設計 ダムの安定計算
はじめに
図に示すような単純形状のダムの安定計算プログラムです。 このような計算は擁壁などの場合も含めよく行うのですが、その都度プログラムを作っていました。 でも簡単な事例をどこかに載せておけば何かと便利なのでここに事例としてストックしておきます。
計算式
| Eccentricity from dam base center | |
| Vertical stress at upstream bottom edge of dam | |
| Vertical stress at downstream bottom edge of dam | |
| Total vertical force | |
| Total horizontal force | |
| Total moment around upstream bottom edge of dam | |
| dam base length in U/D direction | |
| Cohesion of foundation rock | |
| Internal friction angle of foundation rock | |
| Sum of vertical force | |
| Sum of uplift force | |
| Safety factor for sliding | |
| safety factor for floatation |
| Allowable bearing capacity | |
| Ultimate bearing capacity | |
| Safety factor (=1.5: BS 8110-1) |
Dynamic Water Pressure by Westergaard
| dynamic water pressure at depth | |
| horizontal force due to dynamic water pressure | |
| unit weight of water | |
| seismic coefficient | |
| depth from water surface to foundation | |
| depth from water surface to acting point of dynamic water pressure | |
| arm length from foundation to centroid of dynamic pressure force |
Sedimantation Pressure Force
| horizontal force due to sedimentation pressure | |
| unit weight of submerged sedimentation (=12 kN/m3) | |
| sedimentation pressure coefficient (=0.5) | |
| sedimentation depth from foundation to surface of sedimentation | |
| arm length from foundation to centroid of sedimentation pressure force |
安定計算プログラム
このプログラムで扱っている荷重は以下の通りです。堆砂圧は含まれていません。
地震時動水圧については、上流側・下流側とも同じ方向にかけています。 例えば、地震時慣性力が上流側から下流側に作用する場合、上流側動水圧は上流から下流方向、下流側動水圧も上流から下流方向、すなわち堤体を引っ張る方向にかけています。
出力は、画面出力としていますが、計算はJupyter上で行っているので、テキストをコピーしてそのまま報告書に貼り付けられます。
プログラム
import numpy as np # global variants elt=104.0 # dam top elevation elb=70.5 # dam foundation elevation n=0.65 # dpwnstream slope gradiant tw=7.0 # top width of dam bw=tw+(elt-elb)*n # bottom width of dam wc=23.5 # unit weight of concrete (kN/m3) ww=10.0 # unit weight of water (kN/m3) cc=70.0*10 # cohesion of foundation rock (kN/m2) phi=34.0 # internal friction angle of foundation rock (degrees) qu=2*cc*np.cos(np.radians(phi))/(1-np.sin(np.radians(phi))) # ultimate bearing capacity qa=qu/1.5 # allowable bearing capacity (SF=1.5) print('* Basic Design Conditions') print('Dam top level (EL) {0:10.3f} m'.format(elt)) print('Dam foundation level (EL) {0:10.3f} m'.format(elb)) print('Dam height {0:10.3f} m'.format(elt-elb)) print('Dam upstream slope m={0:4.2f}'.format(0)) print('Dam downstream slope n={0:4.2f}'.format(n)) print('Dam top width {0:10.3f} m'.format(tw)) print('Dam bottom width {0:10.3f} m'.format(bw)) print('Unit weight of concrete {0:10.3f} kN/m3'.format(wc)) print('Unit weight of water {0:10.3f} kN/m3'.format(ww)) print('Cohesion of foundation rock {0:10.3f} kN/m2'.format(cc)) print('Internal friction angle of foundation rock {0:10.3f} degrees'.format(phi)) print('Allowable Bearing capacity of foundation rock {0:10.3f} kN/m2'.format(qa)) print() def dambody(): # dam self weight hh=elt-elb w1=wc*tw*hh ; x1=tw/2 ; y1=hh/2 w2=wc*(bw-tw)*hh/2 ; x2=tw+(bw-tw)/3 ; y2=hh/3 wdb=w1+w2 xx=(w1*x2+w2*x2)/wdb yy=(w1*y1+w2*y2)/wdb return wdb,xx,yy def upwp(uwl): # upstream water pressure hh=uwl-elb uwp=ww*hh**2/2 yy=hh/3 return uwp,yy def dnwp(dwl): # downstream water pressure hh=dwl-elb dwp=ww*hh**2/2 yy=hh/3 return dwp,yy def dnww(dwl): # downstream water weight hh=dwl-elb dww=ww*n*hh**2/2 xx=bw-n*hh/3 return dww,xx def updp(uwl,kh): # upstream dynamic water pressure hh=uwl-elb udp=ww*7/12*kh*hh**2 yy=0.4*hh return udp,yy def dndp(dwl,kh): # downstream dynamic water pressure hh=dwl-elb ddp=ww*7/12*kh*hh**2 yy=0.4*hh return ddp,yy def uplift(uwl,dwl): # uplift xd=8.0 # drain location cu=0.2 # uplift coefficient h1=uwl-elb h2=dwl-elb u1=ww*bw*h2 ; x1=bw/2 u2=ww*xd*(h1-h2)*cu ; x2=xd/2 u3=ww*(bw-xd)*(h1-h2)*cu/2; x3=xd+(bw-xd)/3 u4=ww*xd*(h1-h2)*(1-cu)/2 ; x4=xd/3 upl=u1+u2+u3+u4 xx=(u1*x1+u2*x2+u3*x3+u4*x4)/upl return upl,xx def calc(uwl,dwl,kh,icase): wdb,xdb,ydb=dambody() # dam self weight uwp,yuwp=upwp(uwl) # upstream water pressure dwp,ydwp=dnwp(dwl) # downstream water pressure dww,xdww=dnww(dwl) # downstream water weight udp,yudp=updp(uwl,kh) # upstream dynamic water pressure ddp,yddp=dndp(dwl,kh) # downstream dynamic water pressure upl,xupl=uplift(uwl,dwl) # uplift vv=np.zeros(7,dtype=np.float64) hh=np.zeros(7,dtype=np.float64) xx=np.zeros(7,dtype=np.float64) yy=np.zeros(7,dtype=np.float64) mm=np.zeros(7,dtype=np.float64) ss=['Dam self weight', 'U/S water pressure', 'D/S water pressure', 'D/S water weight', 'U/S dynamic pressure', 'D/S dynamic pressure', 'Uplift'] i=0; vv[i]= wdb ; hh[i]= wdb*kh ; xx[i]=xdb ; yy[i]=ydb # Dam self weight i=1; vv[i]= 0 ; hh[i]= uwp ; xx[i]=0 ; yy[i]=yuwp # U/S water pressure i=2; vv[i]= 0 ; hh[i]=-dwp ; xx[i]=0 ; yy[i]=ydwp # D/S water pressure i=3; vv[i]= dww ; hh[i]=0 ; xx[i]=xdww ; yy[i]=0 # D/S water weight i=4; vv[i]= 0 ; hh[i]= udp ; xx[i]=0 ; yy[i]=yudp # U/S dynamic pressure i=5; vv[i]= 0 ; hh[i]= ddp ; xx[i]=0 ; yy[i]=yddp # D/S dynamic pressure i=6; vv[i]=-upl ; hh[i]=0 ; xx[i]=xupl ; yy[i]=0 # Uplift mm=vv*xx+hh*yy ee=np.sum(mm)/np.sum(vv)-bw/2 pu=np.sum(vv)/bw*(1-6*ee/bw) pd=np.sum(vv)/bw*(1+6*ee/bw) fss=(cc*bw+np.sum(vv)*np.tan(np.radians(phi)))/np.sum(hh) fsf=(wdb+dww)/upl if icase==1: scase=' (usual operation)' if icase==2: scase=' (unusual operation)' if icase==3: scase=' (earthquake: SEE)' if icase==4: scase=' (earthquake: SEE)' if icase==5: scase=' (flood: PMF)' print('* Case-'+str(icase)) s='-'*(21+12*5) print(s) print('Conditions: UWL=EL.{0:7.3f}m, DWL=EL.{1:7.3f}m, kh={2:6.3f}'.format(uwl,dwl,kh)+scase) print(s) print('{0:<21s}{1:>12s}{2:>12s}{3:>12s}{4:>12s}{5:>12s}'.format('Load Item','V (kN/m)','H (kN/m)','x (m)','y (m)','M (kN-m/m)')) print(s) for i in range(len(ss)): print('{0:<21s}{1:12.3f}{2:12.3f}{3:12.3f}{4:12.3f}{5:12.3f}'.format(ss[i],vv[i],hh[i],xx[i],yy[i],mm[i])) print('{0:<21s}{1:12.3f}{2:12.3f}{3:>12s}{4:>12s}{5:12.3f}'.format('Sum',np.sum(vv),np.sum(hh),'---','---',np.sum(mm))) print(s) print('{0:<21s}{1:>12s}{2:>12s}{3:>12s}{4:>12s}{5:>12s}'.format('Summary of Result','e (m)','pu (kN/m2)','pd (kN/m2)','FSs','FSf')) # Limit of acting point of resultant if icase==1: elim=bw/6 # middle third if icase==2: elim=bw/4 # middle half if icase==3: elim=bw/2 # within base if icase==4: elim=bw/2 # within base if icase==5: elim=bw/2 # within base se='e_limit={0:.3f}'.format(elim) print('{0:<21s}{1:12.3f}{2:12.3f}{3:12.3f}{4:12.3f}{5:12.3f}'.format(se,ee,pu,pd,fss,fsf)) print(s) print() def main(): kh0=0.367 # horizontal seismic coefficient upmf=101.35; dpmf=100.25 # PMF water level unwl= 98.00; dnwl= 91.37 # Normal water level ulwl= 95.00; dlwl= 87.40 # Dead water level (LWL) uwl=unwl; dwl=dnwl; kh= 0.000; calc(uwl,dwl,kh,1) uwl=unwl; dlw=dlwl; kh= 0.000; calc(uwl,dwl,kh,2) uwl=unwl; dwl=dnwl; kh= kh0 ; calc(uwl,dwl,kh,3) uwl=unwl; dwl=dnwl; kh=-kh0 ; calc(uwl,dwl,kh,4) uwl=upmf; dwl=dpmf; kh= 0.000; calc(uwl,dwl,kh,5) #============== # Execution #============== if __name__ == '__main__': main()
画面出力
* Basic Design Conditions Dam top level (EL) 104.000 m Dam foundation level (EL) 70.500 m Dam height 33.500 m Dam upstream slope m=0.00 Dam downstream slope n=0.65 Dam top width 7.000 m Dam bottom width 28.775 m Unit weight of concrete 23.500 kN/m3 Unit weight of water 10.000 kN/m3 Cohesion of foundation rock 700.000 kN/m2 Internal friction angle of foundation rock 34.000 degrees Allowable Bearing capacity of foundation rock 1755.345 kN/m2 * Case-1 --------------------------------------------------------------------------------- Conditions: UWL=EL. 98.000m, DWL=EL. 91.370m, kh= 0.000 (usual operation) --------------------------------------------------------------------------------- Load Item V (kN/m) H (kN/m) x (m) y (m) M (kN-m/m) --------------------------------------------------------------------------------- Dam self weight 14081.934 0.000 14.258 13.352 200784.914 U/S water pressure 0.000 3781.250 0.000 9.167 34661.458 D/S water pressure 0.000 -2177.785 0.000 6.957 -15150.121 D/S water weight 1415.560 0.000 24.253 0.000 34331.811 U/S dynamic pressure 0.000 0.000 0.000 11.000 0.000 D/S dynamic pressure 0.000 0.000 0.000 8.348 0.000 Uplift -6461.321 0.000 13.844 0.000 -89447.689 Sum 9036.174 1603.465 --- --- 165180.374 --------------------------------------------------------------------------------- Summary of Result e (m) pu (kN/m2) pd (kN/m2) FSs FSf e_limit=4.796 3.892 59.156 568.901 16.363 2.399 --------------------------------------------------------------------------------- * Case-2 --------------------------------------------------------------------------------- Conditions: UWL=EL. 98.000m, DWL=EL. 91.370m, kh= 0.000 (unusual operation) --------------------------------------------------------------------------------- Load Item V (kN/m) H (kN/m) x (m) y (m) M (kN-m/m) --------------------------------------------------------------------------------- Dam self weight 14081.934 0.000 14.258 13.352 200784.914 U/S water pressure 0.000 3781.250 0.000 9.167 34661.458 D/S water pressure 0.000 -2177.785 0.000 6.957 -15150.121 D/S water weight 1415.560 0.000 24.253 0.000 34331.811 U/S dynamic pressure 0.000 0.000 0.000 11.000 0.000 D/S dynamic pressure 0.000 0.000 0.000 8.348 0.000 Uplift -6461.321 0.000 13.844 0.000 -89447.689 Sum 9036.174 1603.465 --- --- 165180.374 --------------------------------------------------------------------------------- Summary of Result e (m) pu (kN/m2) pd (kN/m2) FSs FSf e_limit=7.194 3.892 59.156 568.901 16.363 2.399 --------------------------------------------------------------------------------- * Case-3 --------------------------------------------------------------------------------- Conditions: UWL=EL. 98.000m, DWL=EL. 91.370m, kh= 0.367 (earthquake: SEE) --------------------------------------------------------------------------------- Load Item V (kN/m) H (kN/m) x (m) y (m) M (kN-m/m) --------------------------------------------------------------------------------- Dam self weight 14081.934 5168.070 14.258 13.352 269787.014 U/S water pressure 0.000 3781.250 0.000 9.167 34661.458 D/S water pressure 0.000 -2177.785 0.000 6.957 -15150.121 D/S water weight 1415.560 0.000 24.253 0.000 34331.811 U/S dynamic pressure 0.000 1619.005 0.000 11.000 17809.057 D/S dynamic pressure 0.000 932.455 0.000 8.348 7784.132 Uplift -6461.321 0.000 13.844 0.000 -89447.689 Sum 9036.174 9322.995 --- --- 259775.663 --------------------------------------------------------------------------------- Summary of Result e (m) pu (kN/m2) pd (kN/m2) FSs FSf e_limit=14.388 14.361 -626.316 1254.374 2.814 2.399 --------------------------------------------------------------------------------- * Case-4 --------------------------------------------------------------------------------- Conditions: UWL=EL. 98.000m, DWL=EL. 91.370m, kh=-0.367 (earthquake: SEE) --------------------------------------------------------------------------------- Load Item V (kN/m) H (kN/m) x (m) y (m) M (kN-m/m) --------------------------------------------------------------------------------- Dam self weight 14081.934 -5168.070 14.258 13.352 131782.814 U/S water pressure 0.000 3781.250 0.000 9.167 34661.458 D/S water pressure 0.000 -2177.785 0.000 6.957 -15150.121 D/S water weight 1415.560 0.000 24.253 0.000 34331.811 U/S dynamic pressure 0.000 -1619.005 0.000 11.000 -17809.057 D/S dynamic pressure 0.000 -932.455 0.000 8.348 -7784.132 Uplift -6461.321 0.000 13.844 0.000 -89447.689 Sum 9036.174 -6116.064 --- --- 70585.085 --------------------------------------------------------------------------------- Summary of Result e (m) pu (kN/m2) pd (kN/m2) FSs FSf e_limit=14.388 -6.576 744.629 -116.572 -4.290 2.399 --------------------------------------------------------------------------------- * Case-5 --------------------------------------------------------------------------------- Conditions: UWL=EL.101.350m, DWL=EL.100.250m, kh= 0.000 (flood: PMF) --------------------------------------------------------------------------------- Load Item V (kN/m) H (kN/m) x (m) y (m) M (kN-m/m) --------------------------------------------------------------------------------- Dam self weight 14081.934 0.000 14.258 13.352 200784.914 U/S water pressure 0.000 4758.612 0.000 10.283 48934.399 D/S water pressure 0.000 -4425.312 0.000 9.917 -43884.349 D/S water weight 2876.453 0.000 22.329 0.000 64228.801 U/S dynamic pressure 0.000 0.000 0.000 12.340 0.000 D/S dynamic pressure 0.000 0.000 0.000 11.900 0.000 Uplift -8636.215 0.000 14.320 0.000 -123670.433 Sum 8322.173 333.300 --- --- 146393.332 --------------------------------------------------------------------------------- Summary of Result e (m) pu (kN/m2) pd (kN/m2) FSs FSf e_limit=14.388 3.203 96.041 482.390 77.275 1.964 ---------------------------------------------------------------------------------
作図プログラム
import numpy as np import matplotlib.pyplot as plt import matplotlib.ticker as tick elt=104.0 # dam top elevation elb=70.5 # dam foundation elevation n=0.65 # dpwnstream slope gradiant tw=7.0 # top width of dam bw=tw+(elt-elb)*n # bottom width of dam upmf=101.35; dpmf=100.25 # PMF water level unwl= 98.00; dnwl= 91.37 # Normal water level ulwl= 95.00; dlwl= 87.40 # Dead water level (LWL) def barrow(x1,y1,x2,y2): col='#333333' arst='<->,head_width=3,head_length=10' aprop=dict(arrowstyle=arst,connectionstyle='arc3',facecolor=col,edgecolor=col,shrinkA=0,shrinkB=0,lw=1) plt.annotate('', xy=(x1,y1), xycoords='data', xytext=(x2,y2), textcoords='data', fontsize=0, arrowprops=aprop) def main(): fsz=12 xmin=-20; xmax=40; dx=5 ymin=65; ymax=110; dy=5 plt.figure(figsize=(8,8),facecolor='w') plt.rcParams['font.size']=fsz plt.rcParams['font.family']='sans-serif' tstr='Basic Shape of Concrete Gravity Dam (unit: m)' plt.xlim([xmin,xmax]) plt.ylim([ymin,ymax]) plt.xlabel('Distance (m)') plt.ylabel('Elevation (EL.m)') plt.xticks(np.arange(xmin,xmax+dx,dx)) plt.yticks(np.arange(ymin,ymax+dy,dy)) plt.gca().spines['right'].set_visible(False) plt.gca().spines['top'].set_visible(False) # plt.gca().yaxis.set_ticks_position('left') # plt.gca().xaxis.set_ticks_position('bottom') plt.gca().set_aspect('equal',adjustable='box') plt.title(tstr,loc='left',fontsize=fsz+2) damx=np.array([0,0,tw,bw]) damy=np.array([elb,elt,elt,elb]) plt.fill(damx,damy,facecolor='#eeeeee',edgecolor='#000000',lw=2) # dam body plt.fill([-5,bw+5,bw+5,-5],[elb,elb,elb-2,elb-2],fill=False, hatch='//',lw=0) # hatching of foundation plt.plot([-5,bw+5],[elb,elb],'-',lw=2) # foundation line plt.plot([xmin+5,0],[elt,elt],'-',lw=1) plt.plot([xmin+5,0],[elb,elb],'-',lw=1) plt.plot([xmin+5,0],[upmf,upmf],'-',lw=1) plt.plot([xmin+5,0],[unwl,unwl],'-',lw=1) plt.plot([xmin+5,0],[ulwl,ulwl],'-',lw=1) plt.text(xmin+5.0,elt,'EL.{0:.3f}'.format(elt),va='bottom',ha='left',fontsize=fsz) plt.text(xmin+5.0,elb,'EL.{0:.3f}'.format(elb),va='bottom',ha='left',fontsize=fsz) plt.text(xmin+5.0,upmf,'PMF: EL.{0:.3f}'.format(upmf),va='bottom',ha='left',fontsize=fsz) plt.text(xmin+5.0,unwl,'NWL: EL.{0:.3f}'.format(unwl),va='bottom',ha='left',fontsize=fsz) plt.text(xmin+5.0,ulwl,'LWL: EL.{0:.3f}'.format(ulwl),va='bottom',ha='left',fontsize=fsz) plt.plot([xmax-5,tw+(elt-dpmf)*n],[dpmf,dpmf],'-',lw=1) plt.plot([xmax-5,tw+(elt-dnwl)*n],[dnwl,dnwl],'-',lw=1) plt.plot([xmax-5,tw+(elt-dlwl)*n],[dlwl,dlwl],'-',lw=1) plt.text(xmax-5.0,dpmf,'PMF: EL.{0:.3f}'.format(dpmf),va='bottom',ha='right',fontsize=fsz) plt.text(xmax-5.0,dnwl,'NWL: EL.{0:.3f}'.format(dnwl),va='bottom',ha='right',fontsize=fsz) plt.text(xmax-5.0,dlwl,'LWL: EL.{0:.3f}'.format(dlwl),va='bottom',ha='right',fontsize=fsz) plt.plot([0,0],[elt,elt+3],'-',lw=1,color='#333333') plt.plot([tw,tw],[elt,elt+3],'-',lw=1,color='#333333') x1=0; x2=tw; y1=elt+2.5; y2=y1; barrow(x1,y1,x2,y2) xs=0.5*(x1+x2); ys=y1 plt.text(xs,ys,'{0:.3f}'.format(x2-x1),va='bottom',ha='center',fontsize=fsz) plt.plot([0,0],[elb,elb-4],'-',lw=1,color='#333333') plt.plot([bw,bw],[elb,elb-4],'-',lw=1,color='#333333') x1=0; x2=bw; y1=elb-3.5; y2=y1; barrow(x1,y1,x2,y2) xs=0.5*(x1+x2); ys=y1 plt.text(xs,ys,'{0:.3f}'.format(x2-x1),va='top',ha='center',fontsize=fsz) x1=2; x2=x1; y1=elb; y2=elt; barrow(x1,y1,x2,y2) xs=x1; ys=0.5*(y1+y2) plt.text(xs,ys,'{0:.3f}'.format(y2-y1),va='center',ha='left',fontsize=fsz,rotation=90) ang=np.degrees(np.arctan(1/n)) ys=80.0; xs=bw-(ys-elb)*n+1 plt.text(xs,ys,'1:{0:.2f}'.format(n),va='center',ha='center',fontsize=fsz,rotation=-ang) fnameF='fig_dam.jpg' plt.savefig(fnameF, dpi=200, bbox_inches="tight", pad_inches=0.1) plt.show() #============== # Execution #============== if __name__ == '__main__': main()
