###### Mechanics of Materials

58 SOLVED PROBLEMS

Question: 9.4

## The vehicle engine mounting bracket shown in Fig. 9.39 is made from uniform steel channel section for which Young’s modulus, E = 200 GN/m². It can be assumed for both.channels that the relevant second moment of area, I = 2 × 10^-8 m^4 and cross-sectional area, A = 4× 10^-4m². The bracket can be idealised ...

a) Figure 9.40 shows suitable node, dof. and eleme...
Question: 9.6

## (a) Evaluate the element stiffness matrix, in global coordinates, for the three-node triangular membrane element, labelled a in Fig. 9.43. Assume plane stress conditions, Young’s modulus, E = 200 GN/m², Poisson’s ratio, ν = 0.3, thickness, t = 1 mm, and the same displacement functions as Example9.5 ...

(a) Figure 9.44 shows suitable node labelling for ...
Question: 9.7

## Figure 9.47 shows a 1 mm thick sheet of steel, one edge of which is fully restrained whilst the opposite edge is subjected to a uniformly distributed tension of total value 40 kN. For the material Young’s modulus, E = 200 GN/m² and Poisson’s ratio, ν = 0.3, and plane stress condition can be assumed ...

(a) Advantage can be taken of the single symmetry ...
Question: 9.5

## Derive the stiffness matrix in global coordinates for a three-node triangular membrane element for plane stress analysis. Assume that the elastic modulus, E, and thickness, t, are constant throughout, and that the displacement functions are u(x, y) =α1 + α2x + α3y υ(x, y) = α4 + α5x + α6y ...

With reference to §9.9 and with respect to the nod...
Question: 9.3

## A steel beam is supported and loaded as shown in Fig. 9.36. The relevant second moments of area are such that I^(a) = 2I^(a) = 2 × 10^-5 m^4 and Young’s modulus E for the beam material = 200 GN/m². Using the displacement based finite element method and representing each member by a simple beam ...

(a) Figure 9.37 shows suitable node, dof. and elem...
Question: 9.2

## Figure 9.34 shows the members and idealised support conditions for a roof truss. All three members of which are steel and have the same cross-sectional area such that AE = 12 MN throughout. Using the displacement based finite element method, treating the truss as a pin-jointed plane frame and each ...

(a) Figure 9.35 shows suitable node, dof. and elem...
Question: 9.1

## Figure 9.31 shows a planar steel support structure, all three members of which have the same axial stiffness, such that AE/L = 20 MN/m throughout. Using the displacement based finite element method and treating each member as a rod: (a) assemble the necessary terms in the structural stiffness ...

(a) Figure 9.32 shows suitable node, dof. and elem...
Question: 3.2

## (a) Determine the “shape factor” of a T-section beam of dimensions 100 mm × 150 mm x 12 mm as shown in Fig. 3.38. (b) A cantilever is to be constructed from a beam with the above section and is designed to carry a uniformly distributed load over its complete length of 2 m. Determine the maximum u.d ...

(a)                           \text { Shape...
Question: 4.5

## An initially unstressed short steel cylinder, internal radius 0.2 m and external radius 0.3 m, is subjected to a temperature distribution of the form T =a + b loge r to ensure constant heat flow through the cylinder walls. With this form of distribution the radial and circumferential stresses at ...

$T=a+b\log _{e}r$ 200=a+b\log ...