Basic structural elements

• Axially Loaded Beams
  • The simplest structure
  • Slender beam with rectangular cross section of the length L, the width w, and the height h.
  • Beam with one fixed end
  • Typical one-dimensional problem
  • Stress corresponding to the axial load force T = F/wh creates the strain S = DL/L
    

• Relation for the strain:
  Based on the basic definition of elasticity

Static analysis

• Force, F and the deformation DL are constant along the beam
• Static compliance, c_s, and the spring constant, k:
  beam functioning as a spring element

Dynamic analysis

• Based on the solution of the equation of motion corresponding to this case
• The equation of motion for the longitudinal displacement u_x:

• The solution for displacement u_x corresponding to the boundary conditions shown in preceding figure:

     where   ...    u_L is the displacement at the free end of the beam
                                                     k is the wave number
                                                     x is the coordinate

• Strain along the beam is defined by the relation:

• Stress corresponding to the strain:
  from the Hooke's law

• Mechanical impedance, Z_mL at the point x=L
  supposing the harmonic movement

    where   T_L ... stress at the free end of the beam

      F_L ...  force at the free end of the beam

• Mechanical admittance
  from previous expression by replacing the goniometric functions by its decomposition to the series

         where  m_e   ...  equivalent mass
                                                                   c_n   ...  compliance equivalent to the mode of vibrations n

• Circuit representation of mechanical impedance of an axially loaded beam
  mass, compliance, and optionally resistance connected in series for each mode
  branches corresponding to each mode connected in parallel