Inertial Force Beam

Calculates the residual for the inertial force/moment and the contribution of mass dependent Rayleigh damping and HHT time integration scheme.

Description

This class computes the $var element = document.getElementById("moose-equation-c57bb68d-364a-494f-a59e-2ea88c76f660");katex.render("i^{th}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ component of the inertial force/torque and mass/inertia proportional Rayleigh damping for the beam element due to mass/rotational inertia. Please look at C0TimoshenkoBeam for details.

Input Parameters

IyVariable containing second moment of area about y axis
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing second moment of area about y axis
IzVariable containing second moment of area about z axis
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing second moment of area about z axis
areaVariable containing cross-section area
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing cross-section area
componentAn integer corresponding to the direction the variable this kernel acts in. (0 for disp_x, 1 for disp_y, 2 for disp_z, 3 for rot_x, 4 for rot_y and 5 for rot_z)
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction the variable this kernel acts in. (0 for disp_x, 1 for disp_y, 2 for disp_z, 3 for rot_x, 4 for rot_y and 5 for rot_z)
displacementsThe displacement variables appropriate for the simulation geometry and coordinate system
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacement variables appropriate for the simulation geometry and coordinate system
rotationsThe rotational variables appropriate for the simulation geometry and coordinate system
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The rotational variables appropriate for the simulation geometry and coordinate system
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

AyVariable containing first moment of area about y axis
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing first moment of area about y axis
AzVariable containing first moment of area about z axis
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing first moment of area about z axis
IxVariable containing second moment of area about x axis. Defaults to Iy+Iz
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Variable containing second moment of area about x axis. Defaults to Iy+Iz
accelerationsTranslational acceleration variables
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Translational acceleration variables
alpha0Alpha parameter for mass dependent numerical damping induced by HHT time integration scheme
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Alpha parameter for mass dependent numerical damping induced by HHT time integration scheme
betabeta parameter for Newmark Time integration
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:beta parameter for Newmark Time integration
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
densitydensityName of Material Property or a constant real number defining the density of the beam.
Default:density
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of Material Property or a constant real number defining the density of the beam.
eta0Name of material property or a constant real number defining the eta parameter for the Rayleigh damping.
Default:0
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of material property or a constant real number defining the eta parameter for the Rayleigh damping.
gammagamma parameter for Newmark Time integration
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:gamma parameter for Newmark Time integration
matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
rotational_accelerationsRotational acceleration variables
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Rotational acceleration variables
rotational_velocitiesRotational velocity variables
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Rotational velocity variables
velocitiesTranslational velocity variables
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Translational velocity variables

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystem timeThe tag for the matrices this Kernel should fill
Default:system time
C++ Type:MultiMooseEnum
Options:nontime, system, time
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagstimeThe tag for the vectors this Kernel should fill
Default:time
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Contribution To Tagged Field Data Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshTrueWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:True
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(modules/solid_mechanics/test/tests/beam/dynamic/dyn_timoshenko_small.i)
(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht_ti.i)
(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht.i)
(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small.i)

(modules/solid_mechanics/test/tests/beam/dynamic/dyn_timoshenko_small.i)

# Test for small strain Timoshenko beam vibration in y direction

# An impulse load is applied at the end of a cantilever beam of length 4m.
# The properties of the cantilever beam are as follows:
# Young's modulus (E) = 2e4
# Shear modulus (G) = 1e4
# Shear coefficient (k) = 1.0
# Cross-section area (A) = 1.0
# Iy = 1.0 = Iz
# Length (L)= 4 m
# density (rho) = 1.0

# For this beam, the dimensionless parameter alpha = kAGL^2/EI = 8
# Therefore, the beam behaves like a Timoshenko beam.

# The FEM solution for this beam with 100 elements give first natural period of 0.2731s with a time step of 0.005.
# The acceleration, velocity and displacement time histories obtained from MOOSE matches with those obtained from ABAQUS.

# Values from the first few time steps are as follows:
# time    disp_y                vel_y                 accel_y
# 0.0     0.0                   0.0                   0.0
# 0.005   2.5473249455812e-05   0.010189299782325     4.0757199129299
# 0.01    5.3012872677486e-05   0.00082654950634483  -7.8208200233219
# 0.015   5.8611622914354e-05   0.0014129505884026    8.055380456145
# 0.02    6.766113649781e-05    0.0022068548449798   -7.7378187535141
# 0.025   7.8981810558437e-05   0.0023214147792709    7.7836427272305

# Note that the theoretical first frequency of the beam using Euler-Bernoulli theory is:
# f1 = 1/(2 pi) * (3.5156/L^2) * sqrt(EI/rho) = 4.9455

# This implies that the corresponding time period of this beam (under Euler-Bernoulli assumption) is 0.2022s.

# This shows that Euler-Bernoulli beam theory under-predicts the time period of a thick beam. In other words, the Euler-Bernoulli beam theory predicts a more compliant beam than reality for a thick beam.

[Mesh]
  type = GeneratedMesh
  xmin = 0
  xmax = 4.0
  nx = 100
  dim = 1
  displacements = 'disp_x disp_y disp_z'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_z]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./accel_x]
    type = NewmarkAccelAux
    variable = accel_x
    displacement = disp_x
    velocity = vel_x
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_x]
    type = NewmarkVelAux
    variable = vel_x
    acceleration = accel_x
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./accel_y]
    type = NewmarkAccelAux
    variable = accel_y
    displacement = disp_y
    velocity = vel_y
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_y]
    type = NewmarkVelAux
    variable = vel_y
    acceleration = accel_y
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./accel_z]
    type = NewmarkAccelAux
    variable = accel_z
    displacement = disp_z
    velocity = vel_z
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_z]
    type = NewmarkVelAux
    variable = vel_z
    acceleration = accel_z
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_x]
    type = NewmarkAccelAux
    variable = rot_accel_x
    displacement = rot_x
    velocity = rot_vel_x
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_x]
    type = NewmarkVelAux
    variable = rot_vel_x
    acceleration = rot_accel_x
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_y]
    type = NewmarkAccelAux
    variable = rot_accel_y
    displacement = rot_y
    velocity = rot_vel_y
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_y]
    type = NewmarkVelAux
    variable = rot_vel_y
    acceleration = rot_accel_y
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_z]
    type = NewmarkAccelAux
    variable = rot_accel_z
    displacement = rot_z
    velocity = rot_vel_z
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_z]
    type = NewmarkVelAux
    variable = rot_vel_z
    acceleration = rot_accel_z
    gamma = 0.5
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./fixy1]
    type = DirichletBC
    variable = disp_y
    boundary = left
    value = 0.0
  [../]
  [./fixz1]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value = 0.0
  [../]
  [./fixr1]
    type = DirichletBC
    variable = rot_x
    boundary = left
    value = 0.0
  [../]
  [./fixr2]
    type = DirichletBC
    variable = rot_y
    boundary = left
    value = 0.0
  [../]
  [./fixr3]
    type = DirichletBC
    variable = rot_z
    boundary = left
    value = 0.0
  [../]
[]

[NodalKernels]
  [./force_y2]
    type = UserForcingFunctorNodalKernel
    variable = disp_y
    boundary = right
    functor = force
  [../]
[]

[Functions]
  [./force]
    type = PiecewiseLinear
    x = '0.0 0.005 0.01 1.0'
    y = '0.0 1.0  0.0  0.0'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  line_search = 'none'
  nl_rel_tol = 1e-11
  nl_abs_tol = 1e-11
  start_time = 0.0
  dt = 0.005
  end_time = 0.5
  timestep_tolerance = 1e-6
[]

[Kernels]
  [./solid_disp_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 0
    variable = disp_x
  [../]
  [./solid_disp_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 1
    variable = disp_y
  [../]
  [./solid_disp_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 2
    variable = disp_z
  [../]
  [./solid_rot_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 3
    variable = rot_x
  [../]
  [./solid_rot_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 4
    variable = rot_y
  [../]
  [./solid_rot_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 5
    variable = rot_z
  [../]
  [./inertial_force_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 0
    variable = disp_x
  [../]
  [./inertial_force_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 1
    variable = disp_y
  [../]
  [./inertial_force_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 2
    variable = disp_z
  [../]
  [./inertial_force_rot_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 3
    variable = rot_x
  [../]
  [./inertial_force_rot_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 4
    variable = rot_y
  [../]
  [./inertial_force_rot_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 1.0
    Iy = 1.0
    Iz = 1.0
    Ay = 0.0
    Az = 0.0
    component = 5
    variable = rot_z
  [../]
[]

[Materials]
  [./elasticity]
    type = ComputeElasticityBeam
    youngs_modulus = 2e4
    poissons_ratio = 0.0
    shear_coefficient = 1.0
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalBeamStrain
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    area = 1.0
    Ay = 0.0
    Az = 0.0
    Iy = 1.0
    Iz = 1.0
    y_orientation = '0.0 1.0 0.0'
  [../]
  [./stress]
    type = ComputeBeamResultants
    block = 0
  [../]
  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1.0'
  [../]
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_x
  [../]
  [./disp_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_y
  [../]
  [./vel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = vel_y
  [../]
  [./accel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = accel_y
  [../]
[]

[Outputs]
  exodus = true
  csv = true
  perf_graph = true
[]

(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht_ti.i)

# Test for damped small strain euler beam vibration in y direction

# An impulse load is applied at the end of a cantilever beam of length 4m.
# The properties of the cantilever beam are as follows:
# Young's modulus (E) = 1e4
# Shear modulus (G) = 4e7
# Shear coefficient (k) = 1.0
# Cross-section area (A) = 0.01
# Iy = 1e-4 = Iz
# Length (L)= 4 m
# density (rho) = 1.0
# mass proportional rayleigh damping(eta) = 0.1
# stiffness proportional rayleigh damping(eta) = 0.1
# HHT time integration parameter (alpha) = -0.3
# Corresponding Newmark beta time integration parameters beta = 0.4225 and gamma = 0.8

# For this beam, the dimensionless parameter alpha = kAGL^2/EI = 6.4e6
# Therefore, the behaves like a Euler-Bernoulli beam.

# The displacement time history from this analysis matches with that obtained from Abaqus.

# Values from the first few time steps are as follows:
# time  disp_y                vel_y                accel_y
# 0.0   0.0                   0.0                  0.0
# 0.2   0.019898364318588     0.18838688112273     1.1774180070171
# 0.4   0.045577003505278     0.087329917525455   -0.92596052423724
# 0.6   0.063767907208218     0.084330765885995    0.21274543331268
# 0.8   0.073602908614573     0.020029576220975   -0.45506879373455
# 1.0   0.06841704414745     -0.071840076837194   -0.46041813317992

[Mesh]
  type = GeneratedMesh
  nx = 10
  dim = 1
  xmin = 0.0
  xmax = 4.0
  displacements = 'disp_x disp_y disp_z'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_z]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./accel_x] # These auxkernels are only to check output
    type = TestNewmarkTI
    displacement = disp_x
    variable = accel_x
    first = false
  [../]
  [./accel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = accel_y
    first = false
  [../]
  [./accel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = accel_z
    first = false
  [../]
  [./vel_x]
    type = TestNewmarkTI
    displacement = disp_x
    variable = vel_x
  [../]
  [./vel_y]
    type = TestNewmarkTI
    displacement = disp_y
    variable = vel_y
  [../]
  [./vel_z]
    type = TestNewmarkTI
    displacement = disp_z
    variable = vel_z
  [../]
  [./rot_accel_x]
    type = TestNewmarkTI
    displacement = rot_x
    variable = rot_accel_x
    first = false
  [../]
  [./rot_accel_y]
    type = TestNewmarkTI
    displacement = rot_y
    variable = rot_accel_y
    first = false
  [../]
  [./rot_accel_z]
    type = TestNewmarkTI
    displacement = rot_z
    variable = rot_accel_z
    first = false
  [../]
  [./rot_vel_x]
    type = TestNewmarkTI
    displacement = rot_x
    variable = rot_vel_x
  [../]
  [./rot_vel_y]
    type = TestNewmarkTI
    displacement = rot_y
    variable = rot_vel_y
  [../]
  [./rot_vel_z]
    type = TestNewmarkTI
    displacement = rot_z
    variable = rot_vel_z
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./fixy1]
    type = DirichletBC
    variable = disp_y
    boundary = left
    value = 0.0
  [../]
  [./fixz1]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value = 0.0
  [../]
  [./fixr1]
    type = DirichletBC
    variable = rot_x
    boundary = left
    value = 0.0
  [../]
  [./fixr2]
    type = DirichletBC
    variable = rot_y
    boundary = left
    value = 0.0
  [../]
  [./fixr3]
    type = DirichletBC
    variable = rot_z
    boundary = left
    value = 0.0
  [../]
[]

[NodalKernels]
  [./force_y2]
    type = UserForcingFunctorNodalKernel
    variable = disp_y
    boundary = right
    functor = force
  [../]
[]

[Functions]
  [./force]
    type = PiecewiseLinear
    x = '0.0 0.2 0.4 10.0'
    y = '0.0 0.01  0.0  0.0'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  line_search = 'none'
  l_tol = 1e-11
  nl_max_its = 15
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-10
  start_time = 0.0
  dt = 0.2
  end_time = 5.0
  timestep_tolerance = 1e-6

  # Time integrator
  [./TimeIntegrator]
    type = NewmarkBeta
    beta = 0.4225
    gamma = 0.8
  [../]
[]

[Kernels]
  [./solid_disp_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 0
    variable = disp_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 1
    variable = disp_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 2
    variable = disp_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 3
    variable = rot_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 4
    variable = rot_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 5
    variable = rot_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./inertial_force_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 0
    variable = disp_x
    alpha = -0.3
  [../]
  [./inertial_force_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 1
    variable = disp_y
    alpha = -0.3
  [../]
  [./inertial_force_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 2
    variable = disp_z
    alpha = -0.3
  [../]
  [./inertial_force_rot_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 3
    variable = rot_x
    alpha = -0.3
  [../]
  [./inertial_force_rot_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 4
    variable = rot_y
    alpha = -0.3
  [../]
  [./inertial_force_rot_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 5
    variable = rot_z
    alpha = -0.3
  [../]
[]

[Materials]
  [./elasticity]
    type = ComputeElasticityBeam
    youngs_modulus = 1.0e4
    poissons_ratio = -0.999875
    shear_coefficient = 1.0
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalBeamStrain
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    area = 0.01
    Ay = 0.0
    Az = 0.0
    Iy = 1.0e-4
    Iz = 1.0e-4
    y_orientation = '0.0 1.0 0.0'
  [../]
  [./stress]
    type = ComputeBeamResultants
    block = 0
  [../]
  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1.0'
  [../]
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_x
  [../]
  [./disp_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_y
  [../]
  [./vel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = vel_y
  [../]
  [./accel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = accel_y
  [../]
[]

[Outputs]
  file_base = 'dyn_euler_small_rayleigh_hht_out'
  exodus = true
  csv = true
  perf_graph = true
[]

(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small_rayleigh_hht.i)

# Test for damped small strain euler beam vibration in y direction

# An impulse load is applied at the end of a cantilever beam of length 4m.
# The properties of the cantilever beam are as follows:
# Young's modulus (E) = 1e4
# Shear modulus (G) = 4e7
# Shear coefficient (k) = 1.0
# Cross-section area (A) = 0.01
# Iy = 1e-4 = Iz
# Length (L)= 4 m
# density (rho) = 1.0
# mass proportional rayleigh damping(eta) = 0.1
# stiffness proportional rayleigh damping(eta) = 0.1
# HHT time integration parameter (alpha) = -0.3
# Corresponding Newmark beta time integration parameters beta = 0.4225 and gamma = 0.8

# For this beam, the dimensionless parameter alpha = kAGL^2/EI = 6.4e6
# Therefore, the behaves like a Euler-Bernoulli beam.

# The displacement time history from this analysis matches with that obtained from Abaqus.

# Values from the first few time steps are as follows:
# time  disp_y                vel_y                accel_y
# 0.0   0.0                   0.0                  0.0
# 0.2   0.019898364318588     0.18838688112273     1.1774180070171
# 0.4   0.045577003505278     0.087329917525455   -0.92596052423724
# 0.6   0.063767907208218     0.084330765885995    0.21274543331268
# 0.8   0.073602908614573     0.020029576220975   -0.45506879373455
# 1.0   0.06841704414745     -0.071840076837194   -0.46041813317992

[Mesh]
  type = GeneratedMesh
  nx = 10
  dim = 1
  xmin = 0.0
  xmax = 4.0
  displacements = 'disp_x disp_y disp_z'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_z]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./accel_x]
    type = NewmarkAccelAux
    variable = accel_x
    displacement = disp_x
    velocity = vel_x
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./vel_x]
    type = NewmarkVelAux
    variable = vel_x
    acceleration = accel_x
    gamma = 0.8
    execute_on = timestep_end
  [../]
  [./accel_y]
    type = NewmarkAccelAux
    variable = accel_y
    displacement = disp_y
    velocity = vel_y
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./vel_y]
    type = NewmarkVelAux
    variable = vel_y
    acceleration = accel_y
    gamma = 0.8
    execute_on = timestep_end
  [../]
  [./accel_z]
    type = NewmarkAccelAux
    variable = accel_z
    displacement = disp_z
    velocity = vel_z
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./vel_z]
    type = NewmarkVelAux
    variable = vel_z
    acceleration = accel_z
    gamma = 0.8
    execute_on = timestep_end
  [../]
  [./rot_accel_x]
    type = NewmarkAccelAux
    variable = rot_accel_x
    displacement = rot_x
    velocity = rot_vel_x
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./rot_vel_x]
    type = NewmarkVelAux
    variable = rot_vel_x
    acceleration = rot_accel_x
    gamma = 0.8
    execute_on = timestep_end
  [../]
  [./rot_accel_y]
    type = NewmarkAccelAux
    variable = rot_accel_y
    displacement = rot_y
    velocity = rot_vel_y
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./rot_vel_y]
    type = NewmarkVelAux
    variable = rot_vel_y
    acceleration = rot_accel_y
    gamma = 0.8
    execute_on = timestep_end
  [../]
  [./rot_accel_z]
    type = NewmarkAccelAux
    variable = rot_accel_z
    displacement = rot_z
    velocity = rot_vel_z
    beta = 0.4225
    execute_on = timestep_end
  [../]
  [./rot_vel_z]
    type = NewmarkVelAux
    variable = rot_vel_z
    acceleration = rot_accel_z
    gamma = 0.8
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./fixy1]
    type = DirichletBC
    variable = disp_y
    boundary = left
    value = 0.0
  [../]
  [./fixz1]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value = 0.0
  [../]
  [./fixr1]
    type = DirichletBC
    variable = rot_x
    boundary = left
    value = 0.0
  [../]
  [./fixr2]
    type = DirichletBC
    variable = rot_y
    boundary = left
    value = 0.0
  [../]
  [./fixr3]
    type = DirichletBC
    variable = rot_z
    boundary = left
    value = 0.0
  [../]
[]

[NodalKernels]
  [./force_y2]
    type = UserForcingFunctorNodalKernel
    variable = disp_y
    boundary = right
    functor = force
  [../]
[]

[Functions]
  [./force]
    type = PiecewiseLinear
    x = '0.0 0.2 0.4 10.0'
    y = '0.0 0.01  0.0  0.0'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  l_tol = 1e-11
  nl_max_its = 15
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-10
  start_time = 0.0
  dt = 0.2
  end_time = 5.0
  timestep_tolerance = 1e-6
[]

[Kernels]
  [./solid_disp_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 0
    variable = disp_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 1
    variable = disp_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_disp_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 2
    variable = disp_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 3
    variable = rot_x
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 4
    variable = rot_y
    zeta = 0.1
    alpha = -0.3
  [../]
  [./solid_rot_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 5
    variable = rot_z
    zeta = 0.1
    alpha = -0.3
  [../]
  [./inertial_force_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 0
    variable = disp_x
    alpha = -0.3
  [../]
  [./inertial_force_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 1
    variable = disp_y
    alpha = -0.3
  [../]
  [./inertial_force_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 2
    variable = disp_z
    alpha = -0.3
  [../]
  [./inertial_force_rot_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 3
    variable = rot_x
    alpha = -0.3
  [../]
  [./inertial_force_rot_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 4
    variable = rot_y
    alpha = -0.3
  [../]
  [./inertial_force_rot_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.4225
    gamma = 0.8
    eta = 0.1
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 5
    variable = rot_z
    alpha = -0.3
  [../]
[]

[Materials]
  [./elasticity]
    type = ComputeElasticityBeam
    youngs_modulus = 1.0e4
    poissons_ratio = -0.999875
    shear_coefficient = 1.0
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalBeamStrain
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    area = 0.01
    Ay = 0.0
    Az = 0.0
    Iy = 1.0e-4
    Iz = 1.0e-4
    y_orientation = '0.0 1.0 0.0'
  [../]
  [./stress]
    type = ComputeBeamResultants
    block = 0
  [../]
  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1.0'
  [../]
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_x
  [../]
  [./disp_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_y
  [../]
  [./vel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = vel_y
  [../]
  [./accel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = accel_y
  [../]
[]

[Outputs]
  exodus = true
  csv = true
  perf_graph = true
[]

(modules/solid_mechanics/test/tests/beam/dynamic/dyn_euler_small.i)

# Test for small strain euler beam vibration in y direction

# An impulse load is applied at the end of a cantilever beam of length 4m.
# The properties of the cantilever beam are as follows:
# Young's modulus (E) = 1e4
# Shear modulus (G) = 4e7
# Shear coefficient (k) = 1.0
# Cross-section area (A) = 0.01
# Iy = 1e-4 = Iz
# Length (L)= 4 m
# density (rho) = 1.0

# For this beam, the dimensionless parameter alpha = kAGL^2/EI = 6.4e6
# Therefore, the beam behaves like a Euler-Bernoulli beam.

# The theoretical first and third frequencies of this beam are:
# f1 = 1/(2 pi) * (3.5156/L^2) * sqrt(EI/rho)
# f2 = 6.268 f1

# This implies that the corresponding time period of this beam are 2.858 s and 0.455s

# The FEM solution for this beam with 10 element gives time periods of 2.856 s and 0.4505s with a time step of 0.01.
# A smaller time step is required to obtain a closer match for the lower time periods/higher frequencies.
# A higher time step of 0.05 is used in this test to reduce testing time.

# The time history from this analysis matches with that obtained from Abaqus.

# Values from the first few time steps are as follows:
# time       disp_y            vel_y            accel_y
# 0     0.0                  0.0                0.0
# 0.05  0.0016523559162602   0.066094236650407  2.6437694660163
# 0.1   0.0051691308901533   0.07457676230532  -2.3044684398197
# 0.15  0.0078956772343372   0.03448509146203   4.7008016060883
# 0.2   0.0096592517031463   0.03605788729033  -0.63788977295649
# 0.25  0.011069233444348    0.020341382357756  0.0092295756535376

[Mesh]
  type = GeneratedMesh
  xmin = 0.0
  xmax = 4.0
  dim = 1
  nx = 10
  displacements = 'disp_x disp_y disp_z'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./disp_z]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./rot_z]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_vel_z]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_x]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_y]
  order = FIRST
  family = LAGRANGE
  [../]
  [./rot_accel_z]
  order = FIRST
  family = LAGRANGE
  [../]
[]

[AuxKernels]
  [./accel_x]
    type = NewmarkAccelAux
    variable = accel_x
    displacement = disp_x
    velocity = vel_x
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_x]
    type = NewmarkVelAux
    variable = vel_x
    acceleration = accel_x
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./accel_y]
    type = NewmarkAccelAux
    variable = accel_y
    displacement = disp_y
    velocity = vel_y
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_y]
    type = NewmarkVelAux
    variable = vel_y
    acceleration = accel_y
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./accel_z]
    type = NewmarkAccelAux
    variable = accel_z
    displacement = disp_z
    velocity = vel_z
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./vel_z]
    type = NewmarkVelAux
    variable = vel_z
    acceleration = accel_z
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_x]
    type = NewmarkAccelAux
    variable = rot_accel_x
    displacement = rot_x
    velocity = rot_vel_x
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_x]
    type = NewmarkVelAux
    variable = rot_vel_x
    acceleration = rot_accel_x
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_y]
    type = NewmarkAccelAux
    variable = rot_accel_y
    displacement = rot_y
    velocity = rot_vel_y
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_y]
    type = NewmarkVelAux
    variable = rot_vel_y
    acceleration = rot_accel_y
    gamma = 0.5
    execute_on = timestep_end
  [../]
  [./rot_accel_z]
    type = NewmarkAccelAux
    variable = rot_accel_z
    displacement = rot_z
    velocity = rot_vel_z
    beta = 0.25
    execute_on = timestep_end
  [../]
  [./rot_vel_z]
    type = NewmarkVelAux
    variable = rot_vel_z
    acceleration = rot_accel_z
    gamma = 0.5
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./fixy1]
    type = DirichletBC
    variable = disp_y
    boundary = left
    value = 0.0
  [../]
  [./fixz1]
    type = DirichletBC
    variable = disp_z
    boundary = left
    value = 0.0
  [../]
  [./fixr1]
    type = DirichletBC
    variable = rot_x
    boundary = left
    value = 0.0
  [../]
  [./fixr2]
    type = DirichletBC
    variable = rot_y
    boundary = left
    value = 0.0
  [../]
  [./fixr3]
    type = DirichletBC
    variable = rot_z
    boundary = left
    value = 0.0
  [../]
[]

[NodalKernels]
  [./force_y2]
    type = UserForcingFunctorNodalKernel
    variable = disp_y
    boundary = right
    functor = force
  [../]
[]

[Functions]
  [./force]
    type = PiecewiseLinear
    x = '0.0 0.05 0.1 10.0'
    y = '0.0 0.01  0.0  0.0'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  solve_type = NEWTON

  dt = 0.05
  end_time = 5.0
  timestep_tolerance = 1e-6
[]

[Kernels]
  [./solid_disp_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 0
    variable = disp_x
  [../]
  [./solid_disp_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 1
    variable = disp_y
  [../]
  [./solid_disp_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 2
    variable = disp_z
  [../]
  [./solid_rot_x]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 3
    variable = rot_x
  [../]
  [./solid_rot_y]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 4
    variable = rot_y
  [../]
  [./solid_rot_z]
    type = StressDivergenceBeam
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    component = 5
    variable = rot_z
  [../]
  [./inertial_force_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 0
    variable = disp_x
  [../]
  [./inertial_force_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 1
    variable = disp_y
  [../]
  [./inertial_force_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 2
    variable = disp_z
  [../]
  [./inertial_force_rot_x]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 3
    variable = rot_x
  [../]
  [./inertial_force_rot_y]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 4
    variable = rot_y
  [../]
  [./inertial_force_rot_z]
    type = InertialForceBeam
    block = 0
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    velocities = 'vel_x vel_y vel_z'
    accelerations = 'accel_x accel_y accel_z'
    rotational_velocities = 'rot_vel_x rot_vel_y rot_vel_z'
    rotational_accelerations = 'rot_accel_x rot_accel_y rot_accel_z'
    beta = 0.25
    gamma = 0.5
    area = 0.01
    Iy = 1e-4
    Iz = 1e-4
    Ay = 0.0
    Az = 0.0
    component = 5
    variable = rot_z
  [../]
[]

[Materials]
  [./elasticity]
    type = ComputeElasticityBeam
    youngs_modulus = 1.0e4
    poissons_ratio = -0.999875
    shear_coefficient = 1.0
    block = 0
  [../]
  [./strain]
    type = ComputeIncrementalBeamStrain
    block = '0'
    displacements = 'disp_x disp_y disp_z'
    rotations = 'rot_x rot_y rot_z'
    area = 0.01
    Ay = 0.0
    Az = 0.0
    Iy = 1.0e-4
    Iz = 1.0e-4
    y_orientation = '0.0 1.0 0.0'
  [../]
  [./stress]
    type = ComputeBeamResultants
    block = 0
  [../]
  [./density]
    type = GenericConstantMaterial
    block = 0
    prop_names = 'density'
    prop_values = '1.0'
  [../]
[]

[Postprocessors]
  [./disp_x]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_x
  [../]
  [./disp_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = disp_y
  [../]
  [./vel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = vel_y
  [../]
  [./accel_y]
    type = PointValue
    point = '4.0 0.0 0.0'
    variable = accel_y
  [../]
[]

[Outputs]
  exodus = true
  csv = true
  perf_graph = true
[]

Description
Input Parameters
Input Files