Beam_curved.xls
To determine deflection of a curved beam
By Alex Slocum, 12/29/03, last modified 09/21/04 by Xue'en Yang
Enters numbers in BOLD, Results in RED
Schematic
Beam dimensions and properties Values
If round cross-section (enter "yes" or "no") yes Mo
Outside diameter, Do (mm) 30 F x
Inside diamter, Di (mm) 25
If rectangular cross-section no
Outside height, ho (mm) 20 Fy
Outside width, wo (mm) 10
Inside height, hi (mm) 18
Inside width, wi (mm) 8 Circular cross-section
Radius of curvature, Rr (mm) 100
Angle, phi (degrees) 90
Angle, phi (radius) 1.571
Modulus of elasticity, E (N/mm^2) 200000
Poisson ratio, v 0.29
Shear modulus, G (Pa) 77519 Rectangular cross-section
Moment of inertia, I (mm^4) 20586
Distance farthest fiber to neutral axis, cc (mm) 15
Cross section area, A 216
EA 4.32E+07
GA 1.67E+07
EI 4.12E+09
Loading
Point load, Fx (N) 10
Point load, Fy (N) 76
Moment, Mo (N*mm) 22
X direction displacement (microns) Axial Shear Bending Total
Due to Fx 0.018 0.047 0.865 0.930
Due to Fy 0.000 0.000 9.230 9.230
Due to Mo 0.000 0.000 0.031 0.031
Total 0.018 0.047 10.125 10.190
Y direction displacement (microns) Axial Shear Bending Total
Due to Fx 0.000 0.000 1.214 1.214
Due to Fy 0.138 0.357 14.498 14.992
Due to Mo 0.000 0.000 0.053 0.053
Total 0.138 0.357 15.766 16.260
Slope (milliradians) Bending
Due to Fx 0.014
Due to Fy 0.185
Due to Mo 0.001
Total 0.199
Equations
P F x cos F y sin V F x sin F y cos M F xR 1 co
P P P V V V M
cos sin 0 sin cos 0 R 1 co
F x F y M o F x F y M o F x
U
F x cos F y sin cos
F xsin F y cos sin
F x R 1
2 2 2
x Rd Rd
F x EA GA
0 0 0
axial shear
P F x cos F y sin V F x sin F y cos M F xR 1 co
P P P V V V M
cos sin 0 sin cos 0 R 1 co
F x F y M o F x F y M o F x
U
F x cos F y sin cos
F xsin F y cos sin
F x R 1
2 2 2
x Rd Rd
F x EA GA
0 0 0
axial shear
x R F x 2 sin 2 F y 1 cos R 2 sin 2 1 cos 2 1
2
F x F y
EA 4 2 axial GA 4 2 EI
shear
U
F x cos sin F y sin
F x cos sin F y cos
2 2 2
F xR
y Rd Rd
F y EA GA
0 0 0
axial shear
y R F x 1 cos F y 2 sin 2 R 1 cos 2 2 sin 2
2
1
F x F y
EA 2 4 GA 2 4 EI
axial shear
U F xR 1 cos F yR sin M o
M o
0 EI
Rd
bending
1 2F sin 2F 1 cos M R
R x
EI
R y o
bending
Mo Rsin
R(1-cos)
F x
Fy R
cular cross-section
angular cross-section
Check: Bending (from
Roark) for phi = p/2
Bending ratio
0.865 1
9.230 1
0.031 1
10.125 1
1.214 1
14.498 1
0.053 1
15.766 1
0.014 1
0.185 1
0.001 1
0.199 1
M F xR 1 cos F yR sin M o ds Rd
M M M
0 R 1 cos R sin 1
F x F y M o
F x R 1 cos F y R sin 1 cos M oR 1 cos
2
2 2
Rd Rd
0 EI
shear bending
M F xR 1 cos F yR sin M o ds Rd
M M M
0 R 1 cos R sin 1
F x F y M o
F x R 1 cos F y R sin 1 cos M oR 1 cos
2
2 2
Rd Rd
0 EI
shear bending
1 cos 2 1 3F 6 8sin sin 2 3F 1 2 cos cos
2
R x M o R sin
2
R y
2 EI 4 2
shear bending
F x R sin 1 cos F y R sin M oR sin
2 2 2 2
Rd Rd
0 EI
shear bending
2 sin 2 1 3F 1 2 cos cos 3F 2 sin 2
2
R x M o R 1 cos
2
R y
4 EI 2 4
shear bending