The main spar of an airplane wing can be modeled as a cantilever beam whose web...

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The main spar of an airplane wing can be modeled as a cantilever beam whose web is subjected to a tip load as shown in Figure 2. y↑ L Figure 2: Shear web of wing spar. yt h X It is known from Mechanics of Materials that the stress state at all points in the beam can be approx- imated by My VQ (x, y, z) = =- σzy(x, y, z) = It where yy-0-0-0-0, • M = M(x) is the internal bending moment at, positive when bottom of the beam is in tension, dM(x) dr • V = V(x) = is the internal shear force at , positive when pointing in the positive (respectively negative) y-direction on the positive (respectively negative) -face, ⚫ I is the area moment of inertia about the z-centroidal axis, given here by and I = 12 2 Q = Q(y) = {[()*-*] · 2 (a) (10 points) Determine the shear distribution V(r) and moment distribution M(x) along the web. (b) (12 points) Show that the differential equations of equilibrium are satisfied at all points in the web if body forces are neglected. (c) (8 points) Show that the stress state satisfies the traction boundary conditions on the top and bottom faces, i.e. for y = ±h/2. The main spar of an airplane wing can be modeled as a cantilever beam whose web is subjected to a tip load as shown in Figure 2. y↑ L Figure 2: Shear web of wing spar. yt h X It is known from Mechanics of Materials that the stress state at all points in the beam can be approx- imated by My VQ (x, y, z) = =- σzy(x, y, z) = It where yy-0-0-0-0, • M = M(x) is the internal bending moment at, positive when bottom of the beam is in tension, dM(x) dr • V = V(x) = is the internal shear force at , positive when pointing in the positive (respectively negative) y-direction on the positive (respectively negative) -face, ⚫ I is the area moment of inertia about the z-centroidal axis, given here by and I = 12 2 Q = Q(y) = {[()*-*] · 2 (a) (10 points) Determine the shear distribution V(r) and moment distribution M(x) along the web. (b) (12 points) Show that the differential equations of equilibrium are satisfied at all points in the web if body forces are neglected. (c) (8 points) Show that the stress state satisfies the traction boundary conditions on the top and bottom faces, i.e. for y = ±h/2.

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a To determine the shear distribution Vr and moment distribution Mr along the web we need to solve the differential equations based on the given stres... View the full answer