8.1 Stiffness and Strength of Upholstery Frames
Frames of upholstery furniture belong to 3D structures, made of rails not lying in one plane, or 2D frames, but loaded with forces not lying in the system plane. (Fig. 8.1).
The general principles and proceedings when solving frame 3D systems with the method of forces are the same as with solving 2D frames (see calculations on the side frames of chairs). First, the degree of static indeterminacy ns should be determined:
ns = p + 6t — 6,
p number of support nodes,
t number of necessary cuts of closed contours and 6 number of equilibrium equations for any 3D force system
Therefore, the frame presented in Fig. 8.2 is a 19-fold statically indeterminate system ns = 19.
Then, the frame of references x, y, z should be assumed, so that as many rods lay in the xy plane as possible. The loads of frames should be the most disadvantageous for them (Fig. 8.3). The distribution of evenly distributed load q shown in Fig. 8.3a is caused by the tension of belts or springs. The value of this load, dependent on the strength of the belt tension, number of belts and their length, can be determined from the equation:
© Springer International Publishing Switzerland 2015 J. Smardzewski, Furniture Design,
Fig. 8.1 Examples of upholstery frame constructions: a covered mortise and tenon joints, b bridle joints
Fig. 8.2 Calculation diagram of static indeterminacy of the system
q even distribution on the length of external rails, i number of belts or springs, l length of the rail and
Pn strength of tension of one belt or one spring.
Values of concentrated loads shall be assumed on the basis of standardised data. By discarding the extra nodes and replacing them with the appropriate forces, we obtain the basic system. This system must be unchanging and statically determinate, and at the same time as easy as possible to solve. If a given frame is symmetrical, then in order to simplify the calculations also a symmetrical basic system should be assumed. Symmetrical and asymmetrical overvalues will cause symmetrical and asymmetrical graphs of moments.
Fig. 8.3 Methods of loading upholstery frames: a transverse with upholstery belts, b with force focused on internal rail, c with force focused on external rail
Below are the results of the numerical calculations determining the impact of the construction of upholstery frame on its stiffness and strength. During the calculations, the following assumptions were made as:
• upholstery frame is a flat grid statically loaded,
• operational load in the form of concentrated force P = 1000 N is applied,
• in the middle of the length of the longitudinal external element (rail),
• support of the grid results from the way the base of the case of the sofa is placed and mounted,
• in the calculations, only bending moments Mg and cutting forces T will be considered,
• cross section of the element (rail) is 32 x 50 mm,
• frame elements are made of flat-pressed particle board of module E = 3500 MPa and Poisson’s ratio v = 0.3. The stiffness of these elements does not change on their length and
• construction (carpentry) joints are perfectly stiff.
Calculations were made using the finite elements method, while the basic static schemes, i. e. the support state and frame loads are shown in Fig. 8.4. The main purpose of this analysis was to identify the optimal solution for the frame
Fig. 8.4 Static schemes of frames, along with division into calculation elements: a frame with longitudinal system of internal rails, b frame with transverse system of two internal rails, c frame with transverse system of four internal rails (mm)
construction, by assuming the maximum strength of components and significant material saving as the optimisation criterion. Based on the static schemes of the frames assumed for consideration, it was concluded that the values of reaction forces of the lifter locks are equal. This obviousness results from the fixed dimensional proportions and fixed location of mounting of the fittings.
However, depending on the number of support points for transverse rails on the case of the sofa, the value of reaction at these points is reduced proportionally to the number of supports. This has a significant impact on the distribution of internal forces in elements (Fig. 8.5).
From the given schemes of distribution of internal forces, it is easy to note that the maximum bending moment of the longitudinal front rail of the construction from Fig. 8.4a is 3 to 4 times greater than the corresponding bending moments in constructions as in Fig. 8.4b, c. Therefore, a more correct constructions are structures containing a transverse system of internal rails. Moreover, in all the
Fig. 8.5 Distribution of bending moments and cutting forces in rails: a frame with longitudinal system of internal rails, b frame with transverse system of two internal rails, c frame with transverse system of four internal rails
considered construction variants, the maximum bending moment occurs under concentrated force P that is at half the length of the longitudinal front rail. The difference between the values of maximum moments for frames with a transverse system of internal rails was about 30 % in favour of the system with four internal rails. However, for this reason, one should not expect significant material savings or the possibility of reduction of the coefficient of the cross-sectional strength resulting from them. These savings will not compensate for an increase in expenditure arising from the application of two additional internal rails in the system as shown in Fig. 8.4c. Besides, the construction stiffens slightly, which is shown by the values of displacement of individual nodes illustrated in Fig. 8.6.
The results shows that frame of upholstery furniture with a transverse rails is less prone to damage than a frame with a longitudinal system.
Fig. 8.6 Displacement of nodes of the frames: a frame with longitudinal system of internal rails, b frame with transverse system of two internal rails, c frame with transverse system of four internal rails