Statically Indeterminate 2-D Structures

The majority of side frames of skeletal furniture belong to systems internally statically indeterminate (Fig. 6.22). The degree of static indeterminate of a frame is specified by the equation:

s = (r + h) — 3t,

where

s number of unknowns exceeding the number of equilibrium equations (for 2-D systems, we have three equilibrium equations), r number of passive forces (support rods), t number of shields (rods) of the system, h number of glue-lines joining the shields together.

For example, from Fig. 6.22, we have, respectively,

r = 3; t = 7, h = 21,

hence,

s = (3 + 21) – 3 x 7 = 3.

The system is therefore threefold internally statically indeterminate.

In order to solve the system of the established degree of static indetermination, it is transformed into a statically determinate system, hereafter called the basic sys­tem. A basic system is formed from the statically indeterminate system by removing n nodes (adding the system s degrees of freedom) and placing in their place attributable generalised forces Xb X2, …, X3 called overvalues. In order to obtain the basic system, we can reject both external nodes (reactions) and internal nodes (by cutting the rod). Figure 6.22 shows the side frame of the chair before and after rejecting the internal nodes. Rejecting s nodes causes that in the primary system in place of the deleted nodes, s displacements can be formed, which could not be formed in an indeterminate system. The primary system becomes identical with a given statically indeterminate system if the relative displacements caused by simultaneous action of all supernumerary sizes and active loads are equal to zero. This condition can be written in the form of a so-called canonical system of equations by the method of forces, which takes the form:

X

coefficient dependent on the shape of the cross section of the rod (k =1, 2—rectangle),

static moment of the field of cross section found above the line

parallel to the neutral axis,

moment of inertia of the cross section,

width of the cross section,

internal forces caused by internal load, Xi = 1, located in the point of cutting the construction in thought, internal forces caused by the load, Xk.

Values of integrals are as follows:

M-Mkdx, NiNk dx, TiTk dx, (6.15)

we calculate by graphic integration.

If the function Ф is linear, then the integral UUdx is equal to the product of

the area Q of the function Ф and the ordinate n of the function Ф in cross section, in which lies the centre of gravity of the field of function Ф (Fig. 6.23). To simplify, in Table 6.1, fields of surfaces of the more important 2-D figures and locations of centres of gravity have been put together, and Table 6.2 shows ready equations for calculating certain integrals of this type.

Based on the theory about the reciprocity of displacements, coefficients located symmetrically in relation to the global diagonal of the system of canonical equa­tions are equal to one another, that is,

dik = dki. (6.16)

To calculate the coefficients di0, the following equation is used:

(6.17)

x

where

M0, N0, T0 internal forces caused in the primary system statically determinate by the action of external causes.

Table 6.1 The areas and location of centres of gravity of certain 2-D figures

In many cases, for frame systems, the impact of cutting and normal forces is ignored, by applying the following equation for calculations:

(6.18)

After determining all coefficients Sik Si0, the system of equations should be solved by determining the sought overvalues Xb X2, …, XS. Definitive values of reactions R, bending moments M, cutting forces T and longitudinal forces N, anywhere in the statically indeterminate system, are determined from the general superposition equations:

R = R0 + X1R1 + X2R2 + ••• + XsRs,

M = M0 + X1M1 + X2M2 + ••• + XsMs ;

(6.19)

T = T0 + X1T1 + X2T2 + ••• +XsTs,

N = N0 + X1N1 + X2N2 + • • • +XsNs,

in which values corresponding to static sizes R, M, T, N have been marked by the indexes 1, 2, . , S, determined for particular states Xi = 1, while static sizes caused by an external factor in the statically determinate system are marked by index 0. If for particular supernumeraries Xi only the course of moments M have been deter­mined, and the impact of cutting and normal forces were omitted, then the value of cutting forces, moments and normal forces are calculated by solving the primary statically determinate system loaded by external forces and all the already specified overvalues Xt (Fig. 6.24).

Example

Let us establish the distribution of internal forces M, T, N in the side frame of a chair, as in Fig. 6.24a, made from pinewood (E = 12,000 MPa), whose rails have a constant cross section in the shape of a rectangle with the dimensions b x h=10 x 30 mm. The degree of statistical indetermination of the system is as follows:

s = (3 + 21) – 3 x 7 = 3.

The statically indeterminate system is replaced by a statically determinate one by incising the muntin (Fig. 6.24b). Then, we introduce the virtual load Xj = 1, X2 = 1 and X3 = 1, and we calculate the reactions and course of moments and cutting and normal forces (Fig. 6.25). Therefore,

RX = 0 ) HA = P = 1000 N,

RMa = 0 ) Rb = 2000 N, (6.20)

RY = 0 ) Ra = Rb.

Then, we calculate the coefficients Sik of canonical equation given below:

Xt^n + X21>12 + X3^13 + d10 = 0

Xtd21 + X2 d22 + X3^23 + ^20 = 0 (6.21)

X]d31 + X2S32 + X3^33 + ^30 = 0.

Omitting the negligible share in shearing and normal forces, we obtain:

J12 = J21 = 0 due to the asymmetry of the charts, S13 = S31 = 0 due to the asymmetry of the charts.

^33 — EJ • 2(0.3(-1)(-1) + 0.4(-1) • (-1)) — EJ 1.4; (6.25)

Because the stiffness of all the elements is the same EJ = const, we obtain:

( 0.0347×1 + 31.7 — 0

I 0.054×2 – 0.21×3 – 66 — 0 (6.30)

-0.21×2 + 1.4×3 + 265 — 0.

By solving the system of equations, we obtain the supernumerary values:

x1 = -913.5

X2 = 1166.7 (6.31)

X3 = -14.3.

Graphs of moments in the statically indeterminate system, therefore, have the course as shown in Fig. 6.26.