Generation of simultaneous equations

Table 3.1 is intended to simplify the generation of the matrix [K] in equation (3.7) for which the solution provides the nodal deflections. As an example, when the two edges are free (see Figure 3.1 and equation 3.8), the coefficients in the first and last two rows of [K] are the coefficients of w in equations (3.14) and (3.15). If the edge at node 1 (or n) is simply supported, the first (or last) row and column of the matrix in equation (3.8) are to be deleted (resulting in a smaller matrix). Similarly, if the deflection is prevented by the introduction of a support at any intermediate node j, the order of [K] is reduced by the deletion of the jth row and column. Examples 3.1 and 3.2 are chosen such that all the conditions discussed above and in the previous section are applied.

the nodes of simple beams of span λ loaded by external loading q (force/length2 for a circular wall and

force/length for a beam). For any continuous variation of the load intensity, the following equation may be used: (3.20)

which gives Q1 and Qn at the edge nodes and Qi at a typical node i. This involves the assumption of parabolic

variation of q over any three consecutive nodes. 3.5 Sudden change in thickness

When a circular wall or a beam on elastic foundation has a sudden change in flexural rigidity at a node i, an effective value EIi is to be used in the finite-difference equations in Table 3.1 and 3.2. The effective flexural

rigidity is given by the equation5:

(3.21)

in which the subscripts L and R refer to the two sections just to the left and right of node i. A sudden change in thickness in a circular wall corresponds also to a sudden change in k value; at node i where the change occurs, ki

is to be taken as the average of the values just before and after i.

Figure 3.2 Prestressed concrete circular wall with loading representing condition before completion of the application of the prestress (Example 3.1).

3.6 Examples

Example 3.1 Circular wall, on bearing pads, with linearly varying load

A cricular concrete wall (Figure 3.2a) is subjected to circumferential prestressing providing inward load normal to the surface of linearly varying intensity covering half the wall height. The top edge of the wall is free and the bottom edge is supported on bearing pads providing an elastic support of stiffness K=0.6×10−3E, where E is the modulus of elasticity of the wall material. Find the bending moment and the hoop force variation over the wall height as well as the value of the shearing force at the bottom edge. The wall dimensions are shown in the figure; Poisson’s ratio, v=0.2.

The loading covering half the wall only represents the condition during the application of the prestress. In some cases loading with such partial prestress may produce a larger moment than when the prestress is completed covering the whole height.

Nine equally spaced nodes are shown in Figure 3.2(b), with λ=l/8. At each node the thickness hi and the values

EIi, Ji, Ci are calculated, using equations (3.2) to (3.6). The equivalent concentrated loads Qi are listed in the last

column of the table in Figure 3.2(b) (see equation 3.20).

Applying the appropriate finite-difference equation from Table 3.1 at each of the nodes 1, 2,…, 9 gives a group of simultaneous equations [K]{w]={Q} in which

It can be seen that this matrix is calculated by substitution in equation (3.8), which is applicable when the two edges of the wall are free. To account for the effect of the bearing pad at node 1, the value K=0.6×10–3_{E is}

added to element (1, 1) of the matrix. (Note that K=0.3645EI1/λ3.)

Solution of the simultaneous equations (3.7) gives the deflections at the nodes.

{w}=(−10–3q/λ3/EI1){58.6872, 68.6176, 69.1763, 58.0965, 33.1659, 8.6600, −0.8389, −1.9454, −1.2114}.

Substitution in equations (3.9) and (3.10) gives the hoop force and the bending moment at each of the nine nodes:

{N}=qr{−0.761, −0.852, −0.822, −0.659, −0.358, −0.089, 0.008, 0.018, 0.010}

Figure 3.3 Bending moment and hoop force in a prestressed concrete circular wall in a construction stage when prestressing is applied only on the lower half of the height (see Figure 3.2a).

Applying equation (3.17) (in Table 3.2) at node i=1 and using the values of Q, J and C listed in Figure 3.2(b) gives the reaction at the bearing pad, which is equal to the required shearing force at the bottom edge

This value should, of course, be equal to the product kw1. The variation of the hoop force and the bending

moment over the wall height are depicted in Figure 3.3.

Example 3.2 Circular wall, with bottom edge encastré and top edge hinged

Consider a circular wall with the same dimensions as in Figure 3.2(a) but which has the bottom edge encastré and the top edge hinged. Find the bending moment and the shear at the bottom edge due to a uniform outward load of intensity q. Use the same spacing λ as in the previous example. Assume Poisson’s ratio v=0.2.

The matrix [K] is obtained simply by adding 2J1=2×1.0=2.0 to element (2,2) of the corresponding matrix in the

preceding example and deleting the first and the last rows and columns, This can be seen by applying equations (3.13) and (3.12) at nodes 2 and 8, respectively (see Table 3.1 and Figure 3.2b). In this way the matrix is

reduced to 7×7, which is used to find the deflection at the seven nodes 2 to 8 by solving equation (3.7). The vector {Q} has seven elements each equal to ql/8. Solution of the equations gives:

{w2, w3,…, w8}=(qλ4/EI1){0.30016, 0.57725, 0.71548, 0.77785, 0.82486, 0.83753, 0.65330}.

For the shear at the bottom edge, use equation (3.18) in Table 3.2(c):

with Q1=ql/16, R1=V1=0.14008ql.

The accuracy of the finite-difference solution is improved by reducing λ and increasing the number of nodes. To study the effect of varying λ on the results, the above example is solved with λ=l/10,l/20, l/40, l/60 and l/70,

giving the results in Table 3.3.

Table 3.3 indicates that in this particular example little gain in accuracy is achieved by choosing λ smaller than l/40.

Table 3.3 Answers to Example 3.2 solved several times with varying λ

λ Multiplier

l/8 l/10 l/20 l/40 l/60 l/70

M1 94 104 119 123 123 124 10−4ql2

V1 1401 1436 1511 1535 1540 1541 10−4ql