# What is the stiffness of a beam

The Stiffness is a quantity in technical mechanics that describes the relationship between the load that acts on a body and its elastic deformation. The rigidity of a body depends on its material and its geometry. Depending on the type of load, a distinction is made between different stiffnesses, e.g. B. bending, elongation or torsional stiffness. The reciprocal of stiffness is called compliance.

The rigidity, along with the fracture toughness, strength, ductility, hardness, density and melting temperature, is one of the material properties of a material.

### Stiffnesses

Rigidity always consists of a material and a geometry term. Which material size is used - i.e. shear or elasticity module - depends on the stress that causes the external load. Stiffnesses are noted in such a way that related deformation variables result, e.g. B. Stretching instead of length changes. The reason for this is that the stiffness is a property of the cross-section. However, the cross-section geometry can change over the length of a component, so that the multiplication by the length is not always correct. The spring constant is a special case.

### Tensile stiffness

The tensile stiffness is the product of the material's modulus of elasticity in the direction of loading times the cross-sectional area perpendicular to the direction of loading. It is independent of the shape of the cross-section, it depends on its area S. for example in N

The above formulation applies to free transverse contraction of the cross section. In the case of hindered transverse contraction, the modulus of transverse contraction is used for the modulus of elasticity. The longitudinal expansion ε of the body is the acting normal force F. as well as proportional to the tensile stiffness. ### Bending stiffness

The bending stiffness is the product of the modulus of elasticity and the moment of area of ​​the 2nd degree I. of the cross section. The second-degree moment of area depends essentially on the shape of the cross-section. for example in Nmm2

How strong the deflection or lowering of a flexurally stressed component is at a given load depends not only on the flexural rigidity but also on its length and the storage conditions. The curvature κ of the body is the applied bending moment M.B. as well as the bending stiffness proportional. ### Torsional stiffness

The torsional stiffness is the product of the second polar area moment I.p and the thrust module G of the material. The polar moment of area is related to the axis around which the body is twisted. for example in Nmm2

How much, i.e. by what angle, a body is twisted under a certain load depends not only on the second-order polar area moment, but also on its length and the storage conditions. The drill of the body is the applied torsional moment M.T as well as proportional to the torsional stiffness. ### Spring constant

In practice, it is often not the elongation ε, but the absolute change in length ΔL. of interest. Therefore, in the case of springs, the spring constant is determined by the ratio of the necessary force F. for a certain deflection ΔL. described. The spring constant consists of the stiffness of the spring divided by its length. While the tensile stiffness is independent of the length of the spring, the spring constant is halved when the length of the spring is doubled.

Example: A tension rod with the cross section A. = 100 mm2 and a modulus of elasticity of 210,000 N / mm2 has a stiffness of E.·A. = 2,1·107 N. Is the staff L. = 100 mm long so is its spring constant E.·A. / L. = 210,000 N / mm. The calculation only applies if the cross-section of the bar is constant.

### Rigidity and strength

Rigidity is not to be confused with strength.

• The strength is a measure of the tolerable loads on a material. This limit value (e.g. the tensile strengthR.m) can be looked up in material data sheets.
• The Stiffness of a component depends on the rigidity of the material (E or G module), but also on the size and shape of the cross-sectional area.

How stiff is a component has nothing to do with its strength!