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49-406: Inflexible may refer to: Stiffness , the rigidity of an object, the extent to which it resists deformation in response to an applied force Beardmore Inflexible , a British three-engined all-metal prototype bomber aircraft of the 1920s HMS Inflexible , one of several Royal Navy ships of this name Inflexible -class ship of the line built for

98-1398: A beam therefore correspond to the frequencies at which resonance can occur. A free–free beam is a beam without any supports. The boundary conditions for a free–free beam of length L {\displaystyle L} extending from x = 0 {\displaystyle x=0} to x = L {\displaystyle x=L} are given by: If we apply these conditions, non-trivial solutions are found to exist only if cosh ⁡ ( β n L ) cos ⁡ ( β n L ) − 1 = 0 . {\displaystyle \cosh(\beta _{n}L)\,\cos(\beta _{n}L)-1=0\,.} This nonlinear equation can be solved numerically. The first four roots are β 1 L = 1.50562 π {\displaystyle \beta _{1}L=1.50562\pi } , β 2 L = 2.49975 π {\displaystyle \beta _{2}L=2.49975\pi } , β 3 L = 3.50001 π {\displaystyle \beta _{3}L=3.50001\pi } , and β 4 L = 4.50000 π {\displaystyle \beta _{4}L=4.50000\pi } . The corresponding natural frequencies of vibration are: The boundary conditions can also be used to determine

147-604: A device such as the Cutometer. The Cutometer applies a vacuum to the skin and measures the extent to which it can be vertically distended. These measurements are able to distinguish between healthy skin, normal scarring, and pathological scarring, and the method has been applied within clinical and industrial settings to monitor both pathophysiological sequelae, and the effects of treatments on skin. Euler%E2%80%93Bernoulli beam equation Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory )

196-704: A double clamped beam of length L {\displaystyle L} (fixed at x = 0 {\displaystyle x=0} and x = L {\displaystyle x=L} ) are This implies solutions exist for sin ⁡ ( β n L ) sinh ⁡ ( β n L ) = 0 . {\displaystyle \sin(\beta _{n}L)\,\sinh(\beta _{n}L)=0\,.} Setting β n := n π L {\displaystyle \beta _{n}:={\frac {n\pi }{L}}} enforces this condition. Rearranging for natural frequency gives Besides deflection,

245-424: A fiber with a radial distance z {\displaystyle z} below the neutral axis is ( ρ + z ) d θ . {\displaystyle (\rho +z)d\theta .} Therefore, the strain of this fiber is The stress of this fiber is E z ρ {\displaystyle E{\dfrac {z}{\rho }}} where E {\displaystyle E}

294-400: A positive value of M {\displaystyle M} produces compressive stress at the bottom surface. With this choice of bending moment sign convention, in order to have d M = Q d x {\displaystyle dM=Qdx} , it is necessary that the shear force Q {\displaystyle Q} acting on the right side of the section be positive in

343-440: Is flexibility or compliance , typically measured in units of metres per newton. In rheology , it may be defined as the ratio of strain to stress , and so take the units of reciprocal stress, for example, 1/ Pa . A body may also have a rotational stiffness, k , {\displaystyle k,} given by k = M θ {\displaystyle k={\frac {M}{\theta }}} where In

392-494: Is flexibility or pliability: the more flexible an object is, the less stiff it is. The stiffness, k , {\displaystyle k,} of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single degree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as k = F δ {\displaystyle k={\frac {F}{\delta }}} where, Stiffness

441-467: Is a parameter of interest that represents its firmness and extensibility, encompassing characteristics such as elasticity, stiffness, and adherence. These factors are of functional significance to patients. This is of significance to patients with traumatic injuries to the skin, whereby the pliability can be reduced due to the formation and replacement of healthy skin tissue by a pathological scar . This can be evaluated both subjectively, or objectively using

490-399: Is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams . It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory . It

539-694: Is named after Jacob Bernoulli , who made the significant discoveries. Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750. The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load: d 2 d x 2 ( E I d 2 w d x 2 ) = q {\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left(EI{\frac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}\right)=q\,} The curve w ( x ) {\displaystyle w(x)} describes

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588-470: Is simpler: In the absence of a transverse load, q {\displaystyle q} , we have the free vibration equation. This equation can be solved using a Fourier decomposition of the displacement into the sum of harmonic vibrations of the form where ω {\displaystyle \omega } is the frequency of vibration. Then, for each value of frequency, we can solve an ordinary differential equation The general solution of

637-438: Is the elastic modulus and I {\displaystyle I} is the second moment of area of the beam's cross section. I {\displaystyle I} must be calculated with respect to the axis which is perpendicular to the applied loading. Explicitly, for a beam whose axis is oriented along x {\displaystyle x} with a loading along z {\displaystyle z} ,

686-429: Is the elastic modulus in accordance with Hooke's Law . The differential force vector, d F , {\displaystyle d\mathbf {F} ,} resulting from this stress, is given by This is the differential force vector exerted on the right hand side of the section shown in the figure. We know that it is in the e x {\displaystyle \mathbf {e_{x}} } direction since

735-461: Is the radius of curvature . Therefore, This vector equation can be separated in the bending unit vector definition ( M {\displaystyle M} is oriented as e y {\displaystyle \mathbf {e_{y}} } ), and in the bending equation: The dynamic beam equation is the Euler–Lagrange equation for the following action The first term represents

784-495: Is undesirable, while a low modulus of elasticity is required when flexibility is needed. In biology, the stiffness of the extracellular matrix is important for guiding the migration of cells in a phenomenon called durotaxis . Another application of stiffness finds itself in skin biology. The skin maintains its structure due to its intrinsic tension, contributed to by collagen , an extracellular protein that accounts for approximately 75% of its dry weight. The pliability of skin

833-476: Is usually defined under quasi-static conditions , but sometimes under dynamic loading. In the International System of Units , stiffness is typically measured in newtons per meter ( N / m {\displaystyle N/m} ). In Imperial units, stiffness is typically measured in pounds (lbs) per inch. Generally speaking, deflections (or motions) of an infinitesimal element (which

882-451: Is valid for the fibers in the lower half of the beam. The expression for the fibers in the upper half of the beam will be similar except that the moment arm vector will be in the positive z {\displaystyle z} direction and the force vector will be in the − x {\displaystyle -x} direction since the upper fibers are in compression. But the resulting bending moment vector will still be in

931-425: Is viewed as a point) in an elastic body can occur along multiple DOF (maximum of six DOF at a point). For example, a point on a horizontal beam can undergo both a vertical displacement and a rotation relative to its undeformed axis. When there are M {\displaystyle M} degrees of freedom a M × M {\displaystyle M\times M} matrix must be used to describe

980-425: The − y {\displaystyle -y} direction since e z × − e x = − e y . {\displaystyle \mathbf {e_{z}} \times -\mathbf {e_{x}} =-\mathbf {e_{y}} .} Therefore, we integrate over the entire cross section of the beam and get for M {\displaystyle \mathbf {M} }

1029-436: The z {\displaystyle z} axis pointing upwards, and the y {\displaystyle y} axis pointing into the figure. The sign of the bending moment M {\displaystyle M} is taken as positive when the torque vector associated with the bending moment on the right hand side of the section is in the positive y {\displaystyle y} direction, that is,

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1078-429: The z {\displaystyle z} direction so as to achieve static equilibrium of moments. If the loading intensity q {\displaystyle q} is taken positive in the positive z {\displaystyle z} direction, then d Q = − q d x {\displaystyle dQ=-qdx} is necessary for force equilibrium. Successive derivatives of

1127-1346: The natural frequencies of the beam. Each of the displacement solutions is called a mode , and the shape of the displacement curve is called a mode shape . The boundary conditions for a cantilevered beam of length L {\displaystyle L} (fixed at x = 0 {\displaystyle x=0} ) are If we apply these conditions, non-trivial solutions are found to exist only if cosh ⁡ ( β n L ) cos ⁡ ( β n L ) + 1 = 0 . {\displaystyle \cosh(\beta _{n}L)\,\cos(\beta _{n}L)+1=0\,.} This nonlinear equation can be solved numerically. The first four roots are β 1 L = 0.596864 π {\displaystyle \beta _{1}L=0.596864\pi } , β 2 L = 1.49418 π {\displaystyle \beta _{2}L=1.49418\pi } , β 3 L = 2.50025 π {\displaystyle \beta _{3}L=2.50025\pi } , and β 4 L = 3.49999 π {\displaystyle \beta _{4}L=3.49999\pi } . The corresponding natural frequencies of vibration are The boundary conditions can also be used to determine

1176-464: The Royal Navy in the late 16th century French submarine Inflexible (S615) , a French nuclear submarine LMS Jubilee Class 5727 Inflexible , a steam locomotive constructed in 1936. ST Inflexible , a French tugboat See also [ edit ] Flexibility (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

1225-496: The SI system, rotational stiffness is typically measured in newton-metres per radian . In the SAE system, rotational stiffness is typically measured in inch- pounds per degree . Further measures of stiffness are derived on a similar basis, including: The elastic modulus of a material is not the same as the stiffness of a component made from that material. Elastic modulus is a property of

1274-494: The above equation is where A 1 , A 2 , A 3 , A 4 {\displaystyle A_{1},A_{2},A_{3},A_{4}} are constants. These constants are unique for a given set of boundary conditions. However, the solution for the displacement is not unique and depends on the frequency. These solutions are typically written as The quantities ω n {\displaystyle \omega _{n}} are called

1323-433: The axial stiffness is k = E ⋅ A L {\displaystyle k=E\cdot {\frac {A}{L}}} where Similarly, the torsional stiffness of a straight section is k = G ⋅ J L {\displaystyle k=G\cdot {\frac {J}{L}}} where Note that the torsional stiffness has dimensions [force] * [length] / [angle], so that its SI units are N*m/rad. For

1372-436: The beam's cross section is in the y z {\displaystyle yz} plane, and the relevant second moment of area is where it is assumed that the centroid of the cross section occurs at y = z = 0 {\displaystyle y=z=0} . Often, the product E I {\displaystyle EI} (known as the flexural rigidity ) is a constant, so that This equation, describing

1421-451: The beam. The stresses in a beam can be calculated from the above expressions after the deflection due to a given load has been determined. Because of the fundamental importance of the bending moment equation in engineering, we will provide a short derivation. We change to polar coordinates. The length of the neutral axis in the figure is ρ d θ . {\displaystyle \rho d\theta .} The length of

1470-402: The beam. Typically a value of A 1 = 1 {\displaystyle A_{1}=1} is used when plotting mode shapes. Solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency ω n {\displaystyle \omega _{n}} , i.e., the beam can resonate . The natural frequencies of

1519-638: The bending moment vector exerted on the right cross section of the beam the expression where I {\displaystyle I} is the second moment of area . From calculus, we know that when d w d x {\displaystyle {\dfrac {dw}{dx}}} is small, as it is for an Euler–Bernoulli beam, we can make the approximation 1 ρ ≃ d 2 w d x 2 {\displaystyle {\dfrac {1}{\rho }}\simeq {\dfrac {d^{2}w}{dx^{2}}}} , where ρ {\displaystyle \rho }

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1568-460: The constituent material; stiffness is a property of a structure or component of a structure, and hence it is dependent upon various physical dimensions that describe that component. That is, the modulus is an intensive property of the material; stiffness, on the other hand, is an extensive property of the solid body that is dependent on the material and its shape and boundary conditions. For example, for an element in tension or compression ,

1617-461: The corresponding Euler–Lagrange equation is Now, Plugging into the Euler–Lagrange equation gives or, which is the governing equation for the dynamics of an Euler–Bernoulli beam. When the beam is homogeneous, E {\displaystyle E} and I {\displaystyle I} are independent of x {\displaystyle x} , and the beam equation

1666-407: The coupling stiffness. It is noted that for a body with multiple DOF, the equation above generally does not apply since the applied force generates not only the deflection along its direction (or degree of freedom) but also those along with other directions. For a body with multiple DOF, to calculate a particular direct-related stiffness (the diagonal terms), the corresponding DOF is left free while

1715-407: The deflection w {\displaystyle w} have important physical meanings: d w / d x {\displaystyle dw/dx} is the slope of the beam, which is the anti-clockwise angle of rotation about the y {\displaystyle y} -axis in the limit of small displacements; is the bending moment in the beam; and is the shear force in

1764-550: The deflection of the beam in the z {\displaystyle z} direction at some position x {\displaystyle x} (recall that the beam is modeled as a one-dimensional object). q {\displaystyle q} is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of x {\displaystyle x} , w {\displaystyle w} , or other variables. E {\displaystyle E}

1813-526: The deflection of a uniform, static beam, is used widely in engineering practice. Tabulated expressions for the deflection w {\displaystyle w} for common beam configurations can be found in engineering handbooks. For more complicated situations, the deflection can be determined by solving the Euler–Bernoulli equation using techniques such as " direct integration ", " Macaulay's method ", " moment area method , " conjugate beam method ", "

1862-412: The figure clearly shows that the fibers in the lower half are in tension. d A {\displaystyle dA} is the differential element of area at the location of the fiber. The differential bending moment vector, d M {\displaystyle d\mathbf {M} } associated with d F {\displaystyle d\mathbf {F} } is given by This expression

1911-741: The function that minimizes the functional S {\displaystyle S} . For a dynamic Euler–Bernoulli beam, the Euler–Lagrange equation is ∂ 2 ∂ x 2 ( E I ∂ 2 w ∂ x 2 ) = − μ ∂ 2 w ∂ t 2 + q ( x ) . {\displaystyle {\cfrac {\partial ^{2}}{\partial x^{2}}}\left(EI{\cfrac {\partial ^{2}w}{\partial x^{2}}}\right)=-\mu {\cfrac {\partial ^{2}w}{\partial t^{2}}}+q(x).}

1960-413: The kinetic energy where μ {\displaystyle \mu } is the mass per unit length, the second term represents the potential energy due to internal forces (when considered with a negative sign), and the third term represents the potential energy due to the external load q ( x ) {\displaystyle q(x)} . The Euler–Lagrange equation is used to determine

2009-497: The mode shapes from the solution for the displacement: As with the cantilevered beam, the unknown constants are determined by the initial conditions at t = 0 {\displaystyle t=0} on the velocity and displacements of the beam. Also, solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency ω n {\displaystyle \omega _{n}} . The boundary conditions of

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2058-402: The mode shapes from the solution for the displacement: The unknown constant (actually constants as there is one for each n {\displaystyle n} ), A 1 {\displaystyle A_{1}} , which in general is complex, is determined by the initial conditions at t = 0 {\displaystyle t=0} on the velocity and displacements of

2107-407: The principle of virtual work ", " Castigliano's method ", " flexibility method ", " slope deflection method ", " moment distribution method ", or " direct stiffness method ". Sign conventions are defined here since different conventions can be found in the literature. In this article, a right-handed coordinate system is used with the x {\displaystyle x} axis to the right,

2156-520: The remaining should be constrained. Under such a condition, the above equation can obtain the direct-related stiffness for the degree of unconstrained freedom. The ratios between the reaction forces (or moments) and the produced deflection are the coupling stiffnesses. The elasticity tensor is a generalization that describes all possible stretch and shear parameters. A single spring may intentionally be designed to have variable (non-linear) stiffness throughout its displacement. The inverse of stiffness

2205-428: The sciences, especially structural and mechanical engineering . Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made. The Bernoulli beam

2254-408: The special case of unconstrained uniaxial tension or compression, Young's modulus can be thought of as a measure of the stiffness of a structure. The stiffness of a structure is of principal importance in many engineering applications, so the modulus of elasticity is often one of the primary properties considered when selecting a material. A high modulus of elasticity is sought when deflection

2303-421: The stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or different points) or the same degree of freedom at two different points. In industry, the term influence coefficient is sometimes used to refer to

2352-474: The title Inflexible . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Inflexible&oldid=1035454218 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Stiffness The complementary concept

2401-623: Was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution . Additional mathematical models have been developed, such as plate theory , but the simplicity of beam theory makes it an important tool in

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