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Critical Elastic Moment Formula

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Critical Elastic Moment represents the maximum moment a beam can carry in its elastic range before it becomes unstable due to lateral-torsional buckling. Check FAQs

M cr = (\frac{C b \cdot π}{L}) \cdot \sqrt{((E \cdot I y \cdot G \cdot J) + (I y \cdot C w \cdot (\frac{π \cdot E}{{(L)}^{2}})))}

M_cr - Critical Elastic Moment?C_b - Moment Gradient Factor?L - Unbraced Length of Member?E - Elastic Modulus of Steel?I_y - Y Axis Moment of Inertia?G - Shear Modulus?J - Torsional Constant?C_w - Warping Constant?π - Archimedes' constant?

Critical Elastic Moment Example

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Here is how the Critical Elastic Moment equation looks like with Values.

Here is how the Critical Elastic Moment equation looks like with Units.

Here is how the Critical Elastic Moment equation looks like.

6.7919 = (\frac{1.96 \cdot 3.1416}{12}) \cdot \sqrt{((200 \cdot 5000 \cdot 80 \cdot 21.9) + (5000 \cdot 0.2 \cdot (\frac{3.1416 \cdot 200}{{(12)}^{2}})))}

6.7919N*m = (\frac{1.96 \cdot 3.1416}{12m}) \cdot \sqrt{((200GPa \cdot 5000mm⁴/mm \cdot 80GPa \cdot 21.9) + (5000mm⁴/mm \cdot 0.2 \cdot (\frac{3.1416 \cdot 200GPa}{{(12m)}^{2}})))}

6.7919 = (\frac{1.96 \cdot 3.1416}{12}) \cdot \sqrt{((200 \cdot 5000 \cdot 80 \cdot 21.9) + (5000 \cdot 0.2 \cdot (\frac{3.1416 \cdot 200}{{(12)}^{2}})))}

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Critical Elastic Moment Solution

Follow our step by step solution on how to calculate Critical Elastic Moment?

FIRST Step Consider the formula

M cr = (\frac{C b \cdot π}{L}) \cdot \sqrt{((E \cdot I y \cdot G \cdot J) + (I y \cdot C w \cdot (\frac{π \cdot E}{{(L)}^{2}})))}

Next Step Substitute values of Variables

∴

M cr = (\frac{1.96 \cdot π}{12m}) \cdot \sqrt{((200GPa \cdot 5000mm⁴/mm \cdot 80GPa \cdot 21.9) + (5000mm⁴/mm \cdot 0.2 \cdot (\frac{π \cdot 200GPa}{{(12m)}^{2}})))}

Next Step Substitute values of Constants

∴

M cr = (\frac{1.96 \cdot 3.1416}{12m}) \cdot \sqrt{((200GPa \cdot 5000mm⁴/mm \cdot 80GPa \cdot 21.9) + (5000mm⁴/mm \cdot 0.2 \cdot (\frac{3.1416 \cdot 200GPa}{{(12m)}^{2}})))}

Next Step Convert Units

∴

M cr = (\frac{1.96 \cdot 3.1416}{1200cm}) \cdot \sqrt{((200GPa \cdot 5E-6m⁴/m \cdot 80GPa \cdot 21.9) + (5E-6m⁴/m \cdot 0.2 \cdot (\frac{3.1416 \cdot 200GPa}{{(1200cm)}^{2}})))}

Next Step Prepare to Evaluate

∴

M cr = (\frac{1.96 \cdot 3.1416}{1200}) \cdot \sqrt{((200 \cdot 5E-6 \cdot 80 \cdot 21.9) + (5E-6 \cdot 0.2 \cdot (\frac{3.1416 \cdot 200}{{(1200)}^{2}})))}

Next Step Evaluate

∴

M cr = 6.79190728759447N*m

LAST Step Rounding Answer

∴

M cr = 6.7919N*m

Critical Elastic Moment Formula Elements

Variables

Constants

Functions

Critical Elastic Moment

Critical Elastic Moment represents the maximum moment a beam can carry in its elastic range before it becomes unstable due to lateral-torsional buckling.

Symbol: M_cr

Measurement: Moment of ForceUnit: N*m

Note: Value should be greater than 0.

Formulas to find Critical Elastic Moment

Moment Gradient Factor

Moment Gradient Factor is rate at which moment is changing with length of beam.

Symbol: C_b

Measurement: NAUnit: Unitless

Note: Value should be greater than 0.

Formulas to find Moment Gradient Factor

Unbraced Length of Member

Unbraced Length of Member is the distance between two points along a structural member where lateral support is provided.

Symbol: L

Measurement: LengthUnit: m

Note: Value should be greater than 0.

Formulas to find Unbraced Length of Member

Elastic Modulus of Steel

Elastic Modulus of Steel is a measure of the stiffness of steel. It quantifies the ability of steel to resist deformation under stress.

Symbol: E

Measurement: PressureUnit: GPa

Note: Value should be greater than 0.

Formulas to find Elastic Modulus of Steel

Y Axis Moment of Inertia

Y Axis Moment of Inertia is a geometric property of a cross-section that measures its resistance to bending about the y-axis, also known as the second moment of area about the y-axis.

Symbol: I_y

Measurement: Moment of Inertia per Unit LengthUnit: mm⁴/mm

Note: Value should be greater than 0.

Formulas to find Y Axis Moment of Inertia

Shear Modulus

Shear Modulus is the slope of the linear elastic region of the shear stress–strain curve.

Symbol: G

Measurement: PressureUnit: GPa

Note: Value should be greater than 0.

Formulas to find Shear Modulus

Torsional Constant

Torsional Constant is a geometrical property of a bar's cross-section which is involved in the relationship between the angle of twist and applied torque along the axis of the bar.

Symbol: J

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Torsional Constant

Warping Constant

Warping Constant is a measure of the resistance of a thin-walled open cross-section to warping. Warping refers to the out-of-plane deformation of the cross-section that occurs during torsion.

Symbol: C_w

Measurement: NAUnit: Unitless

Note: Value can be positive or negative.

Formulas to find Warping Constant

Archimedes' constant

Archimedes' constant is a mathematical constant that represents the ratio of the circumference of a circle to its diameter.

Symbol: π

Value: 3.14159265358979323846264338327950288

sqrt

A square root function is a function that takes a non-negative number as an input and returns the square root of the given input number.

Syntax: sqrt(Number)

Other formulas in Beams category

Go Maximum Laterally Unbraced Length for Plastic Analysis

L pd = r y \cdot \frac{3600 + 2200 \cdot (\frac{M 1}{M p})}{F yc}

Go Maximum Laterally Unbraced Length for Plastic Analysis in Solid Bars and Box Beams

L pd = \frac{r y \cdot (5000 + 3000 \cdot (\frac{M 1}{M p}))}{F y}

How to Evaluate Critical Elastic Moment?

Critical Elastic Moment evaluator uses Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus*Torsional Constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))) to evaluate the Critical Elastic Moment, The Critical Elastic Moment formula is defined as the methods given in design codes for determining the slenderness of a section. The elastic critical moment (Mcr) is similar to the Euler (flexural) buckling of a strut in that it defines a buckling load. Critical Elastic Moment is denoted by M_cr symbol.

How to evaluate Critical Elastic Moment using this online evaluator? To use this online evaluator for Critical Elastic Moment, enter Moment Gradient Factor (C_b), Unbraced Length of Member (L), Elastic Modulus of Steel (E), Y Axis Moment of Inertia (I_y), Shear Modulus (G), Torsional Constant (J) & Warping Constant (C_w) and hit the calculate button.

FAQs on Critical Elastic Moment

What is the formula to find Critical Elastic Moment?

The formula of Critical Elastic Moment is expressed as Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus*Torsional Constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))). Here is an example- 6.791907 = ((1.96*pi)/12)*sqrt(((200000000000*5E-06*80000000000*21.9)+(5E-06*0.2*((pi*200000000000)/(12)^2)))).

How to calculate Critical Elastic Moment?

With Moment Gradient Factor (C_b), Unbraced Length of Member (L), Elastic Modulus of Steel (E), Y Axis Moment of Inertia (I_y), Shear Modulus (G), Torsional Constant (J) & Warping Constant (C_w) we can find Critical Elastic Moment using the formula - Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus*Torsional Constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))). This formula also uses Archimedes' constant and Square Root (sqrt) function(s).

Can the Critical Elastic Moment be negative?

No, the Critical Elastic Moment, measured in Moment of Force cannot be negative.

Which unit is used to measure Critical Elastic Moment?

Critical Elastic Moment is usually measured using the Newton Meter[N*m] for Moment of Force. Kilonewton Meter[N*m], Millinewton Meter[N*m], Micronewton Meter[N*m] are the few other units in which Critical Elastic Moment can be measured.

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Critical Elastic Moment Formula

Critical Elastic Moment Example

Critical Elastic Moment Solution

Critical Elastic Moment Formula Elements

Other formulas in Beams category

How to Evaluate Critical Elastic Moment?

FAQs on Critical Elastic Moment

Variable Name