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{{Language_sel|LANG=ger|ARTIKEL=Steifigkeit}}
{{PSM_Infobox}}
Stiffness
__FORCETOC__

==General information==

The term stiffness is often used relatively uncritically in [[Materials Testing|materials]] or [[Polymer Testing|polymer testing]], and stiffness is frequently equated with the [[Elastic Modulus|modulus of elasticity]], even though the units of measurement for moduli and stiffness are different. In [[Materials Testing|materials testing]], there are two fundamental factors that influence the determination of [[Material Value|characteristic values]]. These are the so-called [[Machine Compliance|machine stiffness]] as a property of a [[Material Testing Machine|material testing machine]] and the stiffness of the [[Specimen|test specimen]], both of which influence the determination of characteristic values.

Depending on the type of material testing ([[Tensile Test|tensile]], [[Compression Test|compression]], or [[Bend Test|bending test]]) and the absolute load level, the [[Material Testing Machine|universal testing machine]] must have sufficient resistance to the unavoidable deformation of the [[Load Framework|load frame]] and the force measuring cell (see: [[Electro-mechanical Force Transducer|electro-mechanical force transducer]] and [[Piezoelectric Force Transducer|piezoelectric force transducer]]). The tensile or compressive compliance of the testing machine with an identical load cell can be assumed to be almost the same, disregarding the [[Specimen Clamping|clamping]] and extension elements. In the bending test, the influence of machine compliance is almost negligible, regardless of the direction of travel of the crosshead, since the forces in this test are significantly lower than in the compression or tensile test. A special case arises when torsion or shear tests (see: [[Dynamic-mechanical Analysis (DMA) – Torsional Stress|dynamic-mechanical analysis (DMA) – torsional stress]]) are performed, as the type of load and compliance of the special hybrid materials testing machine is different in this case. Analogous to the testing of materials with defined test specimen geometry, complex constructions such as machines, apparatus, or bridges, for example, also exhibit stiffness against stress-induced deformation, whereby in this case, different load cases may be dominant regardless of the type of [[Stress|stress]] ([[Quasi-static Test Methods|static]], [[Fatigue|oscillating]], or [[Impact Loading Plastics|impact]]).

==Definition of stiffness==

The stiffness of a material is generally understood to be the [[Elastic Modulus|elastic modulus]] or [[Shear Modulus|shear modulus]] for the respective type of stress in the [[Deformation#Elastic deformation|linear-elastic]] or [[Linear-viscoelastic Behaviour|linear-viscoelastic]] ([[Plastics|plastics]]) stress range. For [[Tensile Test|tensile testing]], this is the [[Elastic Modulus|modulus of elasticity]] ''E''t, for compression testing ''E''c, and for bending testing the [[Flexural Modulus|flexural modulus]] ''E''f is used, while for torsion or shear testing the [[Shear Modulus|shear modulus]] ''G'' is used. For larger [[Deformation|deformations]], the elastic modulus is no longer used, but rather the increase in, for example, the stress–strain curve d''σ''/d''ε'' (see: [[Tensile Test|tensile test]]) to describe the stiffness at a given deformation.

The stiffness of a [[Plastic Component|component]] in relation to a specific load case is determined by its dimensions (e.g., wall thickness or outer diameter) and its structural design (e.g., geometric design, open or closed profiles, and ribbing). Due to different geometric parameters, the stiffness of a component or its resistance to mostly elastic deformation under varying [[Stress|loads]] will only match in exceptional cases.

In general, stiffness ''S'' in the sense of technical mechanics or strength theory usually represents the product of the stiffness of the [[Material & Werkstoff|material]] and the construction, which corresponds to a specific geometric variable. As a result, this [[Material Parameter|parameter]] describes the combined resistance of the material and construction to elastic deformation due to an external force or moment (bending or torsional moment).

The stiffness of a structure or component ([[Specimen|test specimen]]) is usually only defined for small elastic [[Deformation|deformations]] and depends on the elastic properties (modulus ''M'') of the material (plastic) as well as on the geometric conditions (''G'') relating to the main types of load on the component or test specimen and is generally calculated using '''Eq. (1)'''. In [[Materials Testing|material]] and [[Polymer Testing|polymer testing]] in particular, the term compliance ''C'' is often used instead of stiffness, which is defined according to '''Eq. (2)''' as the reciprocal of stiffness ''S''.

{|
|- valign="top"
|width="550px"|<math>S= M\cdot G\!</math>
|width="50px"|(1)
|}

{|
|- valign="top"
|width="550px"|<math>C=\frac{1}{S}\!</math>
|width="50px"|(2)
|}

For lightweight constructions, the specific stiffness ''S''s according to '''Eq. (3)''' is often used, as the mass of the component is of great importance here. This parameter is obtained by relating the stiffness to the average [[Density|density]] ''ρ'' of the materials used in the construction, whereby different units of measurement result depending on the application and load case.

{|
|- valign="top"
|width="550px"|<math>S_{s}=\frac{S}{\rho}\!</math>
|width="50px"|(3)
|}

==Tensile stiffnes==

Tensile stiffness, often referred to as elongation stiffness, is the product of the [[Elastic Modulus|modulus of elasticity]] under tensile stress ''E''t of the [[Material & Werkstoff|material]] used in the direction of stress ''x'' and the cross-sectional area ''A''0. This area is perpendicular to the direction of stress and is independent of the shape or geometry of the cross-section. For cross-sections that vary over their length, the smallest cross-sectional area must be used to calculate the tensile stiffness. In the case of tensile test specimens, the plane-parallel part of the test specimen must therefore be used.

In tensile testing, prismatic [1] or [[Multipurpose Test Specimen|multipurpose test specimens]] of type 1A or 1B [2] are sometimes used in polymer testing, whereby the latter have shoulders on both sides for better clamping ('''Fig. 1''') (see: [[Specimen Clamping|specimen clamping]]). In the case of prismatic test specimens, the specimen stiffness ''S''t is constant within the clamping length ''L'' and is calculated according to '''Eq. (4)''' for [[Deformation#Elastic deformation|linear-elastic deformation]]. This equation also applies to shoulder test specimens, but only in the plane-parallel part of the test specimen ('''Fig. 1a''' and '''1b''').

{|
|- valign="top"
|width="550px"|<math>S_{t}=E_{t}\cdot A_{0}= E_{t}\cdot b_{0} \cdot d_{0}</math>
|width="50px"|(4)
|}

If the strain in the tensile test exceeds the [[Linear-viscoelastic Behaviour|linear-viscoelastic range]], initial [[Micro-damage Limit|micro-damage]] occurs, the stress–strain curve bends, and the test specimen cross-section decreases in accordance with the [[Poisson's Ratio|Poisson's ratio]] of the respective material.

For this reason, '''Eq. (5)''' must be used to describe the current [[Specimen|stiffness of the specimen]], which is only valid for the instantaneous strain of the test specimen ('''Fig. 1c'''). This equation therefore only applies to the unimpeded transverse contraction of the test specimen or component, i.e., in the case where no stresses occur in the y and z directions and the current cross-section ''A'' is known (see: [[Uniaxial Stress State|uniaxial]] and [[Multiaxial Stress State|multiaxial stress state]]).

{|
|- valign="top"
|width="550px"|<math>S_{t}=\frac{d\sigma}{d\epsilon}\cdot A=\frac{d\sigma}{d\epsilon}\cdot(b d)</math>
|width="50px"|(5)
|}

[[File:Stiffness-1.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 1''':
|width="600px"|Schematic explanation of tensile stiffness
|}

The compliance under tensile stress ''C''t is calculated as the reciprocal of the tensile stiffness ''S''t according to '''Eq. (6)''' and is expressed in the unit 1/N. Compliance decreases with increasing cross-sectional area and/or increasing [[Elastic Modulus|modulus of elasticity]] ''E''t.

{|
|- valign="top"
|width="550px"|<math>C_{t}=\frac{1}{S_{t}}</math>
|width="50px"|(6)
|}

==Compressive and buckling stiffness==

The stiffness under compressive stress (see also: [[Compression Test Compliance|compression test compliance]]) is calculated in the same way as for tensile stress, but different stress cases arise depending on the test specimen length, cross-section, and storage conditions of the test specimen.

In the case of very long, slender component geometries or test specimens, a special failure case occurs, which is also known as EULER buckling (named after the Swiss mathematician and physicist Leonhard Euler) ('''Fig. 2a''') and represents an instability problem [3]. Depending on the storage conditions, geometry, and modulus of elasticity of the test specimen, lateral buckling (bending buckling) can occur as a result of compressive stress at a critical load. In the case of torsion or bending with torsion, however, twist or bending-twist buckling (tilting) can lead to failure. In the elastic deformation range, the critical buckling load or compressive stress ''F''k is calculated according to '''Eq. (7)''' and the corresponding buckling stress ''σ''k is obtained according to '''Eq. (8)'''.

{|
|- valign="top"
|width="550px"|<math>F_{k}=\frac{\pi^{2}\cdot E_{c}\cdot I_{y} }{s^{2}}</math>
|width="50px"|(7)
|}

{|
|- valign="top"
|width="550px"|<math>\sigma_{k}=\frac{\pi^{2}\cdot E_{c}}{\lambda^{2}}</math>
|width="50px"|(8)
|}

In '''Eq. (7)''', ''E''c is the [[Elastic Modulus|modulus of elasticity]] under compressive stress, ''I''y is the minimum axial moment of area of the section, and ''s'' is the so-called buckling length, which is the product of the buckling length coefficient ''β'' and the test specimen length ''L'' (''s'' = ''β'' ''L''). The value ''λ'' is referred to as the [[Slenderness Ratio|slenderness ratio]] and is calculated according to '''Eq. (9)''', where ''I''y is the so-called smallest radius of inertia according to '''Eq. (10)'''.

{|
|- valign="top"
|width="550px"|<math>\lambda =\beta \cdot L \cdot \sqrt{\frac{A_{0}}{I_{y}}}=\frac{\beta \cdot L}{i_{y}}</math>
|width="50px"|(9)
|}

{|
|- valign="top"
|width="550px"|<math>i_{y}=\sqrt{\frac{I_{y}}{A_{0}}}</math>
|width="50px"|(10)
|}

[[File:Stiffness-2.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 2''':
|width="600px"|Buckling behaviour and compressive stiffness
|}

The support conditions in the [[Compression Test|compression test]] ('''Fig. 2b''') correspond to the second elastic buckling case in '''Fig. 2a''', whereby ''s'' = ''L'' applies. To prevent buckling of the test specimens in the compression test, the [[Slenderness Ratio|slenderness ratio]] ''λ'' should be between 6 and 10, which is why the length ''L'' of a prismatic test specimen is calculated according to '''Eq. (11)''' [4, 5].

{|
|- valign="top"
|width="550px"|<math>L=\frac{\lambda \cdot d_{0}}{3,46}</math>
|width="50px"|(11)
|}

For prismatic test specimens or components, the [[Specimen Compliance|specimen stiffness]] ''S''c is constant within the clamping length ''L'' and is calculated according to '''Eq. (12)''' for [[Deformation#Elastic deformation|linear-elastic deformation]].

{|
|- valign="top"
|width="550px"|<math>S_{c}=E_{c}\cdot A_{0}= E_{c} \cdot b_{0} \cdot d_{0}</math>
|width="50px"|(12)
|}

If the compressive strain in plastics exceeds the [[Linear-viscoelastic Behaviour|linear-viscoelastic range]] in the compression test, [[Micro-Damage Limit|micro-damage]] occurs and the compressive stress–compressive strain curve becomes increasingly non-linear. Therefore, '''Eq. (13)''', which is valid for the instantaneous compressive strain of the test specimen and the true cross-sectional area ''A'', must be used to describe the current stiffness of the specimen ('''Fig. 2c''').

{|
|- valign="top"
|width="550px"|<math>S_{c}=\frac{d\sigma}{d\epsilon}\cdot A=\frac{d\sigma}{d\epsilon}\cdot(b d)</math>
|width="50px"|(13)
|}

Analogous to tensile stress, compressive stiffness ''C''c is also calculated as the reciprocal of stiffness ''S''c according to '''Eq. (14)''' and is specified in the unit of measurement 1/N.

{|
|- valign="top"
|width="550px"|<math>C_{c}=\frac{1}{S_{c}}</math>
|width="50px"|(14)
|}

==Bending stiffness==

In [[Bend Loading|bend loading]], the geometric variable describing stiffness is the axial moment of inertia, regardless of whether [[Bend Test#The three-point bending test method|three-point]] or [[Bend Test#The four-point bending test method|four-point bending]] is involved. However, it is important to note whether the bending of a component or test specimen occurs around the y-axis or z-axis, as is common in [[Impact Test|impact bending tests]], for example. In these cases, the minimum or maximum axial moment of inertia ''I''y or ''I''z must be used ('''Fig. 3'''). In this case, the minimum bending stiffness is calculated as the product of the modulus of elasticity under bending stress and the axial moment of inertia ''I''y (see also: [[Bend Test Compliance|bend test compliance]]).

In the three-point bending test on [[Plastics|plastics]] [6], bending is usually performed around the y-axis of the prismatic test specimen, so that the specimen stiffness is calculated according to '''Eq. (15)'''.

{|
|- valign="top"
|width="550px"|<math>S_{f}=E_{f}\cdot I_{y}=\frac{E_{f}\cdot b \cdot h^{3}}{12}</math>
|width="50px"|(15)
|}

In the case of a non-linear flexural stress–peripheral fibre strain curve (see: [[Bend Loading|bend loading]] and [[Peripheral Fibre Strain|peripheral fibre strain]]), '''Eq. (16)''' is used to describe the current flexural stiffness (see '''Fig. 3c''').

{|
|- valign="top"
|width="550px"|<math>S_{f}=\frac{d\sigma_{f}}{d\varepsilon_{f}}\cdot I_{y}=\frac{d\sigma_{f}}{d\varepsilon_{f}}\cdot \frac{b\cdot h^{3}}{12}</math>
|width="50px"|(16)
|}

[[File:Steifigkeit-3.jpg|500px]]
{|
|- valign="top"
|width="50px"|'''Fig. 3''':
|width="600px"|Explanation of bending stiffness
|}

Information on bending stiffness is more complicated if there is a so-called skew bending at any angle between the y- and z-axes, i.e., two bending moments ''M''bz and ''M''by act simultaneously. In this case, the minimum and maximum area moments of inertia ''I''y and ''I''z from the geometric limits for the bending stiffness.

Analogous to tensile stress, the compliance ''C''f under bending is also calculated as the reciprocal value of the specimen stiffness ''S''f according to '''Eq. (17)''' and has the unit of measurement 1/N mm2.

{|
|- valign="top"
|width="550px"|<math>C_{f}=\frac{1}{S_{f}}</math>
|width="50px"|(17)
|}

==Shear stiffness==

In the case of shear stress (see: [[Shear Modulus|shear modulus]]), depending on the specific load case, more complex stress distributions occur than in the case of ideal [[Uniaxial Stress State|uniaxial stress]]. For example, in the case of single or double-cut shearing or coupling connections in pipes, the shear area ''A''s = ''A''·''γ'' ('''Figs 4a''' and '''4b''') is particularly important, whereas in the case of bending stress due to transverse forces in the cross-section, especially of layered materials (laminates and prepregs), a parabolic stress distribution ''τ''(''h'') is recorded ('''Fig. 4c'''), with a maximum at ''h''/2.

By definition, the shear stiffness is the product of the shear modulus ''G'' and the effective cross-sectional area ''A'' and is calculated according to '''Eq. (18)''', whereby the correction factor ''γ'' is intended to approximately account for the non-uniform parabolic stress distribution ''τ''(''h'') in the cross-section ('''Fig. 4c''').

{|
|- valign="top"
|width="550px"|<math>S_{s}=G\cdot A\cdot \gamma = G\cdot A_{s}</math>
|width="50px"|(18)
|}

The unit of measurement for shear stiffness is given in N, and compliance is determined as the reciprocal value with ''C''s = 1/''S''s.

[[File:Steifigkeit-4.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 4''':
|width="600px"|Explanation of shear stiffness in shear and bending tests
|}

==Torsional stiffness==

The torsional stiffness ''S''to, also referred to as twisting stiffness, corresponds to the product of the shear modulus ''G'' and the torsional moment of inertia ''I''t, and calculation '''Eq. (19)''' only applies in the elastic deformation range for small angles of twist ''φ''t.

{|
|- valign="top"
|width="550px"|<math>S_{to}=G\cdot I_{t}</math>
|width="50px"|(19)
|}

The torsional moment of inertia ''I''t refers to the component or test specimen axis x, around which it is twisted or rotated as a result of an applied torsional moment ''M''t ('''Fig. 5a'''). Depending on the type of torsional stress present–torsion without warping with no normal stress (Neuber theory for closed profiles), torsion with unimpeded warping (Saint-Vernant theory), or warping force torsion—different conditions apply for the exact specification of torsional stiffness.

At the same time, the torsional moment of inertia ''I''t is only identical to the polar moment of inertia ''I''p in special geometric cases, i.e., for closed box, circular, rectangular, or tubular profiles, and can be determined using relatively simple calculation equations according to ''I''t = ''I''p = ''I''y + ''I''z ('''Fig. 5b'''). In all other cases, especially for complex geometries, approximate solutions must be used to specify the torsional moment of inertia.

In the torsion test, the torsional stiffness of a prismatic test specimen is calculated as the product of the [[Shear Modulus|shear modulus]] ''G'' and the polar moment of inertia ''I''p ('''Fig. 5b''') according to '''Eq. (20)'''.

{|
|- valign="top"
|width="550px"|<math>S_{to}=G\cdot I_{p}=\frac{G\cdot b\cdot h}{12}\cdot \left ( b^{2}+h^{2} \right )</math>
|width="50px"|(20)
|}

The specimen compliance ''C''to in the torsion test is also defined as the reciprocal value of the specimen stiffness ''S''to according to '''Eq. (21)''' and has the unit of measurement 1/N mm2.

{|
|- valign="top"
|width="550px"|<math>C_{to}=\frac{1}{S_{to}}</math>
|width="50px"|(21)
|}

[[File:Steifigkeit-5.jpg]]
{|
|- valign="top"
|width="50px"|'''Fig. 5''':
|width="600px"|Explanation of torsional stiffness and torsional moment of inertia
|}

==Application of stiffness==

When dimensioning more complicated components or complex constructions, the basic requirements for stiffness in the various main stress directions must be taken into account. In particular, consideration must also be given to how the requirement profile changes as a result of media, thermal, corrosive, and erosive influences, as well as [[Ageing|ageing]]. The interactions of the main stresses must be taken into account, as well as the superimposition of different load collectives, such as static and vibrating stresses.

From this perspective, a bridge structure, for example, must have sufficient bending stiffness in both main axes to withstand static and dynamic loads, while also ensuring sufficient tensile and compressive stiffness in detail. Connecting elements must also have sufficient shear stiffness. Additional stresses, such as static or dynamic wind loads, must be taken into account by ensuring the necessary torsional stiffness, e.g. by selecting a suitable basic design and ribbing. Since these requirements are sometimes mutually exclusive, such a complex construction always represents a compromise, which, however, should not have any weak points when subjected to stress.

The collapse of the Tacoma Bridge in the USA in 1940, which was built in 1938, shows the effects of choosing the wrong basic design. By selecting a torsionally flexible open profile instead of a closed box profile, strong winds caused torsional vibrations in the bridge, which ultimately led to its collapse [7]. Amateur film footage impressively documents the strong vibrations of this bridge structure.

==See also==

* [[Machine Compliance|Machine compliance]]
* [[Specimen Compliance|Specimen compliance]]
* [[Compression Test Compliance|Compression test compliance]]
* [[Tensile Test Compliance|Tensile test compliance]]
* [[Bend Test Compliance|Bend test compliance]]

'''References'''

{|
|-valign="top"
|[1]
|ISO 527-5 (2021-11): Plastics – Determination of Tensile Properties – Part 5: Test Conditions for Unidirectional Fibre-reinforced Plastic Composites
|-valign="top"
|[2]
|ISO 527-2 (2025-06): Plastics – Determination of Tensile Properties – Part 2: Test Conditions for Moulding and Extrusion Plastics
|-valign="top"
|[3]
|Szabo, I.: Einführung in die Technische Mechanik. Springer, Berlin (1984) 8th Edition, (ISBN 3-540-13293-7; see [[AMK-Büchersammlung|AMK-Library]] under T 15)
|-valign="top"
|[4]
|[[Bierögel, Christian|Bierögel, C.]]: Quasistatische Prüfverfahren. In: [[Grellmann, Wolfgang|Grellmann, W.]], [[Seidler, Sabine|Seidler, S.]] (Eds.): Kunststoffprüfung. Carl Hanser, Munich (2025) 4th Edition, pp. 106–151 (ISBN 978-3-446-44718-9; E-Book: ISBN 978-3-446-48105-3; see [[AMK-Büchersammlung|AMK-Library]] under A 23)
|-valign="top"
|[5]
|ISO 604 (2002-03): Plastics – Determination of Composite Properties
|-valign="top"
|[6]
|ISO 178 (2019-04): Plastics – Determination of Flexural Properties
|}

[[Category:Stiffness Compliance]]

==Weblinks==

{|
|-valign="top"
|[7]
|https://www.youtube.com/watch?v=3mclp9QmCGs
|}

[[Category:Deformation]]

Stiffness - Revision history