beam element example

8/3/2019 Beam Element Example

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Example

Consider a simply-supported beam structure under distributed loadq, as shown inFigure 1.

q

E,I,L

Figure 1 One element model with distributed force q

The whole structure is approximated using one beam finite element. Since nonodal forces exist, the finite element equation can be written as

1

2 2 2

1

3

2

2 2 2

2

12 6 12 6 / 2

6 4 6 2 /12

12 6 12 6 / 2

6 2 6 4 /12

v L L qL

L L L L qLEI

v L L qLL

L L L L qL

=

Since there is only one element, the global equations are the same as the local element

equations. The displacement boundary conditions are v1 = v2 = 0. Introducing these

boundary conditions and unknown reaction forces, we have

1

2 2 2

1

3

2

2 2 2

2

012 6 12 66 4 6 2 /12

012 6 12 6

6 2 6 4 /12

L L F L L L L qLEI

L L F L

L L L L qL

=

The reduced equations corresponding to unknown slops are

2 2 21

3 2 2 22

4 2 /12

2 4 /12

L L LEIq

L L L L

=

Solving this matrix equation yields the solution:

3 3

1 2,

24 24

qL qL

EI EI = =

Thus, the two ends of the structure do not move vertically but rotates with a slop 1 and

2. Displacement along the beam element can be approximated by


2/4

3

4 2

1 2 3 4

3

0

( )24( ) [ ]

0 24

24

qL

qL s sEIv s N N N N

EI

qL

EI

= =

(a)

Displacement v(s) is a quadratic function of parameters with negative coefficient fors2.

The element bending moment and shear force can be calculated as follows:

3

2

2

3

0

24( ) [ 6 12 ( 4 6 ) 6 12 ( 2 6 )]

0 12

24

qL

EI qLEI M s s L s s L s

L

qL

EI

= + + + =

3

3

3

0

24( ) [12 6 12 6 ] 0

0

24

qL

EI EIV s L L

L

qL

EI

= =

Since no shear force appears in the element, this loading condition produces a purebending moment.

The one of the biggest dangers in the finite element analysis is to believe theaccuracy of the solution without verification. Many people simply believe the output

results from the computer. In the truss structure, we have shown that the finite element

solution is exactly the same with the analytical solution. Is that true for the beamelement? Since the analytical solution of the beam structure in Figure 1 is known in the

literature, let us compare the analytical solution to the finite element solution. The

analytical solution of the transverse displacement is given by

4

3 4( ) ( 2 )24

analytical qLv s s s sEI

= +

which is fourth-order function ofs, while the finite element solution in Eq. (a) is the

second-order function ofs. Figure 2 compares the difference between the analytical and

finite element solutions of the transverse displacements. The displacement from the finiteelement solution at the element center is only 80% of that from the analytical solution.


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0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0s

v(s

)

v analytical

v FE

Figure 2 Transverse displacement of the beam element

The deviation of the finite element solution is more significant if the bendingmoment and shear force of the beam structure is compared. From the analytical solution,

the bending moment and shear force of the beam can be calculated by

22( ) ( )

2analytical

qL M s s s=

( ) (2 1)2

analytical

qLV s s=

Notice that the bending moment of the beam finite element was a constant function and

the shear force was zero. Figure 3 compares the bending moment and shear force fromthe analytical and finite element solutions.

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0 0.2 0.4 0.6 0.8 1s

M(s)

M Analytical

M FE

(a) Bending moments


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-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

0 0.2 0.4 0.6 0.8 1s

V(s)

M Analytical

M FE

(b) Shear forces

Figure 3 Error from the one finite element analysis with beam

We have discussed that the finite element solution is not accurate for the beamelement. How can we improve the accuracy of the finite element solution? The

fundamental approach in the finite element method is that if the structure is refined using

more finite elements, then the finite element solution converges to the analytical solution.

beam element example

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