Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - Specific

Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - Specific

Notes: In general, the plastic bending shape factor can be expressed using the following expressions: 𝑘𝑥 = Τ2 ∙ 𝑄𝑥 ∙ ത𝑦 𝐼𝑥 and 𝑘𝑦 = Τ2 ∙ 𝑄𝑦 ∙ ҧ𝑥 𝐼𝑦. Where the plastic bending shape factor is constant for a cross section is expressed as a constant value. The expression for the plastic bending shape factor is also adapted to give the correct centroid distance where appropriate. The Radius of Gyration is not given in this table as it can

be calculated for all shapes about either axis using the expression: 𝜌 = 𝐼/𝐴.

AA-SM-001-000 Section Properties

Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - Specific

Description Area Centroids First Moment of Area Second Moment of Area Plastic Bending Shape Factor

Square

𝐴 = 𝑎2

ത𝑥 =𝑎

2

ത𝑦 =𝑎

2

𝑄𝑥 =𝑎3

4

𝑄𝑦 =𝑎3

4

𝐼𝑥 =𝑎4

12

𝐼𝑦 =𝑎4

12

𝑘𝑥 = 1.50

𝑘𝑦 = 1.50

Rectangle

𝐴 = 𝑏 ∙ 𝑑

ത𝑥 =𝑏

2

ത𝑦 =𝑑

2

𝑄𝑥 =𝑏 ∙ 𝑑2

4

𝑄𝑦 =𝑑 ∙ 𝑏2

4

𝐼𝑥 =𝑏 ∙ 𝑑3

12

𝐼𝑦 =𝑑 ∙ 𝑏3

12

𝑘𝑥 = 1.50

𝑘𝑦 = 1.50

Hollow Rectangle

𝐴 = 𝑏 ∙ 𝑑 − 𝑏𝑖 ∙ 𝑑𝑖

ത𝑥 =𝑏

2

ത𝑦 =𝑑

2

𝑄𝑥 =𝑏 ∙ 𝑑2 − 𝑏𝑖 ∙ 𝑑𝑖

2

4

𝑄𝑦 =𝑑 ∙ 𝑏2 − 𝑑𝑖 ∙ 𝑏𝑖

2

4

𝐼𝑥 =𝑏 ∙ 𝑑3 − 𝑏𝑖 ∙ 𝑑𝑖

3

12

𝐼𝑦 =𝑑 ∙ 𝑏3 − 𝑑𝑖 ∙ 𝑏𝑖

3

12

𝑘𝑥 =3

2∙𝑏 ∙ 𝑑3 − 𝑏𝑖 ∙ 𝑑𝑖

2 ∙ 𝑑

𝑏 ∙ 𝑑3 − 𝑏𝑖 ∙ 𝑑𝑖3

𝑘𝑦 =3

2∙𝑑 ∙ 𝑏3 − 𝑑𝑖 ∙ 𝑏𝑖

2 ∙ 𝑏

𝑑 ∙ 𝑏3 − 𝑑𝑖 ∙ 𝑏3

Circle

𝐴 = 𝜋 ∙ 𝑅2

ത𝑥 = 𝑅

ത𝑦 = 𝑅

𝑄𝑥 =2 ∙ 𝑅 3

6

𝑄𝑦 =2 ∙ 𝑅 3

6

𝐼𝑥 =𝜋 ∙ 𝑅4

4

𝐼𝑦 =𝜋 ∙ 𝑅4

4

𝑘𝑥 = 1.698

𝑘𝑦 = 1.698

Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - SpecificDescription Area Centroids First Moment of Area Second Moment of Area Plastic Bending Shape Factor

Hollow Circle

𝐴 = 𝜋 ∙ 𝑅2 − 𝑟2

ത𝑥 = 𝑅

ത𝑦 = 𝑅

𝑄𝑥 =2 ∙ 𝑅 3 − 2 ∙ 𝑟 3

6

𝑄𝑦 =2 ∙ 𝑅 3 − 2 ∙ 𝑟 3

6

𝐼𝑥 =𝜋 ∙ 𝑅4 − 𝑟4

4

𝐼𝑦 =𝜋 ∙ 𝑅4 − 𝑟4

4

𝑘𝑥 =16 ∙ 2 ∙ 𝑟 ∙ 2 ∙ 𝑅 3 − 2 ∙ 𝑟 3

3 ∙ 𝜋 ∙ 2 ∙ 𝑅 4 − 2 ∙ 𝑟 4

𝑘𝑦 =16 ∙ 2 ∙ 𝑟 ∙ 2 ∙ 𝑅 3 − 2 ∙ 𝑟 3

3 ∙ 𝜋 ∙ 2 ∙ 𝑅 4 − 2 ∙ 𝑟 4

Semi-Circle

𝐴 =𝜋 ∙ 𝑅2

2

ത𝑥 = 𝑅

ത𝑦 =4 ∙ 𝑅

3 ∙ 𝜋

𝑄𝑥 =2 ∙ 0.532 ∙ 𝑅3

6

𝑄𝑦 =2 ∙ 0.532 ∙ 𝑅3

6

𝐼𝑥 =𝜋

8−

8

9 ∙ 𝜋∙ 𝑅4

𝐼𝑦 =𝜋 ∙ 𝑅4

8

𝑘𝑥 = 1.860

𝑘𝑦 = 1.698

Hollow Semi-Circle

𝐴 =𝜋 ∙ 𝑅2 − 𝑟2

2

ത𝑥 = 𝑅

ത𝑦 = 0.4244 ∙ 𝑅 +𝑟2

𝑅 + 𝑟

𝑄𝑥 =2 ∙ 0.532 ∙ 𝑅3 − 𝑟3

6

𝑄𝑦 = ത𝑦 ∙𝐴

2

𝐼𝑥 =𝜋

8∙ 𝑅4 − 𝑟4 − 𝜋 ∙ ത𝑦2 ∙

𝑅2 − 𝑟2

2

𝐼𝑦 =𝜋

8∙ 𝑅4 − 𝑟4

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ 𝑅 − ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Ellipse

𝐴 = 𝜋 ∙ 𝑎 ∙ 𝑏

ത𝑥 = 𝑎

ത𝑦 = 𝑏

𝑄𝑥 =4 ∙ 𝑏 ∙ 𝐴

6 ∙ 𝜋

𝑄𝑦 =4 ∙ 𝑎 ∙ 𝐴

6 ∙ 𝜋

𝐼𝑥 =𝜋 ∙ 𝑎 ∙ 𝑏3

4

𝐼𝑦 =𝜋 ∙ 𝑏 ∙ 𝑎3

4

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Hollow Ellipse

𝐴 = 𝜋 ∙ 𝐴 ∙ 𝐵 − 𝑎 ∙ 𝑏

ത𝑥 = 𝑎

ത𝑦 = 𝑏

𝐼𝑥 =𝜋 ∙ 𝑎 ∙ 𝑏3 − 𝑎𝑖 ∙ 𝑏𝑖

3

4

𝐼𝑦 =𝜋 ∙ 𝑏 ∙ 𝑎3 − 𝑏𝑖 ∙ 𝑎𝑖

3

4

𝑘𝑥 =1.698 ∙ 𝑎 ∙ 𝑏3 − 𝑏 ∙ 𝑎𝑖 ∙ 𝑏𝑖

2

𝑎 ∙ 𝑏3 − 𝑎𝑖 ∙ 𝑏𝑖3

𝑘𝑦 =1.698 ∙ 𝑏 ∙ 𝑎3 − 𝑎 ∙ 𝑏𝑖 ∙ 𝑎𝑖

2

𝑏 ∙ 𝑎3 − 𝑏𝑖 ∙ 𝑎𝑖3

Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - SpecificDescription Area Centroids First Moment of Area Second Moment of Area Plastic Bending Shape FactorIsosceles Triangle

𝐴 =𝑑 ∙ 𝑏

2

ത𝑥 = 0.5 ∙ 𝑏

ത𝑦 =2 ∙ 𝑑

3

𝑄𝑥 =𝑏 ∙ ത𝑦2

3−4 ∙ 𝑏 ∙ 𝑑 ∙ ത𝑦

27

𝑄𝑦 = ത𝑥 − ത𝑥 ∙𝑦

𝑑∙𝑑 ∙ 𝑏

4

𝐼𝑥 =𝑏 ∙ 𝑑3

36

𝐼𝑦 =𝑑 ∙ 𝑏3

48

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Equilateral Triangle

𝐴 = 0.433 ∙ 𝑎2

ത𝑥 = 0.5 ∙ 𝑎

ത𝑦 = 0.577 ∙ 𝑎

𝑄𝑥 =𝑎 ∙ ത𝑦2

3−4 ∙ 𝑎2 ∙ 0.866 ∙ ത𝑦

27

𝑄𝑦 = ത𝑥 − ത𝑥 ∙ത𝑦

0.866 ∙ 𝑎∙0.866 ∙ 𝑎2

4

𝐼𝑥 = 0.01804 ∙ 𝑎4

𝐼𝑦 = 0.01804 ∙ 𝑎4

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Tee Section

𝐴 = 𝑡 ∙ 𝑏 + 𝑡𝑤 ∙ 𝑑

ത𝑥 = 0.5 ∙ 𝑏

ത𝑦 =𝑏 ∙ 𝑡2 + 𝑡𝑤 ∙ 𝑑 ∙ 2 ∙ 𝑡 + 𝑑

2 ∙ 𝑡 ∙ 𝑏 + 𝑡𝑤 ∙ 𝑑

𝑄𝑥 =𝑑 + 𝑡 − ത𝑦

2∙ 𝑑 + 𝑡 − ത𝑦 ∙ 𝑡𝑤

𝑄𝑦 =𝑏2 ∙ 𝑡

8+𝑡𝑤

2 ∙ 𝑑

8

𝐼𝑥 =𝑏 ∙ 𝑑 + 𝑡 3

3−𝑑3 ∙ 𝑏 − 𝑡𝑤

3− 𝐴 ∙ 𝑑 + 𝑡 − ത𝑦 2

𝐼𝑦 =𝑏3 ∙ 𝑡

12+𝑡𝑤

3 ∙ 𝑑

12

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ 𝑑 + 𝑡 − ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Channel Section

𝐴 = 𝑡 ∙ 𝑏 + 2 ∙ 𝑡𝑤 ∙ 𝑑

ത𝑥 = 0.5 ∙ 𝑏

ത𝑦 =𝑏 ∙ 𝑡2 + 2 ∙ 𝑡𝑤 ∙ 𝑑 ∙ 2 ∙ 𝑡 + 𝑑

2 ∙ 𝑡 ∙ 𝑏 + 2 ∙ 𝑡𝑤 ∙ 𝑑

𝑄𝑥 =𝑡2 ∙ 𝑏

8+𝑡𝑤 ∙ 𝑑 ∙ 𝑡 + 𝑑 −

𝑡𝑤 ∙ 𝑑𝑏

2

𝑄𝑦 =𝑏2 ∙ 𝑡

8+𝑡𝑤 ∙ 𝑑 ∙ 𝑏 − 𝑡𝑤

8

𝐼𝑥 =𝑏 ∙ 𝑑 + 𝑡 3

3−𝑑3 ∙ 𝑏 − 2 ∙ 𝑡𝑤

3− 𝐴 ∙ 𝑑 + 𝑡 − ത𝑦 2

𝐼𝑦 =𝑏3 ∙ 𝑑 + 𝑡

12−

𝑏 − 2 ∙ 𝑡𝑤3 ∙ 𝑑

12

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ 𝑑 + 𝑡 − ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Abbott Aerospace – Analysis Method

XL-VIKINGDisplay Your Math in Excel

Taken from: Analysis and Design of Composite and Metallic Flight Vehicle Structures

Section Properties - SpecificDescription Area Centroids First Moment of Area Second Moment of Area Plastic Bending Shape Factor

Wide Flange Beam with Equal Flanges

𝐴 = 2 ∙ 𝑡 ∙ 𝑏 + 𝑡𝑤 ∙ 𝑑

ത𝑥 = 0.5 ∙ 𝑏

ത𝑦 =𝑑

2+ 𝑡

𝑄𝑥 =𝑡𝑤 ∙ 𝑑2

4+ 𝑏 ∙ 𝑡 ∙ 𝑑 + 𝑡

𝑄𝑦 =𝑏2 ∙ 𝑡

2+𝑡𝑤

2 ∙ 𝑑

4

𝐼𝑥 =𝑏 ∙ 𝑑 + 2 ∙ 𝑡 3

12−𝑑3 ∙ 𝑏 − 𝑡𝑤

12

𝐼𝑦 =𝑏3 ∙ 𝑡

6−𝑡𝑤

3 ∙ 𝑑

12

𝑘𝑥 =2 ∙ 𝑄𝑥 ∙ ത𝑦

𝐼𝑥

𝑘𝑦 =2 ∙ 𝑄𝑦 ∙ ത𝑥

𝐼𝑦

Equal Legged Angle

𝐴 = 𝑡 ∙ 2 ∙ 𝑎 − 𝑡

ത𝑥 = 0.7071 ∙ 𝑎

ത𝑦1 =0.7071 ∙ 𝑎2 + 𝑎 ∙ 𝑡 − 𝑡2

2 ∙ 𝑎 − 𝑡

ത𝑦2 =0.7071 ∙ 𝑎2

2 ∙ 𝑎 − 𝑡

𝐼𝑥 =𝑎4 − 𝑏4

12−0.5 ∙ 𝑡 ∙ 𝑎2 ∙ 𝑏2

𝑎 + 𝑏

𝐼𝑦 =𝑎4 − 𝑏4

12

Unequal Legged Angle

𝐴 = 𝑡 ∙ 𝑑 + 𝑏 − 𝑡

ത𝑥 =𝑏2 + 𝑑 ∙ 𝑡 − 𝑡2

2 ∙ 𝑏 + 𝑑 − 𝑡

ത𝑦 =𝑑2 + 𝑏 ∙ 𝑡 − 𝑡2

2 ∙ 𝑏 + 𝑑 − 𝑡

𝐼𝑥 =1

3∙ 𝑏 ∙ 𝑑3 − 𝑏 − 𝑡 ∙ 𝑑 − 𝑡 3 − 𝐴 ∙ 𝑑 − ത𝑦 2

𝐼𝑦 =1

3∙ 𝑑 ∙ 𝑏3 − 𝑑 − 𝑡 ∙ 𝑏 − 𝑡 3 − 𝐴 ∙ 𝑏 − ത𝑥 2