Roark's Formulas for Stress and Strain |
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Properties of sections - table A.1 (15-29) |

A=area (mm^{2}), x/y= distance to extreme fiber (mm), I=moment of inertia (mm^{4}), r=radius of gyration (mm), Z=plastic section modulus (mm^{3}), SF = shape factor |

15. SOLID CIRCLE |
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A solid circle (mm^{2}): |
A |

y_{c} solid circle (mm): |
yc |

x_{c} solid circle (mm): |
xc |

Ix solid circle (mm^{4}): |
Ix |

Iy solid circle (mm^{4}): |
Iy |

rx solid circle (mm): | rx |

ry solid circle (mm): | ry |

Zx solid circle (mm^{3}): |
Zx |

Zy solid circle (mm^{3}): |
Zy |

SFx solid circle: | 1.698 |

SFy solid circle: | 1.698 |

16. HOLLOW CIRCLE |
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A hollow circle (mm^{2}): |
A |

y_{c} hollow circle (mm): |
yc |

x_{c} hollow circle (mm): |
xc |

Ix hollow circle (mm^{4}): |
Ix |

Iy hollow circle (mm^{4}): |
Iy |

rx hollow circle (mm): | rx |

ry hollow circle (mm): | ry |

Zx hollow circle (mm^{3}): |
Zx |

Zy hollow circle (mm^{3}): |
Zy |

SFx hollow circle: | SFx |

SFy hollow circle: | SFy |

17. VERY THIN ANNULUS |
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A very thin annulus (mm^{2}): |
A |

y_{c} very thin annulus (mm): |
yc |

x_{c} very thin annulus (mm): |
xc |

Ix very thin annulus (mm^{4}): |
Ix |

Iy very thin annulus (mm^{4}): |
Iy |

rx very thin annulus (mm): | rx |

ry very thin annulus (mm): | ry |

Zx very thin annulus (mm^{3}): |
Zx |

Zy very thin annulus (mm^{3}): |
Zy |

SFx very thin annulus: | SFx |

SFy very thin annulus: | SFy |

18. SECTOR OF SOLID CIRCLE |
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A sector of solid circle (mm^{2}): |
A |

y_{c1} sector of solid circle (mm): |
yc1 |

y_{c2} sector of solid circle (mm): |
yc2 |

x_{c} sector of solid circle (mm): |
xc |

Ix sector of solid circle (mm^{4}): |
Ix |

Iy sector of solid circle (mm^{4}): |
Iy |

rx sector of solid circle (mm): | rx |

ry sector of solid circle (mm): | ry |

Zx sector of solid circle (mm^{3}): |
Zx |

Neutral axis x - from the vertex (mm): | NAx |

Zy sector of solid circle (mm^{3}): |
Zy |

SFx sector of solid circle: | SFx |

SFy sector of solid circle: | SFy |

alfa1 angle (degrees): | SFy |

19. SEGMENT OF SOLID CIRCLE (IF ALFA <=PI/4 USE CASE 20) |
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A segment of solid circle (mm^{2}): |
A |

y_{c1} segment of solid circle (mm): |
yc |

y_{c2} segment of solid circle (mm): |
yc |

x_{c} segment of solid circle (mm): |
xc |

Ix segment of solid circle (mm^{4}): |
Ix |

Iy segment of solid circle (mm^{4}): |
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) |
---|

A segment of solid circle (mm^{2}): |
A |

y_{c1} segment of solid circle (mm): |
yc |

y_{c2} segment of solid circle (mm): |
yc |

x_{c} segment of solid circle (mm): |
xc |

Ix segment of solid circle (mm^{4}): |
Ix |

Iy segment of solid circle (mm^{4}): |
Iy |

rx segment of solid circle (mm): | rx |

ry segment of solid circle (mm): | ry |

21. SECTOR OF HOLLOW CIRCLE |
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if t/R is small, alfa can exceed pi to form an overlapped annulus |

A sector of hollow circle (mm^{2}): |
A |

y_{c1} sector of hollow circle (mm): |
yc |

y_{c2} sector of hollow circle (mm): |
yc |

x_{c} sector of hollow circle (mm): |
xc |

Ix sector of hollow circle (mm^{4}): |
Ix |

Iy sector of hollow circle (mm^{4}): |
Iy |

rx sector of hollow circle (mm): | rx |

ry sector of hollow circle (mm): | ry |

22. SOLID SEMICIRCLE |
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A solid semicircle (mm^{2}): |
A |

y_{c1} solid semicircle (mm): |
yc1 |

y_{c2} solid semicircle (mm): |
yc2 |

x_{c} solid semicircle (mm): |
xc |

Ix solid semicircle (mm^{4}): |
Ix |

Iy solid semicircle (mm^{4}): |
Iy |

rx solid semicircle (mm): | rx |

ry solid semicircle (mm): | ry |

Zx solid semicircle (mm^{3}): |
Zx1 |

Zy solid semicircle (mm^{3}): |
Zx2 |

SFx solid semicircle: | 1.856 |

SFy solid semicircle: | 1.698 |

NA solid semicircle: | yp1 |

23. HOLLOW SEMICIRCLE |
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A hollow semicircle (mm^{2}): |
A |

y_{c2a} hollow semicircle (mm): |
yc2 |

y_{c2b} hollow semicircle (mm): |
yc2 |

y_{c1} hollow semicircle (mm): |
yc1 |

x_{c} hollow semicircle (mm): |
xc |

Ixa hollow semicircle (mm^{4}): |
Ix |

Ixb hollow semicircle (mm^{4}): |
Ix |

Iya hollow semicircle (mm^{4}): |
Iy |

Iyb hollow semicircle (mm^{4}): |
Iy |

yp hollow semicircle (mm): | yp |

Zx hollow semicircle (mm^{3}): |
Zx |

Zy hollow semicircle (mm^{3}): |
Zy |

24. SOLID ELLIPSE |
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A solid ellipse (mm^{2}): |
A |

y_{c} solid ellipse (mm): |
yc |

x_{c} solid ellipse (mm): |
xc |

Ix solid ellipse (mm^{4}): |
Ix |

Iy solid ellipse (mm^{4}): |
Iy |

rx solid ellipse (mm): | rx |

ry solid ellipse (mm): | ry |

Zx solid ellipse (mm^{3}): |
Zx |

Zy solid ellipse (mm^{3}): |
Zy |

SFx solid ellipse: | 1.698 |

SFy solid ellipse: | 1.698 |

25. HOLLOW ELLIPSE (WALL THICKNESS IS NOT CONSTANT) |
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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. |

A hollow ellipse (mm^{2}): |
A |

y_{c} hollow ellipse (mm): |
yc |

x_{c} hollow ellipse (mm): |
xc |

Ix hollow ellipse (mm^{4}): |
Ix |

Iy hollow ellipse (mm^{4}): |
Iy |

rx hollow ellipse (mm): | rx |

ry hollow ellipse (mm): | ry |

Zx hollow ellipse (mm^{3}): |
Zx |

Zy hollow ellipse (mm^{3}): |
Zy |

SFx hollow ellipse: | SFx |

SFy hollow ellipse: | SFy |

26. HOLLOW ELLIPSE WITH CONSTANT WALL THICKNESS |
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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. |

a/b : | a/b |

maximum allowed wall thickness t_{max} (mm) : |
tmax |

A hollow ellipse-constant WT (mm^{2}): |
A |

y_{c} hollow ellipse-constant WT (mm): |
yc |

x_{c} hollow ellipse-constant WT (mm): |
xc |

Ix hollow ellipse-constant WT (mm^{4}): |
Ix |

Iy hollow ellipse-constant WT (mm^{4}): |
Iy |

Zx hollow ellipse-constant WT (mm^{3}): |
rx |

Zy hollow ellipse-constant WT (mm^{3}): |
ry |

27. HOLLOW SEMIELLIPSE WITH CONSTANT WALL THICKNESS |
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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. |

a/b hollow semiellipse-constant WT (mm): | a/b |

t_{max} hollow semiellipse-constant WT (mm): |
tmax |

D (must be > 0.2 and <=1): | D |

A hollow semiellipse-constant WT (mm^{2}): |
A |

y_{c2} hollow semiellipse-constant WT (mm): |
yc2 |

y_{c1} hollow semiellipse-constant WT (mm): |
yc1 |

x_{c} hollow semiellipse-constant WT (mm): |
xc |

IX hollow semiellipse-constant WT (mm^{4}): |
IX |

Ix hollow semiellipse-constant WT (mm^{4}): |
Ix |

Iy hollow semiellipse-constant WT (mm^{4}): |
one-half the value for Iy in case 26 |

y_{p} vertical distance from bottom to plastic neutral axis (mm): |
ypHSE |

Zx hollow semiellipse-constant WT (mm^{3}): |
Zx |

Zy hollow semiellipse-constant WT (mm^{3}): |
one-half the value for Zy in case 26 |

28. REGULAR POLYGON WITH n SIDES |
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for n=3 see case 9. For n=4, see cases 1 and 13. |

for n=5, Z1=Z2=0.8825 ro_{1}^{3}. For an axis perpendicular |

to axis 1, Z=0.8838 ro_{1}^{3}. The location of this axis is |

0.7007a from that side which is perpendicular to axis 1. |

alfa regular polygon with n sides (degrees): | alfa |

A regular polygon with n sides (mm^{2}): |
A |

ro_{1} regular polygon with n sides (mm): |
ro1 |

ro_{2} regular polygon with n sides (mm): |
ro2 |

y_{1} regular polygon with n sides (mm): |
y1 |

y_{2} regular polygon with n sides (mm): |
y2 |

I1 regular polygon with n sides (mm^{4}): |
I1 |

I2 regular polygon with n sides (mm^{4}): |
I2 |

r1 regular polygon with n sides (mm): | r1 |

r2 regular polygon with n sides (mm): | r2 |

Z regular polygon with n sides (mm^{3}): |
Zx |

Z1 regular polygon with 5 sides (mm^{3}): |
Z1 |

Z2 regular polygon with 5 sides (mm^{3}): |
Z2 |

location axis perp. to axis 1 (n=5)(mm): | LAPA1 |

29. HOLLOW REGULAR POLYGON WITH n SIDES |
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alfa hollow reg. polygon-n sides (degrees): | alfa |

A hollow regular polygon-n sides (mm^{2}): |
A |

ro_{1} hollow regular polygon-n sides (mm): |
ro1 |

ro_{2} hollow regular polygon-n sides (mm): |
ro2 |

y_{1} hollow regular polygon-n sides (mm): |
y1 |

y_{2} hollow regular polygon-n sides (mm): |
y2 |

I1 hollow regular polygon-n sides (mm^{4}): |
I1 |

I2 hollow regular polygon-n sides (mm^{4}): |
I2 |

r1 hollow regular polygon-n sides (mm): | r1 |

r2 hollow regular polygon-n sides (mm): | r2 |