civil engineering - structural analysis - worsak kanok-nukulchai

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CIVIL ENGINEERING - Structural Analysis - Worsak Kanok-Nukulchai

STRUCTURAL ANALYSIS Worsak Kanok-Nukulchai Asian Institute of Technology, Thailand Keywords: Structural System, Structural Analysis, Discrete Modeling, Matrix Analysis of Structures, Linear Elastic Analysis. Contents 1. Structural system 2. Structural modeling. 3. Linearity of the structural system. 4. Definition of kinematics 5. Definitions of statics 6. Balance of linear momentum 7. Material constitution 8. Reduction of 3D constitutive equations for 2D plane problems. 9. Deduction of Euler-Bernoulli Beams from Solid. 10. Methods of structural analysis 11. Discrete modeling of structures 12. Matrix force method 13. Matrix displacement method 14. Trends and perspectives Glossary Bibliography Biographical Sketch To cite this chapter Summary This chapter presents an overview of the modern method of structural analysis based on discrete modeling methods. Discrete structural modeling is suited for digital computation and has lead to the generalization of the formulation procedure. The two principal methods, namely the matrix force method and the matrix displacement method, are convenient for analysis of frames made up mainly of one-dimensional members. Of the two methods, the matrix displacement method is more popular, due to its natural extension to the more generalized finite element method. Using displacements as the primary variables, the stiffness matrix of a discrete structural model can be formed globally as in the case of the matrix displacement method, or locally by considering the stiffness contributions of individual elements. This latter procedure is generally known as the direct stiffness method. The direct stiffness procedure allows the assembly of stiffness contributions from a finite number of elements that are used to model any complex structure. This is also the procedure used in a more generalized method known as the Finite Element Method

©Encyclopedia of Life Support Systems (EOLSS)

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1. Structural System Structural analysis is a process to analyze a structural system in order to predict the responses of the real structure under the excitation of expected loading and external environment during the service life of the structure. The purpose of a structural analysis is to ensure the adequacy of the design from the view point of safety and serviceability of the structure. The process of structural analysis in relation to other processes is depicted in Figure 1.

Figure 1. Role of structural analysis in the design process of a structure. A structural system normally consists of three essential components as illustrated in Figure 2: (a) the structural model; (b) the prescribed excitations; and (c) the structural responses as the result of the analysis process. In all cases, a structure must be idealized by a mathematical model so that its behaviors can be determined by solving a set of mathematical equations.

Figure 2. Definition of a structural system


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A structural system can be one-dimensional, two-dimensional or three-dimensional depending on the space dimension of the loadings and the types of structural responses that are of interest to the designer. Although any real-world structure is strictly three-dimensional, for the purpose of simplification and focus, one can recognize a specific pattern of loading under which the key structural responses will remain in just one or two-dimensional space. Some examples of 2D and 3D structural systems are shown in Figure 3.

Figure 3. Some examples of one, two and three-dimensional structural models. 2. Structural Modeling A structural (mathematical) model can be defined as an assembly of structural members (elements) interconnected at the boundaries (surfaces, lines, joints). Thus, a structural model consists of three basic components namely, (a) structural members, (b) joints (nodes, connecting edges or surfaces) and (c) boundary conditions.


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(a) Structural members: Structural members can be one-dimensional (1D) members (beams, bars, cables etc.), 2D members (planes, membranes, plates, shells etc.) or in the most general case 3D solids. (b) Joints: For one-dimensional members, a joint can be rigid joint, deformable joint or pinned joint, as shown in Figure 4. In rigid joints, both static and kinematics variables are continuous across the joint. For pinned joints, continuity will be lost on rotation as well as bending moment. In between, the deformable joint, represented by a rotational spring, will carry over only a part the rotation from one member to its neighbor offset by the joint deformation under the effect of the bending moment.

Figure 4. Typical joints between two 1D members. (c) Boundary conditions: To serve its purposeful functions, structures are normally prevented from moving freely in space at certain points called supports. As shown in Figure 5, supports can be fully or partially restrained. In addition, fully restrained components of the support may be subjected to prescribed displacements such as ground settlements.


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Figure 5. Boundary conditions of supports for 1D members. 3. Linearity of the Structural System Assumptions are usually observed in order for the structural system to be treated as linear: (a) The displacement of the structure is so insignificant that under the applied loads, the

deformed configuration can be approximated by the un-deformed configuration in satisfying the equilibrium equations.

(b) The structural deformation is so small that the relationship between strain and displacement remains linear.

(c) For small deformation, the stress-strain relationship of all structural members falls in the range of Hooke’s law, i.e., it is linear elastic, isotropic and homogeneous.

As a result of (a), (b) and (c), the overall structural system becomes a linear problem; consequently the principle of superposition holds. 4. Definition of Kinematics a) Motions As shown in Figure 6, the motion of any particle in a body is a time parameter family of its configurations, given mathematically by

^

~ ~( , )x x P t= (1)

where x̂ is a time function of a particle, P, in the body. During a motion, if relative positions of all particles remain the same as the original configuration, the motion is called a “rigid-body motion”. b) Displacement Displacement of a particle is defined as a vector from its reference position to the new position due to the motion. If P0 is taken as the reference position of P at t = t0, then the displacement of P at time t = t1 is


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^ ^

1 1~ ~ ~( , ) ( , ) ( , )u P t x P t x P t= − 0 (2)

c) Deformation The quantitative measurement of deformation of a body can be presented in many forms: (1) Displacement gradient matrix

~u∇

1,1 1,2 1,3

, 2,1 2,2 2

3,1 3,2 3,3

i j

u u u

u u u u u

u u u

⎡ ⎤⎢ ⎥

⎡ ⎤∇ = = ⎢⎣ ⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

,3 ⎥ (3a)

Note that this matrix is not symmetric. (2) Deformation gradient matrix F

,or ij ij i jF I u F uδ= +∇ = + (3b) where I is the identity matrix and ijδ is Kronecker delta. Like the displacement gradient matrix, the deformation gradient matrix is not symmetric.

Figure 6. Motion of a body and position vectors of a particle. (3) Infinitesimal strain tensor (for small strains)

,1 (2ij i j j iu uε = + , ) (4)


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1,1 1,2 2,1 1,3 3,111 12 13

21 22 23 2,2 2,3 3,2

31 32 333,3

1 1( ) (2 2

1 (2

( )

u u u u u

u u u

SYM u

ε ε εε ε εε ε ε

)

)

⎡ ⎤+ +⎢ ⎥⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥=⎢ ⎥ +⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥⎢ ⎥⎣ ⎦

(5)

The strain tensor is symmetric. Its symmetric shear strain components in the opposite off-diagonal positions can be combined to yield the engineering shear strain ijγ as depicted in Figure 7 as,

,ij ij ji i j j iu u ,γ ε ε= + = + (6)

Figure 7. Engineering shear strain ijγ

Since

~ε is symmetric, there is no need to work with all the 9 components, strain tensor

is often rewritten in a vector form as

11 22 33 12 23 31~[ Tε ε ε ε γ γ γ= ] (7)

(4) Rotational Strain tensor, illustrated in Figure 8, is defined as

, ,1 (2ij j i i ju uω = − ) (8a)

or in matrix form:

2,1 1,2 3,1 1,3

1,2 2,1 3,2 2,3~

1,3 3,1 2,3 3,2

1 10 ( ) (2 2

1 ( ) 0 (2 21 1( ) ( ) 02 2

u u u u

u u u u

u u u u

ω

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢= − −⎢⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

)

1 )⎥⎥ (8b)


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Figure 8. Illustration of a rotational strain 12ω Some notes on the properties of

~~andε ω

a) One can easily show that

,i j ij iju ε ω= − (9) and that a displacement gradient is decomposable into 2 parts

~ ~andε ω− .

b) If , ,~

then 0 (pure deformation)i j j iu u ω= =

c) If , , ~

then 0 (pure rotation)i j j iu u ε= − =

5. Definitions of Statics Stress is defined as internal force per unit deformed area, distributed continuously within the domain of a continuum that is subjected to external applied forces. When the deformation is small, stress can be approximated by internal force per un-deformed area based on its original configuration. This type of stress is represented by the Cauchy stress tensor in the form of

11 12 13

21 22 23

31 32 33

ij

σ σ σσ σ σ σ σ

σ σ σ

⎡ ⎤⎢⎡ ⎤= = ⎢⎣ ⎦⎢ ⎥⎣ ⎦

⎥⎥ (10)


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As illustrated in Figure 9, the two subscript indices i and j corresponding to ix and jx refer to the normal vector of the surface and the direction associated with the acting internal force.

Figure 9. Definition of Cauchy Stress Tensor.

6. Balance of Linear Momentum Based on the 2nd law of Newton, the rate of linear momentum change equals the total force applied in the same direction. Thus, at any deformed state BBt of a deformable body, the rate of total linear momentum change on the left-handed side equals to body force, , over the whole body Btb B and traction force τ on the boundary tB∂ respectively.

~ ~t t

t tB B B

d u dv b dv dadt ∂ ~

t

ρ ρ= +∫ ∫ ∫ τ (11)

where tρ is the unit mass of the body at the BBt state and u is the velocity of a particle in the body. In componential form, using the relationship between the Cauchy stress tensor and the traction as illustrated in Figure 10, one can present Eq.(11) in tensor notation as

t t t

t j t j ij iB B B

d u dv b dv n dadt ∂

ρ ρ σ= +∫ ∫ ∫ (12)

where tρ is mass density, BBt is a deformed body at time t, tB∂ is its boundary and n is its unit vector. Eq.(11) is the global balance of linear momentum. It applies to a body at the state of BtB (deformed configuration corresponding to a time station t). Integrals over BBt and tB∂ are not explicit as the deformed configuration itself depends on . This is

therefore a nonlinear problem. ~( )u t


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Figure 10 Relationship between Cauchy stress tensor and traction. For the case of small displacements and infinitesimal strains, 0tB B≈ and 0tB Bδ δ≈ . Then, all integrals can be carried over BBo and 0B∂ , Eq.(11), which governs the body as a whole, can be localized to obtain equations of motion which govern any particle within the body. This is achieved through the use of the Gauss theorem which transforms surface integral to a volume integral, i.e.,

,o o

ij i ij iB B

n da dv∂

σ σ=∫ ∫ (13)

Hence, Eq.(12) is reduced to

,o o o

o j o j ij iB B B

u dv b dv dvρ ρ σ= +∫ ∫ ∫ (14)

Consequently, the tensor form of local balance of linear momentum, also known as the equation of motion is

,o j o j ij iu bρ ρ σ= + (15a) The left-handed side term is inertia force and the two right-handed side terms are respectively the body force and the internal force all in xj direction. 7. Material Constitution So far, the balance of linear momentum is applicable to any solid material. In the present scope, only linear, elastic, isotropic and homogeneous material is considered. In most practical structures under service loads, the material deformation is limited to infinitesimal small strains. Under this situation, its stress-strain relationship can be assumed to vary linearly, as illustrated in Figure 11. The standard Hooke’s three-dimensional stress-strain relationship can be expressed as


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Figure 11. Stress-strain relationship in most structural materials can be assumed linear if strain is limited to a small magnitude.

11 11

22 22

33 33

12 12

23 23

31 31

2 0 02 0 0 0

2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

0σ ελ μ λ λσ ελ λ μ λσ ελ λ λ μσ εμσ εμσ εμ

+⎧ ⎫ ⎧⎡ ⎤⎪ ⎪ ⎪⎢ ⎥+⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥+⎪ ⎪ ⎪=⎨ ⎬ ⎨⎢ ⎥⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎣ ⎦⎩ ⎭ ⎩

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(15c)

or

2ij ij kk ijσ λδ ε με= + (15d) in which the two Lame’s constants

and (1 )(1 2 ) 2(1 )

E Eνλ μν ν ν

= =+ − +

.

8. Reduction of 3D Constitutive Equations for 2D Plane Problems There are two special cases of 3D solid that can be reduced to 2D plane problems, as follows: a) Plane strain problem When the longitudinal dimension of a prismatic solid is relatively long, strains in the longitudinal direction can be negligible as shown in Figure 12.


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Figure 12. Plane strain reduction from 3D prismatic solid From the 3D constitutive equations in Eq.(15c), by prescribing 33 13 23 0ε ε ε= = = , the remaining constitutive equation can be expressed as

11 11

22 22

12 12

2 02 0

0 0 2

σ λ μ λ εσ λ λ μσ μ

+⎧ ⎫ ⎧⎡ ⎤⎪ ⎪ ⎪⎢= +⎨ ⎬ ⎨⎢⎪ ⎪ ⎪⎢ ⎥⎣ ⎦⎩ ⎭ ⎩

εε

⎫⎪⎥⎬⎥⎪⎭

(16a)

or

2ij ij kk ijσ λδ ε με= + (i , j = 1, 2) (16b) Note that while 22 33σ σ+ is not zero due to the Poisson’s ratio effect and can be obtained as

33 11 22( )σ λ ε ε= + (16c) (b) Plane stress problem When the transverse dimension of a solid is relatively small as illustrated in Figure 13, say in x3 direction, the transverse stress components, namely, 33 13,σ σ and 23σ can be assumed negligible. Consequently the only stress components need to be considered in plane stress problems are 11 22 12, ,σ σ σ all acting in the plane. Referring to the third equation of Eq.(15), in views of the zero value of 33σ , one can

obtain 33 11 22(2

)λε ε ελ μ

= − ++

. Then substituting 33ε in terms of 11ε and 22ε in the

first and second equation of Eq.(15) leads to:


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11 11

22 22

12 12

2 02 0

0 0 2

σ λ μ λ εσ λ λ μσ μ

⎡ ⎤+⎧ ⎫ ⎧⎪ ⎪ ⎪⎢= +⎨ ⎬ ⎨⎢⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎩⎣ ⎦

εε

⎫⎪⎥⎬⎥⎪⎭

(17a)

or

2ij ij kk ijσ λδ ε με= + (i,j = 1,2) (17b) where

22

2 1Eλμ νλ

λ μ ν= =

+ − (18)

Note that the form of the constitutive equation for the plane stress problems is exactly the same as that for the plane strain problems, except that λ is used instead of λ . Table 1 summarizes the three sets of equations of which the number is exactly the same as the number of unknowns in both 2D and 3D cases. To solve for the unknowns, one can combine the set of equations to obtain a single governing equation in terms of the displacement unknowns, , as follows: ju

,( )o j o j k kj j kku b u u ,ρ ρ λ μ μ= + + + (19) This governing equation is a second-order partial differential equation with respect to space and time.

Figure 13. Non-zero stress components in 2D plane stress problems


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9. Deduction of Euler-Bernoulli Beams from Solid Thin beam is one of the most common 1D components of a structural system, defined by its bending capability about the beam axis. Thin beam is governed by Euler-Bernoulli beam theory, based on the following assumptions: Assumption 1. Transverse dimensions of a beam are small in comparison with the longitudinal dimension; hence, normal stresses in the two transverse directions are assumed to be negligible as illustrated in Figure 14. Assumption 2. Shear distortion in a thin beam is negligible. Therefore, its plane sections which are normal to the beam axis remain plane and normal to the deformed beam axis as illustrated in Figure 15.

Equations 2D 3D 1 Local Balance of Linear Momentum

( ), 1,o j o j ij iu b jρ ρ σ= + = n 2 3

2 Constitutive Law 2 ( , 1,ij ij kk ij i j n)σ λ δ ε με= + =

3 6

3 Strain-Displacement Relations

, ,1 ( )( ,2ij i j j iu u i j nε = + = 1, )

3 6

Total number of equations 8 15 Unknowns 1 Displacements ( 1,iu i n= ) 2 3

2 Strains ( , 1, )ij i j nε = 3 6

3 Stresses ( , 1, )ij i j nσ = 3 6

Total number of unknowns 8 15

Table 1. Summary Of Equations And Unknowns For 2D/3D Solids

Figure 14. All transverse stress components in thin beam are negligible.


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Figure 15. Plane section remains plane in thin beam (a) Effect of Assumption 1 From the 3D constitutive equations

11 11

22 22

33 33

12 12

23 23

31 31

2 0 02 0 0 0

2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

0σ ελ μ λ λσ ελ λ μ λσ ελ λ λ μσ εμσ εμσ εμ

+⎧ ⎫ ⎧⎡ ⎤⎪ ⎪ ⎪⎢ ⎥+⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥+⎪ ⎪ ⎪=⎨ ⎬ ⎨⎢⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪⎣ ⎦⎩ ⎭ ⎩

⎫⎪⎪⎪⎪⎬⎥⎪⎪⎪⎪⎭

)

(20)

Eliminating 22 33(σ σ+ between the first equation of Eq.(20), i.e.,

11 11 22 33( 2 ) ( )σ λ μ ε λ ε ε= + + + (21) and the summation of the second and the third equations, i.e.,

22 33 11 22 332 2( )( ) 0σ σ λε λ μ ε ε+ = + + + = (22) leads to

11 11(3 2 )( )λ μσ μελ μ+

=+

(23)

Substituting (1 2 )(1 ) 2(1 )

E and Eνλ μν ν ν

= =− + +

in Eq.(23) yields

11 11Eσ ε= (24)


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Note that 22 33 11 11( ) 2λε ε ελ μ−

+ = = −+

νε due to the Poisson’s ratio effect.

(b) Stress resultants

11 11A

N σ= ∫ dA (Normal stress resultant) (25)

12 12A

N σ= ∫ dA (Transverse shear resultant) (26)

11 2A

M x dAσ= ∫ (Bending moment) (27)

(c) Effect of Assumption 2.

Figure 16. Displacement field can be represented by the displacements and rotation of the corresponding plane section based on thin beam. assumptions.

From Figure 16, the longitudinal displacement field at any point in the beam body can be expressed in terms of the longitudinal displacement at the corresponding point on the beam axis and the effect of the rotation of the plane section, i.e.,

1 1 2 1 1 2( , ) ( )u x x U x x θ= − (28) In views of the assumption that plane section remains plane and normal to beam axis, one can show that the rotation of the plane section equal the slope of the beam axis, i.e.,

2

1

dUdx

θ = . Substituting θ with 2

1

dUdx

leads to

2

1 1 2 1 1 2 11

( , ) ( ) ( )dU

u x x U x x xdx

= − (29)


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If the normal strains in the transverse direction are negligible, the transverse displacement field in the beam body is equal to the corresponding transverse displacement of at the beam axis, i.e.,

2 1 2 2 1( , ) ( )u x x U x= (30) (d) Strains resultants Measurements of deformation in a thin beam are made on a small segment of the beam in the form of strain resultants. The main strain resultant is the normal strain of a small segment , i.e.,

110

limX

XΔ →

Δ

1

11 1 21

( )du

x xdx

ε ≡ (31)

Substituting Eq.(29) in Eq.(31) leads to

21

11 1 2 1 2 121 1

( ) ( ) ( )dU d U2x x x xdx dx

ε ≡ − x (32)

The second thin beam assumption of zero shear can be confirmed from

1 2 2 212 1 2 1 2

2 1 2 1 1( )

u u dU dUx x U x

x x x dx dx∂ ∂

γ∂ ∂

⎧ ⎫∂ ⎪ ⎪= + = − + =⎨ ⎬∂ ⎪ ⎪⎩ ⎭

0 (33)

(e) Constitutive equations At the material level, the relationship between the only pair of the stress and strain tensors is given in Eq.(24), i.e., 11 11Eσ ε= . Applying ( )

A

dA∫ on both sides of Eq.(24) in views of

Eq.(32) leads to

21 1 2 1 1

11 11 1 2 2 21 11

( ) ( )( , ) .

A A A

dU x d U dU xN E x x dA E dA x dA EA

dx dxdxε

⎛ ⎞= = + =⎜ ⎟

⎜ ⎟⎝ ⎠

∫ ∫ ∫ (34)

Note that when the origin of / 2

2/ 2

0 . .h

A h

x dA i e bdx−

≡∫ ∫ 2 0= 2x axis is located at the

centroid of the plane section. Applying 2 x d A

on both sides of Eq.(24) in views of Eq.(32) leads to


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2 221 2

11 2 2 2 21 1 1

( ) ( ) 22

A A A

dU d U d UM x dA E x dA E x dA EI

dx dx dxσ= = − =∫ ∫ ∫ (35)

where the sectional moment of inertia, 2

3 2I x dA= ∫ . (f) Beam equation of motion The equations of motion, Eq.(15), are applicable to any particle within a continuum. These equations will be used to derive a set of governing equations for thin beam members. (i) Based on Eq.(15), by applying the cross-sectional integration over both sides of the equation of motion in 1x direction, the longitudinal equation of motion can be derived as follows:

0 1 0 1 11,1 21,2 31,3( )A A A

u dA b dA dAρ ρ σ σ σ= + + +∫ ∫ ∫ (36)

In view of Eq.(29) and the definition of stress resultants in Equations 25 and 26, Eq.(36) can be further expanded as follows:

2 110 1 2 1

1 1[ ] 21

2A A A

dUU x dA F dA d

dx x x∂σ ∂σ

ρ∂ σ

− = + +∫ ∫ A∫ (37)

11 21

0 1 11 2

dN dNAU F

dx dxρ = + + (38)

As is not a function of 21N 2x , the last term on the right-handed side can be omitted, thus,

110 1 1

1

dNAU F

dXρ = + (39)

(ii) Based on Eq.(15), by applying 2()

A

x dA∫ over both sides of the equation, the bending

equation of motion can be derived as follows:

0 1 2 0 1 2 11,1 21,2 2( ) ( ) ( )A A A

u x dA b x dA x dAρ ρ σ σ= + +∫ ∫ ∫ (40)

Substituting Eq.(29) and considering zero value of the first term on the right-handed side yield


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2 11 210 1 2 2 2 2 2

1 1 20

A A A

dUU x x dA x dA x dx b

dx x x∂σ ∂σ

ρ∂ ∂

⎡ ⎤− = + +⎢ ⎥

⎣ ⎦∫ ∫ ∫ (41)

Omitting the zero terms and integrating by parts the last term of Eq.(41) lead to

2 20 2 11 2 21 2 2 21

1 1 2( )

A A A A

dU d dx dA x dA x dx b dAx dx dx

ρ σ σ∂

− = + −∫ ∫ ∫ σ∫ (42)

Using the definitions in Eqs. (26) and (27) and in view of the vanishing second term on the right-handed side, the bending equation of motion can be obtained as

20 3 12

1 1

dU dMI Ndx dx

ρ− = − (43)

(iii) Based on Eq.(15), by applying ()

A

dA∫ in 2x direction over both sides of the

equation, the transverse equation of motion can be derived as follows:

12 22 122 0 2

1 2 3(o )

A A A

u dA b dA dAx x x

∂σ ∂σ ∂σρ ρ

∂ ∂ ∂= + + +∫ ∫ ∫ (44)

Removing the zero terms and defining the vertical distributed unit load 0 2

A

q bρ= ∫ dA

lead to

0 2 121A

dU dA q dAdx

ρ = +∫ σ∫ (45)

Finally, upon substituting the definition of the sectional shear term from Eq.(26), the transverse equation of motion for thin beam can be obtained as

120 2

1

dNAU q

dxρ = + (46)

Table 2 summarizes the three sets of equations that govern the behaviors of thin beam problems. The number of equations is seven which is exactly the same as the number of unknowns. To solve for the unknowns, one can combine the three set of equations to obtain two sets of governing equations in terms of the displacement unknowns as follows:

21

1 021

d U1EA F A

dxρ+ = U (47)


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42

3 0 3411

d U d0 2EI q I

dxdxθρ ρ− = − AU (48)

For static problems, the time-dependent terms can be omitted, and the equations of motions reduce to the equilibrium equations of thin beams as follows:

21

11

0d U

EA Fdx

+ = (49)

4

23 4

1

0d U

EI qdx

− = (50)

10. Methods of Structural Analysis To design safe structures, structural engineers must fully understand the structural behaviors of these structures. In the long past, structural engineers gained the knowledge into the structural behaviors by carrying out experimentations using a physical model of the real structure in the laboratory. Based on the test results, the behaviors of the prototype structure can be understood and generalized. However, the physical modeling has its limitations as it is expensive and time-consuming. Thus, mathematical modeling has been a viable alternative.

Equations Number 1 Local Balance of Linear Momentum

111 0

1

12 0 31

120 2

1

dNF AU

dx

dMN I

dx

dNq AU

dx

ρ 1

ρ θ

ρ

+ =

− =

+ =

3

2 Constitutive Law

11 11

3

N EA

M EI

ε

φ

=

= −

2

3 Strain-Displacement Relations 2


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111

12

221

dUdx

d U

dx

ε

φ

=

=

Total number of equations 7 Unknowns (Figure 17) Number 1 Displacements 1 2,U U 2

2 Strain resultants 11,ε φ 2

3 Stress resultants 11 12, ,M N N 3

Total number of unknowns 7

Table 2. Summary of Equations and Unknowns for Thin Beams

Figure 17. Definition of unknown variables in a thin beam segment As shown in the diagram of Figure 18, mathematical modeling is an idealization of a specific structural behavior that is expressed by mathematical equations. For simple structures, mathematical model is a convenient tool for predicting a specific behavior of the structures. For more complex structural system, a structural model may consist of an assembly of structural members. Analytical solution of its mathematical model requires solving a large-scale boundary value problem. Thus, the solution of large-scale structural system based on mathematical modeling is rather impractical, if not impossible at all. After the advent of computer technology, discrete modeling has become an industrial standard in structural analysis. Like mathematical modeling, the formulation of discrete structural model is based on a mathematical foundation, to enable its large-scale solution with the help of computational tools. Basically, each structural member in the structural model must be discretized so that its key properties can be represented at selected discrete variables associated with the member. The assembly of member contributions leads to a sizable set of algebraic equations associated with these variables, which can be systematically solved using computers.


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Figure 18. Three modeling techniques and their relationships.

11. Discrete Modeling of Structures (a) Definition of discrete structures In discrete structural model, finite number of control points (joints) will be designated and corresponding discrete variables are assigned to these joints, for which the solution will be obtained. Interior (member) variables are treated as dependent variables, which would be determined once the joint variables are known. The difference in the classical continuous model and the discrete model is demonstrated in Figure 19.

Figure 19. Continuous model versus discrete model. Therefore a discrete structure is a network of joints or nodes interlinked by 1D, 2D or 3D deformable members or any of their combinations. Joints are usually established for a member at points of geometric discontinuities and applied loads. (b) Definition of discrete quantities In the continuous system, one deals with a particle within the domain of the continuum. Interesting quantities in a continuum can be categorized as statics and kinematics.


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Statics consist of external force and internal force in the form of stress tensors, while kinematics consist of displacements and deformation in the form of strain tensors. In the discrete system, an element is associated with a set of selected nodes, only through which the element interacts with other elements. The external work done associated with each of these nodes is the product of ‘a vector of force, R ’ and its corresponding ‘vector of displacement, ’. The internal work done of an element is represented by the product of ‘a vector of action, ’ and the corresponding ‘vector of deformation, ’.

rS

v

T

(c) Discrete representation of internal work done: If a structural member is free of interior member loads, the distribution of stresses will follow a specific pattern. Consequently, a set of discrete actions ( ) and deformations ( v ) can be defined to represent the total internal work done, i.e.,

S

0

T

B

dV S vσ ε =∫ (51)

Generally, for a 2D beam member, it can be shown that three quantities are needed for the pair of action and deformation for their product to represent the overall internal work done as

0

1 1

1 2 2 1 2 3 2

3

T

v

dV M M N S S S v

v

θ

σ ε θ

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪= =⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪Δ⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

∫B

(52)

where the three action terms 1 2M M N are the bending moments at the two ends of

the beam and its axial force respectively, while the corresponding three deformation terms 1 2θ θ Δ are the two chord rotations and the axial elongation respectively, as

shown in Figure 20.


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Figure 20 Definition of actions and deformations for 2D bending members

It could be shown that the relationship between and , i.e., , can be obtained as

1 1

2

3 3

1 1 03 6

1 1 06 3

0 0

L LEI GA L EI GA Lv SL LvEI GA L EI GA L

v SLAE

⎡ ⎤+ − +⎢ ⎥′ ′⎧ ⎫ ⎧ ⎫⎢ ⎥⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢= − + +⎨ ⎬ ⎨ ⎬⎢ ′ ′⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭⎢ ⎥⎢ ⎥⎣ ⎦

2S⎥⎥ (53)

The corresponding beam stiffness can be obtained as k 1k f −= 12. Matrix Force Method Matrix Force Method employs the virtual force principle to form a system of flexibility equations. In Figure 21, a diagram shows the relationship between statics R and , and that between the kinematics and , as well as the constitutive equation linking and

. Note that if one defines , one can prove by using virtual principle that

.

Sv r S

v S BR=Tr B v=

If the size of is exactly the same as the number of equilibrium equations, this is the case of statically determinate structures. The solution of S can be obtained explicitly from the equilibrium equations alone, i.e., S

S

BR= . Subsequently, other part of the solution can be determined one after another. If the size of S is greater than the number of equations, this is the case of statically indeterminate structures. The solution of S cannot be determined directly from the equilibrium equations alone. The three sets of equations, namely the equilibrium, the material constitution and the compatibility, must be combined to obtain a system of flexibility equations, r to obtain r . Subsequently other parts of the solution, v F R=


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and can be determined. The formulation procedure for the matrix force method applicable to statically indeterminate structures will be presented next.

S

Figure 21. Notation of quantities and their relationships in matrix force method For the purpose of demonstration, a simple indeterminate structure with 3 degrees of statically indeterminacy under joint loads R , temperature change TΔ and prescribed displacements ˆ ˆx and q will be used in the formulation of the flexibility equations based on the matrix force method. The following step by step formulation procedure is used: (a) Define an associated determinate base structure. A statically determinate base structure will be selected after releasing kinematics corresponding to the selected redundants X as shown on the right-handed side of Figure 22. (b) Decomposition of the structure into 3 base structures. In Figure 23, the indeterminate structure is decomposed into 3 cases of the base structure under (a) applied joint loads, (b) the redundant forces and (c) thermal changes. The sum of the kinematics corresponding to the redundants must be compatible to their real values of the total structure before their release. Thus, there are three compatibility equations to be solved for the three unknown redundants X . The total value of each quantity is the sum of its contributions from the three cases, as listed in Table 3.


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Figure 22. A prototype indeterminate structure with 3 degrees of static indeterminacy and its base structure after 3 redundants are removed.

Figure 23. The superposition of three cases of base structure to satisfy the real kinematics corresponding to the redundants.

Statics Kinematics

Actions ( 0R X TS S S S= + = ) Deformations R X Tv v v v= + +

Forces R Displacements R X Tr r r r= + +

Redundants X Kinematics of X ˆ R X Tx x x x= + + Reactions ( 0R X TQ Q Q Q= + = ) Settlements q̂

Table 3. List of statics and kinematics

(c) Determination of the transformation matrices ( , ,R X R XB B D and D ) and thermal strain ( ) TV As the base structure is statically determinate, actions and reactions can be obtained easily using the equilibrium equation alone. As a result, the transformation matrices


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, ,R X R XB B D and D can be obtained, as well as the deformation due to the free expansion effect of the temperature change, as illustrated in Figure 24.

TV

Figure 24. Analysis of 3 systems of statically determinate structures (d) Determining the unknown redundants X using virtual force method: Applying virtual forces Xδ corresponding to the redundants on the base structure as the virtual force system, Figure 25, one can obtain the equality of external and internal virtual works as:

( ) ( )( )

( )

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

ˆ ˆ

E IT T T

X X

T T T T TX X R X T

T TX X R X T

T T T TX X R X X X T

T TX XR XX X T

q R

W W

X x Q q S v

X x X D q X B v v v

x D q B f S f S v

x D q B f B R B f B X B v

x D q F R F X B v

x x x x

δ δ

δ δ δ

δ δ δ

=

+ =

+ = + +

+ = + +

⎡ ⎤ ⎡ ⎤+ = + +⎣ ⎦ ⎣ ⎦

+ = + +

+ = + X Tx+

(54)


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Figure 25. Application of virtual force to obtain kinematics corresponding to the redundants.

The last equation assures that the prescribed kinematics ˆ ˆqx x+ at the supports are satisfied by the summation of the three cases. If there is no support settlement, both ˆ qˆx and x are zero. Physically the size of X must be adjusted to ensure this condition;

thus, this equation is solved for the value of X , i.e.,

{ } { }ˆ ˆT TXX X XR X TF X x D q F R B v= + − − →⎡ ⎤⎣ ⎦ X (55)

Note the two subscripts of the flexibility matrix T

XR XF B f B= R . The first refers to the resulting displacement vector x while the second refers to the applying force vector R .

Similarly, the two subscripts in the flexibility matrix TXX XF B f B= X refer to the

displacement x and the force vector X respectively. (e) Obtain other parts of the solution. Similar to Eq.(54), one obtains by virtual force principle the following:

ˆTR RR RX Rr D q F R F X B v+ = + + T

T (56) where . andT T

RR R R RX R XF B f B F B f B= =

Once the unknown redundants X are solved, the remaining quantities can be obtained as shown in Table 4. Finally, the step by step procedure of the Matrix Force Method is summarized in Table 5.

Statics Kinematics

Actions R XS B R B X= + Deformations R Xv f B R f B X vT= + +

Forces R Displacements ˆT TRR RX R T Rr F R F X B v D q= + + −

Redundants X Kinematics of X

ˆ ˆT TXR XX X T Xx F R F X B v D q= + + −

Reactions R XQ D R D X= + Settlements q̂

Table 4. Final results of statics and kinematics after X is solved.

1 Define a base structure and identify ˆX and x 2 Define f , , , ,R r S v and Q (if there is settlement ) q̂


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3 Form , , ,R X R XB B D D and (if temperature changes are given) Tv4 Form , ,RR RX XR XXF F F and F 5

Solve for X from { } { }ˆ ˆT TXX X XRF X x D q F R B v= + − −⎡ ⎤⎣ ⎦ X T

6 Obtain the rest from Table 4.

Table 5. Step by step Procedure for Matrix Force Method

13. Matrix Displacement Method Matrix Displacement Method employs the virtual displacement principle to form a system of stiffness equations. Figure 26 shows a diagram of the relationship between kinematics v and and the statics r R and , as well as the constitutive equation linking and . Note that if one defines v a

SS v r= , one can prove by using virtual

principle that . TR a S= In a structure in which the deformation v can be obtained directly from known displacement vector , this structure is referred as a kinematically determinate structure. In usual cases, displacements in a structure are unknowns and therefore most practical structures are kinematically indeterminate.

r

In Matrix Displacement Method, it is necessary to identify the degree of kinematic indeterminacy ( ), also known as the number of degrees of freedom (dof). In frame structures, this refers to the number of independent discrete displacements, i.e., all displacements that vary varied independently. In a 2D frame structure, each rigid joint has 3 degrees of freedom, while in 3D frame structure, each rigid joint has 6 degrees of freedom. If there are internal constraints, some joint degrees of freedom could become inter-dependent. Thus, the size of independent degrees of freedom will reduce. The need to eliminate dependency among joint degrees of freedom is to avoid solving a singular equation system.

KIo

Figure 26 illustrates the deformation modes associated with each degree of freedom independent of the others. When constraints are prescribed in the system, the number of degrees of freedom will be reduced. Figure 27 illustrates the same structure except that three constraints are introduced, i.e., the three members are not allowed to elongate or shorten after assuming that their axial deformation is negligible. Under these constraints, deformation modes in (a) (b) (d) and (e) in Figure 26 are not feasible, while one new deformation mode arises as shown in Figure 27. Theoretically, the degree of kinematic indeterminacy can be calculated from the number of joint degrees of freedom less the number of internal constraints. In contrary to the force method which derives kinematics from statics, the displacement method derives statics from kinematics. The relationships among the displacement r , the deformation v , the action and the nodal force S R are depicted in Figure 28.


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For the purpose of demonstration, a simple kinematically indeterminate structure under joint loads R , temperature change TΔ and prescribed displacements ˆ ˆx and q as shown in Figure 29 will be used in the formulation of the stiffness equations based on the matrix displacement method. The following step by step formulation procedure is used: (a) Define ˆ, , , , , ,R r S v X x Q and q (Figure 29). (b) Decompose the total structure to separate the effects of , r x̂ and TΔ As shown in Figure 30, the independent effects of , r x̂ and TΔ are considered separately and their contributions into each quantity of statics and kinematics will be superposed as listed in Table 6. (c) Derive the matrices and , , , , , ,r x r x T Ta a c c k v q TS

Figure 26. Degree of Kinematic Indeterminacy equals joint degrees of freedom if there is no constraint in the system.


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Figure 27. Degree of Kinematic Indeterminacy equals joint degrees of freedom less number of constraints.

Based on the geometric change, one can obtain the deformed elastic curve as a result of a unit displacement of r by which and can be derived. Similarly for the effect of settlement

ra rcx̂ , one can obtain xa and xc . For the effect of temperature change, r and

x have to be clamped; thus can be obtained while deformation is normally zero. However, if constraint exists by means of allowing for , can be non-zero. Figure 30 presents the symbolic results of all these quantities. The member stiffness matrix

relating to can be obtained from

TS Tv

Tq Tv

k v S 1f − .

Figure 28. Notation of quantities and their relationships in matrix displacement method (d) Determination of by virtual displacement method. r Using the virtual displacement system on the right-handed side of Figure 31, applying virtual displacement rδ corresponding to the external force R , one can obtain the equality of external and internal virtual works as:


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( ) ( )( )( )

ˆ

ˆ

E IT T T

r r

T T T T Tr r r x T

T Tr r r x T

T T T Tr r r r x r

T Tr rr xx r T

q r x T

W W

r R q Q v S

r R r c Q r a S S S

R c Q a k v k v S

TR c Q a k a r a k a x a S

R c Q K r K x a S

R R R R R

δ δ

δ δ δ

δ δ δ

=

+ =

+ = + +

+ = + +

⎡ ⎤ ⎡ ⎤+ = + +⎣ ⎦ ⎣ ⎦

+ = + +

+ = + +

(57)

Figure 29 Quantities needed in the formulation of stiffness equations by Matrix Displacement Method.

Figure 30. The superposition of effects due to three cases of kinematic excitation


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Figure 31. Application of virtual displacement rδ to obtain external force R . The last equation above equates external forces to internal forces as individual effects of the three cases. Note that ij i jK a k a= is a stiffness matrix. The first subscript in the

stiffness matrices refer to the internal force result while the second subscript refers

to the displacement cause. Thus, the term will result in ijK

ˆrxK x xR which corresponds to the contribution to internal force vector R as a result of applying the settlement vector x̂Once the displacement vector is obtained from the system of stiffness equations, i.e.,

. r

{ } { }ˆTrr r xx r TK r R c Q K x a S= + − −⎡ ⎤⎣ ⎦

T (58)

the remaining quantities can be obtained as shown in Table 6. Finally, the step by step procedure of the Matrix Displacement Method is summarized in Table 7. 14. Trends and Perspectives Discrete approach of structural modeling is suited for digital computation and has lead to the generalization of the formulation procedure. The two principal methods, namely the matrix force method and the matrix displacement method, are convenient for analysis of frames made up mainly of one-dimensional members. Of the two methods, the matrix displacement method is more popular, due to its natural extension to the more generalized finite element method.

Kinematics Statics

Deformation r xv a r a x vT= + +

Action r xS k a r k a x ST= + +

Displacement r Force ˆ T Trr xx r T rR K r K x a S c Q= + + −

Settlement x̂ Reaction T T

xr xx x T xX K r K x a S c Q= + + −

Dependent dof r xq q q qT= + + Dependent force Q


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Table 6. Final results of statics and kinematics after is solved. r

1 Identify or degrees of freedom of the problem and select appropriate vector of

KIo

r2 Define k , , ,R r S v , ˆ,X x , Q and q

3 Form and (if temperature changes are given) , , ,r x r xa a c c ,T Tv S4 Form , ,rr rx xr xxK K K and K5

Solve for r from { } { }ˆT Trr r xx r TK r R c Q K x a S= + − −⎡ ⎤⎣ ⎦

6 Obtain the rest from Table 7.

Table 8 Step by step Procedure for Matrix Displacement Method Using displacements as the primary variables, the stiffness matrix of a discrete structural model can be formed globally as in the case of the matrix displacement method, or locally by considering the stiffness contributions of individual elements. This latter procedure is generally known as the direct stiffness method. The direct stiffness procedure allows the assembly of stiffness contributions from a finite number of elements that are used to model any complex structure. The basic procedure for developing the stiffness of an element is to assume appropriate shape functions that represent the displacement field over the element. Based on these assumed displacement fields, the element stiffness can be formulated in terms of nodal degrees of freedom. The direct stiffness process was in fact the main reason how the powerful general-purposed finite element programs can be developed for any discrete structural model. Today, many commercial software packages are widely available for linear elastic analysis based on the stiffness method. These commercial software packages may help reduce the work load of structural engineers, but will not decrease their accountability on the reliability the analysis results. Thus, it is important that structural engineers understand the fundamentals of the analysis, whether the analysis is performed by the engineers themselves or with the help of a computer program. Glossary Base structure: The choice of a statically determinate structure reduced from the

original statically indeterminate structure after removing the redundant kinematics.

Boundary condition:

The imposition of either force or displacement along a boundary of the structure

Cauchy stress: A definition of stress associated with the deformed configuration. Compatibility: The term used to describe the conformity between the strains and

the displacement field of a deformed structure. Constitutive equations:

The equations used to describe the relationship between stresses and strains of a material.

Deformation gradient:

The gradient of deformation in the neighborhood of a particle.


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Deformation: A change in the relative positions of particles of a body. Degrees of freedom:

Displacement variables associated with nodes of a discrete structure.

Direct stiffness method:

A method to directly assemble the contributions of individual members or elements of a structure to form the stiffness matrix of a structure.

Discrete model of structures:

A representation of the structure to be analyzed by a computer method by which the primary variables are determined numerically.

Discrete modeling:

A method to model structures by a finite number of elements (members) that are associated with a set of discrete variables to be determined numerically.

Displacement: The disposition measurement of a particle in a body or a structure. Elastic: Property of a material that rebounds to its original position from

deformation upon the release of the applied force. Engineering shear strain:

A measurement of shear distortion of a body commonly used in engineering.

Equilibrium: A state in which the internal forces balance all external forces in all directions.

Euler-Bernoulli beams:

A beam theory by which the plane section of a beam remains plane and normal to the neutral axis.

Excitation: Any disturbance to the structure including external forces, prescribed displacement, accelerations, etc.

Finite element method:

A method to model structures by a finite number of typical elements associated with a set of discrete variables to be determined numerically.

Infinitesimal strain:

A linearized strain after neglecting the nonlinear high-order terms due to the smallness of the strain.

Internal work done:

Work done by the product of stress and strain.

Isotropic: A description of material property that is invariant to all directions.

Joints: A point where two or more one-dimensional members meet, about which a set of degrees of freedom are specified..

Kinematics: Quantities associated with geometry, the position changes or the deformation of geometry. This term is used in opposition to the term “statics”.

Linear elastic analysis:

A structural analysis that assumes linear stress-strain relationship for all elastic structural members.

Mathematical model:

A mathematical representation of the structure.

Matrix analysis of structures:

A term used to describe structural analysis of discrete structural system using matrix operations.

Matrix displacement method:

An analysis of discrete structural model of which displacements are primary variables.

Matrix force An analysis of discrete structural model of which force parameters


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method: are primary variables. Motion: A time-function of positions occupied by particles of a body. Nodes: Locations of a structure about which the degrees of freedom are

assigned. Plane strain problem:

A solid of prismatic shape and force patterns, for which the longitudinal strain components can be assumed zero, and thus a plane strip of the body can be used in the analysis.

Plane stress problem:

A thin plate structure subjected to in-plane force only, for which the transverse stress components can be assumed zero.

Redundant forces:

The corresponding forces that need to be applied back to the base structure to restore the kinematic conditions of the indeterminate structure.

Rigid joints: Joints between two or more structural members that are considered to be undeformable, and therefore ensure kinematic continuity between connecting members.

Statically determinate structures:

A structure of which the internal forces (stresses) can be obtained directly from the external forces using the equilibrium condition alone.

Statically indeterminate structures:

A structure of which the internal forces (stresses) cannot be obtained directly from the external forces using the equilibrium condition alone. In this case, displacement and deformation (strains) must be determined first before internal forces can be obtained.

Statics: Quantities associated with forces, ether external or internal, including stress and stress resultant. This term is used in opposition to the term “kinemetics”.

Stiffness matrix: The term used to quantify the force vector that is required to produce 1 unit of displacement vector.

Strains resultants:

A measurement of strain for a cross-section of bending members, such as beams, plates and shells.

Stress resultants: A product of stress over the cross-sectional area of bending members. For beams, the stress resultants include axial force, shear force and bending moment.

Structural analysis:

Structural analysis is a process to analyze a structural system in order to predict the responses of the real structure under the excitation of expected loading and external environment during the service life of the structure.

Virtual work done:

Work produced by either statics (force) or kinematics (displacement) in a virtual system over the corresponding kinematics or statics respectively in the real system. The virtual work principle requires that the internal virtual work done equals the external virtual work done.

Bibliography W. Kanok-Nukulchai, (2002), Computer Methods of Structural Analysis, Lecture note, School of Civil Engineering, Asian Institute of Technology, Pathumthani, Thailand. [A comprehensive notes of discrete structural analysis from which the structural analysis section is extracted from].


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A. Ghali and A. M. Neville, (1978), Structural Analysis (A Unified Classical and Matrix Approach), Chapman and Hall, London. [This is a good text in the same approach that readers can study in parallel to this section]. Alan Jennings, (1977), Matrix Computation for Engineers and Scientists, John Wiley and Sons. [A good background book for matrix formulations and many exercises to work on] W. Kanok-Nukulchai, (2002), Analysis Interpretive Computer package - 1993 Version, Asian Institute of Technology. [This is the manual of a symbolic programming tool that can be used to perform some exercises]. Biographical Sketch Worsak Kanok-Nukulchai received his Ph.D. in Structural Engineering and Structural Mechanics from the University of California at Berkeley in 1978 before joining Asian Institute of Technology, where he is currently a professor of Civil Engineering. He has extensive research experiences in Computational Mechanics for the last 25 years and has published widely in the field. In 1997 he was admitted to be a member of the Royal Institute of Thailand. In 1999, he was recognized as the National Distinguished Researcher by Thailand’s National Research Council. He is a Council Member of the International Association of Computational Mechanics, and the current Chairman of the International Steering Committee of a Regional Series of East-Asia Pacific Conferences on Structural Engineering and Construction. To cite this chapter Worsak Kanok-Nukulchai, (2005), STRUCTURAL ANALYSIS, in Civil Engineering, [Eds. Kiyoshi Horikawa, and Qizhong Guo], in Encyclopedia of Life Support Systems (EOLSS), Developed under the Auspices of the UNESCO, Eolss Publishers, Oxford ,UK, [http://www.eolss.net]


civil engineering - structural analysis - worsak kanok-nukulchai

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