1. A heavy straight beam of flexural rigidity

EI and weight

\(w_0\) per unit length rests on a rigid plane foundation. One end is subjected to a moment

\(M_0\) , as shown in Fig.

13.16 . Find the length

a of the segment that separates from the foundation, the reaction at the end and the concentrated contact force at the separation point.

Fig. 13.16 A heavy beam lifted off a support by an end moment

2. Suppose the moment \(M_0\) in Problem 1 and Fig. 13.16 is negative, we then expect the end of the beam to separate from the foundation. Find the length of the separation segment and the concentrated contact force at the separation point.

3. A simply supported beam of length L and flexural rigidity EI is indented by a symmetric rigid body with a fourth power surface defined by the initial gap function \(g_0(x)\!=\!Cx^4\) , where C is a constant. Show that contact occurs at two points separated by a distance a as in Fig. 13.2 and find the relation between a and the indenting force P .

4. Two identical straight beams each of flexural rigidity

EI are glued together. A frictionless wedge of angle

\(2\alpha \) is now driven into the end to separate the joint as shown in Fig.

13.17 . If the cohesion of the glued joint can be defined by an interface energy

\(\varDelta \gamma \) per unit length, use an energy argument to find the relation between the dimensions

a ,

b where

\(z\!=\!a\) defines the location of the apex of the wedge and

\(z\!=\!b\) is the separation point. Find the concentrated moment transmitted between the beams at

\(z\!=\!b\) and show that it is independent of

\(\alpha , a\) and

b . Find also the axial force

F exerted on the wedge.

Fig. 13.17 Two glued beams separated by a wedge

5. Solve Problem 4 for the case where there is friction between the wedge and the beams with friction coefficient f .

6. If two carbon nanotubes are almost parallel, van der Waals forces will tend to make them adhere together. Approximate the solution of this problem by treating each tube as a curved beam of mean radius R and flexural rigidity EI per unit length [along the axis of the tube]. Assume that an arc segment \(2\alpha \) in each beam is in contact [and hence by symmetry is bent into a plane surface] and use an energy argument to find a relation between \(\alpha , EI,\) and the interface energy \(\varDelta \gamma \) per unit area.

7. Use the method of Sect. 13.1.5 to estimate the distribution of contact pressure in the problem of Fig. 13.3 , if the beam cross section is rectangular with height h and width b and the material has density \(\rho \) and Young’s modulus E .

8. A large flexible plate of weight

w per unit area rests on a rigid horizontal foundation. A vertical force

P is applied to the plate, causing a circular region of radius

a to lose contact with the foundation as shown in Fig.

13.18 . Show that

$$ a=\sqrt{\frac{2P}{\pi w}}. $$

Fig. 13.18 A heavy circular plate lifted by a central force

9. A small rigid spherical particle of radius

R is trapped between two identical thin plates, each of thickness

t as shown in Fig.

13.19 . Adhesive forces, with interface energy

\(\varDelta \gamma \) cause the plates to adhere to each other at points distant from the particle. Assuming (i) that the separated region is circular and of radius

a , and (ii) that the contact between the sphere and each plate can be approximated by a concentrated force

P , find the values of

P and

a in terms of

\(R, \varDelta \gamma \) and the stiffness of the plates

D .

Fig. 13.19 Two adhered plates separated by a spherical particle

10. A rigid flat-ended cylinder of radius b is pressed into a circular plate of radius a by a force P that acts through the point \(c,\theta \) . The plate is built in at \(r\!=\!a\) . Assuming that contact occurs for all \(\theta \) at \(r\!=\!b\) , find the force per unit length around this line as a function of \(\theta \) and the angle of tilt of the cylinder. What is the maximum value of c for which this assumption is reasonable?

11. Figure

13.20 shows an elastic membrane of thickness

h and modulus

E that is being pulled away from a rigid plane to which it adheres with interface energy

\(\varDelta \gamma \) . If the applied tensile stress is

\(\sigma \) and it is applied in a direction inclined at an angle

\(\alpha \) to the plane, find the value of

\(\sigma \) for which further decohesion can be expected to occur. Will your solution also apply to the cases (i)

\(\alpha \!=\!0\) and (ii)

\(\alpha \!>\!\pi /2\) ?

Fig. 13.20 An adhered membrane pulled away from a plane support

12. Suppose that the rigid reinforcing ring in Sect. 13.4 has a circular cross section of radius \(b\!\gg \!\varDelta \) . In other words, it is a toroid. Find the extent of the contact area and the contact pressure distribution, including any concentrated forces that may arise.

13. A frictionless rigid conical wedge is driven into a cylindrical tube by an axial force \(F_0\) , as shown in Fig. 13.21 . The wedge angle \(\alpha \!\ll \!1\) .

Assuming that contact occurs only at the end of the tube as shown, find an expression for the radial displacement of the tube

\(u_r\) as a function of

z and

\(F_0\) . Hence, show that there is a minimum angle

\(\alpha _0\) for which contact is restricted to the end and find its value. The cylinder has a radius

a and thickness

t and the material has Young’s modulus

E and Poisson’s ratio

\(\nu \) . What will happen if

\(\alpha \!<\!\alpha _0\) ?

Fig. 13.21 A frictionless conical wedge pressed into a cylindrical shell