## Wll

The required nodal reactions are therefore = 27.27 kN, M\ = 18.18 kNm, Y2 = -20.45 kN and Y3 = -6.82 kN.

Representing these reactions together with the applied moment on a sketch of the deformed beam, Fig. 9.38, and considering force and moment equilibrium, gives

Example 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

Section A-A

channels that the relevant second moment of area, / = 2 x 10~8 m4 and cross-sectional area, A — 4 x 10~~4 m2. The bracket can be idealised as two beams, the common junction of which can be assumed to be infinitely stiff and the other ends to be fully restrained. Using the displacement based finite element method, and representing the constituent members as simple beam elements:

(a) assemble the necessary terms in the structural stiffness matrix;

(b) hence, determine for the condition shown in Fig. 9.39 (i) the nodal displacements with respect to the global coordinates, and (ii) the combined axial and bending extreme fibre stresses at the built-in ends and at the common junction.

### Solution

(a) Figure 9.40 shows suitable node, dof. and element labelling. The structure does not have symmetry or redundant members. The least number of beam elements will be used to

1 kN

1 kN

7777rm

0.10 m minimise the hand calculations which, in this example, is two. Both elements will have the same A, E and /, i.e. {A, E, I)ia) = (A, E, I)(b) = A, E, /, but will have different lengths, i.e. L(a) and Lih).

With reference to §9.8, the element stiffness matrix inclusive of axial terms and in global coordinates is appropriate, namely:

A cos2 a + (12/ sin2 a)/L2, (A — 12//L2)cosasina, —(6/sina)L,

—A cos2 a — (12/ sin2 a)/L2, —(A — 12//L2)cosasinoi, . —(6/sina)/L,

—(A — 12//L2)cosasina, —A sin2 a — (12/cos2a)/L2, (6/cosor)/L,

A cos2 a + (12/ sin2 a)/L2, (A — 12//L2)cosasina, (6/ sin a)/L,

A cos2 a + (12/ sin2 a)/L2, (A — 12//L2)cosasina, (6/ sin a)/L,

Evaluating, for both elements, only those stiffness terms essential for the analysis:

Element a

L(a) = o.lm, otla) = 180°, cosoi(a) = -1. sina(al = 0

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