2.8 Loading

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The loading essentially consists of the terms in Equation (2.23). Here, we focus on centrifugal

loading and temperature loading.

2.8.1 Centrifugal loading

Centrifugal loading is a body force that is activated when a body rotates at an angular

speed ω about an axis. The force in a point q is proportional to the distance from the axis

and the square of the angular speed and is directed away from and orthogonal to the axis

(Figure 2.19).

Consider two points on the axis p1 and p2. Let

e := p2 − p1

p2 − p1 (2.227)

be a unit vector on the axis. Then, the point p obtained by dropping q orthogonally on the

axis satisfies

p = p1 + [(q − p1) · e]e. (2.228)

Accordingly, the centrifugal force in q satisfies

f = (q − p)ω2 (2.229)

= {(q − p1) − [(q − p1) · e]e} ω2. (2.230)

r

f

p

p1

p2

q

ω

Figure 2.19 Definition of the centrifugal axis

104 LINEAR MECHANICAL APPLICATIONS

The components of f in the reference configuration, satisfying

f = f KGK, (2.231)

can be directly substituted in Equation (2.23).

2.8.2 Temperature loading

Temperature loading acts as residual stress and corresponds to the term βKL(θ)T in

Equation (2.23). Indeed, imagine you heat a sphere while completely suppressing the expansion:

a compressive stress builds up, which will lead to the expansion of the sphere if you

relax the constraint. Accordingly, the residual stress −βKL(θ)T is related to the expansion

of the material. Indeed, for isotropic linear elastic materials, it was shown in Section 1.14.3

that (Equation (1.447))

βKL(θ ) = [3λ(θ) + 2μ(θ)]α(θ)GKL. (2.232)

More generally, defining the anisotropic expansion coefficient by

EKL = αKL(θ)T (2.233)

the corresponding stress needed to avoid this expansion for a linear material yields

(Equation (1.420))

SKL = −_KLMN(θ)αMN(θ)T (2.234)

leading to

βKL(θ ) = _KLMN(θ)αMN(θ ). (2.235)

In Equation (2.23), thermal loading is the integral of the negative thermal stress. Indeed,

an increase in temperature has the same effect as a pulling force (assuming that the body

expands as the temperature increases). Numerical integration requires the knowledge of

the temperature at the integration points. If the temperatures are given in the nodes, an

interpolation has to be performed to obtain the integration point values. Usually, the shape

functions that are used to interpolate the displacements are also used to interpolate the

temperature (Equation (2.11)):

T (X) =

N

_

i=1

ϕi(ξ, η, ζ )T (Xi ). (2.236)

Because of Equation (2.234), and assuming that the material properties do not vary wildly

within an element, the same interpolation is used for the thermal stresses too. However,

because of the fact that the strains are obtained through differentiation of the displacements

(Equation (2.4)), the degree of the mechanical stress interpolation pattern is one less than for

the thermal stress interpolation. This leads to numerical problems unless reduced integration

is used for the interpolation of the temperatures. Indeed, the number of reduced integration

points is such that interpolating polynomials have a degree that is one less than the degree

of the shape functions: linear for quadratic brick elements and constant for linear elements.

LINEAR MECHANICAL APPLICATIONS 105

Accordingly, the interpolated temperature and the thermal stress is at most trilinear in

quadratic elements with reduced integration and constant in linear elements with reduced

integration.

For fully integrated bricks, the temperature integration has to be reduced. This is now

illustrated for the 20-node brick.

First, the temperature is calculated at the reduced integration points:

T

∗

j

=

20

_

k=1

ϕR

kj Tk, j = 1, . . . , 8. (2.237)

This temperature is linearly extrapolated to the nodes of the element:

T

∗∗

i

=

8

_

j=1

aijT

∗

j, i= 1, . . . , 20 (2.238)

and finally the linearly extrapolated temperature is interpolated at the full integration points:

T

∗∗∗

l

=

20

_

i=1

ϕilT

∗∗

i , l= 1, . . . , 27. (2.239)

Substituting Equation (2.237) and Equation (2.238) into Equation (2.239) yields a linear

relationship:

T

∗∗∗

l

=

20

_

k=1

cklTk, l= 1, . . . , 27 (2.240)

where

ckl =

20

_

i=1

8

_

j=1

ϕilaijϕR

kj ,

k = 1, . . . , 20

l = 1, . . . , 27.

(2.241)

ϕil and ϕR

kj are the values of the shape functions for the 20-node brick element at the

full and reduced integration points, respectively, and aij are the trilinear functions, which

are also used for the 8-node brick element (_A_

−1 in Equation (2.121)). Equation (2.240)

replaces Equation (2.236) for the temperature interpolation in the 20-node brick element

with full integration. For the concrete coefficients, the reader is referred to the CalculiX

code (CalculiX 2003).

For the fully integrated linear element, the reduced temperature interpolation leads to

a constant temperature at the full integration points, which is equal to the mean of the

temperature at the nodes. Thus, we get in this case for Equation (2.240)

T

∗∗∗

l

= 1

8

8

_

k=1

Tk l = 1, . . . , 8. (2.242)

The reduced integration for the temperature in fully integrated elements ensures that the

thermal stress and the mechanical stress are modeled with interpolation functions of the

same degree.

106 LINEAR MECHANICAL APPLICATIONS

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