Structural beams in quake

The configuration of a simple cantilever beam is presented in Figure 1.. A 300-kN point load is applied at the free right end.. The beam is fixed at the left end and the rotation at the

Structural Beams in QUAKE/W

This example looks at the behavior of some simple structural beams The purpose is to verify that the QUAKE/W results match hand-calculated values

The configuration of a simple cantilever beam is presented in Figure 1 A 300-kN point load is applied at the free right end The beam is fixed at the left end and the rotation at the left end is specified as zero The small circle at the left end indicates that a rotation-type boundary condition has been specified

Distance - m

Figure 1 Configuration of a simple cantilever beam

Figure 2 Circle indicates rotation boundary condition at the left end

In QUAKE/W, beams do not have any mass, but a QUAKE/W analysis requires that there be some mass somewhere in the problem In this case, a row of elements has been included below the beam with a small unit weight and a small stiffness (small G modulus), so as to not to affect the beam stiffness

Applying the 300-kN point load causes the beam at the free end to vibrate, as in Figure 3, but eventually

to settle down at a displacement equal to 1 m This matches hand-calculations as follows:

1.0

Displacement - cantilever end

Time (sec)

-0.2

-0.4

-0.6

-0.8

-1.0

-1.2

-1.4

-1.6

-1.8

0.0

Figure 3 Displacement at free end of cantilever

The moment distribution should be linear between zero at the free end, to 3000 kN-m (300 kN times

10 m) at the fixed end This is confirmed by the graph in Figure 4

Moment distribution

-500

0

500

1000

1500

2000

2500

3000

3 Simple beam with a point load

This case is a simple 10 m long beam fixed at the left end, and on a roller at the right end, with a 1000 kN point load at the mid-length point

Figure 5 A simple beam with a point load

Applying the load causes the beam to vibrate slightly, but then it settles down at a deflection at the mid-point equal to 0.208 m This matches hand-calculates values, as follows:

1000 10

0.208

PL



Displacement - mid beam

Time (sec)

-0.1

-0.2

-0.3

-0.4

0.0

0.0 0.1 0.2 0.3 0.4

Figure 6 Deflections at the mid-point of the beam with a point load

The moment distribution should be linear between the ends and the mid-point, and vary between zero at the ends and a maximum 2500 kN-m at the mid-point (1000 /2 kN times 10/2 m = 2500) The shear in the beam should be 500 kN The resulting QUAKE/W graphs in Figure 7 and Figure 8 confirm that this is the case

Moment distribution

X (m) -2500

0

2500

Figure 7 Moment distribution for beam with point load at the middle

Moment distribution

X (m)

-10000

-20000

0

10000

20000

Figure 8 Shear in beam with point load at the middle

4 Simple beam with a uniform load

This analysis is a repeat of the previous case, but with a uniformly distributed load of 100 kN per metre

As with the other two cases, when the load is applied, the beam vibrates, but then settles down at a maximum displacement at the mid-point equal to 0.13 m This again matches a hand-calculated value:

0.13

wL

Displacement - mid beam

Time (sec)

-0.04

-0.06

-0.09

-0.11

-0.13

-0.15

-0.18

-0.20

-0.22

-0.02

0.0 0.1 0.2 0.3 0.4

Figure 9 Deflections at the mid-point of the beam with a uniform load

The maximum moment should be 1250 kN-m (w*L2/8 = 100*10*10/8), and the maximum shear should

be 500 kN (w*L/2) This matches the QUAKE/W output, as shown in Figure 10 and Figure 11

Moment distribution

X (m)

-200

-400

-600

-800

-1000

-1200

-1400

0

Figure 10 Moment distribution for beam with uniform load

Moment distribution

X (m)

-10000

-20000

0

10000

20000

Figure 11 Shear distribution for beam with uniform load

The agreement between hand-calculated values and the QUAKE/W results indicates that the beam