## U

where [A'] is the required assembled stiffness matrix.

### Example 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, v = 0.3, and plane stress condition can be assumed.

40 kN

40 kN

(a) Taking advantage of any symmetry, using two triangular membrane elements and hence the assembled stiffness matrix derived for the previous Example, 9.6, determine the nodal displacements in global coordinates.

(b) Determine the corresponding element principal stresses and their directions and illustrate these on a sketch of the continuum.

### Solution

(a) Advantage can be taken of the single symmetry by modelling only half of the continuum. Figure 9.48 shows suitable node and dof. labelling, and division of the upper half of the continuum into two triangular membrane elements. Reference to the previous Example, 9.6, will reveal that the assembled stiffness matrix derived in answering this question can, conveniently, be utilised in solving the current example.

To simulate the clamped edge, dofs. 5 to 8 need to be suppressed, i.e. «3 = V3 = «4 = V4 = 0. Additionally, whilst node number 2 should be unrestrained in the jc-direction, freedom in the ^-direction needs to be suppressed to simulate the symmetry condition, i.e. v2 = 0. Applying these boundary conditions and hence partitioning the structural stiffness matrix result from Example 9.6, gives the reduced equations as

xn

0 0