2 LAYERS 3 LAYERS 4 LAYERS

𝑖 = 1, 2 𝑖 = 1, 2, 3 𝑖 = 1, 2, 3, 4

FLEXURAL STIFFNESS

𝐸𝐼𝑓 =1 + 𝛼 + 𝜋2𝛼𝛽

1 + 𝜋2𝛽𝐸𝐼𝑠

𝛼 =𝐷𝑓

𝐷2 𝛽 =

𝐷2

𝐷𝑄𝐿2 𝐷𝑓 = ∑ 𝐸𝐼𝑖

𝑖

= ∑ 𝐸𝑏𝑡𝑖

3

12𝑖

𝐷2 = 𝐸𝐼𝑠 𝐷𝑄 = 𝐴𝐺𝑖𝑛𝑡

𝐼𝑠 =𝑏𝑡𝑑2

2 𝐼𝑠 =

𝑏𝑡1𝑑2

2 𝐼𝑠 = 𝑏 (

𝑡1𝑑2

2+

𝑡2𝑎22

2)

𝐴 =𝑏𝑑2

𝑡𝑖𝑛𝑡 𝐴 =

𝑏𝑑2

2 𝑡𝑖𝑛𝑡 𝐴 =

𝑏𝑑

𝑡𝑖𝑛𝑡∙

𝑡1𝑑2 + 𝑡2𝑎22

3 𝑡1𝑑 + 𝑡2𝑎2

TORSIONAL STIFFNESS

𝐺𝐽𝑡 = 𝐺 (∑𝑏𝑡𝑖

3

3+ 𝐽𝑐𝑜𝑚𝑝

𝑖

)

𝐽𝑐𝑜𝑚𝑝 = 𝐽𝑠 (1 −2

𝜆𝑏𝑡𝑎𝑛ℎ

𝜆𝑏

2) 𝐽𝑐𝑜𝑚𝑝 = 𝐽𝑠 (1 −

𝛾1

𝜆1𝑏𝑡𝑎𝑛ℎ

𝜆1𝑏

2−

𝛾2

𝜆2𝑏𝑡𝑎𝑛ℎ

𝜆2𝑏

2)

𝐽𝑠 = 4 𝐼𝑠

𝜆 = √𝐺𝑖𝑛𝑡

𝐺∙

2

𝑡𝑖𝑛𝑡𝑡 𝜆 = √

𝐺𝑖𝑛𝑡

𝐺∙

1

𝑡𝑖𝑛𝑡𝑡1 𝜆1 = √

𝐺𝑖𝑛𝑡

𝐺∙

3 𝑡1 + 𝑡2 − 𝛿

2 𝑡𝑖𝑛𝑡𝑡1𝑡2

𝜆2 = √𝐺𝑖𝑛𝑡

𝐺∙

3 𝑡1 + 𝑡2 + 𝛿

2 𝑡𝑖𝑛𝑡𝑡1𝑡2

𝛿 = √9 𝑡12 − 2 𝑡1𝑡2 + 𝑡2

2

휀 =3 𝑡1

2𝑑2 + 4 𝑡1𝑡2(𝑎1𝑎2 − 𝑎12) + 𝑡2

2𝑎22

𝑡1𝑑2 + 𝑡2𝑎22

𝛾1 = 1 +휀

𝛿 𝛾2 = 1 −

휀

𝛿

5 LAYERS

𝑖 = 1, 2, 3, 4, 5

FLEXURAL STIFFNESS

𝐸𝐼𝑓 =1 + 𝛼 + 𝜋2𝛼𝛽

1 + 𝜋2𝛽𝐸𝐼𝑠

𝛼 =𝐷𝑓

𝐷2 𝛽 =

𝐷2

𝐷𝑄𝐿2 𝐷𝑓 = ∑ 𝐸𝐼𝑖

𝑖

= ∑ 𝐸𝑏𝑡𝑖

3

12𝑖

𝐷2 = 𝐸𝐼𝑠 𝐷𝑄 = 𝐴𝐺𝑖𝑛𝑡

𝐼𝑠 = 𝑏 (𝑡1𝑑2

2+ 2 𝑡2𝑎2

2)

𝐴 =𝑏𝑑

4 𝑡𝑖𝑛𝑡∙

𝑡1𝑑2 + 4 𝑡2𝑎22

𝑡1𝑑 + 𝑡2𝑎2

TORSIONAL STIFFNESS

𝐺𝐽𝑡 = 𝐺 (∑𝑏𝑡𝑖

3

3+ 𝐽𝑐𝑜𝑚𝑝

𝑖

)

𝐽𝑐𝑜𝑚𝑝 = 𝐽𝑠 (1 −𝛾1

𝜆1𝑏𝑡𝑎𝑛ℎ

𝜆1𝑏

2−

𝛾2

𝜆2𝑏𝑡𝑎𝑛ℎ

𝜆2𝑏

2)

𝐽𝑠 = 4 𝐼𝑠

𝜆1 = √𝐺𝑖𝑛𝑡

𝐺∙

2 𝑡1 + 𝑡2 − 𝛿

2 𝑡𝑖𝑛𝑡𝑡1𝑡2

𝜆2 = √𝐺𝑖𝑛𝑡

𝐺∙

2 𝑡1 + 𝑡2 + 𝛿

2 𝑡𝑖𝑛𝑡𝑡1𝑡2

𝛿 = √4 𝑡12 + 𝑡2

2

휀 =2 𝑡1

2𝑑2 + 4 𝑡1𝑡2(𝑎22 + 2 𝑎1𝑎2 − 𝑎1

2) + 4 𝑡22𝑎2

2

𝑡1𝑑2 + 4 𝑡2𝑎22

𝛾1 = 1 +휀

𝛿 𝛾2 = 1 −

휀

𝛿