Structural beams in quake

GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com Structural Beams in QUAKE/W Introduction 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 Cantilever beam The configuration of a simple cantilever beam is presented in Figure 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 10 Distance - m Figure Configuration of a simple cantilever beam Figure Circle indicates rotation boundary condition at the left end In QUAKE/W, beams 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 m This matches hand-calculations as follows:  max  PL3 300  103   1.0 3EI  108  103 QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com Displacement - cantilever end 0.0 -0.2 Y-Displacement (m) -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 0.0 0.2 0.4 0.6 0.8 1.0 Time (sec) Figure 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 Moment distribution 3000 2500 Moment (kN-m) 2000 1500 1000 500 -500 10 X (m) Figure Moment distribution in the cantilever beam QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com 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 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 midpoint equal to 0.208 m This matches hand-calculates values, as follows:  max PL3 1000  103    0.208 48 EI 48  108  103 Displacement - mid beam 0.0 Y-Displacement (m) -0.1 -0.2 -0.3 -0.4 0.0 0.1 0.2 0.3 0.4 Time (sec) Figure 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 and Figure confirm that this is the case QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com Moment distribution Moment (kN-m) 2500 -2500 10 X (m) Figure Moment distribution for beam with point load at the middle Moment distribution 20000 Shear Force (kN) 10000 -10000 -20000 10 X (m) Figure Shear in beam with point load at the middle QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com 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:  max  5wL4  100  104   0.13 384 EI 384  108  103 Displacement - mid beam -0.02 -0.04 Y-Displacement (m) -0.06 -0.09 -0.11 -0.13 -0.15 -0.18 -0.20 -0.22 0.0 0.1 0.2 0.3 0.4 Time (sec) Figure 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 QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com Moment distribution -200 Moment (kN-m) -400 -600 -800 -1000 -1200 -1400 10 X (m) Figure 10 Moment distribution for beam with uniform load Moment distribution 20000 Shear Force (kN) 10000 -10000 -20000 10 X (m) Figure 11 Shear distribution for beam with uniform load Closing remarks The agreement between hand-calculated values and the QUAKE/W results indicates that the beam element formulation in QUAKE/W is correct and functions as intended The sign conventions for moments and shears in a finite element formulation come from the sign convention for numbering elements In QUAKE/W, the node numbering sequence around and element is always counter-clockwise The consequence is that the sign on the moments and shears depends on which side of the beam is connected to elements Both signs are possible for the same beam In the end, the sign does not really matter, provided that the deflections and rotations are correct and consistent QUAKE/W Example File: Structural Beams in Quake.docx (pdf) (gsz) Page of ... m = 2500) The shear in the beam should be 500 kN The resulting QUAKE/ W graphs in Figure and Figure confirm that this is the case QUAKE/ W Example File: Structural Beams in Quake. docx (pdf) (gsz)... distribution in the cantilever beam QUAKE/ W Example File: Structural Beams in Quake. docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com Simple beam with a point... -10000 -20000 10 X (m) Figure Shear in beam with point load at the middle QUAKE/ W Example File: Structural Beams in Quake. docx (pdf) (gsz) Page of GEO-SLOPE International Ltd, Calgary, Alberta,