Critical Elastic Moment Solution

STEP 0: Pre-Calculation Summary
Formula Used
Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2))))
Mcr = ((Cb*pi)/L)*sqrt(((E*Iy*G*J)+(Iy*Cw*((pi*E)/(L)^2))))
This formula uses 1 Constants, 1 Functions, 8 Variables
Constants Used
pi - Archimedes' constant Value Taken As 3.14159265358979323846264338327950288
Functions Used
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., sqrt(Number)
Variables Used
Critical Elastic Moment - (Measured in Kilonewton Meter) - Critical Elastic Moment is similar to the Euler (flexural) buckling of a strut in that it defines a buckling load.
Moment Gradient Factor - Moment Gradient Factor is rate at which moment is changing with length of beam.
Unbraced Length of Member - (Measured in Centimeter) - Unbraced Length of Member is defined as the distance between adjacent Points.
Elastic Modulus of Steel - (Measured in Gigapascal) - Elastic Modulus of Steel is a measure of the stiffness of steel. It quantifies the ability of steel to resist deformation under stress.
Y Axis Moment of Inertia - (Measured in Meter⁴ per Meter) - Y Axis Moment of Inertia is defined as the moment of inertia of cross-section about YY.
Shear Modulus in Steel Structures - (Measured in Gigapascal) - Shear Modulus in Steel Structures is the slope of the linear elastic region of the shear stress–strain curve.
Torsional constant - The 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.
Warping Constant - The Warping Constant is often referred to as the warping moment of inertia. It is a quantity derived from a cross-section.
STEP 1: Convert Input(s) to Base Unit
Moment Gradient Factor: 1.96 --> No Conversion Required
Unbraced Length of Member: 12 Meter --> 1200 Centimeter (Check conversion here)
Elastic Modulus of Steel: 200 Gigapascal --> 200 Gigapascal No Conversion Required
Y Axis Moment of Inertia: 5000 Millimeter⁴ per Millimeter --> 5E-06 Meter⁴ per Meter (Check conversion here)
Shear Modulus in Steel Structures: 80 Gigapascal --> 80 Gigapascal No Conversion Required
Torsional constant: 21.9 --> No Conversion Required
Warping Constant: 0.2 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
Mcr = ((Cb*pi)/L)*sqrt(((E*Iy*G*J)+(Iy*Cw*((pi*E)/(L)^2)))) --> ((1.96*pi)/1200)*sqrt(((200*5E-06*80*21.9)+(5E-06*0.2*((pi*200)/(1200)^2))))
Evaluating ... ...
Mcr = 0.00679190728759447
STEP 3: Convert Result to Output's Unit
6.79190728759447 Newton Meter --> No Conversion Required
FINAL ANSWER
6.79190728759447 6.791907 Newton Meter <-- Critical Elastic Moment
(Calculation completed in 00.020 seconds)

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NSS College of Engineering (NSSCE), Palakkad
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13 Beams Calculators

Critical Elastic Moment
Go Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2))))
Limiting Laterally Unbraced Length for Inelastic Lateral Buckling
Go Limiting Length for Inelastic Buckling = ((Radius of gyration about minor axis*Beam Buckling Factor 1)/(Specified Minimum Yield Stress-Compressive Residual Stress in Flange))*sqrt(1+sqrt(1+(Beam Buckling Factor 2*Smaller Yield Stress^2)))
Specified Minimum Yield Stress for Web given Limiting Laterally Unbraced Length
Go Specified Minimum Yield Stress = ((Radius of gyration about minor axis*Beam Buckling Factor 1*sqrt(1+sqrt(1+(Beam Buckling Factor 2*Smaller Yield Stress^2))))/Limiting Length for Inelastic Buckling)+Compressive Residual Stress in Flange
Beam Buckling Factor 1
Go Beam Buckling Factor 1 = (pi/Section Modulus about Major Axis)*sqrt((Elastic Modulus of Steel*Shear Modulus in Steel Structures*Torsional constant*Cross Sectional Area in Steel Structures)/2)
Limiting Laterally Unbraced Length for Inelastic Lateral Buckling for Box Beams
Go Limiting Length for Inelastic Buckling = (2*Radius of gyration about minor axis*Elastic Modulus of Steel*sqrt(Torsional constant*Cross Sectional Area in Steel Structures))/Limiting buckling moment
Critical Elastic Moment for Box Sections and Solid Bars
Go Critical Elastic Moment = (57000*Moment Gradient Factor*sqrt(Torsional constant*Cross Sectional Area in Steel Structures))/(Unbraced Length of Member/Radius of gyration about minor axis)
Beam Buckling Factor 2
Go Beam Buckling Factor 2 = ((4*Warping Constant)/Y Axis Moment of Inertia)*((Section Modulus about Major Axis)/(Shear Modulus in Steel Structures*Torsional constant))^2
Limiting Laterally Unbraced Length for Full Plastic Bending Capacity for Solid Bar and Box Beams
Go Limiting Laterally Unbraced Length = (3750*(Radius of gyration about minor axis/Plastic Moment))/(sqrt(Torsional constant*Cross Sectional Area in Steel Structures))
Maximum Laterally Unbraced Length for Plastic Analysis
Go Laterally Unbraced Length for Plastic Analysis = Radius of gyration about minor axis*(3600+2200*(Smaller Moments of Unbraced Beam/Plastic Moment))/(Minimum Yield Stress of Compression Flange)
Maximum Laterally Unbraced Length for Plastic Analysis in Solid Bars and Box Beams
Go Laterally Unbraced Length for Plastic Analysis = (Radius of gyration about minor axis*(5000+3000*(Smaller Moments of Unbraced Beam/Plastic Moment)))/Yield Stress of Steel
Limiting Laterally Unbraced Length for Full Plastic Bending Capacity for I and Channel Sections
Go Limiting Laterally Unbraced Length = (300*Radius of gyration about minor axis)/sqrt(Flange Yield Stress)
Limiting Buckling Moment
Go Limiting buckling moment = Smaller Yield Stress*Section Modulus about Major Axis
Plastic Moment
Go Plastic Moment = Specified Minimum Yield Stress*Plastic modulus

Critical Elastic Moment Formula

Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2))))
Mcr = ((Cb*pi)/L)*sqrt(((E*Iy*G*J)+(Iy*Cw*((pi*E)/(L)^2))))

What is Buckling of a section?

Buckling is the event where a beam spontaneously bends from straight to curved under a compressive load. Also, it describes the relation between the force and the distance between the two ends of the beam, the force-strain curve.

What are the causes of Lateral Buckling & measures to prevent it?

The applied vertical load results in compression and tension in the flanges of the section. The compression flange tries to deflect laterally away from its original position, whereas the tension flange tries to keep the member straight.
The best way to prevent this type of buckling from occurring is to restrain the flange under compression, which prevents it from rotating along its axis. Some beams have restraints such as walls or braced elements periodically along their lengths, as well as on the ends.

How to Calculate Critical Elastic Moment?

Critical Elastic Moment calculator uses Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))) to calculate 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 Mcr symbol.

How to calculate Critical Elastic Moment using this online calculator? To use this online calculator for Critical Elastic Moment, enter Moment Gradient Factor (Cb), Unbraced Length of Member (L), Elastic Modulus of Steel (E), Y Axis Moment of Inertia (Iy), Shear Modulus in Steel Structures (G), Torsional constant (J) & Warping Constant (Cw) and hit the calculate button. Here is how the Critical Elastic Moment calculation can be explained with given input values -> 6.8E-6 = ((1.96*pi)/12)*sqrt(((200000000000*5E-06*80000000000*21.9)+(5E-06*0.2*((pi*200000000000)/(12)^2)))).

FAQ

What is 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 and is represented as Mcr = ((Cb*pi)/L)*sqrt(((E*Iy*G*J)+(Iy*Cw*((pi*E)/(L)^2)))) or Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))). Moment Gradient Factor is rate at which moment is changing with length of beam, Unbraced Length of Member is defined as the distance between adjacent Points, Elastic Modulus of Steel is a measure of the stiffness of steel. It quantifies the ability of steel to resist deformation under stress, Y Axis Moment of Inertia is defined as the moment of inertia of cross-section about YY, Shear Modulus in Steel Structures is the slope of the linear elastic region of the shear stress–strain curve, The 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 & The Warping Constant is often referred to as the warping moment of inertia. It is a quantity derived from a cross-section.
How to calculate 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 is calculated using Critical Elastic Moment = ((Moment Gradient Factor*pi)/Unbraced Length of Member)*sqrt(((Elastic Modulus of Steel*Y Axis Moment of Inertia*Shear Modulus in Steel Structures*Torsional constant)+(Y Axis Moment of Inertia*Warping Constant*((pi*Elastic Modulus of Steel)/(Unbraced Length of Member)^2)))). To calculate Critical Elastic Moment, you need Moment Gradient Factor (Cb), Unbraced Length of Member (L), Elastic Modulus of Steel (E), Y Axis Moment of Inertia (Iy), Shear Modulus in Steel Structures (G), Torsional constant (J) & Warping Constant (Cw). With our tool, you need to enter the respective value for Moment Gradient Factor, Unbraced Length of Member, Elastic Modulus of Steel, Y Axis Moment of Inertia, Shear Modulus in Steel Structures, Torsional constant & Warping Constant and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
How many ways are there to calculate Critical Elastic Moment?
In this formula, Critical Elastic Moment uses Moment Gradient Factor, Unbraced Length of Member, Elastic Modulus of Steel, Y Axis Moment of Inertia, Shear Modulus in Steel Structures, Torsional constant & Warping Constant. We can use 1 other way(s) to calculate the same, which is/are as follows -
  • Critical Elastic Moment = (57000*Moment Gradient Factor*sqrt(Torsional constant*Cross Sectional Area in Steel Structures))/(Unbraced Length of Member/Radius of gyration about minor axis)
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