Ebook The finite element method in engineering (4th edition) Part 2

(BQ) Part 2 book The finite element method in engineering has contents: Basic equations and solution procedure; basic equations and solution procedure, analysis of plates, analysis of three dimensional problems, dynamic analysis, formulation and solution procedure,...and other contents.

APPLICATION TO SOLID MECHANICS PROBLEMS

in this book.

In this chapter, the general equations of solid and structural mechanics are presented.The displacement method (or equivalently the principle of minimum potential energy) isused in deriving the finite element equations The application of these equations to severalspecific cases is considered in subsequent chapters

8.2 BASIC EQUATIONS OF SOLID MECHANICS

In 3-dimensional In 2-dimensional In 1-dimensional

The unknown quantities, whose number is equal to the number of equations available, invarious problems are given below:

In 3-dimensional In 2-dimensional In 1-dimensional

Thus, we have as many equations as there are unknowns to find the solution of any stressanalysis problem In practice, we will also have to satisfy some additional equations, such

as external equilibrium equations (which pertain to the overall equilibrium of the bodyunder external loads), compatibility equations (which pertain to the continuity of strainsand displacements), and boundary conditions (which pertain to the prescribed conditions

on displacements and/or forces at the boundary of the body)

Although any analytical (exact) solution has to satisfy all the equations stated viously, the numerical (approximate) solutions, like the ones obtained by using the finiteelement method, generally do not satisfy all the equations However, a sound understand-ing of all the basic equations of solid mechanics is essential in deriving the finite elementrelations and also in estimating the order of error involved in the finite element solution

pre-by knowing the extent to which the approximate solution violates the basic equations,including the compatibility and boundary conditions Hence, the basic equations of solidmechanics are summarized in the following section for ready reference in the formulation

of finite element equations

8.2.2 Equations

(i) External equilibrium equations

If a body is in equilibrium under specified static loads, the reactive forces and momentsdeveloped at the support points must balance the externally applied forces and moments

In other words, the force and moment equilibrium equations for the overall body (overall

or external equilibrium equations) have to be satisfied If φx , φ y, and φz are the bodyforces, Φx, Φ y, and Φz are the surface (distributed) forces, Px , P y, and Pz are the external

concentrated loads (including reactions at support points such as B, C, and D in Figure 8.1), and Qx , Q y , and Qz are the external concentrated moments (including reactions at

support points such as B, C, and D in Figure 8.1), the external equilibrium equations can

BASIC EQUATIONS OF SOLID MECHANICS 281

Figure 8.1 Force System for Macroequilibrium for a Body.

For moment equilibrium:

where S is the surface and V is the volume of the solid body.

(ii) Equations of internal equilibrium:

Due to the application of loads, stresses will be developed inside the body If we consider

an element of material inside the body, it must be in equilibrium due to the internalstresses developed This leads to equations known as internal equilibrium equations.Theoretically, the state of stress at any point in a loaded body is completely defined

in terms of the nine components of stress σxx, σyy, σzz, σxy , σyx, σyz , σzy , σzx, and σxz,where the first three are the normal components and the latter six are the components

of shear stress The equations of internal equilibrium relating the nine components ofstress can be derived by considering the equilibrium of moments and forces acting on the

elemental volume shown in Figure 8.2 The equilibrium of moments about the x, y, and z

axes, assuming that there are no body moments, leads to the relations

σ yx = σxy , σ zy = σyz , σ xz = σzx (8.3)These equations show that the state of stress at any point can be completely defined by

the six components σxx, σyy, σzz, σxy, σyz , and σzx The equilibrium of forces in x, y,

Figure 8.2 Elemental Volume Considered for Internal Equilibrium (Only the Components of

Stress Acting on a Typical Pair of Faces Are Shown for Simplicity)

and z directions gives the following differential equilibrium equations:

For a two-dimensional problem, there will be only three independent stress components

(σxx, σyy, σxy) and the equilibrium equations, Eqs (8.4), reduce to

In one-dimensional problems, only one component of stress, namely σxx, will be there and

hence Eqs (8.4) reduce to

∂σ xx

BASIC EQUATIONS OF SOLID MECHANICS 283

(iii) Stress–strain relations (constitutive relations) for isotropic materials

Three-dimensional case In the case of linearity elastic isotropic three-dimensionalsolid, the stress–strain relations are given by Hooke’s law as

ε0 is the vector of initial strains, E is Young’s modules, and v is Poisson’s ratio of

the material In the case of heating of an isotropic material, the initial strain vector isgiven by

where α is the coefficient of thermal expansion, and T is the temperature charge.

Sometimes, the expressions for stresses in terms of strains will be needed By includingthermal strains, Eqs (8.7) can be inverted to obtain

where the matrix [D] is given by

of plane stress In plane stress distribution, it is assumed that

where z represents the direction perpendicular to the plane of the plate as shown in

Figure 8.3, and the stress components do not vary through the thickness of the plate (i.e.,

in z direction) Although these assumptions violate some of the compatibility conditions,

they are sufficiently accurate for all practical purposes provided the plate is thin In thiscase, the stress–strain relations, Eqs (8.7) and (8.10), reduce to

BASIC EQUATIONS OF SOLID MECHANICS 285

In the case of plane stress, the component of strain in the z direction will be nonzero and

is given by (from Eq 8.7)

distribution, it is assumed that w = 0 and (∂w/∂z) = 0 at every cross section Here, the dependent variables are assumed to be functions of only the x and y coordinates

provided we consider a cross section of the body away from the ends In this case, thethree-dimensional stress–strain relations given by Eqs (8.7) and (8.10) reduce to

z

z y

⎫

⎬

BASIC EQUATIONS OF SOLID MECHANICS 287

(iv) Stress–strain relations for anisotropic materials

The stress–strain relations given earlier are valid for isotropic elastic materials The term

“isotropic” indicates that the material properties at a point in the body are not a function

of orientation In other words, the material properties are constant in any plane passingthrough a point in the material There are certain materials (e.g., reinforced concrete,fiber-reinforced composites, brick, and wood) for which the material properties at anypoint depend on the orientation also In general, such materials are called anisotropicmaterials The generalized Hooke’s law valid for anisotropic materials is given in thissection The special cases of the Hooke’s law for orthotropic and isotropic materials willalso be indicated

For a linearly elastic anisotropic material, the strain–stress relations are given by thegeneralized Hooke’s law as [8.7, 8.8]

BASIC EQUATIONS OF SOLID MECHANICS 289

where the matrix [C] is symmetric and is called the compliance matrix Thus, 21 dent elastic constants (equal to the number of independent components of [C]) are needed

indepen-to describe an anisotropic material Note that subscripts 1, 2, and 3 are used instead of

x, y, and z in Eq (8.37) for convenience.

Certain materials exhibit symmetry with respect to certain planes within the body

In such cases, the number of elastic constants will be reduced from 21 For an orthotropicmaterial, which has three planes of material property symmetry, Eq (8.37) reduces to

Here, E11, E22, and E33 denote the Young’s modulus in the planes defined by axes 1,

2, and 3, respectively; G12, G23, and G13 represent the shear modulus in the planes 12,

23, and 13, respectively; and v12, v13, and v23 indicate the major Poisson’s ratios Thus,nine independent elastic constants are needed to describe an orthotropic material underthree-dimensional state of stress For the specially orthotropic material that is in a state

of plane stress, σ3 = σ23 = σ13 = 0 and Eq (8.38) reduces to

which involves four independent elastic constants The elements of the compliance matrix,

in this case, can be expressed as

The stress–strain relations can be obtained by inverting the relations given byEqs (8.37), (8.38), and (8.40) Specifically, the stress–strain relations for a speciallyorthotropic material (under plane stress) can be expressed as

section, we assume the deformations to be small so that the strain–displacement relationsremain linear

To derive expressions for the normal strain components εxx and εyy and the shear strain

component εxy, consider a small rectangular element OACB whose sides (of lengths dx and dy) lie parallel to the coordinate axes before deformation When the body undergoes

deformation under the action of external load and temperature distributions, the element

OACB also deforms to the shape O  A  C  B  as shown in Figure 8.5 We can observe that

the element OACB has two basic types of deformation, one of change in size and the other

of angular distortion

Since the normal strain is defined as change in length divided by original length, the

strain components εxx and εyy can be found as

ε xx =

change in length of the fiber OA that lies

in the x directon before deformation original length of the fiber OA

BASIC EQUATIONS OF SOLID MECHANICS 291

change in length of the fiber OB that lies

in the y directon before deformation original length of the fiber OB

The expressions for the remaining normal strain component εzz and shear strain

components εyz and εzx can be derived in a similar manner as

In the case of two-dimensional problems, Eqs (8.44)–(8.46) are applicable, whereas

Eq (8.44) is applicable in the case of one-dimensional problems

In the case of an axisymmetric solid, the strain–displacement relations can bederived as

where u and w are the radial and the axial displacements, respectively.

(vi) Boundary conditions

Boundary conditions can be either on displacements or on stresses The boundary ditions on displacements require certain displacements to prevail at certain points onthe boundary of the body, whereas the boundary conditions on stresses require that thestresses induced must be in equilibrium with the external forces applied at certain points

con-on the boundary of the body As an example, ccon-onsider the flat plate under inplane loadingshown in Figure 8.6

In this case, the boundary conditions can be expressed as

u = v = 0 along the edge AB

(displacement boundary conditions)

b

Figure 8.6 A Flat Plate under Inplane Loading.

BASIC EQUATIONS OF SOLID MECHANICS 293

and

σ yy = σxy = 0 along the edges BC and AD

σ xx = −p, σyy = σxy = 0 along the edge CD

(stress boundary conditions)

It can be observed that the displacements are unknown and are free to assume any valuesdictated by the solution wherever stresses are prescribed and vice versa This is true of allsolid mechanics problems

For the equilibrium of induced stresses and applied surface forces at point A of

Figure 8.7, the following equations must be satisfied:

y

y

x z

(b) Equillibrium of internal stresses and surface forces around point A

Figure 8.7 Forces Acting at the Surface of a Body.

where x, y, and z are the direction cosines of the outward drawn normal (AN) at point A; and Φx, Φy, and Φz are the components of surface forces (tractions) acting at

point A in the directions x, y, and z, respectively The surface (distributed) forces Φx,

Φy, and Φz have dimensions of force per unit area Equation (8.51) can be specialized totwo- and one-dimensional problems without much difficulty

(vii) Compatibility equations

When a body is continuous before deformation, it should remain continuous after mation In other words, no cracks or gaps should appear in the body and no part shouldoverlap another due to deformation Thus, the displacement field must be continuous aswell as single-valued This is known as the “condition of compatibility.” The condition ofcompatibility can also be seen from another point of view For example, we can see from

defor-Eqs (8.44)–(8.49) that the three strains εxx, εyy, and εxy can be derived from only two

displacements u and v This implies that a definite relation must exist between εxx, εyy, and εxy if these strains correspond to a compatible deformation This definite relation iscalled the “compatibility equation.” Thus, in three-dimensional elasticity problems, thereare six compatibility equations [8.2]:

automati-FORMULATIONS OF SOLID AND STRUCTURAL MECHANICS 295

8.3 FORMULATIONS OF SOLID AND STRUCTURAL MECHANICS

As stated in Section 5.3, most continuum problems, including solid and structural ics problems, can be formulated according to one of the two methods: differential equationmethod and variational method Hence, the finite element equations can also be derived

mechan-by using either a differential equation formulation method (e.g., Galerkin approach)

or variational formulation method (e.g., Rayleigh–Ritz approach) In the case of solidand structural mechanics problems, each of the differential equation and variationalformulation methods can be classified into three categories as shown in Table 8.1

The displacement, force, and displacement–force methods of differential equation mulation are closely related to the principles of minimum potential energy, minimumcomplementary energy, and stationary Reissner energy formulations, respectively We usethe displacement method or the principle of minimum potential energy for presenting thevarious concepts of the finite element method because they have been extensively used inthe literature

for-8.3.1 Differential Equation Formulation Methods

(i) Displacement method

As stated in Section 8.2.1, for a three-dimensional continuum or elasticity problem, thereare six stress–strain relations [Eq (8.10)], six strain–displacement relations [Eqs (8.44)–(8.49)], and three equilibrium equations [Eqs (8.4)], and the unknowns are six stresses

(σij), six strains (εij ), and three displacements (u, v, and w) By substituting Eqs (8.44)–

(8.49) into Eqs (8.10), we obtain the stresses in terms of the displacements By stituting these stress–displacement relations into Eqs (8.4), we obtain three equilibrium

sub-equations in terms of the three unknown displacement components u, v, and w Now these equilibrium equations can be solved for u, v, and w Of course, the additional requirements

such as boundary and compatibility conditions also have to be satisfied while finding the

solution for u, v, and w Since the displacements u, v, and w are made the final unknowns,

the method is known as the displacement method

(ii) Force method

For a three-dimensional elasticity problem, there are three equilibrium equations, Eqs

(8.4), in terms of six unknown stresses σij At the same time, there are six compatibility

equations, Eqs (8.52)–(8.57), in terms of the six strain components εij Now we take

any three strain components, for example, εxy , εyz , and εzx, as independent strains and

Table 8.1 Methods of Formulating Solid and Structural Mechanics Problems

Differential equation formulation methods Variational formulation methods

Displacement Force Displacement- Principle Principle Principlemethod method force method of minimum of minimum of station-

(mixed method) potential complemen- ary Reissner

energy tary energy energy

write the compatibility equations in terms of εxy , εyz , and εzx only By substituting theknown stress–strain relations, Eq (8.10), we express the three independent compatibility

equations in terms of the stresses σij By using these three equations, three of the stresses

out of σxx, σyy, σzz, σxy, σyz , and σzx can be eliminated from the original equilibriumequations Thus, we get three equilibrium equations in terms of three stress componentsonly, and hence the problem can be solved Since the final equations are in terms of stresses(or forces), the method is known as the force method

(iii) Displacement–force method

In this method, we use the strain–displacement relations to eliminate strains from thestress–strain relations These six equations, in addition to the three equilibrium equations,

will give us nine equations in the nine unknowns σxx, σyy, σzz, σxy, σyz , σzx, u, v, and w.

Thus, the solution of the problem can be found by using the additional conditions such ascompatibility and boundary conditions Since both the displacements and the stresses (orforces) are taken as the final unknowns, the method is known as the displacement–forcemethod

8.3.2 Variational Formulation Methods

(i) Principle of minimum potential energy

The potential energy of an elastic body πp is defined as

where π is the strain energy, and Wp is the work done on the body by the externalforces The principle of minimum potential energy can be stated as follows: Of all possible

displacement states (u, v, and w) a body can assume that satisfy compatibility and given

kinematic or displacement boundary conditions, the state that satisfies the equilibriumequations makes the potential energy assume a minimum value If the potential energy,

π p, is expressed in terms of the displacements u, v, and w, the principle of minimum

potential energy gives, at the equilibrium state,

It is important to note that the variation is taken with respect to the displacements in

Eq (8.61), whereas the forces and stresses are assumed constant The strain energy of alinear elastic body is defined as

FORMULATIONS OF SOLID AND STRUCTURAL MECHANICS 297

¯Φy

¯Φz

⎫

⎬

⎭ = vector of displacements, and S1 is the surface of the body

on which surface forces are prescribed Using Eqs (8.63) and (8.64), the potential energy

of the body can be expressed as

“stiffness” method of finite element analysis

(ii) Principle of minimum complementary energy

The complementary energy of an elastic body (πc) is defined as

π c = complementary strain energy in terms of stresses (˜π) − work done by the applied

loads during stress changes ( ˜W p)

The principle of minimum complementary energy can be stated as follows: Of all ble stress states that satisfy the equilibrium equations and the stress boundary conditions,the state that satisfies the compatibility conditions will make the complementary energyassume a minimum value

possi-If the complementary energy πc is expressed in terms of the stresses σij, the principle

of minimum complementary energy gives, for compatibility,

δπ c(σxx , σ yy , , σ zx) = δ˜π(σxx , σ yy , , σ zx)

− δ ˜ W p(σxx , σ yy , , σ zx) = 0 (8.66)

It is important to note that the variation is taken with respect to the stress components in

Eq (8.66), whereas the displacements are assumed constant The complementary strainenergy of a linear elastic body is defined as

By using the strain–stress relations of Eqs (8.7), the complementary strain energy, in

the presence of known initial strain ε0, can be expressed as∗

∗ The correctness of this expression can be verified from the fact that the partial derivative of ˜π

with respect to the stresses should yield the strain–stress relations of Eq (8.7).

The work done by applied loads during stress change (also known as complementary work)

If we use the principle of minimum complementary energy in the finite element analysis,

we assume a simple form of variation for the stress field within each element and derive

conditions that will minimize the functional I (same as πc in this case) The resultingequations are the approximate compatibility equations, whereas the equilibrium equationsare identically satisfied This approach is called the “force” or “flexibility” method of finiteelement analysis

(iii) Principle of stationary Reissner energy

In the case of the principle of minimum potential energy, we expressed πp in terms of

displacements and permitted variations of u, v, and w Similarly, in the case of the principle of minimum complementary energy, we expressed πc in terms of stresses and

permitted variations of σxx , , σ zx In the present case, the Reissner energy (πR) is expressed in terms of both displacements and stresses and variations are permitted in  U

and σ The Reissner energy for a linearly elastic material is defined as

π R =



V [(internal stresses) × (strains expressed in terms of

displacements) − complementary energy in terms of stresses] · dV

− work done by applied forces

FORMULATIONS OF SOLID AND STRUCTURAL MECHANICS 299

The variation of πR is set equal to zero by considering variations in both displacementsand stresses:

gives equilibrium equationsand boundary conditions

The principle of stationary Reissner energy can be stated as follows: Of all ble stress and displacement states the body can have, the particular set that makes theReissner energy stationary gives the correct stress–displacement and equilibrium equa-tions along with the boundary conditions To derive the finite element equations using theprinciple of stationary Reisssner energy, we must assume the form of variation for bothdisplacement and stress fields within an element

possi-(iv) Hamilton’s principle

The variational principle that can be used for dynamic problems is called the Hamilton’sprinciple In this principle, the variation of the functional is taken with respect to time

The functional (similar to πp, πc, and πR) for this principle is the Lagrangian (L)

defined as

L = T − π p = kinetic energy − potential energy (8.73)

The kinetic energy (T ) of a body is given by

⎭ is the vector of velocity components

at any point inside the body Thus, the Lagrangian can be expressed as

Thus, Hamilton’s principle can be stated as

δ

 t2

8.4 FORMULATION OF FINITE ELEMENT EQUATIONS (STATIC ANALYSIS)

We use the principle of minimum potential energy for deriving the equilibrium equationsfor a three-dimensional problem in this section Since the nodal degrees of freedom are

treated as unknowns in the present (displacement) formulation, the potential energy πp has

to be first expressed in terms of nodal degrees of freedom Then the necessary equilibrium

equations can be obtained by setting the first partial derivatives of πp with respect to each

of the nodal degrees of freedom equal to zero The various steps involved in the derivation

of equilibrium equations are given below

Step 1: The solid body is divided into E finite elements.

Step 2: The displacement model within an element “e” is assumed as

⎫

⎬

where  Q (e) is the vector of nodal displacement degrees of freedom of the element, and

[N] is the matrix of shape functions.

Step 3: The element characteristic (stiffness) matrices and characteristic (load) vectorsare to be derived from the principle of minimum potential energy For this, the potential

energy functional of the body πp is written as (by considering only the body and surfaceforces)

where V (e) is the volume of the element, S1(e) is the portion of the surface of the element

over which distributed surface forces or tractions, ¯ Φ, are prescribed, and ¯ φ is the vector

of body forces per unit volume

FORMULATION OF FINITE ELEMENT EQUATIONS (STATIC ANALYSIS) 301

The strain vector ε appearing in Eq (8.78) can be expressed in terms of the nodal displacement vector  Q (e) by differentiating Eq (8.77) suitably as

In Eqs (8.78) and (8.82), only the body and surface forces are considered However,

generally some external concentrated forces will also be acting at various nodes If P ∼ c

denotes the vector of nodal forces (acting in the directions of the nodal displacement

⎪ is the vector of nodal displacements of the entire structure or body,

and M is the total number of nodal displacements or degrees of freedom.

Note that each component of the vector  Q (e) , e = 1, 2, , E, appears in the global nodal displacement vector of the structure or body, Q

∼ Accordingly,  Q (e) for each element

may be replaced by Q

∼ if the remaining element matrices and vectors (e.g., [B], [N], ¯Φ, and

¯φ) in the expression for π (e)

p are enlarged by adding the required number of zero elementsand, where necessary, by rearranging their elements In other words, the summation of

Eq (8.83) implies the expansion of element matrices to “structure” or “body” size followed

by summation of overlapping terms Thus, Eqs (8.82) and (8.83) give

Equation (8.84) expresses the total potential energy of the structure or body in terms of

the nodal degrees of freedom, Q

∼ The static equilibrium configuration of the structures can

be found by solving the following necessary conditions (for the minimization of potentialenergy):

=

FORMULATION OF FINITE ELEMENT EQUATIONS (STATIC ANALYSIS) 303



vector of elementnodal forcesproduced by bodyforces,



P b (e) =



V (e) [N] T ¯φdV = element load vector due to body forces (8.90)

Some of the contributions to the load vector P ∼ may be zero in a particular problem

In particular, the contribution of surface forces will be nonzero only for those elementboundaries that are also part of the boundary of the structure or body that is subjected

to externally applied distributed loading

The load vectors  P i (e) ,  P s (e) , and  P b (e) given in Eqs (8.88)–(8.90) are called

kinemati-cally consistent nodal load vectors [8.3] Some of the components of  P i (e) ,  P s (e) , and  P b (e)

may be moments or even higher order quantities if the corresponding nodal displacementsrepresent strains or curvatures These load vectors are called “kinematically consistent”because they satisfy the virtual work (or energy) equation That is, the virtual work done

by a particular generalized load Pj when the corresponding displacement δQj is ted (while all other nodal displacements are prohibited) is equal to the work done by the

permit-distributed (nonnodal) loads in moving through the displacements dictated by δQj andthe assumed displacement field

Step 4: The desired equilibrium equations of the overall structure or body can now beexpressed, using Eq (8.86), as

P b (e) = assembled (global) nodal load vector (8.93)

Steps 5 and 6: The required solution for the nodal displacements and element stressescan be obtained after solving Eq (8.91)

The following observations can be made from the previous derivation:

1 The formulation of element stiffness matrices, [K (e)], and element load vectors,



P i (e) ,  P s (e) , and  P b (e), which is basic to the development of finite element equations[Eq (8.91), requires integrations as indicated in Eqs (8.87)–(8.90) For some ele-ments, the evaluation of these integrals is simple However, in certain cases, it isoften convenient to perform the integrations numerically [8.4]

2 The formulae for the element stiffness and load vector in Eqs (8.87)–(8.90) remainthe same irrespective of the type of element However, the orders of the stiffnessmatrix and load vector will change for different types of elements For example, in

the case of a triangular element under plane stress, the order of [K (e) ] is 6 × 6 and

of  Q (e) is 6 × 1 For a rectangular element under plane stress, the orders of [K (e)]

and  Q (e)˙ are 8×8 and 8×1, respectively It is assumed that the displacement model

is linear in both these cases

3 The element stiffness matrix given by Eq (8.87) and the assembled stiffness matrix

given by Eq (8.92) are always symmetric In fact, the matrix [D] and the product [B] T [D][B] appearing in Eq (8.87) are also symmetric.

4 In the analysis of certain problems, it is generally more convenient to compute

the element stiffness matrices [k (e) ] and element load vectors p i (e) , p s (e) , and p b (e)

in local coordinate systems∗ suitably set up (differently for different elements) for

minimizing the computational effort In such cases, the matrices [k (e)] and vectors

p i (e) , p s (e) , and p b (e) have to be transformed to a common global coordinate systembefore using them in Eqs (8.92) and (8.93)

5 The equilibrium equations given by Eq (8.91) cannot be solved since the

stiff-ness matrices [K (e) ] and [K ∼] are singular, and hence their inverses do not exist

∗When a local coordinate system is used, the resulting quantities are denoted by lowercase letters

as [k (e) ], p i (e) , p s (e) , and p b (e) instead of [K (e) ],  P i (e) ,  P s (e) , and  P b (e).

REFERENCES 305

The physical significance of this is that a loaded structure or body is free to undergounlimited rigid body motion (translation and/or rotation) unless some support orboundary constraints are imposed on the structure or body to suppress the rigidbody motion These constraints are called boundary conditions The method ofincorporating boundary conditions was considered in Chapter 6

6 To obtain the (displacement) solution of the problem, we have to solve Eq (8.91)after incorporating the prescribed boundary conditions The methods of solving theresulting equations were discussed in Chapter 7

8.3 L.R Calcote: The Analysis of Laminated Composite Structures, Van Nostrand

Reinhold, New York, 1969

8.4 A.K Gupta: Efficient numerical integration of element stiffness matrices, International

Journal for Numerical Methods in Engineering, 19, 1410–1413, 1983.

8.5 R.J Roark and W.C Young: Formulas for Stress and Strain, 6th Ed., McGraw-Hill,

New York, 1989

8.6 G Sines: Elasticity and Strength, Allyn & Bacon, Boston, 1969.

8.7 R.F.S Hearman: An Introduction to Applied Anisotropic Elasticity, Oxford University

Press, London, 1961

8.8 S.G Lekhnitskii: Anisotropic Plates (translation from Russian, 2nd Ed., by S.W Tsai

and T Cheron), Gordon & Breach, New York, 1968

8.1 Consider an infinitesimal element of a solid body in the form of a rectangularparallelopiped as shown in Figure 8.2 In this figure, the components of stressacting on one pair of faces only are shown for simplicity Apply the moment

equilibrium equations about the x, y, and z axes and show that the shear stresses are symmetric; that is, σyx = σxy , σzy = σyz, and σxz = σzx.

8.2 Determine whether the following state of strain is physically realizable:

ε xx = c(x2+ y2), εyy = cy2, ε xy = 2cxy, εzz = εyz = εzx = 0

where c is a constant.

8.3 When a body is heated nonuniformly and each element of the body is allowed toexpand nonuniformly, the strains are given by

ε xx = εyy = εzz = αT, εxy = εyz = εzx = 0 (E1)

where α is the coefficient of thermal expansion (constant), and T = T (x, y, z) is the temperature Determine the nature of variation of T (x, y, z) for which Eqs (E1)are valid

8.4 Consider the following state of stress and strain:

σ xx = x2, σ yy = y2, ε xy = −2xy, σzz = εxz = εyz = 0Determine whether the equilibrium equations are satisfied

8.5 Consider the following condition:

σ xx = x2, σ yy = y2, ε xy = −2xy, σzz = εxz = εyz = 0Determine whether the compatibility equations are satisfied

8.6 Consider the following state of strain:

ε xx = c1x, ε yy = c2, ε zz = c3x + c4y + c5, ε yz = c6y, ε xy = εzx = 0

where ci, i = 1, 2, , 6 are constants Determine whether the compatibility

equations are satisfied

8.7 The internal equilibrium equations of a two-dimensional solid body, in polarcoordinates, are given by

where a, b, and p denote the inner radius, outer radius, and internal

pres-sure, respectively, determine whether this state of stress satisfies the equilibriumequations

8.8 Determine whether the following displacement represents a feasible deformationstate in a solid body:

u = ax, v = ay, w = az

where a is a constant.

8.9 Consider a plate with a hole of radius a subjected to an axial stress p The state

of stress around the hole is given by [8.5]

Determine whether these stresses satisfy the equilibrium equations stated inProblem 8.7.

8.10 Consider a uniform bar of length l and cross-sectional area A rotating about a pivot point O as shown in Figure 8.8 Using the centrifugal force as the body

force, determine the stiffness matrix and load vector of the element using a lineardisplacement model:

u(x) = N1(x)q1+ N2(x)q2

where N1(x) = 1−(x/l), N2(x) = (x/l) and the stress–strain relation, σxx = Eεxx, where E is the Young’s modulus.

ANALYSIS OF TRUSSES, BEAMS,

AND FRAMES

9.1 INTRODUCTION

The derivation of element equations for one-dimensional structural elements is considered

in this chapter These elements can be used for the analysis of skeletal-type systems such asplanar trusses, space trusses, beams, continuous beams, planar frames, grid systems, andspace frames Pin-jointed bar elements are used in the analysis of trusses A truss element

is a bar that can resist only axial forces (compressive or tensile) and can deform only in theaxial direction It will not be able to carry transverse loads or bending moments In planartruss analysis, each of the two nodes can have components of displacement parallel to

X and Y axes In three-dimensional truss analysis, each node can have displacement

components in X, Y , and Z directions Rigidly jointed bar (beam) elements are used in the

analysis of frames Thus, a frame or a beam element is a bar that can resist not only axialforces but also transverse loads and bending moments In the analysis of planar frames,each of the two nodes of an element will have two translational displacement components

(parallel to X and Y axes) and a rotational displacement (in the plane XY ) For a space

frame element, each of the two ends is assumed to have three translational displacement

components (parallel to X, Y , and Z axes) and three rotational displacement components (one in each of the three planes XY , Y Z, and ZX) In the present development, we

assume the members to be uniform and linearly elastic

9.2 SPACE TRUSS ELEMENT

Consider the pin-jointed bar element shown in Figure 9.1, in which the local x axis is

taken in the axial direction of the element with origin at corner (or local node) 1 A lineardisplacement model is assumed as

Figure 9.1 A Space Truss Element.

where q1 and q2 represent the nodal degrees of freedom in the local coordinate system

(unknowns), l denotes the length of the element, and the superscript e denotes the element

number The axial strain can be expressed as

1

l

%

(9.5)

SPACE TRUSS ELEMENT 311

The stress–stain relation is given by

σ xx = Eεxx

{σ xx }

1 × 1 = [D] 1 × 1 {ε xx }

where [D] = [E], and E is the Young’s modulus of the material The stiffness matrix of

the element (in the local coordinate system) can be obtained, from Eq (8.87), as

where A is the area of cross section of the bar To find the stiffness matrix of the bar in

the global coordinate system, we need to find the transformation matrix In general, theelement under consideration will be one of the elements of a space truss Let the (local)

nodes 1 and 2 of the element correspond to nodes i and j, respectively, of the global system as shown in Figure 9.1 The local displacements q1 and q2 can be resolved into

components Q 3i−2 , Q 3i−1 , Q 3i and Q 3j−2 , Q 3j−1 , Q 3j parallel to the global X, Y , Z,

axes, respectively Then the two sets of displacements are related as

where the transformation matrix [λ] and the vector of nodal displacements of element e

in the global coordinate system,  Q (e), are given by

where (Xi , Y i , Z i) and (Xj , Y j , Z j ) are the global coordinates of nodes i and j, respectively, and l is the length of the element ij given by

l =3

(Xj − X i)2+ (Yj − Y i)2 + (Zj − Z i)241/2

(9.12)Thus, the stiffness matrix of the element in the global coordinate system can be obtained,using Eq (6.8), as

Consistent Load Vector

The consistent load vectors can be computed using Eqs (8.88)–(8.90):

p i (e) = load vector due to initial (thermal) strains =



V (e) [B] T [D]ε0dV



(9.15)

The only surface stress that can exist is px and this must be applied at one of the nodal

points Assuming that px is applied at node 1, the load vector becomes

p S (e)1 =



S (e)1[N] T {p x } dS1 = p0

10

 

S1

dS1 = p0A1

10



(9.16)

where px = p0 is assumed to be a constant and the subscript 1 is used to denote the node

The matrix of shape functions [N] reduces to

10

since the stress is located at node 1

Similarly, when px = p0 is applied at node 2, the load vector becomes

p S (e)2 =



S (e)2[N] T {p x } dS1 = p0

01

 

S2

dS1 = p0· A2

01



(9.17)

SPACE TRUSS ELEMENT 313

The total consistent load vector in the local coordinate system is given by

p (e) = p i (e) + p b (e) + p S (e)1 + p S (e)2 (9.18)This load vector, when referred to the global coordinate system, will be



where [λ] is given by Eq (9.9).

9.2.1 Computation of Stresses

After finding the displacement solution of the system, the nodal displacement vector  Q (e)

of element e can be identified The stress induced in element e can be determined, using

Eqs (9.6), (9.4), and (9.8), as

where [B] and [λ] are given by Eqs (9.5) and (9.9), respectively.

Example 9.1 Find the nodal displacements developed in the planar truss shown in

Figure 9.2 when a vertically downward load of 1000 N is applied at node 4 The pertinentdata are given in Table 9.1

Figure 9.2 Geometry of the Planar Truss of Example 9.1.

Solution The numbering system for the nodes, members, and global displacements is

indicated in Figure 9.2 The nodes 1 and 2 in the local system and the local x direction

assumed for each of the four elements are shown in Figure 9.3 For convenience, the global

node numbers i and j corresponding to the local nodes 1 and 2 for each element and the direction cosines of the line ij (x axis) with respect to the global X and Y axes are given

in Table 9.2

Table 9.1.

Member Cross-sectional

“e” A (e) cm2 l (e) cm E (e) N/cm2

SPACE TRUSS ELEMENT 315

The stiffness matrix of element e in the global coordinate system can be computed from [obtained by deleting the rows and columns corresponding to the Z degrees of freedom

Global degrees of freedom ponding to different columnsHence,

0.75 2.50

−2.25

2.50

−0.75

2.50 0.75

2.50

0.25 2.50

0.75 2.50

0.25 2.50

2 − 1

2 − 1

21

12

− 1

2 − 1

2

12

12