Belytschko T. - Introduction (779635), страница 101
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We will then examine CB beamelements more thoroughly from a theoretical viewpoint.The CB approach is subsequently employed for the development of shellelements. Again, we begin with the implementation, illustrating how many of thetechniques developed for continuum elements in the previous chapters can be applieddirectly to shells. The CB shell theory developed here is a synthesis of variousapproaches reported in the literature but also incorporates a new treatment of changes inthickness due to large deformations and conservation of matter. As part of this treatment,the methodologies for describing large rotations in three dimensions are described.Two of the pitfalls of CB shell elements are then examined: shear and membranelocking. These phenomena are examined in the context of beams but the insights gainedare applicable to shell elements. Methods for circumventing these difficulties by meansof assumed strain fields are described and examples of elements which alleviate shear andmembrane locking are given.We conclude with a description of 4-node quadrilateral shell elements thatevaluate the internal nodal forces with one stack of quadrature points, often called onepoint quadrature elements.
These elements are widely used in explicit methods and largescale analysis. Several elements of this genre are reviewed and compared and thetechniques for consistently controlling the hourglass modes which result from theunderintegration are described.9.2TWO DIMENSIONAL BEAMS9-2T.
Belytschko, Chapter 9, Shells and Structures, December 16, 19989.2.1. Governing Equations and Assumptions.In this Section the CB theoryis developed for beams. In addition, we develop a beam element based on classical beamtheory.The governing equations for structures are identical to those for continua:1. conservation of matter2. conservation of linear and angular momentum3.
conservation of energy4. constitutive equations5. strain-displacement equationsThe key feature which distinguishes structures from continua is that assumptions aremade about the motion and the state of stress in the element. In other words, the motionis constrained so that it satisfies certain hypothesis which are based on experimentalobservations on the motion of thin structures and shells. The assumptions on the motionare called kinematic assumptions, the assumptions on the stress field are called kineticassumptions.The major kinematic assumption concerns the motion of the normals to themidline (also called reference line) of the beam. In linear structural theory, the midline isusually chosen to be the loci of the centroids of the cross-sections of the beam. However,the selection of a reference line has no effect on the response of a CB element: any linewhich corresponds approximately to the shape of the beam may be chosen as thereference line.
The choice of reference line only effects the values of the resultantmoments; the stresses and the overall response are not affected. We will use the termsreference line and midline interchangeably, noting that even when the term midline isused the precise location of this line relative to the cross-section of the beam is irrelevantin a CB element. The plane defined by the normals to the midline is called the normalplane.
Fig. 9.2 shows the reference line and normal plane for a beam.ynreference linePCxnormal planePPCCMindlin - Reissnerassumption (exaggerated)Euler Bernonlli assumption9-3T. Belytschko, Chapter 9, Shells and Structures, December 16, 1998Figure 9.2. Motion in an Euler-Bernoulli bean and a shear (Mindlin-Reissner) beam; in the Euler-Bernoullibeam, the normal plane remains plane and normal, whereas in the shear beam the normal plane remainsplane but not normal.Two types of beam theory are widely used: Euler-Bernoulli beam theory andshear beam theory. The kinematic assumptions of these theories are:1.
in Euler-Bernoulli beam theory the planes normal to the midline areassumed to remain plane and normal; this is also called engineering beamtheory while the corresponding shell theory is called the Kirchhoff-Loveshell theory;2. in shear beam theory the planes normal to the midline are assumed toremain plane; this is also called Timoshenko beam theory, and thecorresponding shell theory is called the Mindlin-Reissner shell theory;Euler-Bernoulli beams, as we shall see shortly, do not admit any transverse shear,whereas beams governed by the second assumption do admit transverse shear. Themotions of an Euler-Bernoulli beam are a subset of the motions encompassed by shearbeam theory.For the purpose of describing the consequences of these kinematic assumptions,we consider a straight beam along the x-axis in two dimensions as shown in Fig.
9.2. Letthe x-axis coincide with the midline and the y-axis with the normal to the midline. Weconsider only the instant when the beam is in the configuration described, so thefollowing equations do not constitute a nonlinear theory. We will first express thekinematic assumptions mathematically and develop the rate-of-deformation tensor; therate-of-deformation will have the same properties as the linear strain since the equationsfor the rate-of-deformation can be obtained by replacing velocities by displacements inthe linear strain equations.
The aim of the following is to illustrate the consequences ofthe kinematic assumptions on the strain field, not to construct a theory which is worthimplementing.9.9.2. Timoshenko (Shear Beam) Theory. We first describe the shear beamtheory. This beam thoery is usually called Timoshenko beam theory. The majorassumption of this theory is that the normal planes are assumed to remain plane, i.e. flat.Thus the planes normal to the midline rotate as rigid bodies. Consider the motion of apoint P whose orthogonal projection on the midline is point C.
If the normal planerotates as a rigid body, the velocity of point P relative to the velocity of point C is givenbyv CP = ω × r(9.2.1a)where ω is the angular velocity of the plane and r is the vector from C to P. In twodimensions, the only nonzero component of the angular velocity vector of the plane is thez-component, so ω = θ˙ ez ≡ωe z . Since r = yey , the relative velocity isv CP = ω × r =− yωe x .(9.2.1b)The velocity of any point along the midline is only a function of x, sov M ( x) = vxM ( x)e x + vyM ( x)e y(9.2.1c)9-4T.
Belytschko, Chapter 9, Shells and Structures, December 16, 1998The velocity of any point in the beam is then given by adding the relative velocity(9.2.1b) to the midline velocityv = v M ( x ) +ω × r = v M ( x) − yωe x(9.2.1d)The x-component of the total velocity is obtained form the above:v x( x, y) = vMx ( x ) − yω ( x)(9.2.2)where vxM ( x) is the x-component of the velocity of the midline and θ˙ (x ) is the angularvelocity of the normal to the midline. The y-component of the velocity is equivalent tothat of the midline through the depth of the beam, sovy ( x, y ) = v yM ( x)(9.2.3)Applying the definition of the rate-of-deformation Dij = sym(v i, j ) , see Section 3.3.2,shows that the rate-of-deformation for a Timoshenko beam is given byDxx = vMx, x − yω , x , D yy = 0, D xy =(1 Mv −ω2 y,x)(9.2.4a-c)It can be seen that the only nonzero components of the rate-of-deformation are the axialcomponent, Dxx , and the shear component, Dxy , the latter is called the transverse shear.It can be seen immediately from (9.2.2) and (9.2.3) that the dependent variablesand θ ( x ) need only be C 0 for the rate-of-deformation to be finite throughout thebeam.
Thus the standard isoparametric shape functions can be used in the construction ofshear beam finite elements. Theories for which the interpolants need only be C 0 areoften called C 0 structural theories.viM ( x )9.2.3. Euler-Bernoulli Theory.In the Euler-Bernoulli or engineering beamtheories, the kinematic assumption is that the normal remains normal and straight.Therefore the angular velocity of the normal is given by the rate of change of the slope ofthe midlineω = vyM, xBy examining Eq. (9.2.4c) it can be seen that the above is equivalent to requiring theshear rate-of-deformation Dxy to vanish, which implies that the angle between the normaland the midline does not change, i.e.
the normal remains normal. The axial displacementis then given byvx ( x, y ) = vxM ( x ) − yvMy, x ( x )The rate-of-deformation in Euler-Bernoulli (or engineering) beam theory is given by9-5T. Belytschko, Chapter 9, Shells and Structures, December 16, 1998MDxx = vMx, x − yv y , xx , Dyy = 0, Dxy = 0Two features are noteworthy in the above:1. the transverse shear vanishes;2.
the second derivative of the velocity appears in the expression for the rate-ofdeformation tensor, so the velocity field must be C1 for the rate-ofdeformation to be well-defined.Whereas in the Timoshenko beam, two dependent variables are needed, only a singledependent variable is needed for the Euler-Bernoulli beam. Similar reductions in thenumber of unknowns take place in the corresponding shell theories: a Kirchhoff-Loveshell theory only has three dependent variables, whereas a Mindlin-Reissner theory hasfive dependent variables (six are often used in practice; this is discussed in Section 9.4.This type of structural theory is often called a C1 theory because of the need for C1approximations.
The requirement for C1 approximations is the biggest disadvantage ofEuler-Bernoulli and Kirchhoff-Love theories, since C1 approximations are difficult toconstruct in more than one dimension. For this reason, C1 structural theories are seldomused in software except for beams. Beam elements are often based on Euler-Bernoullitheory because C1 interpolants are easily constructed in one dimension. Theories whichrequire C1 interpolants are often called C1 structural theories.Transverse shear is of significance only in thick beams.
However Timoshenkobeams Mindlin-Reissner shells are frequently used even when transverse shear is notphysically important. For thin beams, the transverse shears in Timoshenko beams also goto zero in well-behaved elements.<b>Текст обрезан, так как является слишком большим</b>.
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