671017.pdf

x

y

p

A B C D E

Figure 1: Cantilever beam.

Figure 2: Computing of bending moment with stress.

Table 1: Deflection error with element CPS4.

Mesh

Deflection error (%)

BCDE

1×32 −28.57 −28.69 −28.74 −28.76

4×32 −27.20 −27.31 −27.34 −27.36

8×32 −27.12 −27.23 −27.27 −27.29

16 × 32 −27.10 −27.22 −27.25 −27.27

32 × 32 −27.10 −27.21 −27.25 −27.26

1×64 −10.67 −10.83 −10.89 −10.92

4×64 −8.45 −8.61 −8.66 −8.68

8×64 −8.32 −8.48 −8.54 −8.57

16 × 64 −8.29 −8.45 −8.54 −8.54

32 × 64 −8.28 −8.45 −8.50 −8.53

64 × 64 −8.28 −8.44 −8.50 −8.53

1 × 128 −4.70 −4.87 −4.93 −4.97

4 × 128 −2.12 −2.31 −2.37 −2.40

8 × 128 −1.97 −2.17 −2.23 −2.27

16 × 128 −1.94 −2.13 −2.20 −2.23

32 × 128 −1.93 −2.13 −2.19 −2.23

64 × 128 −1.93 −2.13 −2.19 −2.22

128 × 128 −1.92 −2.13 −2.19 −2.22

2×4 −95.94 −95.94 −95.93 −95.92

2×8 −85.66 −85.68 −85.68 −85.67

2×16 −60.03 −60.08 −60.09 −60.09

2×32 −27.50 −27.60 −27.63 −27.65

2×64 −8.93 −9.08 −9.12 −9.15

2 × 128 −2.69 −2.86 −2.91 −2.94

2 × 256 −0.99 −1.16 −1.22 −1.25

investigate the computational methods for bending moment
and the influences of element type and mesh partition. Hence,
no interface element was introduced, that is, the pile was
assumedtobefullyattachedtothesoil,andthesoilandpile
were both assumed to have linear elastic behavior.

00

5 10152025 30

10

30

60

120

− 250

− 200

− 150

− 100

− 50

Bendi

ng

moment

(kN m )

Analytical solution

x(m)

Figure 3: Calculation versus analytical solution of bending

moment.

Table 2: Deflection error with element CPS8R.

Mesh

Deflection error (%)

BCD E

1×32 0.15 0.05 0.01 −0.01

2×32 0.33 0.15 0.09 0.05

4×32 0.38 0.18 0.11 0.07

8×32 0.40 0.19 0.12 0.08

1×64 0.27 0.12 0.07 0.04

2×64 0.36 0.17 0.10 0.07

4×64 0.39 0.18 0.12 0.08

8×64 0.40 0.19 0.12 0.08

2. Cantilever Beam Example

2.1. Analytical Solution.The cantilever beam example is

shown inFigure 1.Thewidthofthesquarebeamis1m.The

length퐿is 30 m. A distributed load푝 = 0.5kPa is applied to

the beam. The analytical solution equations are

푀=

1

2

푝(퐿−푥)^2 , (1a)

휔=

푝퐿^4

2퐸퐼

(

푘^2

2

−

푘^3

3

+

푘^4

12

), (1b)

where푀=bending moment,푥=the position coordinate,

푘=푥/퐿,퐸=the Young’s modulus,퐼=the moment of inertia,

휔=−푢푦is the deflection of the beam, and푢푦=displacement

in the푦direction.

The beam parameters are taken as Young’s modulus퐸=

2×10^4 MPa and Poisson’s ratio]= 0.17in the computation.

The element used in the FEM is the 4-node first-order plane

stress element (CPS4). The following two methods were used

to calculate the bending moment.