Consider a lD elastic bar problem defined on 0 4 The domain

Consider a l-D elastic bar problem defined on [0, 4]. The domain is devided into 4 linear 2-node elements as follows. Write down element shape functions N (x) for each element (e = 1, 2, 3), Give a sketch of the shape functions N_i(x) = (f = 1, 2, 3, 4, 5) on the entire domain [0, 4]. Given the nodal displacement matrix d = [4 8 6 2]^T, find the displacement solutions at x = 1.5 and x = 3.6. Given the nodal displacement matrix d = [4 8 6 2]^T, find the strain (defined as du/dx) solutions at x = 0.4 and x = 2.3.

Solution

Using the nodal-based smoothing operation, the strains to be used in Equation (23) is assumedto be the
smoothed
strain for node
k
dened by

e
k


e
(
x
k
)
=


k
e
(
x
)

W
(
x

x
k
)
d

where

W

=

W

W

W

is a diagonal matrix of smoothing function

W

. For simplicity, the smoothingfunction is taken as

W

(

x

x

k

)

=

1

/

A

k

,

x

k

0

,

x

/

k

where

A

k

=

k

d

is the area of smoothing domain for node

k

.Substituting into Equation and integrating by parts, the smoothed strain canbe calculated using

e

k

=

1

A

k

k

e

(

x

)

d

=

1

A

k

k

L

n

u

(

x

)

d

=

e

k

(

u

)

where

k

is the boundary of the smoothing domain for node

k

, and

L

n

is the matrix of the outwardnormal vector on

k

. Equation states the fact that the assumed strain

e

k

is a function of theassumed displacement

u

.Substituting Equation into Equation , the smoothed strain can be expressed in thefollowing matrix form of nodal displacements:

e

k

=

i

N

in

B

i

(

x

k

)

d

i

where

N

in

is the number of nodes in the inuence domain of node

k

(including node

k

). Whenlinear shape functions are used, it is the number of nodes that is directly connected to node

k

inthe triangular mesh (see Figure 1). In Equation , the

B

i

(

x

k

)

is termed as the

smoothed

strainmatrix that is calculated using

B

i

(

x

k

)

=

b

ix

(

x

k

)

00

b

iy

(

x

k

)

b

iy

(

x

k

)

b

ix

(

x

k

)

Using the Gauss integration along each segment of boundary

k

, we have

b

il

=

1

A

k  N

s

m

=

1

N

g

n

=

1

w

n

i

(

x

mn

)

n

l

(

x

m

)

(

l

=

x

,

y

)

(30)where

N

s

is the number of segments of the boundary

k

,

N

g

is the number of Gauss points usedin each segment,

w

n

is the corresponding weight number of Gauss integration scheme, and

n

l

isthe unit outward normal corresponding to each segment on the smoothing domain boundary. In theLC-PIM using linear shape functions,

n

g

=

1 is used. The entries in sub-matrices of the stiffnessmatrix

K

in Equation are then expressed as

K

ij

=

N

k

=

1

K

ij

(

k

where the summation means an assembly process as we practice in the FEM, and

K

ij

(

k

)

is thestiffness matrix associated with node

k

that is computed using

K

ij

(

k

)

=

k

B

T

i

D

B

j

d

=

B

T

i

D

B

j

A

k

The entries (in sub-vectors of nodal forces) of the force vector

f

in Equation can be simplyexpressed as

f

i

=

k

N

in

f

i

(

k

)

The above integration is also performed by a summation of integrals over smoothing domains;hence,

f

i

is an assembly of nodal force vectors at the surrounding nodes of node

k

:

f

i

(

k

)

=

t

(

k

)

U

i

ˆ

t

d

+

(

k

)

U

i

b

d

Note again that the force vector obtained in LC-PIM is the same as that in the FEM, if the sameorder of shape functions is used. Therefore, it is shown again that there is no difference between