Formulas - Search term: Spring st...

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Bending Stiffness in x

D_{11} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{11, i}

Bending Stiffness in y

D_{22} = \sum_{i = 1}^{n} \frac{z_{m a x, i}^{3} - z_{m i n, i}^{3}}{3} d_{22, i}

Center of Stiffness in x

z_{0, x} = \frac{\int_{- t / 2}^{t / 2} E_{x}^{"} (z) z d z}{\int_{- t / 2}^{t / 2} E_{x}^{"} (z) d z} = \frac{I_{1}}{I_{0}}

Center of Stiffness in y-Direction

z_{0, x} = \frac{\int_{- t / 2}^{t / 2} E_{y}^{"} (z) z d z}{\int_{- t / 2}^{t / 2} E_{y}^{"} (z) d z} = \frac{I_{1}}{I_{0}}

EC5 - Decompose Stiffness in Position

ε (z) = \{\begin{matrix} ε_{x} \\ ε_{y} \\ γ_{x y} \end{matrix}\} = \{\begin{matrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix}\} + z \{\begin{matrix} \frac{\partial φ_{y}}{\partial x} \\ - \frac{\partial φ_{x}}{\partial y} \\ \frac{\partial φ_{y}}{\partial y} - \frac{\partial φ_{x}}{\partial x} \end{matrix}\}

Eccentric Oriented Shear Stiffness G12

G_{12}^{"} = (G_{x z, i} - G_{y z, i}) \sin (φ - ß_{i}) \cos (φ - ß_{i})

Equivalent Stiffness by Layer (in x)

χ_{i} = d_{11, i} \frac{{(z_{m a x, i} - z_{0, x})}^{2}}{2} + \sum_{k = i + 1}^{n} d_{11, k} \frac{{(z_{m a x, i} - z_{0, x})}^{2} - {(z_{m i n, i} - z_{0, x})}^{2}}{2}

Ideal Shear Stiffness of Stiffening Bracing

s_{id} = E_{D} \cdot A_{D} \cdot \sin {(α)}^{2} \cdot \cos (α)

Ideal Shear Stiffness of Stiffening Bracing

s_{id} = E_{D} \cdot A_{D}' \cdot \sin {(α)}^{2} \cdot \cos (α)

Ideal Spring Stiffness

C_{ideal} = \frac{2 \cdot P_{e}}{L}

Mass Matrix, Damping Matrix, and Stiffness Matrix

M \frac{d^{2} u}{d t^{2}} + C \frac{d u}{d t} + K (u) u = f

Normal Stiffness Eccentric D67

D_{67} = \sum_{i = 1}^{n} t_{i} d_{12, i}

Normal Stiffness Eccentric D68

D_{68} = \sum_{i = 1}^{n} t_{i} d_{13, i}

Normal Stiffness Eccentric D78

D_{78} = \sum_{i = 1}^{n} t_{i} d_{23, i}

Normal Stiffness x D66

D_{66} = \sum_{i = 1}^{n} t_{i} d_{11, i}

Normal Stiffness y D77

D_{77} = \sum_{i = 1}^{n} t_{i} d_{22, i}

Oriented Shear Stiffness G11

G_{11, i}^{"} = \cos^{2} (φ - ß_{i}) G_{x z, i} + \sin^{2} (φ - ß_{i}) G_{y z, i}

Oriented Shear Stiffness G22

G_{22, i}^{"} = \sin^{2} (φ - ß_{i}) G_{x z, i} + \cos^{2} (φ - ß_{i}) G_{y z, i}

Recalculated Shear Stiffness D44

D_{44} = \cos^{2} (φ) D_{44}^{"} + \sin^{2} (φ) D_{55}^{"}

Recalculated Shear Stiffness D45

D_{45} = \sin (φ) \cos (φ) (D_{44}^{"} - D_{55}^{"})

Recalculated Shear Stiffness D55

D_{55} = \sin^{2} (φ) D_{44}^{"} + \cos^{2} (φ) D_{55}^{"}

Relation Between Elastic Modulus, Stiffness Modulus, and Defomation Modulus of Soil

E_{s} = E \cdot \frac{1 - ν}{(1 + ν) \cdot (1 - 2 ν)}

E_{v} = E \cdot \frac{1}{(1 - ν^{2})}

\Rightarrow E_{v} = E_{s} \cdot \frac{(1 + ν) \cdot (1 - 2 ν)}{(1 - ν) \cdot (1 - ν^{2})}

Rotational Spring Stiffness for Ultimate Limit State

C_{φ, ULS} = K_{d} \cdot I_{p}

Rotational Stiffness of Stiffener with Different Distances to Next Stiffener

c_{θ} = \frac{E \cdot t^{3} \cdot (b_{1} + b_{2})}{4 \cdot b_{1} \cdot b_{2}}

Rotational Stiffness of Stiffener with Same Distances to Next Stiffener

c_{θ} = \frac{E \cdot t^{3}}{3 \cdot b}

Shear Panel Stiffness

S_{prov} = K_{y} \cdot \frac{L^{2}}{π^{2}}

Shear Stiffness in x-Direction

D_{44, c a l c}^{"} = \frac{1}{\int_{- t / 2}^{t / 2} \frac{G_{22}^{"} (z)}{G_{11}^{"} (z) G_{22}^{"} (z) - {G_{12}^{"}}^{2} (z)} {(\frac{\int_{z}^{t / 2} E_{x}^{"} (\bar{z}) (\bar{z} - z_{0, x}) d \bar{z}}{\int_{- t / 2}^{t / 2} E_{x}^{"} (\bar{z}) {(\bar{z} - z_{0, x})}^{2} d \bar{z}})}^{2} d z}

Shear Stiffness Matrix

[\begin{matrix} G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix}]

Spring Constant

C_{u, x} = \frac{E \cdot A}{l}

Stability Bracing of Beams (Required Stiffness)

β_{br} = \frac{1}{Φ} \cdot (\frac{10 \cdot M_{r} \cdot C_{d}}{L_{br} \cdot h_{o}}) (LRFD)

β_{br} = Ω \cdot (\frac{10 \cdot M_{r} \cdot C_{d}}{L_{br} \cdot h_{o}}) (ASD)