## A

The FE mathematical idealization of a MoM member. The model is one-dimensional in x. The two end joints are the site of end quantities joint forces and displacements, that interconnect members. The internal quantities characterize the stresses and deformations in the member. Figure 6.2. The FE mathematical idealization of a MoM member. The model is one-dimensional in x. The two end joints are the site of end quantities joint forces and displacements, that interconnect members. The...

## Jud

Typical finite element geometries in one through three dimensions. Figure 7.3. Typical finite element geometries in one through three dimensions. Just like members in the truss example, one can take finite elements of any kind one at a time. Their local properties can be developed by considering them in isolation, as individual entities. This is the key to the modular programming of element libraries. In the Direct Stiffness Method, elements are isolated by disconnection and...

## E [ ux Uyi UX Uxn Uyn T

Displacement Interpolation The displacement field u(e) (x, y) over the element is interpolated from the node displacements. We shall assume that the same interpolation functions are used for both displacement components.3 Thus (x, y) J2 Nje)(x, y) Uxi, uy(x, y) N e)(x, y) Uyi, where Nje x, y) are the element shape functions. In matrix form where Nje x, y) are the element shape functions. In matrix form ' N e) 0 N(e) 0 N(e) 0 0 N(e) 0 N(e) 0 N(e) _ The minimum conditions on N(e (x, y)...

## Exercise

N 20 Construct by hand the free-free master stiffness matrix of (3.20) using the freedom-pointer technique (3.28). Note start from K initialized to the null matrix, then cycle over e 1, 2, 3. 3 This scheme is recommended to do matrix multiplication by hand. It is explained in B.3.2 of Appendix B. 4 A matrix is singular if its determinant is zero cf. C.2 of Appendix C for a refresher in that topic.

## Exercises

The previous Chapter introduced the TPE-based variational formulation of finite elements, which was illustrated for the bar element. This Chapter applies that technique to a more complicated one-dimensional element the plane beam described by the Bernoulli-Euler mathematical model. Mathematically, the main difference of beams with respect to bars is the increased order of continuity required for the assumed transverse-displacement functions to be admissible. Not only must these functions be...

## Info

In variational mathematics this is called a duality pairing. Table 7.1. Physical Significance of Vectors u and f in Miscellaneous FEM Applications Structures and solid mechanics Heat conduction Acoustic fluid Potential flows General flows Electrostatics Magnetostatics Electric potential Magnetic potential Displacement potential Particle velocity The physical significance of the vectors u and f varies according to the application being modeled, as illustrated in Table 7.1. If the relation...

## L

Terminology in Bernoulli-Euler model of plane beam. 4. Strain energy. The internal strain energy of the beam member accounts only for bending moment deformations. All other effects, notably transverse shear and axial force effects, are ignored. 5. Linearization. Transverse deflections, rotations and deformations are considered so small that the assumptions of infinitesimal deformations apply. 6. Elastic behavior. The beam is fabricated of material assumed to be elastic and...

## O

Hermitian shape functions of plane beam element These shape functions are conveniently written in terms of the dimensionless natural coordinate Figure 13.8. Hermitian shape functions of plane beam element These shape functions are conveniently written in terms of the dimensionless natural coordinate which varies from 1 at node i (x 0) to +1 at node j (x i) i being the element length Nve) 4 (1 - S)2(2 + S), N< e 8 (1 - S)2(1 + S), Nj 4 (1 + )2(2 ), Nj 881(1 + )2(1 ). These...

## P m

Where u contains all degrees of freedom and each ap is a row vector with same length as u. To incorporate the MFCs into the FEM model one selects a weight wp > 0 for each constraints and constructs the so-called Courant quadratic penalty function or penalty energy P Pp > with Pp uT (2 aTp a p u - w p aTpbp) 1 uT K( p)u - uTf p), (10.12) where we have called K(p) wpa Tap and fp) wpaT bt. P is added to the potential energy function n 2 uT Ku uT f to form the augmented potential energy na n +...

## Remark

Paraphrasing an old joke about mathematicians, one may define a computational mechanician as a person who searches for solutions to given problems, an applied mechanician as a person who searches for problems that fit given solutions, and a theoretical mechanician as a person who can prove the existence of problems and solutions. Several branches of computational mechanics can be distinguished according to the physical scale of the focus of attention Nanomechanics deals with phenomena at the...

WHERE THIS MATERIAL FITS 1-3 1.1.1. Computational 1.1.2. Statics vs. Dynamics 1-4 1.1.3. Linear vs. 1.1.4. Discretization 1.1.5. FEM Variants 1.2. WHAT DOES A FINITE ELEMENT LOOK LIKE 1-6 1.3. THE FEM ANALYSIS PROCESS 1-7 1.3.1. The Mathematical FEM 1-8 1.3.2. The Physical 1.3.3. Synergy of Physical and Mathematical FEM 1-9 1.4. INTERPRETATIONS OF THE FINITE ELEMENT METHOD 1-11 1.4.1. Physical Interpretation 1-11 1.4.2. Mathematical 1.6. *WHAT IS NOT COVERED 1-13

## Third Stage Sivb

The Apollo short stack. 11.2.1. Condensation by Explicit Matrix Operations To carry out the condensation process, the assembled stiffness equations of the superelement are partitioned as follows Kbb Kbi ub _ fb ( _ Kib Kii J u _ fi where subvectors ub and u,- collect boundary and interior degrees of freedom, respectively. Take the second matrix equation If K,, is nonsingular we can solve for the interior freedoms Replacing into the first matrix equation of (11.2) yields the...

## U T g

Here the constraint gap vector g is nonzero and T is the same as before. To get the modified system applying the shortcut rule of Remark 9.4, premultiply both sides of (9.25) by TtK, replace Ku by f, and pass the data to the RHS Ku f, in which K TT KT, f Tt (f - Kg). Upon solving (9.26) for u, the complete displacement vector is recovered from (9.25). For the MFC (9.22) this technique gives the system

## Uxl Uy Uy

Whereas the known applied forces are When solving the stiffness equations by hand, the simplest way to account for support conditions is to remove equations associated with known joint displacements from the master system. To apply (3.21) we have to remove equations 1, 2 and 4. This can be systematically accomplished by deleting or striking out rows and columns number 1, 2 and 4 from K and the corresponding components from f and u. The reduced three-equation system is

## N uw

The Tonti diagram for the governing equations of the Bernoulli-Euler beam model. Figure 13.6. The Tonti diagram for the governing equations of the Bernoulli-Euler beam model. where as usual Uand Ware the internal and external energies, respectively. As previously explained, in the Bernoulli-Euler model U includes only the bending energy U f aedV Mk dx f EIk2 dx f EI vf dx f vEIv dx. J V J 0 Jo Jo Jo The external work W accounts for the applied transverse force The three functionals...

## Idealization

The idealization process for a simple structure. The physical system, here a roof truss, is directly idealized by the mathematical model a pin-jointed bar assembly. For this particular structure, the idealization coalesces with the discrete model. Figure 1.6. The idealization process for a simple structure. The physical system, here a roof truss, is directly idealized by the mathematical model a pin-jointed bar assembly. For this particular structure, the idealization coalesces with...

## Distributed Force On Nodes Finite Element

In many thin structures modeled as continuous bodies the appearance of skinny elements is inevitable on account of computational economy reasons. An example is provided by the three-dimensional modeling of layered composites in aerospace and mechanical engineering problems. A physical interface, resulting from example from a change in material, should also be an interelement boundary. That is, elements must not cross interfaces. See Figure 8.3. In two-dimensional FE modeling, if you have a...