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Torsional constant

Torsional constant for square and rectangular hollow sections

Torsional constant

$$ J_{RHS} = {t_{RHS} ^ 3 \cdot {h_{RHS} \over 3} + 2 \cdot k_{RHS} \cdot A_{h_{RHS}}} \; \; , {mm ^ 4} $$

Web thickness of single monorail beam

Determine minimum web thickness of single monorail beam as per AS 1418.18:2001 Section 5.12.3.2

Minimum web thickness

$$ T_{w_{AS1418}} = {\sqrt{ \left({ 240 \cdot {C_F \over B_F} + 60 }\right) \cdot {D \over \left({ 2 \cdot B_F }\right)} \cdot {N_W \over f_y }}} \; \; , {mm} $$

Mid-contour length

Length of mid-contour used in calculating torsion constant for hollow sections

$$ h_{RHS} = {2 \cdot \left({ \left({ b_{RHS} - t_{RHS} }\right) + \left({ d_{RHS} - t_{RHS} }\right) }\right) - \left({ R_{o_{RHS}} + R_{i_{RHS}} }\right) \cdot \left({ 4 - \pi }\right)} \; \; , {mm} $$

Integration constant

Integration constant used to calculate torsional constant for square or rectangular hollow section

Integration constant

$$ k_{RHS} = {{( 2 \cdot A_{h_{RHS}} \cdot t_{RHS} ) \over h_{RHS}}} \; \; $$

Mid-contour area

Area enclosed by mid-contour used in calculating properties of square or rectangular hollow section

Area

$$ A_{h_{RHS}} = {\left({ b_{RHS} - t_{RHS} }\right) \cdot \left({ d_{RHS} - t_{RHS} }\right) - \left({ {( R_{o_{RHS}} + R_{i_{RHS}} ) \over 2 }}\right) ^ 2 \cdot \left({ 4 - \pi }\right)} \; \; , {mm ^ 2} $$

Inner corner radius

Inner corner radius of square or rectangular hollow section

$ t_{RHS} $ > $ 3 $

$$ R_{i_{RHS}} = {1.5 \cdot t_{RHS}} \:, {mm} $$

$$ R_{i_{RHS}} = {1 \cdot t_{RHS}} \:, {mm} $$

Outer corner radius

Outer corner radius of square or rectangular hollow section

$ t_{RHS} $ > $ 3 $

$$ R_{o_{RHS}} = {2.5 \cdot t_{RHS}} \:, {mm} $$

$$ R_{o_{RHS}} = {2 \cdot t_{RHS}} \:, {mm} $$

Hollow section thickness

Thickness of square or rectangular hollow section

$$ t_{RHS} = {9} \; \; , {mm} $$

Width of hollow section

Width of hollow section

$$ b_{RHS} = {150} \; \; , {mm} $$

Depth of hollow section

Depth of square or rectangular hollow section

$$ d_{RHS} = {250} \; \; , {mm} $$

Poisson's ratio

Poisson's ratio

$$ \nu = {0.25} \; \; $$

Young's modulus of elasticity

Young's modulus

$$ E = {200000} \; \; , {MPa} $$

Shear modulus of elasticity

Shear modulus

$$ G = {80000} \; \; , {MPa} $$

Bending stress

Calculate normal stress in simply supported beam, with known section modulus, due to bending caused by 2 symmetrical point loads.

Bending stress

$$ \sigma_{bend_{2p}} = {10 ^ 3 \cdot {M_{B_2p} \over W_{y_{section}}}} \; \; , {MPa} $$

Bending stress

Calculate normal stress in a section, with known section modulus, due to bending caused by distributed load

Bending stress

$$ \sigma_{bend_{d_{max}}} = {10 ^ 3 \cdot {M_{B_{d_{max}}} \over W_{y_{section}}}} \; \; , {MPa} $$

Section modulus of circular section

Section modulus of circular solid section with known diameter

Section modulus

$$ W_{y_{circle}} = {{( \pi \cdot D_{circle_{mm}} ^ 3 ) \over 32}} \; \; , {mm ^ 3} $$

Bending stress

Calculate normal stress in a solid circular section, with known diameter, due to bending moment.

Bending stress

$$ \sigma_{bend_{circle_p}} = {10 ^ 3 \cdot {M_{B_p} \over W_{y_{circle}}}} \; \; , {MPa} $$

Equivalent stress

Calculate von Mises stress

von Mises stress

$$ \sigma_{vonMises} = {\sqrt{ \sigma_x ^ 2 - \sigma_x \cdot \sigma_y + \sigma_y ^ 2 + 3 \cdot \tau_{xy} ^ 2 }} \; \; , {MPa} $$

Reaction force

Reaction force A due to point load P with hinge support at B

Reaction in A

$$ R_{p_{A_1}} = {P \cdot {b \over \left({ a + b }\right)}} \; \; , {N} $$

Maximum bending deflection

Maximum deflection of simply supported beam subject to two equal loads at equal distance from supports.

Maximum deflection

$$ \Delta_{max_{2p}} = {{( P \cdot l ^ 2 \cdot a ) \over \left({ 24 \cdot E \cdot I }\right)} \cdot \left({ 3 - 4 \cdot \left({ {a \over l} }\right) ^ 2 }\right)} \; \; , {mm} $$