Elliptical And Semi Elliptical Beam Buckling Load

Calculate elliptical beam cross section properties for solid and hollow beams.

For hollow ellipse sections the internal and external sections can be assumed elliptical (the wall thickness is not constant), or the wall thickness can be assumed constant (the mid wall is assumed elliptical but the internal and external surfaces are not elliptical). Refer to the figure for more details. Section properties which can not be calculated are set to zero.

Tool Input

• modptype : Material Type
  • αu : User Defined Thermal Expansion Coefficient
  • Eu : User Defined Elastic Modulus
  • Gu : User Defined Shear Modulus
  • ρpu : User Defined Density
• axstype : Cross Section Area Type
• mltype : Unit Mass And Unit Weight Type
• mmtype : Added Mass Type (Submerged Beams Only)
  • Cmu : User Defined Added Mass Coefficient
• axistype : Bending Axis Type
• a : External a Axis
• b : External b Axis
• ta : Wall Thickness At a Axis
• tb : Wall Thickness At b Axis
• L : Length
• ρc : Contents Fluid Density
• ρd : Displaced Fluid Density

Tool Output

• α : Thermal Expansion Coefficient
• ρb : Beam Density
• Ac : Contents Cross Section Area
• Ad : Displaced Cross Section Area
• Ax : Beam Cross Section Area
• Cm : Added Mass Coefficient
• E : Elastic Modulus
• EA : E x A
• EAα : E x A x alpha
• EI : E x I
• G : Shear Modulus
• I : Moment Of Inertia
• Ip : Polar Moment Of Inertia
• J : Mass Moment Of Inertia
• L/r : Slenderness Ratio
• M : Total Beam Mass (Including Contents)
• SF : Shape Factor
• SG : Specific Gravity (Submerged Beams Only)
• Ya : Distance From Outer Fibre To Centroid
• Zp : Plastic Modulus
• Zs : Section Modulus
• ai : Internal a Axis
• bi : Internal b Axis
• m : Mass Per Unit Length (Including Added Mass)
• ma : Added Unit Mass
• mb : Beam Unit Mass
• mc : Contents Fluid Unit Mass
• md : Displaced Fluid Unit Mass
• r : Radius Of Gyration
• w : Unit Weight (Including Contents And Buoyancy)

Calculate beam buckling load from beam effective length for general beams (user defined properties).

Beam end types include: free fixed (cantilever), guided fixed, pinned fixed, fixed fixed (built in or fixed), pinned pinned (simply supported), and guided pinned beam ends. The beam end conditions are of the form left end - right end (for example Pin-Fix is left end pinned and right end fixed). All distances are measured from the left end of the beam.

The buckling load can be calculated using either the Euler equation (suitable for long beams), the Johnson equation (suitable for short beams), or the buckling load equation can be determined from the transition length. The buckling load is positive. The axial load is negative in compression.

The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1).

Use the Result Plot option to plot the buckling load versus nominal length. Use the Result Table option to display the buckling load versus end type. Refer to the figures and help pages for more details.

Tool Input

• modptype : Material Property Type
  • αu : User Defined Thermal Expansion Coefficient
  • Eu : User Defined Elastic Modulus
• eitype : E x I Type
  • Iu : User Defined Section Modulus
  • ru : User Defined Radius Of Gyration
  • EIu : User Defined E x I
• eaatype : E x A x alpha Type
  • EAαu : User Defined E x A x alpha
• loadtype : Axial Load Type
  • Td : User Defined Operating Temperature
  • Tin : User Defined Installation Temperature
  • Fin : User Defined Preload
  • Fau : User Defined Axial Load
• fbtype : Buckling Load Type
  • Fbu : User Defined Buckling Load
• endtype : End Type
• leftype : Effective Length Type
  • feu : User Defined Effective Length Factor
• AX : Cross Section Area
• Lo : Nominal Length
• Sy : Yield Stress

Tool Output

• α : Thermal Expansion Coefficient
• E : Elastic Modulus
• EAα : E x A x alpha (E x AX x α)
• EI : E x I
• Fa : Axial Load
• Fa/Fb : Axial Load Over Buckling Load Ratio (< 1)
• Fb : Buckling Load
• I : Moment Of Inertia
• Le : Effective Length
• Le/r : Slenderness Ratio
• Lt : Transition Length (Short to Long Beam)
• r : Radius Of Gyration

Calculate beam buckling elastic modulus, shear modulus, bulk modulus, density, and thermal expansion coefficient.

The table values of Poisson ratio and bulk modulus are calculated from the elastic modulus and shear modulus. Use the Result Table option to display a table of properties versus material type.

Tool Input

• modptype : Material Type
  • Eu : User Defined Elastic Modulus
  • Gu : User Defined Shear Modulus
  • Ku : User Defined Bulk Modulus
  • νu : User Defined Poisson Ratio
  • ρu : User Defined Density
  • αu : User Defined Thermal Expansion Coefficient

Tool Output

• α : Thermal Expansion Coefficient
• ν : Poisson Ratio
• ρ : Density
• E : Elastic Modulus
• G : Shear Modulus
• K : Bulk Modulus

Calculate semi elliptical beam buckling load for solid and hollow beams.

For hollow ellipse sections the internal and external sections are assumed elliptical (the wall thickness is not constant). For hollow sections with constant wall thickness, the mid wall is assumed elliptical (the internal and external surfaces are not elliptical).

Beam end types include: free fixed (cantilever), guided fixed, pinned fixed, fixed fixed (built in or fixed), pinned pinned (simply supported), and guided pinned beam ends. The beam end conditions are of the form left end - right end (for example Pin-Fix is left end pinned and right end fixed). All distances are measured from the left end of the beam.

The buckling load can be calculated using either the Euler equation (suitable for long beams), the Johnson equation (suitable for short beams), or the buckling load equation can be determined from the transition length. The buckling load is positive. The axial load is negative in compression.

The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1).

Use the Result Plot option to plot the buckling load versus nominal length. Use the Result Table option to display the buckling load versus end type. Refer to the figures and help pages for more details.

Tool Input

• modptype : Material Type
  • αu : User Defined Thermal Expansion Coefficient
  • Eu : User Defined Elastic Modulus
• axstype : Cross Section Area Type
• axistype : Bending Axis Type
• loadtype : Axial Load Type
  • Td : User Defined Operating Temperature
  • Tin : User Defined Installation Temperature
  • Fin : User Defined Preload
  • Fau : User Defined Axial Load
• fbtype : Buckling Load Type
  • Fbu : User Defined Buckling Load
• endtype : End Type
• leftype : Effective Length Type
  • feu : User Defined Effective Length Factor
• a : External a Axis
• b : External b Axis
• ta : Wall Thickness At a Axis
• tb : Wall Thickness At b Axis
• Lo : Nominal Length
• Sy : Yield Stress

Tool Output

• α : Thermal Expansion Coefficient
• AX : Beam Cross Section Area
• E : Elastic Modulus
• EAα : E x A x alpha
• EI : E x I
• Fa : Axial Load
• Fa/Fb : Axial Load Over Buckling Load Ratio (< 1)
• Fb : Buckling Load
• I : Moment Of Inertia
• L/r : Slenderness Ratio
• Le : Effective Length
• Lt : Transition Length (Short to Long Beam)
• r : Radius Of Gyration