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CHAPTER 3 PREREQUISITES AND FUNDAMENTAL CONCEPTS For analysing any civil Engineering structure, we are interested in evaluating stresses and strain due to imposed loading. To understand finite element method, fundamental concepts regarding equilibrium of a structural system and linear elastic theory are mandatory. In this chapter, we overview the analysis of stresses and strains considering equilibrium of a solid body. Transformation of displacements, stresses and strains will be illustrated for two-dimensional (2D) and three-dimensional (3D) cases.
Basic equation from linear elasticity theory
When the body is subjected to an external load, stress is induced in the body. These external forces are two types: body force which acts through the volume of the body, e.g. gravitational force, centrifugal forces and the other forces are surface forces e.g. hydrostatic pressure which acts over the surface of the body.
A three-dimensional body occupying a volume V and having a surface S is shown in the Figure 3.1. Points in the body are located in x, y and z coordinates. The boundary is constrained on some region, where displacement is specified. On the part of the boundary, distributed force per unit area is T, also called tractions, is applied.
Figure 3.1 Three-dimensional body under the action of forces
SU
y
z
x
v Pi
V
S u = 0
ST
T
w
u
fydV
fz dV
fxdV i
(x,y,z)
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The displacement of a point x (=[x, y, z]T), is given by three components of its displacements
u Twvu ,, [3.1]
The distributed force per unit volume, for example, the weight per unit volume is
f Tzyx fff ,, [3.2]
The body force acting on the elemental volume is shown in Figure 3.1. The surface traction T can be written as
T Tzyx TTT ,, [3.3]
The example of traction is distributed contact force and action of pressure. A load P acting at a point i is represented by its three components
Pi Tzyx PPP ,, [3.4]
Figure 3.2 shows the components that define the general state of stress in a three-dimensional elemental volume dV. Normal stresses x, y, and z act on a plane normal to the axis given by the subscript. Shear stresses xy, xz, yx, yz, zx, and zy act on a plane normal to the axis given by the first subscript. The second subscript designates the direction in which the shear stress acts. For static equilibrium, the shear stresses acting on mutually perpendicular planes must be equal:
zyyzzxxzyxxy ,,
Figure 3.2 Sign conventions, notations for stresses on a solid
x
xy
xz
dxx
xx
dxxxz
xz
dxxxy
xy
dzz
zz
dzzzx
zx
dzzzy
zy
y
yz
yx
zy
z
zx
dyyyx
yx
dyy
yy
dyyyz
yz
x
y
z
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Thus the general state of stress is completely defined by six components.
yz
xz
xy
z
y
x
[3.5]
Summation of all the forces in x direction gives
0
0
dxdydzfdzdxdyz
dydxdzy
dxdydzx
dxdydzfdxdydzz
dydx
dxdzdyy
dxdzdydzdxx
dydzF
xxzxyx
xxz
xzxz
xyxyxy
xxxx
[3.6]
Dividing by dxdydz, the equation of equilibrium in x direction will be as below:
0
xxzxyx fzyx [3.7]
Similarly considering the static equilibrium of the elemental block subjected to the body force vector field, the following set of differential equations are obtained which govern the stress distribution within the solid,
0
0
0
zzyzxz
yyzyxy
xxzxyx
fzyx
fzyx
fzyx
[3.8]
In some practical situations, the general state of stress can be reduced to simpler forms. These simplified stress states are described in the following sections.
Plane Stress
In some structures, stresses across the thickness are negligible. Assuming that the z axis is in the direction of the thickness, only the x and y faces of the element are subjected to stress. In this case, the general three-dimensional stress model reduces to two dimensions in which z, xz, and yz are all zero. This two-dimensional state of stress is called plane stress and is defined by 3 components:
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xy
y
x
[3.9]
In the case of two-dimensional stress, the static equilibrium equations reduce to,
0
xxyx fyx
0
y
yxy fyx
[3.10]
Triaxial Stress, Biaxial Stress, and Uniaxial Stress
Triaxial stress refers to a condition where only normal stresses act on an element and all shear stresses (xy, xz, and yz) are zero. An example of a triaxial stress state is hydrostatic pressure acting on a small element submerged in a liquid or soil element inside the half space. A two-dimensional state of stress in which only two normal stresses are present is called biaxial stress. Likewise a one-dimensional state of stress in which normal stresses act along one direction only is called a uniaxial stress.
Pure Shear Pure shear refers to state of stress in which an element is subjected to plane shearing stresses only, as shown in Figure 3.3. Pure shear occurs in elements of a circular shaft under a torsion load.
Figure 3.3 Element in pure shear
Stress-Strain Relationship Most structural materials exhibit a linear relationship between stress and strain at low stress levels as shown in Figure 3.4. This linear elastic region is represented by a straight line on the stress-strain diagram and ends at a point called the proportional limit. For a uniaxial state of stress, normal stress acts only in the x-direction, the linear stress-strain relationship is given by Hooke's Law:
xx E [3.11]
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The constant E is equal to the slope of the stress-strain line, and is called the Elastic Modulus, or Young's Modulus. Hooke's Law also holds for shear stresses and shear strains in the linearly elastic range:
xyxy G [3.12] Where, the constant G is called the Shear Modulus or the Modulus of Rigidity.
Figure 2.4 Stress-strain diagram for a material exhibiting elastic-plastic behavior
It is found experimentally that an axial tensile loading induces a lateral strain corresponding to a reduction in a material specimen's cross-sectional area. Similarly, an axial compressive load causes a lateral strain associated with an increase in the cross-sectional area. When the axial stress is removed, the lateral strain disappears along with the axial strain. The ratio of the lateral strain (due to expansion/contraction of the cross-section) to the axial strain is known as Poisson's Ratio and abbreviated using the constant . For most metals, Poisson's ratio is value between 0.25 and 0.35. In other materials, Poisson's ratio can vary from 0.1 (for some concretes) to 0.5 (for some rubber materials). The derivation of a generalized Hooke's Law in three dimensions for isotropic materials requires the following assumptions:
1. Normal stresses only produce normal strains and do not produce shear strains. 2. Shear stresses only produce shear strains and do not produce normal strains. 3. Material deformations are small, and thus the principle of superposition applies under
multi-axial stressing.
Figure 3.5 shows a two-dimensional element in a homogenous, isotropic material subjected to a biaxial state of stress. The normal stress x causes the element to elongate x /E along the x axis. At the same time, the normal stress y induces a stress of -y in the x direction,
Strain
Figure 3.4 Typical Stress Strain relationship
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which causes the element to contract by -y / E. Applying superposition, the resulting strain x is equal to x /E -y / E, as shown in Figure 3.5.
Figure 3.5 Element deformation due to biaxial stress
A generalized Hooke's Law can be established by extending the previous analysis to include normal strains in the y and z directions and including the stress-strain relationships for pure shear ( = G). The generalized Hooke's Law, which applies to linearly elastic, homogenous, isotropic materials, is thus given by
[3.13] where as previously stated, the constant E is the elastic modulus, G is the shear modulus, and is Poisson's ratio. These three material properties can be shown to be related by the following expression
[3.14]
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Eliminating the constant G from the stress-strain equations, the generalized Hooke's Law can be expressed in matrix form as
or more simply C [3.16]
where [C] is called the compliance matrix. Stresses may be written as a function of the strains by inverting the compliance matrix. The result is
[3.17] which can be expressed as
D [3.18] where [D] is generally referred to as the constitutive matrix.
Stress-strain relationships such as these are known as constitutive equations. There are a total of 36 elastic constants in the compliance and constitutive matrices. However, the vast majority of engineering materials are conservative and it can be shown that conservative materials have constitutive and compliance matrices that are symmetric. In this case, there is a maximum of 21 elastic constants that are actually independent in the generalized Hooke's law. For an isotropic material, the constants must be identical in all directions and the number of independent elastic constants reduces to 2 (for example, elastic modulus E, and Poisson's Ratio .
The stress strain relationship for special cases (plane stress and plane strain cases) is as below:
Plane Stress In this case, the general three-dimensional stress model reduces to two dimensions in which z, xz, and yz are all zero. The Hookes law relation gives us
(3.15)
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yxzxyxyyxyyxx EEEEEE
,12,, [3.19]
The inverse relation is given by
xy
y
x
xy
y
x E
2100
0101
)1( 2 [3.20]
The constitutive matrix is given by
2100
0101
)1( 2
ED [3.21]
and the stress-strain matrix can be written as D [3.22] Plane Strain
If one dimension is very large compared to the others, the strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition. In this case, though all stresses are non-zero, the stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir. The stress strain relationship can be obtained directly from the three dimensional relationship as
xy
y
x
xy
y
x E
22100
0101
)21)(1( [3.23]
The constitutive matrix is given by
22100
0101
)21)(1(
ED [3.24]
and the stress-strain matrix D [3.25] Thermal Stress and Strain
A change in uniform temperature applied to an unconstrained, three-dimensional elastic element produces an expansion or contraction of the element. Free thermal expansion produces normal strains that are related to the change in temperature by
[3.26] where is the coefficient of linear thermal expansion, which is widely tabulated for structural materials. Thermal strain is handled in the same manner as strain due to an applied
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load. Applying superposition, the thermal strains can be directly added to the stress-strain equations:
The temperature strain is represented as an initial strain: TTTT 0000 [3.27]
The stress relations then becomes 0 D [3.28] In plane stress case, TTT 00 [3.29] In plane strain case, TTT 010 , Constraint that 0z results additional strain in x and y direction of magnitude of T
Common engineering solids usually have thermal expansion coefficients that do not vary significantly over the range of temperatures where they are designed to be used, so where extremely high accuracy is not required, calculations can be based on a constant, average, value of the coefficient of expansion. Strain Displacement Relationship
The strains can be represented in vector form that corresponds to three dimensional stresses Tyzxzxyzyx as Tyzxzxyzyx , where, s are normal strains and s are shearing strains
Figure 3.6 Deformed Elemental Shape
Figure 3.6 gives the deformation of the dx-dy face for small deformation, Normal strains are
yv
xu
yx
,
Shearing strain yu
xv
xy
X
Y
dy
dx
u
v dxxuu
dyyvv
yyu
xxv
xv
yu
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Considering other faces, the strain vector can be written as
T
yw
zv
xw
zu
xv
yu
zw
yv
xu
[3.30]
In matrix operator format, the strain-displacement relations for 3D and 2D cases can be written as
vu
xy
y
xand
wvu
xz
yz
xy
z
y
x
xy
y
x
zx
yz
xy
z
y
x
0
0
0
0
0
00
00
00
[3.31]
Symbolically, strain-displacement in general form can be written as u [3.32] Potential Energy
The total energy of an elastic body is defined as the some of total strain energy (U) and the work potential:
WPpotentialworkUEnergyStrain Strain Energy
The work done by external forces in deforming an elastic body is stored within the body in the form of strain energy. Strain energy is a form of potential energy. The total work done by combined stresses on an elastic element like that shown in Figure 3.7 is simply the sum of the work done by each individual component of stress. This approach is valid because, for example, the normal stress x does no work in the y or z directions. Similarly, the shear stress xy does no work associated with strains xz, yz. Figure 3.7 shows the equilibrium of an elastic element of dimension dx, dy, and dz, subject to a normal stress x. The strain energy in the element is calculated by
Figure 3.7 Deformation due to normal stress
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[3.33] where du/dx = x and x dy dz) is the force acting in the x direction.
Since dx dy dz represents the volume of the element, the strain energy density, Uo (strain energy per unit volume) due to a normal stress x can be expressed by the following:
[3.34]
Integrating the strain energy equation gives:
[3.35]
Strain energy density represents the area below the stress-strain curve in the same way that work represents the area below a force-displacement curve. The strain energy associated with shear deformation xy (as shown in Figure 3.8) can be calculated in a similar manner and is given by
[3.36]
Figure 3.8 Deformation due to pure shear
As previously stated, the total strain energy associated with a general state of stress can be calculated by simply adding the strain energy due to each of the individual stress components:
yzyzxzxzxyxyzzyyxxU 21
0 [3.37]
Hence the total strain energy for the general elastic body is given by:
dvUv
TT 2
1 [3.38]
The work potential WP is given by
i
iTi
s
T
v
T PuTdsufdvuWP [3.39]
The total potential energy for the general elastic body
dvv
T 21
ii
Ti
s
T
v
T PuTdsufdvu [3.40]
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Principle of minimum potential energy
For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is minimum, the equilibrium state is stable. i.e. the actual displacement field that satisfies the governing equations is that which renders stationary. For that
0 WPU (3.41)