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I Section Beam Natural Vibration Frequency

Calculate the damped and undamped natural vibration frequency for I section beams (simply supported, fixed, and cantilever beams).

For beams with axial load the axis with minimum stiffness (I1 or I2) should be used unless the beam is constrained to deflect on an alternative axis (buckling normally occurs on the minimum stiffness axis). Use the general beam calculators for cases where vibration and buckling are not parallel. The buckling load can be calculated using either the Euler equation (suitable for long beams), or the Johnson equation (suitable for short beams). The buckling load is dependent on the end type, and is used for mode 1 vibration only.

Added mass should be included for submerged or wet beams. The added mass coefficient can be calculated in accordance with DNVGL RP F105. The submerged natural frequency is calculated for still water conditions, with no vortex shedding. For beams on a soft foundation such as soil, use the effective length factor to allow for movement at the beam ends. For defined beam ends such as structures, the effective length factor should be set to one. The axial load is calculated from temperature.

For longitudinal and torsional vibration, the natural frequency is independent of the cross section, and the general beam calculators can be used.

The mode factor k is dependent on the mode number, and the beam end type. The k factors have been taken from the Shock and Vibration handbook. The damping factor should be set to zero for undamped vibration or set greater than zero and less than or equal to one for damped vibration. The I section is assumed to be symmetrical along axis 1 and 2, with flanges of equal length and thickness.

Use the Result Table and Result Plot options to display tables and plots. Refer to the figures and help pages for more details about the tools.

References :

Shock And Vibration Handbook, Cyril M Harris, McGraw Hill
Roark's Formulas For Stress And Strain, Warren C Young, McGraw Hill

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CALCULATOR : Beam Longitudinal Natural Vibration Frequency (General Beam) [FREE]   ±

Calculate beam damped and undamped longitudinal natural vibration frequency from beam elastic modulus, density and length.

The longitudinal natural frequency is independent of the cross section profile. Select the end type, and vibration mode number (modes 1 to 8). The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes. The damping factor = 0 for undamped vibration, and 1 for critically damped vibration.

Use the Result Table option to display the natural frequency versus either mode number, or end type. Use the Result Plot option to display the natural frequency versus beam length and mode number, or beam length and end type.

Tool Input

  • modptype : Material Property Type
    • Eu : User Defined Elastic Modulus
    • ρpu : User Defined Density
  • endtype : End Type
  • MN : Vibration Mode Number
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • L : Length

Tool Output

  • ρ : Density
  • E : Elastic Modulus
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
CALCULATOR : Beam Torsional Natural Vibration Frequency (General Beam) [FREE]   ±

Calculate beam damped and undamped torsional natural vibration frequency from beam shear modulus, density and length.

The torsional natural frequency is independent of the cross section profile. Select the end type, and vibration mode number (modes 1 to 8). The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes. The damping factor = 0 for undamped vibration, and 1 for critically damped vibration.

Use the Result Table option to display the natural frequency versus either mode number, or end type. Use the Result Plot option to display the natural frequency versus beam length and mode number, or beam length and end type.

Tool Input

  • modptype : Material Property Type
    • Gu : User Defined Shear Modulus
    • ρpu : User Defined Density
  • endtype : End Type
  • MN : Vibration Mode Number
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • L : Length

Tool Output

  • ρ : Density
  • G : Shear Modulous
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
CALCULATOR : Beam Added Mass Coefficient (General Beam) [FREE]   ±

Calculate general beam added mass coefficient and added mass from gap height and characteristic length.

Added mass is included in the unit mass for submerged beams to account for the fluid which is displaced by the beam. The added mass coefficient is calculated in accordance with DNVGL RP F105. The equation is suitable for undamped vibration of circular pipes in still fluid. For circular pipes the diameter should be used as the characteristic length. For other profile shapes the width can be used as the characteristic length. The method may not be valid for other profile shapes (engineering judgement is required). Refer to the help pages for more details.

Reference : DNVGL RP F105 Free Spanning Pipelines (Download From DNVGL website)

Tool Input

  • cmtype : Added Mass Coefficient Type
    • Cmu : User Defined Added Mass Coefficient
  • mb : Beam Mass Per Unit Length
  • ρe : External Fluid Density
  • W : Beam Characteristic Length
  • G : Gap Height
  • AX : Beam Cross Section Area

Tool Output

  • Cm : Added Mass Coefficient
  • G/W : Gap Over Characteristic Length Ratio
  • m : Total Mass Per Unit Length
  • ma : Added Mass Per Unit Length
CALCULATOR : Beam Lateral Natural Vibration Frequency (General Beam) [FREE]   ±

Calculate beam damped and undamped lateral natural vibration frequency for general beams (user defined properties - with axial load or no axial load). Beam unit mass bending stiffness modulus and axial load are user defined.

Select the load type, end type, and vibration mode number (modes 1 to 5 for beams with no axial load, or mode 1 for beams with axial load. The end conditions are: pinned ends (simply supported beams), fixed ends, free fixed ends (cantilever beams), pinned fixed ends, and for beams with no load, also pinned free ends, and free ends (unsupported beams). For beams with axial load the natural frequency equals zero for compressive axial loads greater than or equal to the buckling load.

The buckling load can be calculated using either the Euler equation (suitable for long beams), or the Johnson equation (suitable for short beams). Buckling normally occurs on the axis with lowest stiffness modulus. The buckling stiffness modulus and the vibration stiffness modulus can be defined independently for cases where vibration is not parallel to buckling.

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). The damping factor = 0 for undamped vibration, and 1 for critically damped vibration. The natural frequency equals zero for critical damping.

Use the Result Table option to display the natural frequency versus either mode number, or end type. Use the Result Plot option to display the natural frequency versus beam length and mode number, beam length and end type, or axial load and end type. The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes. Refer to the figures and help pages for more details.

Tool Input

  • eitype : Bending Modulus Type
    • EIvu : User Defined Vibration Bending Modulus (E x I)
    • EIbu : User Defined Buckling Bending Modulus (E x I)
  • loadtype : Axial Load Type
    • Fau : User Defined Axial Load
  • fbtype : Buckling Load Type
  • endtype : End Type
  • MN : Mode Number
  • leftype : Effective Length Type
    • feu : User Defined Effective Length Factor
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • AX : Cross Section Area
  • m : Unit Mass
  • Lo : Nominal Length
  • SY : Yield Stress

Tool Output

  • EIb : Buckling Bending Modulus (E x I)
  • EIv : Vibration Bending Modulus (E x I)
  • Fa : Axial Load
  • Fa/Fb : Axial Load Over Buckling Load Ratio (> -1)
  • Fb : Buckling Load
  • Le : Effective Length
  • Lt : Transition Length (Short to Long Beam)
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
CALCULATOR : Beam Vibration Yield Stress [FREE]   ±

Calculate beam yield stress (SMYS) and tensile stress (SMTS).

Select one of the API, ASME or DNV stress table options. Use the Result Table option to display the stress values for the selected stress table.

Tool Input

  • syutype : Stress Table Type
  • mattype : Material Type
    • SMYSu : User Defined Specified Minimum Yield Stress
    • SMTSu : User Defined Specified Minimum Tensile Stress

Tool Output

  • SMTS : Specified Minimum Tensile Stress
  • SMTS/SMYS : Tensile Stress Over Yield Stress Ratio
  • SMYS : Specified Minimum Yield Stress
  • SMYS/SMTS : Yield Stress Over Tensile Stress Ratio
CALCULATOR : Beam Vibration Material Property [FREE]   ±

Calculate beam 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
CALCULATOR : Beam Lateral Natural Vibration Frequency (I Section Beam) [PLUS]   ±

Calculate I section beam lateral natural vibration frequency for straight sided I beams, and tapered I beams.

The lateral natural vibration frequency can be calculated either with axial load (mode 1 only), or with no axial load (modes 1 to 5). For compressive axial loads, the natural frequency tends to zero as the axial load tends to the buckling load. The buckling load can be calculated using either the Euler equation (suitable for long beams), or the Johnson equation (suitable for short beams). For tension loads, the natural frequency increases with increasing tension. The axial load can either be calculated from delta temperature, or user defined. Added mass can be included in the unit mass for submerged beams to account for the fluid which is displaced by the beam. The submerged natural frequency is calculated for still water conditions.

Use the Result Table option to table either the natural frequency versus mode number, or the natural frequency versus end type. Use the Result Plot option to plot the natural frequency versus either beam length and mode number, beam length and end type, or axial load and end type.

The I section is assumed to be symmetrical along axis 1 and 2. The webs or legs are assumed to be equal length and thickness. Tapered I beams have tapered webs. 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
    • ρpu : User Defined Density
  • mltype : Unit Mass And Unit Weight Type
  • mmtype : Added Mass Type (Submerged Beams Only)
    • Cmu : User Defined Added Mass Coefficient
  • axstype : Cross Section Area Type
  • axistype : Bending Axis Type
  • leftype : Effective Length Type
    • feu : User Defined Effective Length Factor
  • 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
  • endtype : Beam End Type
  • MN : Vibration Mode Number
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • d : Beam Outside Height
  • b : Beam Outside Width
  • tw : Web Wall Thickness
  • tf : Flange Wall Thickness
  • tc : Flange Crotch Wall Thickness
  • Lo : Nominal Length
  • SY : Yield Stress
  • ρd : Displaced Fluid Density

Tool Output

  • α : Thermal Expansion Coefficient
  • ρb : Beam Density
  • AX : Cross Section Area
  • Ad : Displaced Cross Section Area
  • Cm : Added Mass Coefficient
  • 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
  • Le : Effective Length
  • Le/r : Slenderness Ratio
  • Lt : Transition Length (Short to Long Beam)
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
  • m : Mass Per Unit Length (Including Contents And Added Mass)
  • ma : Added Unit Mass
  • mb : Beam Unit Mass
  • md : Displaced Fluid Unit Mass

CALCULATOR : Beam Cross Section Properties (I Section Beam) [FREE]   ±

Calculate I section beam cross section properties for straight sided I beams, and tapered I beams.

The I section is assumed to be symmetrical along axis 1 and 2. The webs or legs are assumed to be equal length and thickness. Tapered I beams have tapered webs. Refer to the figure for more details.

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
  • mltype : Unit Mass And Unit Weight Type
  • mmtype : Added Mass Type (Submerged Beams Only)
    • Cmu : User Defined Added Mass Coefficient
  • axstype : Cross Section Area Type
  • axistype : Bending Axis Type
  • d : Beam Outside Height
  • b : Beam Outside Width
  • tw : Web Wall Thickness
  • tf : Flange Wall Thickness
  • tc : Flange Crotch Wall Thickness
  • L : Length
  • ρd : Displaced Fluid Density

Tool Output

  • α : Thermal Expansion Coefficient
  • ρb : Beam Density
  • 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 (I / Ya)
  • m : Mass Per Unit Length (Including Contents And Added Mass)
  • ma : Added Unit Mass
  • mb : Beam Unit Mass
  • md : Displaced Fluid Unit Mass
  • r : Radius Of Gyration
  • w : Unit Weight (Including Contents And Buoyancy)
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