Questions and Answers of Elasticity Theory Applications

Using the local speed of sound found in Problem 16, calculate the Mach number if that same airplane has a velocity of 342 kts. TAS.
Most helicopter rotors have symmetrical airfoils becausea. they produce more lift than cambered airfoils.b. they can produce both upward and downward lift.c. they do not develop pitching moments.d.
The lift theory most often used in helicopter aerodynamics isa. the gyroscopic precession theory.b. the momentum theory.c. Bernoulli’s theory.d. the settling with power theory.
A helicopter can hover near the ground with less power than it can hover away from the ground becausea. the air is denser near the ground.b. a high‐pressure air “bubble” is produced below it.c.
The primary stress in rotor blades in flight isa. tension.b. bending.c. torsion.d. shear.
As a helicopter moves forward from a hover to a speed of about 15 kts., the power required is reduced becausea. the efficiency of the rotor is improved.b. the helicopter is moving into a region of
If a helicopter is flying at high speed and the engine fails, the pilot should enter an autorotation by slowing the helicopter by using back cyclic then reducing blade angles with the collective.a.
Using material symmetry through 180 rotations about each of the three coordinate axes, explicitly show the reduction of the elastic stiffness matrix to nine independent components for orthotropic
A transversely isotropic material with an x3-axis of symmetry was specified by the elasticity matrix given in Eq. (11.2.12). Under an arbitrary q rotation about the x3-axis given by relation
Determine the displacement field for the beam problem in Exercise 8.2. To determine the rigid body motion terms, choose fixity conditions:Note that with our approximate Saint-Veran solution, we
The solution to the illustrated two-dimensional cantilever beam problem is proposed using the Airy stress function ∅ = C1x2 + C2x2y + C3y3 + C4y5 þ C5x2y3, where Ci are constants. First
The following stress function: 13 xy = Cxy + C + C3= 6 + C4- 5 tys -+C5-9 + C6 20
The cantilever beam shown in the figure is subjected to a distributed shear stress τ, ox/l on the upper face. The following Airy stress function is proposed to solve this problem:
For the pure beam bending problem solved in Example 8.2, calculate and plot the in-plane displacement field given by relation (8.1.22)2. Use the vector distribution plotting scheme illustrated in
For the beam problem in Example 8.3, the boundary conditions required that the resultant normal force vanish at each end (x =± l). Show, however, that the normal stress on each end is not zero, and
Develop the general displacement solution (8.3.9) for the axisymmetric case.Equation 8.3.9 Ur = Ug = v) 1/2 [ - ( + 1 ) a E r + A sino + B cose 4r0 E -a + 2(1-v)a3r log r- (1+v)a3r +
Consider the axisymmetric problem of an annular disk with a fixed inner radius and loaded with uniform shear stress τ, over the outer radius. Using the Airy stress function term a4θ, show that
For the axisymmetric problem of Example 8.6, explicitly develop the displacement solution given by relation (8.4.5).Data from example 8.6Equation 8.4.5 The first example to be investigated involves a
Consider the annular ring loaded with a sinusoidal distributed pressure as illustrated. Show that this problem can be solved using the Airy function = a log r+ ar + (a2r + a22r+ + a23r + a24) cos 20
A long composite cylinder is subjected to the external pressure loading as shown in the following figure. Assuming idealized perfect bonding between the two materials, the normal stress and
Numerically generate and plot the fields of stress (σr, σθ) and displacement (ur) within the composite cylinder of Exercise 8.20 for the specific case with material (1) = steel and material (2) =
Resolve Exercise 8.20 for the case where material: (1) Is rigid and material. (2) Is elastic with modulus E and Poisson’s ratio v.Data from exercise 8.20A long composite cylinder is subjected to
For the case of a thin-walled tube under internal pressure, verify that the general solution for the hoop stress (8.4.3)2 will reduce to the strength of materials relation (see Appendix D, Section
Consider the cut-and-weld problem in which a small wedge of angle a is removed from an annular ring as shown in the figure. The ring is then to be joined back together (welded) at the cut section.
Using superposition of the stress field (8.4.15), develop solution (8.4.18) for the equal but opposite biaxial loading on a stress-free hole shown in Fig. 8.15A. Also justify that this solution will
An elastic circular plug of radius a with properties E1 and v1 is perfectly bonded and embedded in an infinite elastic medium with properties E2 and v2. The composite is loaded with a uniform
An infinite elastic medium contains a perfectly bonded rigid plug and is loaded with uniform far-field biaxial stress T as shown. Using the results from Exercise 8.22, determine the stress and
Show that the stress function:gives the solution to the problem of an elastic half-space loaded by a uniformly distributed shear over the free surface (x≤ 0), as shown in the figure. Identify
Show that the Flamant solution given by Eqs. (8.4.31) and (8.4.32) can also be used to solve the more general wedge problem as shown.Equation 8.4.31Equation 8.4.32 X y Y 0 X
The in-plane rotation in polar coordinates is given by relation Using this form with the Flamant displacement solution (8.4.43), show thatand thus conclude that ωz is an even function of θ that
For the Flamant problem with only normal loading, explore the strain energy in a semicircular area of radius R centered at the loading point. Refer to the discussion in Section 5.8, and use relation
Determine the stress field solution (8.4.47) for the problem of a half-space under a concentrated surface moment as shown in Fig. 8.23. It is recommended to use the superposition and limiting process
Show that the problem of a half-space carrying a concentrated surface moment (see Fig. 8.23) can also be solved using the Airy function form∅ = a4θ+ b24 sin 2θ.Fig 8.23 d P M y
Working in polar coordinates, show that an Airy stress function of the form ∅ = a2r2 + a21r2 cos 2θ (where a2 and a21 are constants) solves the illustrated problem of a shear loaded wedge
For the problem of a half-space under uniform normal loading as shown in Fig. 8.24, show that the maximum shear stress can be expressed by:Plot the distribution of lines of constant maximum shear
Generalize the integral superposition methods used in the examples shown in Section 8.4.9and Exercise 8.40. In particular, show that the stress solution for a half-space carrying general normal and
Using the formulation and boundary condition results of the thin notch crack problem shown in Fig. 8.29, explicitly develop the stress components given by relations (8.4.56) and (8.4.57).Fig
Consider the crack problem shown for the anti-plane strain case with u = v = 0, w = w(x,y). From Section 7.4, the governing equation for the unknown displacement component with zero body force was
For the anti-plane stain crack problem solved in Exercise 8.44, plot contours of the displacement and stress fields.Data from exercise 8.44Consider the crack problem shown for the anti-plane strain
For the mode III crack problem in Exercise 8.44, explore the strain energy in a circular area of radius R centered at the crack tip. Refer to the discussion in Section 5.8 and use relation (6.1.10)
Using strength of materials theory (see Appendix D), the bending stress sq for curved beams is given by σθ =–M(r– B)/[rA(R– B)], where A = b –a, B = (b– a)/log(b/a), R = (a + b)/2. For
For the disk under diametrical compression (Fig. 8.35), plot the distribution of the two normal stresses σx and σy along the horizontal diameter (y = 0, –R< x < R ).Fig 8.35 P 0 T 1 02 P X
The behavior of granular materials has often been studied using photo elastic models of circular particles as shown in the following figure. This provides the full-field distribution of local contact
Solve the rotating disk problem of Example 8.11 for the case of an annular disk with inner radius a and outer radius b being stress free. Explicitly show that for the case b >> a, the maximum stress
Assume that we are dealing with a two-dimensional deformation problem in the x,y-plane where all field variables depend only on x and y. For the following strain fields (A and B are constants),
Show that the general plane strain edge dislocation problem shown in Fig. 15.3 can be solved using methods of Chapter 10 with the two complex potentials:where b = bx + iby. In particular, verify the
Justify that the edge dislocation solution (15.1.2) provides the required multivalued behavior for the displacement field. Explicitly develop the resulting stress fields given by (15.1.3) and
Show that the screw dislocation displacement field (15.1.5) gives the stresses (15.1.6) and (15.1.7).Equation 15.1.5Equation 15.1.6Equation 15.1.7 u = v=0 b 2 W = tan -1 X
For the edge dislocation model, consider a cylinder of finite radius with axis along the dislocation line (z-axis). Show that although the stress solution gives rise to tractions on this cylindrical
The stress field (15.1.7) for the screw dislocation produces no tangential or normal forces on a cylinder of finite radius with axis along the dislocation line (z-axis). However, show that if the
Show that the strain energy (per unit length) associated with the screw dislocation model of Example 15.2 is given by:where Ro is the outer radius of the crystal and Rc is the core radius of the
Using similar notation as Exercise 15.6, show that the strain energy associated with the edge dislocation model of Example 15.1 can be expressed by:Note that this energy is larger than the value
For the Kelvin state as considered in Example 15.4, explicitly justify the displacement and stress results given in relations (15.2.8) and (15.2.10).Data from example 15.4Equation 15.2.8Equation
Verify that the displacements and stresses for the center of compression are given by (15.2.21)and (15.2.22).Equation 15.2.21Equation 15.2.22 u Xi 2R 1 (3xxj - dij R R
A fiber discontinuity is to be modeled using a line of centers of dilatation along the x1-axis from 0 to a. Show that the displacement field for this problem is given by:Fig 5.11 where R = 1 - - -
For the isotropic self-consistent crack distribution case in Example 15.12, show that for the case v = 0.5, relation (15.3.4)3 reduces to:Verify the total loss of moduli at ε = 9/16. Using these
Develop the compatibility relations for couple-stress theory given by (15.4.12). Next, using the constitutive relations, eliminate the strains and rotations, and express these relations in terms of
Explicitly justify that the stress-stress function relations (15.4.14) are a self-equilibrated form.Equation 15.4.14 Ox = Txy mxz 22 2 2 22 32 2 myz Tyx 22 22 + 2 22 32 + 20
For the couple-stress theory, show that the two stress functions satisfy: = 0= +-zd'0=+
Using the general stress relations (15.4.25) for the stress concentration problem of Example 15.13, show that the circumferential stress on the boundary of the hole is given by:Verify that this
Starting with the general relations (15.5.6), verify that the two-dimensional plane stress constitutive equations for elastic materials with voids are given by (15.5.9).Equation 15.5.6Equation 15.5.9
For elastic materials with voids, using the single strain-compatibility equation, develop the stress and stress function compatibility forms (15.5.10) and (15.5.11).Equation 15.5.10Equation 15.5.11
Compare the hoop stress σθ(r,π/2) predictions from elasticity with voids given by relation (15.5.18) with the corresponding results from classical theory. Choosing N = 1/2 and L = 2, for the
For the doublet mechanics Flamant solution in Example 15.15, develop contour plots (similar to Fig. 15.22) for the microstresses p1 and p2. Are there zones where these microstresses are tensile?Data
Consider the gradient elasticity problem under a one-dimensional deformation field of the form u = u(x), v = w = 0. Using constitutive form (15.7.3), determine the stress components. Next show that
Starting with the given displacement form (15.7.8), explicitly justify relations (15.7.9) – (15.7.12) in the gradient dislocation Example 15.16Equation 15.7.8Equation 15.7.9Equation 15.7.12Data
Using integral tables verify that the gradient Flamant stress integral solution in Example 15.17 reduces to the classical elasticity form as per equations (15.7.20).Data from example 15.17Equation
Starting with the general linear form (16.2.1), verify the interpolation relations (16.2.4) and (16.2.5).Equation 16.2.1Equation 16.2.4Equation 16.2.5 u(x,y) = C+Ccx + c3y
For the constant strain triangular element, show that the stiffness matrix is given by (16.3.16).Equation 16.3.16 [K] = he 4Ae [CH+YC66 BIYI C12+B1Y1C66 YC2+ BC66 818 C11 +Y1Y2C66 B2Y1 C12 +1Y2C66
For the case of element-wise constant body forces, verify that the body force vector is given by relation (16.3.17) for the linear triangular element.Equation 16.3.17 he Ae {F} = !e {F&Fy E FEFy}! 3
Verify boundary relation (16.3.19) for the linear triangular element with constant boundary tractions Tnx and Tny.Equation 16.3.19 ~ LT {F} a 7 dS = he he T12 T12 VT VT VT VT 437" V3T y dS he L12 2
For Example 16.1, show that the element stiffness equations for the isotropic case are given by relations (16.4.1) and (16.4.2).Data from example 16.1Equation 16.4.1Equation 16.4.2 Consider the plane
Verify the nodal displacement solution given by (16.4.5) in Example 16.1.Equation 16.4.5Data from example 16.1 U V U3 V3 0.492 0.081 0.441 -0.030 Tx 10- m
Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the stress concentration problem under biaxial loading given in Exercise 8.25. Compare the stress concentration factor from
Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the stress concentration problem under shear loading given in Exercise 8.26. Compare the stress concentration factor from
Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the curve beam problem shown in Fig. 8.32. At the fixed section, compare numerical stress results (σx)with analytical
Using the MATLAB PDE Toolbox (or equivalent), develop an FEM solution for the torsion of a cylinder of circular section with circular keyway as shown in Exercise 9.23. Verify the result of Exercise
Verify the traction relation (16.6.7) for the plane strain case.Equation 16.6.7 Pij = Tikink = 4n(1 - v)r [(1-2 -dij+rinj-rjni+2 r
From strain energy arguments in Section 6.1, it was found that ∂σij/∂ekl = ∂σkl/σeij. Show that these results imply that Cij = Cji, therefore justifying that only 21 independent elastic
Verify the inequality restrictions on the elastic moduli for orthotropic, transversely isotropic, cubic and isotropic materials given by Table 11.1.Table 11.1 Table 11.1 Positive definite
For the orthotropic case, show that by using arguments of a positive definite strain energy function, v2ij i/Ej). Next, using typical values for E1 and E2 from Table 11.2, justify that this theory
Similar to Example 11.1, consider the hydrostatic compression of a material with cubic symmetry. Determine the general expressions for the strains. We now specify that the material is aluminum with
For the torsion of cylinders discussed in Section 11.4, show that with σx = σy = σz = τ xy = 0, the compatibility equations yield:Data from section 11.4As our first example, consider the torsion
In terms of the stress function Ψ, the torsion problem was governed by Eq. (11.4.9):Equation 11.4.9 S44xx - 2S45xy + S554 yy = -2 Show that the homogeneous counterpart of this equation may be
Explicitly justify relationships (11.5.3) between the compliances of the plane stress and plane strain theories.Equation 11.5.3 B11 B22 B66 = S11 S33-S2 $33 S22 S33-S23 S33 S66 S33 - S36 S33 B12 B16
Investigate case 2 (μ1 = μ2) in Eq. (11.5.10) and determine the general form of the Airy stress function. Show that this case is actually an isotropic formulation.Equation 11.5.10 Case 1: = a +i
Determine the roots of the characteristic equation (11.5.7) for S-Glass/Epoxy material with properties given in Table 11.2. Justify that they are purely imaginary.Equation 11.5.7Table 11.2 S112S16M+
Recall that for the plane anisotropic problem, the Airy stress function was found to be: o=F1(21) + F1(21) + F2(22) + F2(22) where Zi = x + y and z2 = x + 2y. Explicitly show that the in-plane
For the plane stress case, in terms of the two complex potentials ∅1 and ∅2, compute the two in-plane displacements u and v and thus justify relations (11.5.14).Equation 11.5.14 u(x,y) = 2Rep1
Determine the polar coordinate stresses and displacements in terms of the complex potentials ∅1 and ∅2, as given by Eqs. (11.5.16) and (11.5.17).Equation 11.5.16Equation 11.5.17 2Re [(sinu cose)
For the plane problem with an orthotropic material, show that the characteristic Eq. (11.5.7) reduces to the quadratic equation in μ2:Justify the isotropic case where β1,2 = 1. Finally, determine
Consider an anisotropic monoclinic material symmetric about the x,y-plane (see Fig. 11.2) and subject to an anti-plane deformation specified by u = v = 0, w = w(x,y). Show that in the absence of body
For Example, 11.5, consider the case of only a normal boundary load (X = 0), and assume that the material is orthotropic with μi = iβi (see Exercise 11.15). Show that the resulting stress field is
Consider the case of the pressurized circular hole in an anisotropic sheet. Using orthotropic material properties given in Table 11.2 for Carbon/Epoxy, compute and plot the boundary hoop stress σθ
Investigate the case of a circular hole of radius a in Example 11.7. Use orthotropic material properties given in Table 11.2 for Carbon/Epoxy with the 1-axis along the direction of loading. Compute
Consider the elliptical hole problem in Example 11.7. By letting a→0, determine the stress field for the case where the hole reduces to a line crack of length 2b. Demonstrate the nature of the
The potentials: were proposed to solve the plane extension of an anisotropic panel containing a crack of length 2a (see Fig. 11.12). Recall that the constants A1 and A2 correspond to the uniform
Construct a contour plot of the crack tip stress component σy from solution (11.6.7). This result could be compared with the equivalent isotropic problem from Exercise 10.24.Equation 11.6.7Data
Consider the case of a crack problem in an anisotropic monoclinic material under anti-plane deformation as described in Exercise 11.16. Following relation (11.6.6), choose the complex potential form
Explicitly develop the governing Navier equation (11.7.5) for the polar orthotropic problem.Verify that its solution is given by (11.7.6) and show how this leads to the stress solution (11.7.7).

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