• Energy principles (see, for example, Washizu, 1974; Reddy, 1984; Liu, 2009).

• Weighted residual methods (see, for example, Crandall, 1956; Finlayson and Scriven, 1966; Finlayson, 1972; Zienkiewicz and Taylor, 2000; Liu, 2009; Liu and Zhang, 2013).

3.3.1 Hamilton’s principle

Hamilton’s principle is a simple yet powerful tool that can be used to derive discretized dynamic system equations. It states simply that,

Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional stationary.

An admissible displacement must satisfy the following conditions:

Condition (a) ensures that the displacements are compatible (continuous) in the problem domain. As will be seen in Chapter 11, there are situations when incompatibility can occur at the edges between elements. Condition (b) ensures that the displacement constraints are satisfied; and condition (c) requires the displacement history to satisfy the constraints at the initial and final times.

Mathematically, Hamilton’s principle states:

(3.1)

The Langrangian functional, L, is obtained using a set of admissible time histories of displacements, and it consists of

(3.2)

where T is the kinetic energy, Π is the potential energy (for our purposes, it is the elastic strain energy), and W_f is the work done by the external forces. It is clear that the Langrangian functional L is some kind of potential energy of the entire system, and thus the kinetic energy and the work done by the external forces results in deduction in paternal energy.

The kinetic energy of the entire problem domain is defined in the integral form

(3.3)

where V represents the whole volume of the solid, and U is the set of admissible time histories of displacements.

The strain energy in the entire domain of elastic solids and structures can be expressed as

(3.4)

where ε are the strains obtained using the set of admissible time histories of displacements. The work done by the external forces over the set of admissible time histories of displacements can be obtained by

(3.5)

where S_f represents the surface of the solid on which the surface forces are prescribed (see Figure 2.2).

Hamilton’s principle allows one to simply assume any set of displacements, as long as it satisfies the three admissible conditions. The assumed set of displacements will not usually satisfy the strong form of governing system equations unless we are extremely lucky, or the problem is extremely simple and we know the exact solution (in this case we do not need to solve the problem). Application of Hamilton’s principle will conveniently guarantee a combination of this assumed set of displacements to produce the most accurate (in the energy norm measure) solution for the system that is governed by the strong form of the system equations.

The power of Hamilton’s principle (or any other variational principle) is that it provides freedom of choice, opportunity, and possibility. For practical engineering problems, one usually does not have to pursue the exact solution, which in most cases is usually unobtainable, because we now have a choice of quite conveniently obtaining a good approximation using Hamilton’s principle, by assuming the likely form, pattern or shape of the solutions. Hamilton’s principle thus provides the foundation for the FEMs. Furthermore, the simplicity of Hamilton’s principle (or any other energy principle) manifests itself in the use of scalar energy quantities. Engineers and scientists like working with scalar quantities when it comes to numerical calculations, as they do not need to worry about the direction. All the mathematical tools required to derive final discrete system equations are basic operations of integration, differentiation, and variation, all of which are standard linear operations. Another plus point of Hamilton’s principle is that the final discrete system equations produced are usually a set of linear algebraic equations that can be solved using conventional methods and standard computational routines. The following demonstrates how the finite element equations can be established using Hamilton’s principle and its simple operations.

3.3.2 Minimum total potential energy principle

For static problems, Hamilton’s principle reduces to the well-known minimum total potential energy principle, which may be stated as,

Of all the admissible displacements the most accurate solution makes the total potential energy minimum.

An admissible displacement must satisfy the following conditions:

Mathematically, the “most” accurate solution is found using

(3.1a)

The total potential energy, Π_t, is obtained using the assume admissible displacement, and it consists of

(3.2a)

It can be proven for stable materials of a well-posed solid mechanics problem that the stationary point given in (3.1a) is a minimum point of the total potential energy for the solid (see, e.g., Liu, 2009).

3.4.1 Domain discretization

The solid or structure is divided into N_e elements with a set of N_n nodes. The procedure is often called meshing, which is usually performed using so-called pre-processors. This is especially true for complex geometries. Figure 3.1 shows an example of a mesh for a two-dimensional solid.

The pre-processor generates unique numbers for all the elements and nodes for the solid or structure in a proper manner. An element is formed by connecting its nodes in a pre-defined consistent fashion to create the connectivity of the element. All the elements together form the entire domain of the problem without any gap or overlapping. It is possible to have different types of elements with different numbers of nodes, as long as they are compatible (no gaps and overlapping; the admissible condition (a) required by Hamilton’s principle) on the boundaries between different elements. In fact, meshes of mixed triangular and quadrilateral elements are frequently used to overcome the difficulties of creating preferable pure quadrilateral elements. The density of the mesh depends upon the accuracy requirement of the analysis and the computational resources available. Generally, a finer mesh will yield results that are more accurate, but will increase the computational cost. As such, the mesh is usually not uniform, with a finer mesh being used in the areas where the displacement gradient is larger or where the accuracy is critical to the analysis. The purpose of the domain discretization is to make it easier in assuming the pattern of the displacement field.

3.4.2 Displacement interpolation

The FEM formulation has to be based on a coordinate system. In formulating FEM equations for elements, it is often convenient to use a local coordinate system that is defined for an element in reference to the global coordination system that is usually defined for the entire structure, as shown in Figure 3.4. Based on the local coordinate system attached on an element, the displacement within the element is now assumed simply by polynomial interpolation using the displacements at its nodes (or nodal displacements) as

(3.6)

where the superscript h stands for approximation, n_d is the number of nodes forming the element, and d_i is the nodal displacement at the ith node, which is the unknown the analyst wants to find out, and can be expressed in a general form of

(3.7)

where n_f is the number of Degrees Of Freedom (DOF) at a node. For 3D solids, n_f = 3, and

(3.8)

Note that the displacement components can also consist of rotations for structures of beams and plates. The vector d_e in Eq. (3.6) is the displacement vector for the entire element, and has the form of

(3.9)

Therefore, the total DOF for the entire element is n_d × n_f.

In Eq. (3.6), N is a matrix of shape functions for the nodes in the element, which are pre-defined to assume the shapes of the displacement variations with respect to the coordinates. It has the general form of

(3.10)

where N_i is a sub-matrix of shape functions for displacement components, which is arranged as

(3.11)

where N_ik is the shape function for the kth displacement component, DOF at the ith node. For 3D solids, n_f = 3, and often we use N_i1 = N_i2 = N_i3 = N_i. Note that it is not always necessary to use the same shape function for all the displacement components at a node. For example, we often use different shape functions for translational and rotational displacements.

Note that this approach of assuming the displacements is often called the displacement method. Most of the commercially available FEM packages use this approach. There are, however, FEM approaches that assume the stresses instead, but they will not be covered in this book.

3.4.3 Standard procedure for constructing shape functions

Consider an element with n_d nodes at x_i; (i = 1, 2, …, n_d), where x^T = {x} for one-dimensional problems, x^T = {x y} for two-dimensional problems, and x^T = {x y z} for three-dimensional problems. We should have n_d shape functions for each displacement component for an element. In the following, we consider only one displacement component in the explanation of the standard procedure for constructing the shape functions. The standard procedure is applicable for any other displacement components. First, the displacement component is approximated in the form of a linear combination of n_d linearly-independent basis functions p_i(x), i.e.

(3.12)

where u^h is the approximation of the displacement component, p_i(x) is the basis function of monomials in the space coordinates x, and α_i is the coefficient for the monomial p_i(x). Vector α is defined as

(3.13)

The p_i(x) in Eq. (3.12) is built with n_d terms of one-dimensional monomials; based on the Pascal’s triangle shown in Figure 3.2 for two-dimensional problems; or the well-known Pascal’s pyramid shown in Figure 3.3 for three-dimensional problems. A basis of complete order of p in the one-dimensional domain has the form

(3.14)

A basis of complete order of p in the two-dimensional domain is provided by

(3.15)

and that in the three-dimensional domain can be written as

(3.16)

As a general rule, the n_d terms of p_i(x) used in the basis should be selected from the constant term to higher orders symmetrically from the Pascal triangle shown in Figure 3.2 or Figure 3.3. Some higher order terms can be selectively included in the polynomial basis if there is a need in specific circumstances.

The coefficients α_i in Eq. (3.12) can be determined by enforcing the displacements calculated using Eq. (3.12) to be equal to the nodal displacements at the n_d nodes of the element. At node i we can have

(3.17)

where d_i is the nodal value of u^h at x = x_i. Equation (3.17) can be written in the following matrix form:

(3.18)

where d_e is the vector that includes the values of the displacement component at all the n_d nodes in the element:

(3.19)

and P is given by

(3.20)

which is called the moment matrix. The expanded form of P is

(3.21)

For two-dimensional polynomial basis functions, we may have

(3.22)

Using Eq. (3.18), and assuming that the inverse of the moment matrix P exists, we can then have

(3.23)

Substituting Eq. (3.23) into Eq. (3.12), we then obtain

(3.24)

or in matrix form

(3.25)

where N(x) is a matrix of shape functions N_i (x) defined by

(3.26)

where is the ith column of matrix , and

(3.27)

The derivatives of the shape functions can be obtained very easily, as all the functions involved are polynomials. The lth derivative of the shape functions is simply given by

(3.28)

3.4.4 Properties of the shape functions

To prove this, we write a set of shape functions in the following general form of

(3.60)

where we ensure inclusion of the complete linear basis functions of p₂(x) = x. The other basis function p_i(x) (i = 1, 3, …, n_d) in Eq. (3.53) can be monomials or any other type of basis function as long as all the basis functions are linearly-independent.

Consider a linear field of u(x) = x, we should have the nodal vector as follows:

(3.61)

Substituting the above equation into Eq. (3.24), we obtain

(3.62)

At the n_d nodes of the element, we have n_d equations:

(3.63)

Using the delta function property of the shape functions, we have

(3.64)

Hence, Eq. (3.63) becomes

(3.65)

Or in matrix form,

(3.66)

Note that the coefficient matrix of Eq. (3.66) is the moment matrix that has a full rank. We thus have

(3.67)

Substituting the previous equation back into Eq. (3.62), we obtain

(3.68)

This proves the property of linear field reproduction.

The delta function property (Property 3) ensures convenient imposition of the essential boundary conditions (admissible condition (b) required by Hamilton’s principle), because the nodal displacement at a node is independent of that at any other nodes. The constraints can often be written in the form of a so-called Single Point Constraint (SPC). If the displacement at a node is fixed, all one needs to do is to remove corresponding rows and columns without affecting the other rows and columns.

The proof of Property 4 gives a convenient way to confirm the partitions of unity property of shape functions. As long as the constant (unit) basis is included in the basis functions, the shape functions constructed are partitions of unity. Properties 4 and 5 are essential for the FEM to pass the standard patch test, used for decades in the FEM for validating the elements. In the standard patch test, the patch is meshed with a number of elements, with at least one interior node. Linear displacements are then enforced on the boundary (edges) of the patch. A successful patch test requires the FEM solution to produce the linear displacement (or constant strain) field at any interior node. Therefore, the property of reproduction of a linear field of shape function provides the foundation for passing the patch test. Note that the property of reproducing the linear field of the shape function does not guarantee successful patch tests, as there could be other sources of numerical error, such as numerical integration, which can cause failure.

Lemma 1 seems to be redundant, since we already have Property 4. However, Lemma 1 is a very convenient property to use for checking the property of partitions of unity of shape functions that are constructed using other shortcut methods, rather than the standard procedure. Using Lemma 1, one only needs to make sure whether the shape functions satisfy Eq. (3.48).

Lemma 2 is another very convenient property to use for checking the property of partitions of unity of shape functions. Using Lemma 2, we only need to make sure that the constructed n_d shape functions are of the delta function property, and they are linear combinations of the same n_d basis functions that are linearly-independent and contain the constant basis function. The conformation of full rank of the moment matrix of the basis functions can sometimes be difficult. In this book, we usually assume that the rank is full for the normal elements, as long as the basis functions are linearly-independent. In usual situations, one will not be able to obtain the shape functions if the rank of the moment matrix is not full. If we somehow obtained the shape functions successfully, we can usually be sure that the rank of the corresponding moment matrix is full.

Lemma 3 is a very convenient property to use for checking the property of linear field reproduction of shape functions. Using Lemma 3, we only need to make sure that the constructed n_d shape functions are of the delta function property, and they are linear combinations of the same n_d basis functions that are linearly-independent and contain the linear basis function.

3.4.5 Formulation of finite element equations in local coordinate system

Once the shape functions are constructed, the finite element (FE) equation for an element can be formulated using the following process. By substituting the interpolation of the nodes, Eq. (3.6), and the strain–displacement equation, say Eq. (2.5), into the strain energy term (Eq. (3.4)), we have

(3.69)

where the subscript e stands for the element. Note that the volume integration over the global domain has been changed to that over the elements. This can be done because we assume that the assumed displacement field satisfies the compatibility condition (see Section 3.3) on all the edges between the elements. Otherwise, techniques discussed in Chapter 11 are needed. In Eq. (3.69), B gives the strain in the element using the element nodal displacements, and is often called the strain–displacement matrix or simply strain matrix. It is defined by

(3.70)

where L is the differential operator that is defined for different problems in Chapter 2. This implies that the strain matrix is determined once the shape functions are obtained. For 3D solids, L is given by Eq. (2.7). By denoting

(3.71)

which is called the element stiffness matrix, Eq. (3.69) can be rewritten as

(3.72)

Note that the stiffness matrix k_e is symmetrical, because

(3.73)

which shows that the transpose of matrix k_e is itself. In deriving the above equation, c = c^T has been employed. Making use of the symmetry of the stiffness matrix, only half of the terms in the matrix need to be evaluated and stored.

By substituting Eq. (3.6) into Eq. (3.3), the kinetic energy can be expressed as

(3.74)

By denoting

(3.75)

which is called the mass matrix of the element, Eq. (3.74) can be rewritten as

(3.76)

It is obvious that the element mass matrix is also symmetrical.

Finally, to obtain the work done by the external forces, Eq. (3.6) is substituted into Eq. (3.5):

(3.77)

where the surface integration is performed only for elements on the force boundary of the problem domain. By denoting

(3.78)

and

(3.79)

Eq. (3.77) can then be rewritten as

(3.80)

F_b and F_s are the nodal forces acting on the nodes of the elements, which are equivalent to the body forces and surface forces applied on the element in terms of the work done on a virtual displacement. These two nodal force vectors can then be added up to form the total nodal force vector f_e:

(3.81)

Substituting Eqs. (3.72), (3.76), and (3.80) into Lagrangian functional L (Eq. (3.2)), we have

(3.82)

Applying Hamilton’s principle (Eq. (3.1)), we have

(3.83)

Note that the variation and integration operators are interchangeable, hence we obtain

(3.84)

To explicitly illustrate the process of deriving Eq. (3.84) from Eq. (3.83), we use a two-degree of freedom system as an example. Here, we show the procedure for deriving the second term in Eq. (3.84):

In Eq. (3.84), the variation and differentiation with time are also interchangeable, i.e.,

(3.85)

Hence, by substituting Eq. (3.85) into Eq. (3.84), and integrating the first term by parts, we obtain

(3.86)

Note that in deriving Eq. (3.86) as above, the condition δd_e = 0 at t₁ and t₂ have been used, which leads to the vanishing of the first term on the right-hand side. This is because the initial condition at t₁ and final condition at t₂ have to be satisfied for any d_e (admissible conditions (c) required by Hamilton’s principle), and no variation at t₁ and t₂ is allowed. Substituting Eq. (3.86) into Eq. (3.84) leads to

(3.87)

To have the integration in Eq. (3.87) as zero for an arbitrary integrand, the integrand itself has to vanish, i.e,

(3.88)

Due to the arbitrary nature of the variation of the displacements, the only insurance for Eq. (3.88) to be satisfied is

(3.89)

Equation (3.89) is the FEM equation for an element, while k_e and m_e are the stiffness and mass matrices for the element, and f_e is the element force vector of the total external forces acting on the nodes of the element. All these element matrices and vectors can be obtained simply by integration for the given shape functions of displacements.

3.4.6 Coordinate transformation

The element equation given by Eq. (3.89) is often formulated based on the local coordinate system defined on an element. In general, the structure is divided into many elements with different orientations (see Figure 3.4). To assemble all the element equations to form the global system equations, a coordinate transformation has to be performed for each element in order to convert its element equation in reference to the global coordinate system which is defined on the whole structure.

The coordinate transformation gives the relationship between the displacement vector d_e based on the local coordinate system and the displacement vector D_e for the same element, but based on the global coordinate system. It is therefore a transformation of a vector between two (local and global) coordinate systems, which can be written in general in the form of

(3.90)

where T is the transformation matrix that can have different forms depending upon the type of element, and will be discussed in detail in later chapters when different elements are formulated. The transformation matrix can also be applied to the force vectors between the local and global coordinate systems:

(3.91)

in which F_e stands for the force vector at node i on the global coordinate system. Substitution of Eq. (3.90) into Eq. (3.89) leads to the element equation based on the global coordinate system:

(3.92)

where

(3.93)

(3.94)

(3.95)

In the cases when the element equations are formulated based on the global coordinate system (for example for the linear triangular or tetrahedral elements), the above coordinate transformation is not necessary, or the T matrix becomes an identity matrix.

3.4.7 Assembly of global FE equation

The FE equations for all the individual elements can be assembled together to form the global FE system equation:

(3.96)

where K and M are the global stiffness and mass matrix, D is a vector of all the displacements at all the nodes in the entire problem domain, and F is a vector of all the equivalent nodal force vectors. The process of assembly is one of simply adding up the contributions from all the elements connected at a node. The detailed process will be demonstrated in Chapter 4 using example problems. It may be noted here that the assembly of the global matrices can be skipped by combining assembling with the procedure of equation solving. This means that the assembling of a term in the global matrix is done only when the equation solver is operating on this term. Such a procedure can drastically reduce the need for storage during the computation.

3.4.8 Imposition of displacement constraints

The global stiffness matrix K in Eq. (3.96) does not usually have a full rank, because displacement constraints (supports) are not yet imposed, and it is non-negative definite or positive semi-definite. Physically, an unconstrained solid or structure is capable of performing rigid movements. Therefore, if the solid or structure is free of support, Eq. (3.96) gives the behavior that includes the rigid body dynamics, if it is subjected to dynamic forces. If the external forces applied are static, the displacements cannot be uniquely determined from Eq. (3.96) for any given force vector. It is physically meaningless to try to determine the static displacements of an unconstrained solid or structure that can move freely.

For constrained solids and structures, the constraints can be imposed by simply removing the rows and columns corresponding to the constrained nodal displacements. We shall demonstrate this method in example problems in later chapters. After such treatments of constraints (and if the constraints are sufficient), the stiffness matrix K in Eq. (3.96) will be of full rank, and will be Positive Definite (PD). Since we have already proven that K is symmetric, K is of a symmetric positive definite property.

3.4.9 Solving the global FE equation

By solving the global FE equation, displacements at the nodes can be obtained. The strain and stress in any element can then be retrieved using Eq. (3.6) in Eqs. (2.5) and (2.8).

• Jacobi’s method;

• Given’s method and householder’s method;

• the bisection method (using Sturm sequences);

• inverse iteration;

• QR method;

• subspace iteration;

• Lanczos’ method.

3.7.1 Central difference algorithm

We first re-write the system equation (3.106) in the form

(3.108)

where F^residual is the residual force vector, and

(3.109)

is defined as the internal force at time t. The acceleration, , can be simply obtained by

(3.110)

In practice, the above equation does not usually require the solving of the matrix equation, since lumped masses are usually used which form a diagonal mass matrix (Petyt, 1990). The solution to Eq. (3.110) is thus trivial, and the matrix equation is the set of independent equations for each degree of freedom i as follows:

(3.111)

where is the residual force, and m_i is the lumped mass corresponding to the ith DOF. We now introduce the following finite central difference equations:

(3.112)

(3.113)

(3.114)

By eliminating D_t+Δt from Eqs. (3.112) and (3.114), we have

(3.115)

To explain the time stepping procedure, refer to Figure 3.5, which shows an arbitrary plot of either displacement or velocity against time. The time stepping/marching procedure in the central difference method starts at t = 0, and computes the acceleration using Eq. (3.110):

(3.116)

For given initial conditions, D₀ and are known. Substituting D₀, , and into Eq. (3.115), we find . Considering a half of the time step and using the central difference Eqs. (3.112) and (3.113), we have

(3.117)

(3.118)

The velocity, at t = −Δt/2 can be obtained by Eq. (3.117) by performing the central differencing at t = −Δt/2 and using values of and D₀:

(3.119)

After this, Eq. (3.118) is used to compute using and :

(3.120)

Then, Eq. (3.117) is used once again to compute D_Δt using and D₀:

(3.121)

Once D_Δt is determined, Eq. (3.118) at t = Δt/2 can be used to obtain by assuming that the acceleration is constant over the step Δt

(3.122)

and using (that is the prescribed initial velocity). We have,

(3.123)

At the next step in time, is again computed using Eq. (3.110). The above process is then repeated. The time marching is continued until it reaches the final desired time.

Note that in the above process, the solutions (displacement, velocity, and acceleration) are obtained without solving any matrix form of system equation, but repeatedly using Eqs. (3.110), (3.117), (3.118), and (3.122). The central difference method is therefore an explicit method. The time marching in explicit methods is therefore extremely fast, and the coding is also very straightforward. It is particularly suited for simulating highly nonlinear, large deformation, contact, and extremely fast events of mechanics.

The central difference method, like most explicit methods, is conditionally stable. This means that if the time step, Δt, becomes too large to exceed a critical time step, Δt_cr, then the computed solution will become unstable and might grow without limit. The critical time step Δt_cr should be the time taken for the fastest stress wave in the solids/structure to across the smallest element in the mesh. Therefore, the time steps used in the explicit methods are typically 100 to 1000 times smaller than those used with implicit methods (that are outlined in the next subsection). The need to use a small time step, and especially its dependence on the smallest element size, makes the explicit codes lose out to implicit codes for some of the problems, especially for those of slow dynamic events.

3.7.2 Newmark’s method (Newmark, 1959)

Newmark’s method is the most widely used implicit algorithm. The example software used in this book, ABAQUS, also uses the Newmark’s method as its implicit solver except that the equilibrium equation defined in Eq. (3.106) is modified with the introduction of an operator defined by Hilber et al. (1978). In this book, we will introduce the standard Newmark’s method as follows. It is first assumed that

(3.124)

(3.125)

where β and γ are constants chosen by the analyst. Equations (3.124) and (3.125) are then substituted into the system equation, Eq. (3.106) to give

(3.126)

If we group all the terms involving on the left and shift the remaining terms to the right, we can write

(3.127)

where

(3.128)

and

(3.129)

The accelerations can then be obtained by solving the matrix system equation, Eq. (3.127):

(3.130)

Note that the above equation involves matrix inversion, and hence it is analogous to solving a matrix equation. This makes it an implicit method, meaning that one needs to solve a set of linear algebraic equations to obtain a solution at the current time step.

The algorithm normally starts with a prescribed initial velocity and displacements, D₀ and The initial acceleration can then be obtained by substituting D₀ and into Eq. (3.106), if is not prescribed initially. Then Eq. (3.130) can be used to obtain the acceleration at the next time step, The displacements and velocities can then be calculated using Eqs. (3.124) and (3.125), respectively. The procedure then repeats to march forward in time until arriving at the final desired time. Because at each time step, the matrix system Eq. (3.127) has to be solved, which can be very time consuming, the implicit algorithm is a very slow time stepping process.

Newmark’s method, like most implicit methods, is unconditionally stable if γ ≥ 0.5 and β ≥ (2γ + 1)²/16. Unconditionally stable methods are those in which the size of the time step, Δt, will not affect the stability of the solution, but rather it is governed by accuracy considerations. The unconditionally stable property allows the implicit algorithms to use significantly larger time steps when the external excitation is of a slow time variation. When the external excitation changes too fast with time, the time step needs to be sufficiently small to capture the time variation features of the excitation.

3.8.1 Summary of shape function properties

The properties of the shape functions are summarized in Table 3.1.

3.8.2 Sufficient requirements for FEM shape functions

Properties 3 and 4 are the minimum requirements for shape functions workable for the FEM. In mesh-free methods (Liu, 2009), Property 3 is not a necessary condition for shape functions. Property 5 is a sufficient requirement for shape functions workable for the FEM for solid mechanics problems. It is possible for shape functions that do not possess Property 5 to produce convergent FEM solutions. In this book, however, we generally require all the FEM shape functions to satisfy Properties 3, 4, and 5. These three requirements are called the sufficient requirements in this book for FEM shape functions; they are the delta function property, partitions of unity, and linear field reproduction.

3.8.3 Recap of FEM procedure

In FEMs, the displacement field U is expressed by displacements at nodes using shape functions defined over elements. Once the shape functions are found, the mass matrix and force vector can be obtained using Eqs. (3.75), (3.78), (3.79), and (3.81). The stiffness matrix can also be obtained using Eq. (3.71), once the shape functions and the strain matrix have been found. Therefore, to develop FE equations for any type of structure components, all one needs to do is formulate the shape function N and then establish the strain matrix B. The other procedures are very much the same. Hence, in the following chapters, the focus will be mainly on the derivation of the shape function and strain matrix for various types of elements for solids and structural components.

1. What is the difference between the strong and weak forms of system equations?

2. What are the conditions that a summed displacement has to satisfy in order to apply the Hamilton’s principle?

3. Briefly describe the standard steps involved in the FEM.

4. Do we have to discretize the problem domain in order to apply the Hamilton’s principle?

5. What is the purpose of dividing the problem domain into elements?

6. For a function defined in [0, l] as

where l is a given constant, x is an independent variable, and a_i (i = 0,1,2) are variable constants (but independent of x). Show that:

a. The variation operator and definite integral operator are exchangeable, i.e.,

b. The variation operator and the differential operator are exchangeable, i.e.,

7. What are the properties of a shape function? Can we use shape functions that do not possess these properties?

8. Answer the following questions with reference to the two meshes, (a) and (b), shown above:

a. Which mesh will yield more accurate results?

b. Which will be more computationally expensive?

c. Suggest a way of meshing which will yield relatively accurate results and at the same time be less computationally expensive than B?

9. Why is there a need to perform coordinate transformation for each element?

10. Describe how element matrices can be assembled together to form the global system matrix.

11. Consider a 1 DOF system with a rigid block of mass m that can move only in the horizontal direction, as shown in the above figure. The system is at the static equilibrium:

a. Give the expressions for the strain potential energy, work done by external forces, and the total potential energy.

b. Derive the equilibrium equations for the system using the minimum potential energy principle.

c. Show that the total potential energy of the 1DOF system is at a minimum at the equilibrium position.

12. Consider a 1DOF system with a rigid block of mass m that can move only in the horizontal direction, as shown in the previous figure. The system is at a dynamic equilibrium. Derive the dynamic equilibrium equation for this system.

13. Consider a 2DOF system with two rigid blocks that can move only in the horizontal direction, as shown in the above figure. The system is at the static equilibrium:

a. Give the expressions for the strain potential energy, work done by external forces, and the total potential energy.

b. Drive the equilibrium equations for the system using the minimum potential energy principle.

14. Consider again a 2DOF system with two rigid blocks that can move only in the horizontal direction, as shown in the previous figure. The system is at a dynamic equilibrium.

a. Give the expressions for the strain potential energy, work done by external forces, and the total potential energy.

b. Drive the equilibrium equations for the system using the minimum potential energy principle.

15. Consider again the simplest problem of a continuum 1D bar of uniform cross-section, as shown in Figure 2.18. The bar is fixed at the left end and is of length l and section area A. It is subjected to a uniform body force f_x and a concentrated force F_s at the right end. The Young’s modulus of the material is E. Using the method of minimum potential energy, obtain the distribution of the displacement for the following cases:

a. f_x = 0 and F_s = constant;

b. f_x = constant and F_s = constant.