BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko

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1 BEAM THEORIES The difference between Euler-Bernoulli and Timoschenko Uemuet Goerguelue Two mathematical models, namely the shear-deformable (Timoshenko) model and the shearindeformable (Euler-Bernoulli) model, are presented. Since the Timoshenko beam theory is higher order than the Euler-Bernoulli theory, it is known to be superior in predicting the transient response of the beam. The superiority of the Timoshenko model is more pronounced for beams with a low aspect ratio. It is shown that use of an Euler-Bernoulli based controller to suppress beam vibration can lead to instability caused by the inadvertent excitation of unmodelled modes. BEAM189 is an element suitable for analyzing slender to moderately stubby/thick beam structures. This element is based on Timoshenko beam theory. Shear deformation effects are included. This element is well-suited for linear, large rotation, and/or large strain nonlinear applications. BEAM189 includes stress stiffness terms, by default, in any analysis with NLGEOM,ON. The provided stress stiffness terms enable the elements to analyze flexural, lateral, and torsional stability problems (using eigenvalue buckling or collapse studies with arc length methods). The beam elements are based on Timoshenko beam theory, which is a first order shear deformation theory: transverse shear strain is constant through the cross section; that is, cross sections remain plane and undistorted after deformation. BEAM188/BEAM189 elements can be used for slender or stout beams. Due to the limitations of first order shear deformation theory, only moderately "thick" beams may be analyzed. The accuracy in modeling composite shells is goverened by the first order shear deformation theory (usually referred to as Mindlin/Reissner shell theory). In Euler Bernoulli beam theory, shear deformations are neglected, and plane sections remain plane and normal to the longitudinal axis. In the Timoshenko beam theory, plane sections still remain plane but are no longer normal to the longitudinal axis. The difference between the normal to the longitudinal axis and the plane section rotation is the shear deformation. These relations are shown in figure.

2 It can be seen in figure IV-2 that in the Euler - Bernoulli beam the deformation at a section, dvo/dx, is just the rotation due to bending only, since the plane section remains normal to the longitudin alaxis. However, in the Timoshenko beam the section deformation is the sum of two contributions: one is due to bending, dvb/dx, and the other is the shear deformation, dvs/dx. By considering an infinitetesimal length of the beam, as shown in figure IV-3, it is seen that the shear deformation in Timoshenko beam theory, dvs/dx, is the same as the shear strain related to pure shear. For linear elastic materials, Hooke s law for shear applies and: Where is equal to the shear stress applied to the element and G is the shear modulus of elasticity for the material. In the Timoshenko beam theory, the shear stress is assumed constant over the cross section. The shear force, V, is related to the shear stress through: where As is equal to the shear area of the section. Combining these two equations:

3 While this equation only applies to linearly elastic materials, it will be the basis for the formulation of the non-linear shear force - shear strain relation. In this study, it is assumed that V and are interrelated though the shear area of the section multiplied by a value which accounts for the non-linear response of Reinforced Concrete to shear force. The existing force based elements available in FEAP account for the deformation resulting from bending alone, however they do not include the shear deformations. The shear deformations will be added according to the following formulation. Shear deflection effects are often significant in the lateral deflection of short beams. The significance decreases as the ratio of the radius of gyration of the beam cross-section to the beam length becomes small compared to unity. Shear deflection effects are activated in the stiffness matrices of ANSYS beam elements by including a nonzero shear deflection constant (SHEAR_) in the real constant list for that element type. The shear deflection constant is defined as the ratio of the actual beam cross-sectional area to the effective area resisting shear deformation. The shear constant should be equal to or greater than zero. The element shear stiffness decreases with increasing values of the shear deflection constant. A zero shear deflection constant may be used to neglect shear deflection. Shear deflection constants for several common sections are as follows: rectangle (6/5), solid circle (10/9), hollow (thin-walled) circle (2), hollow (thin-walled) square (12/5). Shear deflection constants for other cross-sections can be found in structural handbooks.

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7 Theme of session The Timoshenko beam element is formulated using an iso-parametric formulation. Timoshenko beam element The difference between the Timoshenko beam and the technical, Bernoulli, beam is that the former includes the effect of the shear stresses on the deformation. A constant shear over the beam height is assumed. Setting the shear angle, γ, to zero leads to the Bernoulli beam theory. Then the slope of the centre axis, -w', is the same as the rotation, θ, of the cross-section. The shear strain is related to shear stress and transverse force is Figure. Timoshenko beam element However, the interpolation of rotation and element deflection can now be done independently as which gives The element integrals can now be computed. The derivatives in B are easy to evaluate (they are ±1/L) and the above the N-terms denotes a special treatment of this term discussed below. Shear locking and underintegration Solving the element integral exactly creates an element that locks, gives too small deformation, when it becomes too thin. This is due to the problem of representing the Bernoulli solution which is correct for slender beams. The same interpolation functions are used for rotation and beam deflection and a slender beam wants

8 We solve this and create a better beam element by setting θ constant by fixing See the old archives and sheldon site for more on Beam elements.sheldon given a tip on Beam elements in his site.below some of the comments on Beam elements given by our friends in Xansys and sheldon's tip on Beam elements for your reference, For beam elements, generally the cross sectional dimensions should be less than 1/20th or 1/30th of the length of the member,where the distance between the supports defines the length of the member,the physical dimensions and characteristics determine whether beam elements can be used or not. Beam elements are based on two theories (ie)euler-bernoulli Beam theory and Timoshenko theory. In Euler-Bernoulli beams, transverse shear stress is not taken into account wheras in Timoshenko beams transverse shear stresses are taken into account.the reason why transverse shear stress is not taken into account in Euler - Bernoulli beams is bending is assumed to behave in such a way that cross section normal to the neutral axis remain normal to the neutral axis after bending.in case of Timoshenko beams initially cross sction in normal to the neutral axis but does not remain normal after bending. Actually Euler - Bernoulli Beam elements give good results for normal stress, because they are capable of capturing bending dominated deformation fields,if a beam is not slender and it goes into bending dominated deformation then Timoshenko elements are weak to capture normal stress and classical beam elements are weak to capture shear deformation. I think the superiority of classical elements to Timoshenko beam elements comes from cubic Hermitian shape functions. Classical beams are very good for thin beam applications whereas timoshenko beams for good for thick beams. Hope this helps.

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