Leaf Springs and Torsion Bars- النوابض الورقية وأعمدة اللي
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Leaf Springs and Torsion Bars- النوابض الورقية وأعمدة اللي

Leaf Springs and Torsion Bars

Leaf Springs

Acts as:


- an elastic element

- guide

- damper

+ they are easy to manufacturer and convincement to repair




-         large metal content (energy stored in unit volume in spring or torsion bar is 4 times more than a leaf spring; that lead to increase metal content).

-         sufficient unsprung mass

-         short service life.




If it is desired to maintain uniform bending stresses over the length of the beam then the width of the cantilever needs to vary linearly with the location as illustrated in Figure i, (σb = M y/I = 12 F L y / b h3). The concept used in producing compact cantilever springs of uniform bending stress is to chop the triangular form illustrated in Figure i, into a number of strips and recombine them as illustrated in Figure ii. The multi-leaf spring shown and the single triangular section beam both have the same stress and deflection characteristics with the exceptions that the multi-leaf spring is subject to additional damping due to friction between the leaves and that the multi-leaf spring can carry a full load in only one direction due to the tendency for the leaves to separate. Leaf separation can be partially overcome by the provision of clips around the leaves.



The deflection of a triangular leaf spring is given by


          F is force (N)

          L is the length (m)

          E is the Young’s modulus (N/m2)

          I is the second moment of area (m4)

For a rectangular cross-section,


Where b is width (m); and h is thickness (m).


The spring rate is given by


The corresponding bending stress (for cantilever, semi-elliptic and full-elliptic) is given by



For semi-elliptic beam the maximum deflection at the center is given by




For a full-elliptic beam the maximum deflection at the center is given by




* in the above equation the length b can be changed by (n b) where n is the number of leafs and b is the width of single leaf. The leaf width b is chosen from the available range of rolled products. It is desired that the following condition should hold


          6 < b/h < 10

          6 <   n   < 14


The spring length can be determined using the equations for spring stiffness and stress to obtain




          δ = is the total deflection (δst + δd)


- Effect of adding an extra full length number of leaves (ne) to the graduate length number of leaves (ng):



That means the extra leaves will have more stress than graduate ones.



Torsion beam suspension (torsion bars)


Definition: A steel bar that is twisted to support the weight of the vehicle. Torsion bars are used in place of coil or leaf springs on some vehicles, and allow ride height to be adjusted to compensate for sage that occurs over time.

The bars are usually solid of circular cross section although hollow tubes and rectangular bars are used.


τall = 1000- 1050 MPa

The angle of twist and stiffness of a torsion bar are expressed as


The torsion-bar working length L (without spline ends) is determined by the θ equation. It is recommended to choose the diameters and lengths of spline ends depending on the torsion-bar diameter

          dsp = (1.2 -1.3)d; and Lsp= (0.6-1.2)d


Suspension Formulae




Where b is the width of spring blade (m), L is the distance between the eyes of the spring when laden (m), t is the thickness of the blade (m), n is the number of blades, and E is the modulus of elasticity, which (modified to allow for internal friction) is 159 x 106 kN/m2.

    For a torsion bar, the spring rate is given as the twisting moment per angular deflection. When a lever is added, this can be converted into a rate for the vertical deflection of the end of the lever.


    Spring rate (torsion bar, for deflection at end of lever)




Where G is the modulus of rigidity, which is 78.5 x 106 kN/m2 in this case, d is the diameter of the torsion bar (m), l is the effective length of the torsion bar (m), i.e. half the length of the bar for an anti-roll bar, and e is the length of the lever (m).



 Spring rate (coil spring)




Where G is the modulus of rigidity, which is 81.5 x 106 kN/m2 in this case, d is the wire diameter (m), n is the number of free coils, and D is the mean coil diameter (m). To find the number of free coils it is necessary to subtract the number of dead coils form the total number of coils. The dead coils are those that provide the abutment and so cannot be deflected, usually 1.5 to 2 coils.  

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