HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant cross secton beam, constant Young s modulus along the beam, smple loadng scenaros, etc. In ths homework, you wll numercally examne beam bendng usng fnte dfferences and n so dong, you wll be able to perform an advanced beam desgn problem. Fgure 1. A General Beam The classcal beam bendng equaton gves a relatonshp between the deflecton w at any x locaton and the appled (dstrbuted) load q (the force per unt length): 2 ( ) (1) x 2 E(x)I(x) 2 w x 2 = q(x) Usng the product rule for hgher order dervatves, ths becomes a formula we can dscretze more readly usng standard fnte dfference expressons: (2) 2 (EI) x 2 2 w x 2 + 2 (EI) x 3 w x 3 + (EI) 4 w x 4 = q(x) 1
2 DUE: WEDS FEB 21ST 2018 Ths equaton s the governng beam bendng equaton that you wll solve n ths homework. As you can see, ths equaton allows the product of the moment of nerta and the Youngs modulus of the beam to vary along the length. The shear force, Q n the beam can be wrtten as: ( ) (3) E(x)I(x) 2 w x x 2 = Q(x) Whch can be re-wrtten usng the product rule as: (4) 2 (EI) x 2 2 w x 2 + EI 3 w x 3 = Q(x) Ths shear force equaton s useful for examnng the nternal loadng of the beam as well as for applyng boundary condtons/pont loads on the wng. The bendng moment, M along the beam can be expressed as: (5) EI 2 w x 2 = M(x) Ths bendng moment expresson s smlarly useful for observng the loadng wthn the beam as well as applyng moments to the beam. Note, that ths moment equaton s also used n the calculaton of the stress dstrbuton through the beam and s therefore a crtcal part of determnng whether a beam wll survve a certan load. Fnally, the slope of the beam at any locaton s useful for certan boundary condtons and s gven as: (6) w x = θ In addton to the above relatonshps, the stress at a gven cross sectonal locaton of the beam can be calculated usng the followng expresson: (7) σ = M(x)z I where z s the dstance from the neutral axs to the locaton at whch the stress s beng measured.
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION 3 2. Questons Please answer the questons below. All questons should be handed n on paper. The computer code, once complete, should be handed n usng the plagarsm detecton software turntn: class ID: 17441044 Enrollment key: pdes (1) Consder the governng equaton (Eq. 2), fnd/dentfy (you do not need to derve) at least second order fnte dfference equaton expressons to be used to approxmate each of the dervatves n the equaton. You may fnd the followng webste useful for fndng the dervatves: web.meda.mt.edu/~crtaylor/calculator.html (2) Draw a fgure showng the dscretzed beam and wrte the fnte dfference approxmaton that represents the governng dfferental equaton at a sngle node,, along the beam. (3) Based on the fnte dfference expresson of the governng equaton, llustrate the matrx structure (.e. the entres of [A][w] = [q] that you expect when the equatons for all non-boundary nodes of the beam are dscretzed). (4) Descrbe/show how you wll mplement each of the followng boundary condtons: (a) Prescrbed deflecton value at a node of the beam. w(x) = δ(s) (b) Prescrbed beam slope at a node of the beam. ( ) dw = s(x) dx (c) A prescrbed beam bendng moment at a node of the beam. ( ) EI 2 w x 2 = M(x) (d) Prescrbed beam shear force at a node of the beam. ( 2 ) (EI) x 2 2 w x 2 + EI 3 w x 3 = Q(x)
4 DUE: WEDS FEB 21ST 2018 (5) You wll be solvng a cantlever beam problem. Clearly show how you wll handle the boundary condtons below n your fnte dfference method (provde dscrete equatons and sketches to show nodes beng referenced): (a) At x = 0, w(x) = 0 or zero deflecton at fxed end. (b) Atx = 0, θ(x) = 0 or a zero slope at the fxed end. (c) At x = L, M(L) = 0 or a zero bendng moment. (d) At x = L, Q tp = 0 or a zero shear force. (6) Clearly wrte the pseudo code that you wll use to solve the beam bendng problem. Please nclude adequate detal so that someone could understand your approach. (7) In order to determne the maxmum stress n the beam, you wll need to calculate the bendng moment at dfferent locatons n the beam (after you have solved for the deflecton). Please ndcate how you wll calculate the bendng moment at each node n the beam and also clearly ndcate (asssted wth a sketch), the calculaton of the maxmum stress n the beam cross secton. (8) Wrte a computer code (n matlab or another programmng language) based on second order fnte dfference approxmatons to solve a cantlever beam bendng problem. Assume that the beam has zero slope and deflecton at x = 0 and zero bendng moment and shear force at x = L. You may assume a unformly loaded beam (q(x) = 100N/m), E = 1e10 and I = 0.0001m 4, L = 30m. Compare your deflecton results wth theoretcal values. (9) Show that your code converges at the expected rate by approprately plottng the L2-norm of the error vs. the nter-grd spacng. Use the most refned soluton as the correct answer. Show a sample hand-calc of how you determned the error and L2 norm of the error. (10) Applcaton of your code to real world: In ths problem you wll apply your code to analyze an arcraft wng. In ths model, we wll nvoke symmetry assumptons and consder only half of the wng or a cantlever beam. Arcraft wngs are used to generate lft that holds the arcraft aloft. that the lft dstrbuton actng on the cantlever beam s gven by: ( x 2 q(x) = 4.2 10 4 1 L) Assume
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION 5 where the unts are [N/m]. Fgure 2. The Wng Problem The wng has: At x = 0, w(x) = 0 or zero deflecton. Atx = 0, θ(x) = 0 or a zero slope at the fxed end. At x = L, M(L) = 0 or a zero bendng moment. At x = L, Q tp = 0 or a zero shear force. The heght of the beam vares from the wng root to the wng chord accordng to the wng desgn and s assumed to be a lnear change n heght (measured n meters these data are based on a Boeng 777 lke arcraft): H(x) = 1.75 1.7 x 30 Assume that the cross secton of the wng beam s a box wth cap and web thcknesses t h (x) and t b (x) respectvely as shown n fgure 2 (values whch you wll determne). Assume also that the wdth, B, of the box beam (n meters) vares lnearly as follows: B(x) = 2 C B x 30 Where C B < 2.0 s a constant you wll determne. The followng propertes apply to the box beam: Box Beam Materal: 7075-T6 Alumnum E = 71.7GPa, ˆσ = 572 MPa ρ = 2810kg/m 3 Wng span = 60m, cantlever beam length = 30m The second moment of area, I = 1 12 BH3 1 12 bh3
6 DUE: WEDS FEB 21ST 2018 Queston: Determne the box beam thcknesses, t h > 0 and t b > 0 as well as the constant C B that produce the best desgn. A desgn s consdered optmal when t produces the lghtest weght wng whle ensurng the stress s below the ultmate stress of the materal, σ < ˆσ. (11) Bonus: Wrte and prove/test a general beam bendng computer program so that you can apply forces, appled loads, deflectons and slopes at desred locatons of the beam. HINTS: (a) A prescrbed appled beam bendng moment at a node of the beam can be handled by recognzng that the appled moment wll cause a jump n the beam nternal bendng moment at that pont: ( ) EI 2 w x 2 = M appled (x) (b) Smlarly, a prescrbed force at a node of the beam wll cause a change n the beam nternal shear force at that pont: ( ) 2 (EI) x 2 2 w x 2 + EI 3 w x 3 = F Appled (x) In addton, you wll want to setup some method to allow the user to prescrbe the locaton and type of boundary condton to be appled on the beam. Be aware that boundary condtons n ths general case could occur anywhere along the beam. Another consderaton s that, snce boundary condtons can appear at any node, the A-matrx should be defned to allow soluton of the governng equaton at every node before the A-matrx s updated to ensure the satsfacton of boundary condtons. In order to prove your code works, solve the wng problem as a full wng. Apply zero deflecton boundary condton and a zero slope at the center pont of the wng, a free end (bendng moment and shear forces equal zero) at the tps of the wng. Also nclude an engne half-way along the length of each wng sem-span wth a weght of 98, 000N.