 4 Optimal Design o Square BEAM 4.1Topological Optimiation The goal o topological optimiation is get a basic shape o the beam, which we wish to provide a proper shape with light weight to carr more loads. Both the 2-D and 3-D topological optimiation has been accomplished to obtain the best distribution o the materials, the element tpes been used in the optimiation are PLANE82 and SOLID95 o ANASYS. The tendenc o the weight distribution was used to strengthen on the beam. 4.2 Geometric Optimiation An optimal design was perorm utiliing the APDL o ANSYS and the stress limit as the state variables; the arch height and the thickness as the design variables; the qualit o the square beam as the obective unction to optimie the design The height o the beam's mouth The height o the beam's arch Iteration times Fig. 10. The heights var with iterations From ig. 10, we can get: or getting a high ratio o loads/weight in three-point bending test, we might decrease the height o each side but the height o arch can be a little higher. Thickness o bottom Thickness o webs 6 Thickness o top Ó Iteretion times Fig. 11. The thicknesses var with iterations From the ig above, adding the thickness o the top and decrease the thickness o the webs can improve the ratio the loads/weight in three-point bending test. In the end, a composite square beam was made with the result date o optimal design, which is listed below. Fig. 9. Densit distribution (80% mass removed)

8 Table. 3. The optimied parameters o square beam The height o mouth The height o arch The thickness o bottom The thickness o webs The thickness o top The thickness o reinorcement 20mm 40mm 2mm 2.5mm 4mm 2mm The optimal design square beam was tested and the broken orm is shown as below. top o the beam is mainl bearing a stress o compress and the bottom o the beam is mainl bearing stress o etrude. (A real composite unidirectional plate alwas has a etrude strength twice bigger than the compress strength o it). In some ied load condition, the stiness o composite made o iber cloth can be treated as isotropic material. However, the compress strength o composite is much lower than its tensile, which mean there need more material i the component bear compress stress onl. Acknowledgement This paper is unded b the International Echange Program o Harbin Engineering Universit or Innovation-oriented Talents Cultivation. Fig. 12. The tested square beam Table. 4. The ratio o load/weight o the optimied beam The weight o the beam The maimum load o the beam The ratio o load/weight 697g 39002N 55.99N/g The square beam was broken under the load o 8769lb, which means 39022N and better than the initial one. 5 Conclusions Topological optimiation can help us conirm a basic shape and then geometric optimiation can inall determine the eact shape o the structure. In the square beam with three-point bending test, the thickness o the bottom is not ar-o the thickness o the webs o the square beam, which has reinorcements in the middle and the side o it. However, the thickness o the top is beond twice o the bottom. It s not complicated to eplain, or a square beam with three-point bending load, that the Reerences [1] S. Guanlin and H. Gengkai Mechanics o Composite Materials. 1st edition, Tsinghua Universit Press, [2] S. SPRINGER Mechanics o Composite structures. 1st edition, Cambridge Universit Press, [3] Z. Lei The Design and Static Analsis o Composite Bo Girder Bridge Section, National Deense Science and Technolog Universit, 2011 [4] Goldberg J. Theor o prismatic olded plate structures. International Assoc. or Bridge and Structural Engineering Publications.1957 (17) : [5] Vlasov VZ. Thin-walled elastic beams. Jerusalam, Israel : Israel Program or Scientiic Translation;1961. [6] X. Xiangdong. An Eperimental and Theoretical Stud on Ultimate Strength o a Bo Girder. Journal o Ship Mechanics, Vol. 4, No. 5. pp36-43, 2000 [7] S.Chang. Numerical simulation techniques or structural reliabilit analsis o bo girders. Journal o Railwa Science and Engineering, Vol. 4, No. 6, pp25-29, 2007 [8] Giovanni Belingardi, Alem Tekalign Beene. Geometrical optimiation o bumper beam proile made o pultruded composite b numerical simulation. Composite Structures 103(2013) [9] Theor reerence or ANSYS.