Roark's Formulas for Stress and Strain
Properties of sections - table A.1 (15-29)
A=area (mm2), x/y= distance to extreme fiber (mm), I=moment of inertia (mm4), r=radius of gyration (mm), Z=plastic section modulus (mm3), SF = shape factor
15. SOLID CIRCLE
solid circle
distance R (mm): 
 
A solid circle (mm2): A
yc solid circle (mm): yc
xc solid circle (mm): xc
Ix solid circle (mm4): Ix
Iy solid circle (mm4): Iy
rx solid circle (mm): rx
ry solid circle (mm): ry
Zx solid circle (mm3): Zx
Zy solid circle (mm3): Zy
SFx solid circle: 1.698
SFy solid circle: 1.698
16. HOLLOW CIRCLE
hollow circle
distance R (mm): 
distance Ri (mm): 
 
A hollow circle (mm2): A
yc hollow circle (mm): yc
xc hollow circle (mm): xc
Ix hollow circle (mm4): Ix
Iy hollow circle (mm4): Iy
rx hollow circle (mm): rx
ry hollow circle (mm): ry
Zx hollow circle (mm3): Zx
Zy hollow circle (mm3): Zy
SFx hollow circle: SFx
SFy hollow circle: SFy
17. VERY THIN ANNULUS
very thin annulus
distance R (mm): 
thickness t (mm): 
 
A very thin annulus (mm2): A
yc very thin annulus (mm): yc
xc very thin annulus (mm): xc
Ix very thin annulus (mm4): Ix
Iy very thin annulus (mm4): Iy
rx very thin annulus (mm): rx
ry very thin annulus (mm): ry
Zx very thin annulus (mm3): Zx
Zy very thin annulus (mm3): Zy
SFx very thin annulus: SFx
SFy very thin annulus: SFy
18. SECTOR OF SOLID CIRCLE
sector of solid circle
distance R (mm): 
angle alfa (degrees): 
 
A sector of solid circle (mm2): A
yc1 sector of solid circle (mm): yc1
yc2 sector of solid circle (mm): yc2
xc sector of solid circle (mm): xc
Ix sector of solid circle (mm4): Ix
Iy sector of solid circle (mm4): Iy
rx sector of solid circle (mm): rx
ry sector of solid circle (mm): ry
Zx sector of solid circle (mm3): Zx
Neutral axis x - from the vertex (mm): NAx
Zy sector of solid circle (mm3): Zy
SFx sector of solid circle: SFx
SFy sector of solid circle: SFy
alfa1 hoek (graden): SFy
19. SEGMENT OF SOLID CIRCLE (IF ALFA <=PI/4 USE CASE 20)
segment of solid circle1
distance R (mm): 
alfa (degrees): 
 
A segment of solid circle (mm2): A
yc1 segment of solid circle (mm): yc
yc2 segment of solid circle (mm): yc
xc segment of solid circle (mm): xc
Ix segment of solid circle (mm4): Ix
Iy segment of solid circle (mm4): Iy
rx segment of solid circle (mm): rx
ry segment of solid circle (mm): ry
20. SEGMENT OF SOLID CIRCLE (IF ALFA>PI/4 USE CASE 19)
segment of solid circle2
distance R (mm): 
alfa (degrees): 
 
A segment of solid circle (mm2): A
yc1 segment of solid circle (mm): yc
yc2 segment of solid circle (mm): yc
xc segment of solid circle (mm): xc
Ix segment of solid circle (mm4): Ix
Iy segment of solid circle (mm4): Iy
rx segment of solid circle (mm): rx
ry segment of solid circle (mm): ry
21. SECTOR OF HOLLOW CIRCLE
if t/R is small, alfa can exceed pi to form an overlapped annulus
sector of hollow circle
distance R (mm): 
thickness t (mm): 
alfa (degrees): 
 
A sector of hollow circle (mm2): A
yc1 sector of hollow circle (mm): yc
yc2 sector of hollow circle (mm): yc
xc sector of hollow circle (mm): xc
Ix sector of hollow circle (mm4): Ix
Iy sector of hollow circle (mm4): Iy
rx sector of hollow circle (mm): rx
ry sector of hollow circle (mm): ry
22. SOLID SEMICIRCLE
solid semicircle
distance R (mm): 
 
A solid semicircle (mm2): A
yc1 solid semicircle (mm): yc1
yc2 solid semicircle (mm): yc2
xc solid semicircle (mm): xc
Ix solid semicircle (mm4): Ix
Iy solid semicircle (mm4): Iy
rx solid semicircle (mm): rx
ry solid semicircle (mm): ry
Zx solid semicircle (mm3): Zx1
Zy solid semicircle (mm3): Zx2
SFx solid semicircle: 1.856
SFy solid semicircle: 1.698
NA solid semicircle: yp1
23. HOLLOW SEMICIRCLE
hollow semicircle
distance R (mm): 
distance Ri (mm): 
 
A hollow semicircle (mm2): A
yc2a hollow semicircle (mm): yc2
yc2b hollow semicircle (mm): yc2
yc1 hollow semicircle (mm): yc1
xc hollow semicircle (mm): xc
Ixa hollow semicircle (mm4): Ix
Ixb hollow semicircle (mm4): Ix
Iya hollow semicircle (mm4): Iy
Iyb hollow semicircle (mm4): Iy
yp hollow semicircle (mm): yp
Zx hollow semicircle (mm3): Zx
Zy hollow semicircle (mm3): Zy
24. SOLID ELLIPSE
solid ellipse
distance a (mm): 
distance b (mm): 
 
A solid ellipse (mm2): A
yc solid ellipse (mm): yc
xc solid ellipse (mm): xc
Ix solid ellipse (mm4): Ix
Iy solid ellipse (mm4): Iy
rx solid ellipse (mm): rx
ry solid ellipse (mm): ry
Zx solid ellipse (mm3): Zx
Zy solid ellipse (mm3): Zy
SFx solid ellipse: 1.698
SFy solid ellipse: 1.698
25. HOLLOW ELLIPSE (WALL THICKNESS IS NOT CONSTANT)
For this case the inner and outer perimeters are both ellipses and the wall thickness
is not constant. For a cross section with a constant wall thickness see case 26.
hollow ellipse
distance a (mm): 
distance ai (mm): 
distance b (mm): 
distance bi (mm): 
 
A hollow ellipse (mm2): A
yc hollow ellipse (mm): yc
xc hollow ellipse (mm): xc
Ix hollow ellipse (mm4): Ix
Iy hollow ellipse (mm4): Iy
rx hollow ellipse (mm): rx
ry hollow ellipse (mm): ry
Zx hollow ellipse (mm3): Zx
Zy hollow ellipse (mm3): Zy
SFx hollow ellipse: SFx
SFy hollow ellipse: SFy
26. HOLLOW ELLIPSE WITH CONSTANT WALL THICKNESS
The midthickness perimeter is an ellipse (shown dashed) a/b between 0.2 and 5
Cusps will form in the perimeter at the ends of the major axis if tmax is exceeded.
hollow ellipse with constant wall thickness
distance a (mm): 
distance b (mm): 
thickness t (mm): 
 
a/b : a/b
maximum allowed wall thickness tmax (mm) : tmax
A hollow ellipse-constant WT (mm2): A
yc hollow ellipse-constant WT (mm): yc
xc hollow ellipse-constant WT (mm): xc
Ix hollow ellipse-constant WT (mm4): Ix
Iy hollow ellipse-constant WT (mm4): Iy
Zx hollow ellipse-constant WT (mm3): rx
Zy hollow ellipse-constant WT (mm3): ry
27. HOLLOW SEMIELLIPSE WITH CONSTANT WALL THICKNESS
The midthickness perimeter is an ellipse (shown dashed) a/b between 0.2 and 5
Cusps will form in the perimeter at the ends of the major axis if tmax is exceeded.
hollow semiellipse with constant wall thickness
distance a (mm): 
distance b (mm): 
wall thickness t (mm): 
 
a/b hollow semiellipse-constant WT (mm): a/b
tmax hollow semiellipse-constant WT (mm): tmax
D (must be > 0.2 and <=1): D
A hollow semiellipse-constant WT (mm2): A
yc2 hollow semiellipse-constant WT (mm): yc2
yc1 hollow semiellipse-constant WT (mm): yc1
xc hollow semiellipse-constant WT (mm): xc
IX hollow semiellipse-constant WT (mm4): IX
Ix hollow semiellipse-constant WT (mm4): Ix
Iy hollow semiellipse-constant WT (mm4): one-half the value for Iy in case 26
yp vertical distance from bottom to plastic neutral axis (mm): ypHSE
Zx hollow semiellipse-constant WT (mm3): Zx
Zy hollow semiellipse-constant WT (mm3): one-half the value for Zy in case 26
28. REGULAR POLYGON WITH n SIDES
regular polygon with n sides
for n=3 see case 9. For n=4, see cases 1 and 13.
for n=5, Z1=Z2=0.8825 ro13. For an axis perpendicular
to axis 1, Z=0.8838 ro13. The location of this axis is
0.7007a from that side which is perpendicular to axis 1.
distance a (mm): 
number of sides n: 
 
alfa regular polygon with n sides (degrees): alfa
A regular polygon with n sides (mm2): A
ro1 regular polygon with n sides (mm): ro1
ro2 regular polygon with n sides (mm): ro2
y1 regular polygon with n sides (mm): y1
y2 regular polygon with n sides (mm): y2
I1 regular polygon with n sides (mm4): I1
I2 regular polygon with n sides (mm4): I2
r1 regular polygon with n sides (mm): r1
r2 regular polygon with n sides (mm): r2
Z regular polygon with n sides (mm3): Zx
Z1 regular polygon with 5 sides (mm3): Z1
Z2 regular polygon with 5 sides (mm3): Z2
location axis perp. to axis 1 (n=5)(mm): LAPA1
29. HOLLOW REGULAR POLYGON WITH n SIDES
hollow regular polygon with n sides
distance a (mm): 
number of sides n: 
wall thickness t (mm): 
 
alfa hollow reg. polygon-n sides (degrees): alfa
A hollow regular polygon-n sides (mm2): A
ro1 hollow regular polygon-n sides (mm): ro1
ro2 hollow regular polygon-n sides (mm): ro2
y1 hollow regular polygon-n sides (mm): y1
y2 hollow regular polygon-n sides (mm): y2
I1 hollow regular polygon-n sides (mm4): I1
I2 hollow regular polygon-n sides (mm4): I2
r1 hollow regular polygon-n sides (mm): r1
r2 hollow regular polygon-n sides (mm): r2