Chapter21 - AAE 221 – Aerospace Structures II – Spring...

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Unformatted text preview: AAE 221 – Aerospace Structures II – Spring 2004 Chapter 2.1 – Beam Bending and Extension 0. Obtain the bending moment and shear force diagrams for the following cases, where P is a force, M0 a moment and w a force per unit length. In (i) the pin joint acts like a simple support between the two segments (i.e. shear force, but no bending moment, is transmitted). P L (a) a L (b) P w L (c) w P a L/2 (d) L/2 2P w L/2 L (e) (f) a P L w L/4 P M0 3L/4 L (g) L (h) P L/2 L (i) L 1. (a) Calculate the area moments and product of inertia (with respect to axes passing through the centroid) for the bimaterial cross-section sketched in Fig. 1, where E1 = 3.0x107 psi (Steel), and E2 = E3 = 1.0x107 psi (Aluminum). Assume that a = 10t. The subscript corresponds to the rectangular subsections, as indicated in the figure below Fig. 1 (b) Calculate the axial load NT and bending moment MT for the cross-section of part (a) if the temperature changes associated with each of the respective idealized rectangles (subsections 1, 2, and 3) are such that (aDT )1 = -1.0 ¥ 10 -3 , (aDT )2 = 0 , (aDT )3 = +1.0 ¥ 10 -3 Compute the induced axial stress at the two following points (see Fig. 1) Point A y=2a, z=a/2 and comment on your solution. : Point B y=3a, z=a+5t/2 2. A bar with a varying cross-section (Fig. 2) is loaded by an axial force P applied at the center of the end of the bar. Neglecting the effect of gravity, determine and plot (in a non-dimensional way) the beam displacements. Don’t forget to label the axes appropriately. Fig. 2 3. Derive and plot the deflections of a simply supported beam subjected to a shearing traction q(x,y) = qo (given in Pa) acting uniformly along the top surface of the beam and to a concentrated moment at the end equal to M = qoa3 (in Nm). Assume a Young’s modulus of E and neglect the effect of gravity. The beam cross-section is rectangular, as indicated in Fig. 3. Fig. 3 4. Consider the bimaterial cantilever beam shown in Fig. 4. The beam is made of steel (r=7800 kg/m3 , E = 210 GPa), and aluminum (r=2800 kg/m3 , E = 70 GPa). Calculate and plot the deflections of the beam under its own weight, assuming that the acceleration of gravity g = 10.0 m/s2. Also compute and plot the shear (Vz) and moment (My) diagrams. Fig. 4 5. Determine the deflections of a uniform, homogenous, symmetric cantilever/simply supported beam of length L, cross-section A and moment of inertia I, subjected to a distributed partial span linearly increasing load qo (in N/m) and to a point load qoL as indicated in Fig. 5. Obtain the variation of the induced bending moment My(x) along the beam. Plot (in a non-dimensional manner) the w-deflection and the induced moment distribution along the beam. Fig. 5 6. Solve the problem described in Fig. 6. The beam is assumed to be uniform and homogenous, and has a triangular cross-section, as indicated in Fig. 6. Its length is L and its Young’s modulus is E. The only external loading is due to gravity. (a) Obtain the general non-dimensional expression of the deflections, and plot your results for the following special values of the spring stiffness (K = 5 Nm/rad; k = 10 N/m). (b) Assuming that k=0, plot (once again in a non-dimensional fashion) the deflection v(L) at the end of the beam as a function of K for K=0 to K very large (i.e., approaching infinity). Comment on your solution. What happens when K=0 and when K is infinite? Fig. 6 7. To simulate the elastic response of the support in the case of a beam placed on an elastic foundation and subjected to a varying load q(x) (in N/m), the following model is often used. Fig. 7 The elastic foundation is represented by a continuous distribution of springs with stiffness k. Assuming that the two ends of the homogeneous symmetric beam (of length L) are simply supported, write the BVP (do not solve it) describing its equilibrium (Hint: the GDE will be slightly different from the “classical case”, but can be derived very easily from a simple physical reasoning). 8. Consider the problem presented in Figure 8. The beam is supposed to be homogenous and linearly elastic. Its cross-section is constant. Fig. 8 The loading consists exclusively of • A partial span distributed axial load (applied along the axis of the beam) which varies from po at x=0 to 0 at x=L/2; (po in N/m) • A concentrated axial force (also applied on the axis of the beam) po =poL applied at x=3L/4. Compute the axial displacement of the beam. Put your solution in a non-dimensional form and plot it for various representative values of the spring stiffness k (adequately non-dimensionalized). Comment on your solution. 9. Determine the deflection function of a uniform, homogenous, simply supported beam of length L that is loaded with a distributed load, fo, acting downward. Obtain the variation of the bending moment along the beam. Fig. 9 10. A symmetric homogenous cantilever beam of length L, has a concentrated force Fo, at L/4, a concentrated moment Mo, at 3L/4 and a linear and rotational spring (with stiffness k and kq respectively) at L. (a) find the deflection as a function of x. (b) find the bending moment distribution (c) plot the deflectin for Mo=FoL, k=Fo/(2L), kq = FoL/2 Fig. 10 11. Determine the deflections in the x, y, z directions and find the stresses at points a, b, c for the composite cantilever beam illustrated below. Note that we have two different loadings: a distributed line-load of 10 lb/in in the y direction on the top of the beam, and a point load of 100 lb in the x direction acting on the corner (x = 100 in, y = 2 in, and z = 1 in). Do your results agree with your expectations? Fig. 11 12. Consider the problem of a doubly clamped beam on an elastic foundation represented below. The effect of the elastic foundation can be modeled by introducing a continuous distribution of linear springs (with constant stiffness k) along the whole span of the beam. This type of beam satisfies a slightly different differential equation than a usual beam. But this differential equation can be derived very simply from the “usual beam theory” covered in class. (a) Derive the differential equation for this special beam and specify the boundary conditions for the particular problem illustrated below. (b) Calculate the deflection w, as a function of x(i.e., solve the differential equation) and plot the result (assume k=fo/L for the plot). Fig. 12 13. Write and solve the BVP describing the equilibrium of the following symmetric and homogenous beam (length L, Young’s modulus E, cross-section A, moment of inertia I). Plot the deflection (in a non-dimensional manner as a function of x/L). Obtain and plot the variation of the bending moment My(x). Fig. 13 14. Consider a homogenous beam of length L, with a varying cross-section subjected to an axial load P in Fig. 14. Fig. 14 The cross-section is assumed to vary quadratically with x, as in Ê Ê x ˆˆ A(x ) = Ao Á1 - e Á ˜ ˜ Á ˜ Ë L ¯¯ Ë 2 (a) Write and solve the BVP describing the deflection of this beam (b) Represent (in a non-dimensional way) the u-displacement versus x/L (c) Compute and represent the axial stress versus x/L 15. Determine the deflections of a uniform, homogenous, symmetric cantilever beam of length L subjected to a distributed load fo and a concentrated force F (Fig. 15). Obtain the variation of the bending moment along the beam. Plot (in a non-dimensional manner) the v-deflection and the moment distribution along the beam. Fig. 15 16. Solve the problem described below in Fig. 16. The beam is assumed to be symmetric and homogenous. Its length is L, Young’s modulus E, and moment of inertia I. Obtain and plot the deflections (you will need to define a non-dimensional number a = EI/KL3 characterizing the stiffness of the spring) for various values of the spring stiffness. Plot also the variation of the deflection of the end of the beam (x=L) as a function of a. Comment on your results (for example what happens when the spring becomes stiff? What happens when k tends to zero?). Finally, plot (in a non-dimensional way) the moment distribution along the beam. Find the expression of the axial stress along the bottom side of the beam, assuming that the cross section is rectangular and is defined by -a/2 £ y £ a/2 ; -b/2 £ x £ b/s. Fig. 16 17. Determine the deflection in the y direction and find the axial stress at points a and c for the composite cantilever beam subjected to a distributed line-load of 10 lb/in in the y direction on the top of the beam. Fig. 17 18. Consider the homogenous symmetric prismatic cantilever beam subjected to a linearly varying load as shown in Figure 1 (note: qo is in N/m) (a) Set up the boundary value problem, keeping only the relevant equations. (b) Solve for the displacement w and plot the results (after writing your solution in a nondimensional form) (c) Derive (in a non-dimensional form) and plot the induced bending moment My(x) Fig. 18 19. Consider a cantilever beam of varying cross-sectional area, where the width b is constant and the height a(x) varies linearly from a at x = 0 to a/4 at x = L. (a) Set up the boundary value problem. (b) Solve for the axial displacement u(x) and plot. Fig. 19 20. Consider the cantilever heterogeneous I-beam shown in Figure 4. The loading on the beam consists of a uniform “tangential pressure” Po (in Pa) applied along the top surface of the beam, of a point load Q at the end of the beam (at the center of the top edge) and of the weight of the beam. The beam is cantilever at x=0 and is free at x=L. Set up (but do not solve) the boundary value problem. Fig. 20 21. Determine the deflections of a uniform, homogenous, symmetric cantilever/simply supported beam of length L, cross-section A, and moment of inertia I, subjected to a distributed partial span load fo (Fig. 21). Obtain the variation of the induced moment distribution along the beam. Plot (in a non-dimensional manner) the w deflection and the induced distribution along the beam for various values of the rotational spring stiffness K (adequately non-dimensionalized), including K = 0, and K Æ • . Comment on your solution (especially on these two special cases). Fig. 21 22. Solve the problem described in Fig. 22. The beam is assumed to be uniform, symmetric and homogenous. Its length is L, Young’s modulus E, cross-sectional area A and moment of inertia I. Obtain the general non-dimensional expression of the deflections, and plot your results for the following values of the spring stiffness (K = EL/2L; K = EI/4L4). Fig. 22 23. Determine (without solving it) the BVP describing the equilibrium deflections of the following cantilever bimaterial beam subjected to a uniform pressure p (units = Pa) applied on its top surface and an “end moment” Mo (units = Nm). Include the effect of gravity in your equations. Justify any simplification you might make. Compute all the cross-sectional quantities (position of centroid, moments of inertia moments of inertia,...) needed in the expression of the BVP. Fig. 23 The cross-section is made of 2 materials. A is equal to 10t. The top flange (subscript 1) is three times stiffer than the vertical portion (subscript 2), and its density is twice as high. Choose material #2 as the reference material. ...
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This note was uploaded on 01/18/2010 for the course AE 322 taught by Professor Lambros during the Spring '04 term at University of Illinois at Urbana–Champaign.

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