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Angle α_- Between Local Axis x (or y) and Principal Axis 1 (or 2) for Stresses on Negative Surface Side

α_{-} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, -}}{σ_{x, -} - σ_{y, -}}) \in (- 90 °, 90 °]

Angle α₊ Between Local Axis x (or y) and Principal Axis 1 (or 2) for Stresses on Positive Surface Side

α_{+} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, +}}{σ_{x, +} - σ_{y, +}}) \in (- 90 °, 90 °]

Angle α_m Between Local Axis x and Principal Axis 1 for Membrane Stresses

α_{m} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, m}}{σ_{x, m} - σ_{y, m}}) \in (- 90 °, 90 °]

Bending Moment m_x (Creating Stresses in Direction of Local x-Axis)

m_{x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{x} z d z

Bending Moment m_y (Creating Stresses in Direction of Local y-Axis)

m_{y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{y} z d z

Bending Normal Stress According to Reuleaux

σ_{b} = \frac{δ}{D} \cdot E

σ_{x} (x, z) = - κ (x) Ez

Chord prestress coefficient kₚ EN 1993-1-8

k_{p} = 1 + 0.3 \cdot n_{p} \cdot (1 - n_{p})

n_{p} = \frac{f_{p}}{f_{y}}

f_{p} = \frac{N_{p}}{A_{0}} - |M_{0}) W_{0}|

Contact Stress σ_z ("Soil Pressure") in Direction of Surface Axis z

σ_{z} = \frac{F_{z}}{A}

Critical Bending Moment

M_{crit} = \frac{π \cdot \sqrt{E_{0.05} \cdot I_{Z} \cdot G_{0.05} \cdot I_{tor}}}{l_{ef}}

Critical Bending Moment

M_{crit} = \frac{π \cdot \sqrt{E_{0.05} \cdot I_{Z} \cdot G_{0.05} \cdot I_{tor}}}{l_{ef}}

Critical Bending Moment

M_{crit} = η \cdot M_{d}

Critical Bending Moment

M_{crit} = \frac{(2 \cdot e \cdot c^{*} - a_{z}^{*} \cdot B^{*}) + \sqrt{(^{2 \cdot e \cdot c^{*} - a_{z}^{*} \cdot B^{*}) 2} + 4 \cdot 1.0 \cdot (B^{*} \cdot T^{*} - e^{*} \cdot {c^{*}}^{2})}}{2}

c^{*} = K_{y} \cdot \frac{L^{2}}{π^{2} \cdot n^{2}}

a_{z}^{*} = \frac{a_{z}}{n^{2}}

B^{*} = E_{0.05} \cdot I_{Z} \cdot \frac{π^{2} \cdot n^{2}}{L^{2}} + c^{*}

T^{*} = G_{0.05} \cdot I_{T} + c^{*} \cdot e^{2}

n = 1, 2, 3 . . .

Critical Buckling Design Value for Bending Member, FbE

F_{bE} = \frac{1.20 \cdot E_{\min}'}{{R_{B}}^{2}}

F_{bE} = \frac{1.20 \cdot 510, 000 psi}{{9.65}^{2}}

F_{bE} = 6, 577 psi

Critical Buckling Design Value for Compression Member in Major Axis, FcEx

F_{cEx} = F_{cE 1} = \frac{0.822 \cdot E_{\min}'}{[\frac{l_{e 1}}{d_{1}}) 2]}

F_{cEx} = F_{cE 1} = \frac{0.822 \cdot 510, 000 psi}{[\frac{36.0 in}{3.5 in}) 2]}

F_{cEx} = F_{cE 1} = 3, 963 psi

Critical Buckling Design Value for Compression Member in Minor Axis, FcEy

F_{cEy} = F_{cE 2} = \frac{0.822 \cdot E_{\min}'}{[\frac{l_{e 2}}{d_{2}}) 2]}

F_{cEy} = F_{cE 2} = \frac{0.822 \cdot 510, 000 psi}{[\frac{36.0 in}{1.5 in}) 2]}

F_{cEy} = F_{cE 2} = 728 psi

Critical Compressive Force

N_{cr} = \frac{π^{2} \cdot E I}{l_{cr}^{2}}

q_{crit} = \frac{M_{crit} \cdot 8}{L^{2}}

P_{e} = \frac{π ² \cdot E \cdot I}{L ²}

Critical Load Factor for Frame with Restrained Base

f = \frac{ρ + 1}{ρ + 4} \cdot 24 \cdot \frac{E I_{S}}{R \cdot l^{2}}

Critical Load Including Partial Safety Factor

{(\frac{π}{L_{cr, y}})}^{2} * E * I_{y} * \frac{1}{γ_{M}} = N_{cr}

\Leftrightarrow {(\frac{π}{3, 540.5 cm})}^{2} * 21, 000 kN / cm ² * 8, 356 {cm}^{4} * \frac{1}{1.1} = 125.6 kN

Design of Concrete Compressive Stress and Concrete Compression Strut, Method of Damage Equivalent Stress Range According to 6.8.7

E_{c d, m a x, e q u} + 0.43 \sqrt{1 - R_{e q u}} \leq 1

R_{e q u} = E_{c d, m i n, e q u} / E_{c d, m a x, e q u}

E_{c d, m i n, e q u} = σ_{c d, m i n, e q u} / f_{c d, f a t}

E_{c d, m a x, e q u} = σ_{c d, m a x, e q u} / f_{c d, f a t}

Design of Concrete Compressive Stress and Concrete Compression Struts, Simplified Analysis Method According to 6.8.7 (2)

\frac{σ_{c, m a x}}{f_{c d, f a t}} \leq 0.5 + 0.45 \frac{σ_{c, m i n}}{f_{c d, f a t}} \begin{array}{rcl} \leq & 0.9 & for f_{ck} \leq 50 N / mm ² \\ \leq & 0.8 & for f_{ck} \geq 50 N / mm ² \end{array}

Design Value of Shear Stress acc. to DIN EN 1992-1-1, 6.2.5 (1)

v_{Edi} = β \cdot \frac{V_{Ed}}{z \cdot b_{i}} = 1.0 \cdot \frac{800 \cdot 10^{3}}{1152 \cdot 400} = 1.736 N / {mm}^{2}

where

β = 1.0

z = 0.9 \cdot d = 0.9 \cdot 1280 = 1152 mm

Equivalent Stress According to Tresca

σ_{v} = \sqrt{(^{σ_{x} - σ_{y}) 2} + 4 {τ_{x y}}^{2}}

Equivalent Stress According to von Mises

σ_{v} = \sqrt{k_{σ, x} \cdot σ^{2} + k_{τ} \cdot τ^{2}}

Equivalent Stress from Basic Stresses According to von Mises

σ_{v} = \sqrt{σ_{x}^{2} + σ_{y}^{2} - σ_{x} σ_{y} + 3 (τ_{x y}^{2})}

Equivalent Stress σ_{v, Bach}

σ_{v, Bach} = max [|σ_{1} - ν (σ_{2} + σ_{3})); |σ_{2} - ν (σ_{3} + σ_{1})); |σ_{3} - ν (σ_{1} + σ_{2}))]|||

Equivalent Stress σ_{v, Rankine,m}

σ_{v, Rankine, m} = \frac{1}{2} |σ_{x, m} + σ_{y, m}) + \frac{1}{2} \sqrt{(^{σ_{x, m} - σ_{y, m}) 2} + 4 {τ_{xy, m}}^{2}}|

Equivalent Stress σ_v,Bach,- on Negative Surface Side

σ_{v, Bach, -} = \max (\frac{1 - ν}{2} |σ_{x, -} + σ_{y, -}) + \frac{1 + ν}{2} \sqrt{(^{σ_{x, -} - σ_{y, -}) 2} + 4 {τ_{xy, -}}^{2}}; ν |σ_{x, -} + σ_{y, -})||)