Stresses in Construction Elements

A large part of the furniture of a skeletal as well as case structure is based on load-bearing solutions called frames. They are more often exposed to loads than other construction components of the product. Therefore, the construction solution for this part of the product should ensure its sufficient strength. Stiffness of the frame and stresses that occur in its construction elements are dependent on the positioning of the elements in relation to the base, on the positioning of the legs and bars, as well as on the geometry of elements, i. e. variability in their cross section. With a certain linear variable stiffness of the construction components of the frame (legs), it is necessary to change the smallest cross section of the loaded element and the method and location of mounting the bar. It is also easy to identify dangerous places in these elements with the changing dimensions of their cross sections. To this end, we will load the front edge of the table, as it is shown in Fig. 6.110. For a construction without a muntin, the calculation scheme is the statically determinate frame, loaded as in Fig. 6.110a. The distribution of bending moments, being the result of the load, is shown in Fig. 6.111. Assuming that the chair leg is similar to a roller, i. e. it is characterised by a constant stiffness, and then, the biggest bending stresses will occur at the place of its connection to the rail, which can be written as

Fig. 6.111 Distribution of bending moments in the frame of a table without rail

whereby

where

ffg normal stresses from bending, Mg bending moment,

F concentrated force.

The value of these stresses in cross sections в-в at any height bp_p depends rectilinearly on the length b (Fig. 6.112a). By selecting at the same time the loaded joint, the value of the bending moment shifted by it can be reduced, using the side bars as shown in Fig. 6.110b.

In many constructions of case furniture and frames of case furniture, legs with variable cross-sectional geometry are designed, both in the form of slant cones and

Fig. 6.113 Calculation scheme for a leg with variable cross section

legs turned with profile, whose cross sections change from top to bottom. Therefore, for the scheme shown in Fig. 6.112 and with the leg convergence above 6°, for which the hypothesis of permanent cross sections has been rejected, bending stresses along the length of the leg change in a disproportionate manner. By con­sidering the following geometric scheme (Fig. 6.113) and assuming proportions determining the positioning of the examined cross section, we introduce the relations:

The value of these stresses in cross sections в-в at any height bp_p depends rectilinearly on the length b (Fig. 6.112a). By selecting at the same time the loaded joint, the value of the bending moment shifted by it can be reduced, using the side bars as shown in Fig. 6.110b.

In many constructions of case furniture and frames of case furniture, legs with variable cross-sectional geometry are designed, both in the form of slant cones, as well as legs turned with profile, which cross sections change from top to bottom. Therefore, for the scheme shown in Fig. 6.112 and with the leg convergence above 6°, for which the hypothesis of permanent cross sections has been rejected, bending
stresses along the length of the leg change in a disproportionate manner. By con­sidering the following geometric scheme (Fig. 6.113) and assuming proportions determining the positioning of the examined cross section, we introduce the relations:

bp-l

b

where

d diameter of the cross section at the leg base,

do diameter of the cross section at the leg bottom.

By introducing the following geometric dependencies,

1 d — d° _ ^ dp-p — d° /6 297t

2 b 2 bp—p’ 1 ;

diameter of the examined cross section at the height bp_p amounts to:

dp—p = d(y(1 — x) + x). (6.298)

For the examined cross section p-p and the appropriate size of legs bp-p, the value of bending stresses is expressed by the equation:

10Fbsina

[d(y(1 — x) + x)]3

Assuming (for example) that the bottom diameter of the chair leg is two times smaller than that of the top, i. e. x = 0.5, then the change in bending stresses on its length (Fig. 6.112b) will amount to: for

y = 0.25, rp—p = 1.024 • (10/3)Fb sin a = 1.024 • ag, y = 0.50, rp—p = 1.185 • rg, y = 0.75, rp—p = 1.119 • rg, y = 1.00, rp—p = 1.0 • rg.

The most important bending stresses appear in this case halfway up the leg (y = 0.5), rather than in the upper part, that is at the point of the connection with the rail of the chair. The discussed dependencies, for the full variability of dimensions of the leg and its two diameters, are shown in Fig. 6.114. A significant effect of the

change in the geometry of the chair leg on the course of bending stresses must be noted. Maximum bending stresses move in the direction of the upper cross section in case of the increment of the diameter of the lower cross section. Therefore, it can be concluded that at the relation do/d < 0.5, the greatest bending stresses focus not at the place of acting of the biggest bending moment, but in the lower cross sections of the leg and in a way that clearly lowers its strength. With these construction solutions, mounting the lateral bar at will is questionable. Because when making a mortise for the tenon or a hole for the dowel in a clearly excessively loaded place, the strength of the already weak cross section is reduced.

Therefore, when considering the scheme from Fig. 6.110, the chair construction can be narrowed down to an overstiff three-time internally statically indeterminate frame, in which the distribution of bending moments can be determined on the basis of energetic methods known in the construction theory. By using for this purpose the Menabrea’s theorem:

where

xi overvalues,

U elastic energy of the entire system,

Mg bending moment,

E linear elasticity module,

J moment of inertia of the cross section.

the values of the subsequent overvalues xi are determined, and graphs of bending moments are made for constructions with various locations of the side bar at the height of the leg (c/b = Z = 0.19, 0.56, 0.72), as in Fig. 6.115. It can be noticed that for the scheme from Fig. 6.110a, the bending moment acting on a connection in a chair without bar with standard load depends only on its geometry. From Fig. 6.115, it results that if this bar is located lower than the seat, it causes a reduction of the value of the bending moment acting on both the leg connections with the rail and with the bar itself. It is accompanied by a reduction of the values of bending stresses. Therefore, the question arises: Will this relation be confirmed for schemes of linear variable stiffness? Therefore, another considerations must be made.

When evaluating bending stress in full cross section of the legs at the points of mounting the bar, as previously, a concurrent profile must be assumed, that is x = 0.5. Because maximum bending moments in these locations for each type of construction from Fig. 6.115 amount to, respectively,

therefore,

for f =0.19, of = 1.092 … og, for f = 0.56, of = 1.178 … og,

and as it can be seen, they evidently exceed the value of bending stresses at the place of connection of the rear leg with the rail. Therefore, the proposed conver­gence of the leg, expressed by the proportion do/d = 0.5, does not provide its rational connection to the bar without compromising the strength of construction. The full relation between the geometry of the leg of a chair and connection of a bar and the value of bending stresses at the place of its connection is shown in Fig. 6.116. It can be seen that the value of bending stresses clearly increases along with the reduction of the bottom diameter of the leg below the proportion x < 0.6. Lowering the position of the bar in relation to the seat additionally worsens the working conditions of the construction, causing an increase of stresses at the places of their connections with the legs.