###### Aircraft Structures for Engineering Students

331 SOLVED PROBLEMS

3 SOLVED PROBLEMS

Question: p.6.14

## The following interpolation formula is suggested as a displacement function for deriving the stiffness of a plane stress rectangular element of uniform thickness t shown in Fig. P.6.14. ...

For a = 1, b = 2 u=\frac{1}{8}\left[(1-x)(2...
Question: p.6.13

## A constant strain triangular element has corners 1(0,0), 2(4,0) and 3(2,2) and is 1 unit thick. If the elasticity matrix [D] has elements D11 = D22 = a, D12 = D1 = b, D13 = D23 = D31 = D32 = 0 and D33 = c derive the stiffness matrix for the element. ...

Assume displacement functions \begin{aligne...
Question: p.6.12

## A rectangular element used in plane stress analysis has corners whose coordinates in metres referred to an Oxy axes system are 1(−2, −1), 2(2, −1), 3(2, 1), 4(−2, 1). The displacements of the corners (in metres) are u1 = 0.001 u2 = 0.003 u3 = −0.003 u4 = 0 v1 = −0.004 v2 = −0.002 v3 = 0.001 v4 ...

Suitable displacement functions are: \begin...
Question: p.6.11

## A square element 1234, whose corners have coordinates x, y (in metres) of (−1, −1), (1, −1), (1, 1), and (−1, 1), respectively, was used in a plane stress finite element analysis. The following nodal displacements (mm) were obtained:u1 = 0.1 u2 = 0.3 u3 = 0.6 u4 = 0.1 v1 = 0.1 v2 = 0.3 v3 = 0.7 ...

From the first of Eqs (6.96) \left.\begin{a...
Question: p.6.9

## It is required to form the stiffness matrix of a triangular element 123 for use in stress analysis problems. The coordinates of the element are (1, 1), (2, 1), and (2, 2), respectively. (a) Assume a suitable displacement field explaining the reasons for your choice. (b) Form the [B] matrix. ...

(a) There are six degrees of freedom so that the d...
Question: p.6.8

## It is required to formulate the stiffness of a triangular element 123 with coordinates (0, 0), (a, 0), and (0, a), respectively, to be used for ‘plane stress’ problems. (a) Form the [B] matrix. (b) Obtain the stiffness matrix [K^e]. Why, in general, is a finite element solution not an exact ...

(a) The element is shown in Fig. S.6.8. The displa...
Question: p.6.7

## The symmetrical rigid jointed grillage shown in Fig. P.6.7 is encastré at 6, 7, 8 and 9 and rests on simple supports at 1, 2, 4 and 5. It is loaded with a vertical point load P at 3. Use the stiffness method to find the displacements of the structure and hence calculate the support reactions ...

The forces acting on the member 123 are shown in F...
Question: p.6.6

## Given that the force–displacement (stiffness) relationship for the beam element shown in Fig. P.6.6(a) may be expressed in the following form: ...

The stiffness matrix for each element of the beam ...
Question: p.6.5

## The frame shown in Fig. P.6.5 has the planes xz and yz as planes of symmetry. The nodal coordinates of one quarter of the frame are given in Table P.6.5(i). In this structure the deformation of each member is due to a single effect, this being axial, bending or torsional. The mode of deformation ...

Referring to Fig. P.6.5, $u_2$ = 0 fr...
Question: p.6.4

## The symmetrical plane rigid jointed frame 1234567, shown in Fig. P.6.4, is fixed to rigid supports at 1 and 5 and supported by rollers inclined at 45° to the horizontal at nodes 3 and 7. It carries a vertical point load P at node 4 and a uniformly distributed load w per unit length on the span 26. ...

The uniformly distributed load on the member 26 is...
Question: 24.3

## A three-flange wing section is stiffened by the wing rib shown in Fig. P.24.3. If the rib flanges and stiffeners carry all the direct loads while the rib panels are effective only in shear, calculate the shear flows in the panels and the direct loads in the rib flanges and stiffeners. ...

A three-flange wing section is statically determin...
Question: 20.5