```python
# Geometric Properties of Hollow Rectangle
# E.Durham     5-Jul-2019
```




```python
from sympy import *
# from sympy import symbols
```


```python
# define symbols
b, d, t = symbols('b d t')
```


```python
A = 2*t*(d+b-2*t) # Area
I_x = (b*d**3 - (b-2*t) * (d-2*t)**3) / 12 # Second Moment of Inertia for Major Axis
I_y = (d*b**3 - (d-2*t) * (b-2*t)**3) / 12 # Second Moment of Inertia for Minor Axis
S_x = (b*d**3 - (b-2*t) * (d-2*t)**3) / (6*d) # Elastic Section Modulus for Major Axis
S_y = (d*b**3 - (d-2*t) * (b-2*t)**3) / (6*b) # Elastic Section Modulus for Minor Axis
r_x = sqrt((b*d**3 - (b-2*t) * (d-2*t)**3) / (24*t * (d+b-2*t))) # Radius of Gyration for Major Axis
r_y = sqrt((d*b**3 - (d-2*t) * (b-2*t)**3) / (24*t * (b+d-2*t))) # Radius of Gyration for Minor Axis
Z_x = b*t*(d-t) + 2*t * ((d/2)-t)**2 # Plastic Section Modulus for Major Axis
Z_y = d*t*(b-t) + 2*t * ((b/2)-t)**2 # Plastic Section Modulus for Minor Axis
J = (2*t*(d-t)**2 * (b-t)**2) / (d+b-2*t) # St. Venant's Torsional Constant
```


```python
# Specify Depth, d, Width, b, AND Wall Thickness, t in identical units
# Example: 
# if case values are: d = 3.00, b = 2.00 and t = 0.250
# then use case_values = [(d, 3.00), (b, 2.00), (t, 0.250)]
# Enter actual case values below per above
case_values = [(d, 3.00), (b, 2.00), (t, 0.250)]
```


```python
print('Geometric Properties:')
print('Given:')
print('Depth, ', case_values[0], 'unit')
print('Width, ', case_values[1], 'unit')
print('Wall, ',  case_values[2], 'unit')
print('A =', A.subs(case_values).evalf(3), 'unit**2')
print('I_x =', I_x.subs(case_values).evalf(3), 'unit**4')
print('S_x =', S_x.subs(case_values).evalf(3), 'unit**3')
print('Z_x =', Z_x.subs(case_values).evalf(3), 'unit**3')
print('r_x =', r_x.subs(case_values).evalf(3), 'unit')
print('I_y =', I_y.subs(case_values).evalf(3), 'unit**4')
print('S_y =', S_y.subs(case_values).evalf(3), 'unit**3')
print('Z_y =', Z_y.subs(case_values).evalf(3), 'unit**3')
print('r_y =', r_y.subs(case_values).evalf(3), 'unit')
print('J =', J.subs(case_values).evalf(3), 'unit**4')
```

    Geometric Properties:
    Given:
    Depth,  (d, 3.0) unit
    Width,  (b, 2.0) unit
    Wall,  (t, 0.25) unit
    A = 2.25 unit**2
    I_x = 2.55 unit**4
    S_x = 1.70 unit**3
    Z_x = 2.16 unit**3
    r_x = 1.06 unit
    I_y = 1.30 unit**4
    S_y = 1.30 unit**3
    Z_y = 1.59 unit**3
    r_y = 0.759 unit
    J = 2.57 unit**4



```python
# Display formulas in proper math formatting
init_printing()
print('Area , A ='); A

```


```python
print('Second Moment of Inertia, I ='); I_x
```


```python
print('Elastic Section Modulus, S ='); S_x
```


```python
print('Plastic Section Modulus, Z ='); Z_x
```


```python
print('Radius of Gyration, r ='); r_x
```


```python
print("St. Venant's Torsional Constant, J ="); J
```

## Test Values and Expected Results
### RT 3 x x 1/4 from Aluminum Design Manual, 2015 page V-39
d = 3
b = 2
t = 0.25
A = 2.25
I_x = 2.55
S_x = 1.70
Z_x = 2.16
r_x = 1.06
I_y = 1.30
S_y = 1.30
Z_y = 1.59
r_y = 0.759
J = 2.57

## Glossary of Torsional Terms [1]
### HSS Shear Constant
The shear constant, $C_{RT}$, is used for calculating the maximum shear stress due to an applied shear force.  
For hollow structural section, the maximum shear stress in the cross section is given by:  
$\tau_{max} = \frac{V Q}{2 t I}$  
where $V$ is the applied shear force, $Q$ is the statical moment of the portion of the section lying outside the neutral axis taken about the neutral axis, $I$ is the moment of inertia, and $t$ is the wall thickness.  
The shear constant is expressed as the ratio of the applied shear force to the maximum shear stress [3]:  
$C_{RT} = \frac{V}{\tau_{max}} = \frac{2 t I}{Q}$  
### HSS Torsional Constant
The torsional constant, $C$, is used for calculating the shear stress due to an applied torqu. It is expressed as the ratio of the applied torque, $T$, to the shear stress in the cross section, $\tau$:  
$C = \frac{T}{\tau}$  

### St. Venant Torsional Constant
The St. Venant torsional constant, $J$, measures the resistance of a structural member to *pure* or *uniform* torsion. It is used in calculating the buckling moment resistance of laterally unsupported beams and torsional-flexural buckling of compression members in accordance with CSA S16.  

For open cross section, the general formula is given by Galambos (1968):  
$J = \sum(\frac{b't^3}{3})$  
where $b'$ are the plate lengths between points of intersection on their axes, and $t$ are the plate thicknesses. Summation includes all component plates. It is noted that the tabulated values in the Handbook of Steel Construction are based on net plate lengths instead of lengths between intersection points, a mostly conservative approach.  

The expressions for $J$ given herein do not take into account the flange-to-web fillets. Formulas which account for this effect are given by El Darwish and Johnston (1965).

For thin-walled closed sections, the general formula is given by Salmon and Johnson (1980):  
$J = \frac{4 A_o^2}{\int_s ds/t}$  

where $A_o$ is the enclosed area by the walls, $t$ is the wall thickness, $ds$ is a length element along the perimeter. Integration is performed over the entire perimeter $S$.

### Warping Torsional Constant
The warping torsional constant, $C_w$, measures the resistance of a structural member to *nonuniform* or *warping* torsion. It is used in calculating the buckling moment resistance of laterally unsupported beams and torsional-flexural buckling of compression members in accordance with CSA Standard S16.

For open section, a general calculation method is given by Galambos (1968). For section in which all component plates meet at a single poitn, such as angles and T-sections, a calculation method is given by Bleich (1952). For hollow structural sections (HSS), warping deformations are small, and the warping torsional constant is generally taken as zero.  



### References for Torsional Constants
[1] CISC, 2002. Torsional Section Properties of Steel Shapes.  
Canadian Institute of Steel Construction, 2002

[2] Seaburg, P.A. and Carter, C.J. 1997. Torsional Analysis of Structural Steel Members.  
American Institute of Steel Construction, Chicago, Ill.

[3] Stelco. 1981. Hollow Structural Sections - Sizes and Section Properties, 6th Edition.  
Stelco Inc., Hamilton, Ont.

[4] CISC 2016. Handbook of Steel Construction, 11th Edition, page 7-84
www.cisc-icca.ca

### Revision History
0.1 - 2019-07-08 - E.Durham - added graphic image
0.0 - 2019-07-05 - E.Durham - Created initial notebook

9986924094bb5e5eb990849713f8120fa9d80c20
ipynb
Jupyter Notebook
Properties_of_Geometric_Sections/Hollow_Rectangle/Hollow_Rectangle.ipynb
Ewdy/Structural-Design
b4bf079c2153e7fcac9f09e8b8aafd8400e411c1
MIT
2019-07-19T11:43:26.000Z
2022-02-24T22:26:03.000Z
Properties_of_Geometric_Sections/Hollow_Rectangle/Hollow_Rectangle.ipynb
Ewdy/Structural-Design
b4bf079c2153e7fcac9f09e8b8aafd8400e411c1
MIT
2019-07-19T11:31:47.000Z
2019-07-26T12:11:39.000Z
Properties_of_Geometric_Sections/Hollow_Rectangle/Hollow_Rectangle.ipynb
Ewdy/Structural-Design
b4bf079c2153e7fcac9f09e8b8aafd8400e411c1
MIT
2021-12-03T21:07:44.000Z
2021-12-03T21:07:44.000Z
Qwen/Qwen-72B
1. YES
2. YES