Stiffness and Strength Analysis of Skeletal Furniture

6.1 Properties of Skeletal Furniture

Constructions of skeletal furniture belong to the group of multiple statically inde­terminate spatial systems. The analytical solution to the distribution of internal forces or displacements of nodes from the point of view of accounting is a very laborious task. A simple stool with a bar can be a system that is 30-fold internally statically indeterminate, while an armchair with armrests usually constitutes a system 62-fold indeterminate. This means that in order to calculate it, a system of 62 equations with 62 unknowns must be built. Therefore, in solving spatial systems numerical methods are applied.

However, due to the symmetry of the construction of skeletal furniture (Fig. 6.1) and the symmetry of load, analytical solutions can be reduced to stiffness-strength calculations of side frames of this furniture. The side frame of a furniture piece can be created from beam elements connected together in a stiff or articulated way. If we connect four items together articulately, then instead of a construction we shall obtain a mechanism (Fig. 6.2b) that does not bring external loads. In this situation, the applied joints provide the free rotation of each of them in respect of one another, thereby causing significant changes of the angles contained between the compo­nents of the system. Using perfectly stiff connections in the location of the articulate joints, we transform the mechanism into a construction, which sustains slight deformations caused only by the deformation of its components, while the nodes still remain undeformed (Fig. 6.2a).

Of course, not all systems containing articulate joints are mechanisms. Among the many furniture constructions, there are also those which are built from three-element systems. If, in fact, three of the structural elements of the furniture are connected articulately (Fig. 6.3a-c), then we shall obtain a skeleton shifting the external load, while the only type of force occurring in these elements shall be axial forces of compressing or stretching nature. By using two three articulated frames, a garden chair like in Fig. 6.3b can be built. In the discussed constructions, the

© Springer International Publishing Switzerland 2015 319

J. Smardzewski, Furniture Design,

DOI 10.1007/978-3-319-19533-9_6

Fig. 6.1 Symmetry of construction of skeletal furniture

Fig. 6.2 Deformations of subassemblages of furniture with nodes that are a stiff, b articulate

Fig. 6.3 Side frames of furniture: a-c trusses, d, e frames

Fig. 6.4 The impact of how chairs are supported on deformations of side frames: a unmovable support under the front leg, b movable support under the front leg

stiffness of frames depends on the arrangement of the elements in the shape of a triangle and not on the stiffness of joints. Constructing such joints is not a com­plicated task. Therefore, below we will deal with frame constructions, in which particular elements are connected together in a stiff way (Fig. 6.3d, e).

The simplest side frame construction of a chair is a beam of a curved axis (Fig. 6.4). The stiffness and strength of such a frame depends not only on the stiffness and strength of elements and structural nodes, but also on the way the tested construction is supported. The overstiffness of the system by an unmovable support of the front and rear leg (Fig. 6.4a) results in the bending of both these elements and a double overbending of the rail. This obviously improves the overall stiffness of the structure.

In a system that is externally statically determinate (Fig. 6.4b), only the support leg and rail of the seat are subject to bending. As a consequence, this system is far more susceptible and less durable from the previous one.

It should also be noted that a greater number of double-bent structural elements have a positive effect on the increase of stiffness of the system. And so, the side frame of the chair from Fig. 6.5a is characterised by a four times smaller stiffness in relation to the stiffness of the frame illustrated in Fig. 6.5b. Similarly, the con­struction of the table presented in Fig. 6.5c has a more than four times smaller stiffness than a table frame reinforced by an additional bar (Fig. 6.5d). The cause of this regularity is the fact that in systems internally statically determinate (Fig. 6.5a, c), on the length of each beam, the signs of normal stresses, deriving from bending, are not changed. However, in internally statically indeterminate systems (Fig. 6.5b, c), the elements that make up the frame, subject to double bending, change signs of normal stresses on the same side of the beam. In this way, the construction increases its stiffness.

From the engineering point of view, designing skeletal furniture should consist in, among others, assigning reaction of forces of the base, internal forces acting on particular elements and structural nodes, as well as determining displacements of selected construction points (Fig. 6.6).

By analysing the construction of side frames of skeletal furniture, as flat systems, usually externally statically determinate (Fig. 6.6), we are dealing with three reactions (Fig. 6.7a): force Fx parallel to the X-axis, force Fy parallel to the Y-axis, and the moment Mxy in the XY plane. Each of these reactions appears in the case of support having the character of fixing (Fig. 6.7b). Unmovable support (Fig. 6.7c)

Fig. 6.7 Any flat system of forces: a Fx, Fy vectors of forces, respectively, parallel to the system of coordinates, Sx, Sy, linear displacements in the direction of the forces, Mxy the vector of the moment of forces perpendicular to the plane XY, фху rotation in the plane XY, b fixing, c unmovable support, d movable support in the direction of the X-axis

causes only reactions in the form of axial forces and enables free rotation in relation to the point of support, while a movable support (Fig. 6.7d) generates a vector of reaction force directed vertically, at the same time ensuring freedom of rotation and shifting in the horizontal direction.

In spatial systems, we are dealing with six reactions (Fig. 6.8a): the forces Fx, Fy, Fz parallel, respectively, to the axes X, Y, Z, moments Mxy, Myz, Mxz in the plane XY, YZ, XZ. Each of these reactions appears in the case of support having the character of fixing (Fig. 6.8b). Unmovable support (Fig. 6.8c) causes only reactions in the form of axial forces, at the same time enabling free rotation in relation to the point of support. Shifting support (Fig. 6.8d) generates one vector of force reaction directed vertically, while enabling the freedom of rotation in relation to each of the axes and shifts in the plane ZX.

By shifting the above systems on particular constructional solutions of skeletal furniture frames, it should be noted that in the cross sections of the rods of products in which articulate connections were used, only axial forces occur (Fig. 6.9). This

Fig. 6.8 Any spatial system of forces: a Fx, Fy, Fz vectors of forces, respectively, parallel to the system of coordinates, Sx, Sz, linear displacements in the direction of the forces, Mxy, Mxz, Myz, vectors of moments of forces perpendicular to the planes XY, XZ, YZ, <pxy, spxz, <pyz, rotations, respectively, in the plane XY, XZ, YZ, b fixing, c unmovable support, d movable support in the direction of the X and Z axes

Fig. 6.9 Truss: a static scheme, b internal forces in rods

means that components in such constructions should not be too slender in order not to lose their own stability.

In each of the cross sections of flat frames internally statically determinate, we release three overvalues (Fig. 6.10). Their presence causes that the constructor, in conducting stiffness-strength calculations, should take into account both the work of elements and structural nodes under load causing shearing and bending.

In spatial constructions, the shearing of any component in thought causes the release of six overvalues (Fig. 6.11). Each cross section may therefore be com­pressed, sheared, bent and twisted. Taking this into account, in the process of dimensioning cross sections and connectors of particular joints, the lowest strength of wood and glue joints determining strength in the entire construction is taken into account.