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ELASTIC ENERGY

ELASTIC ENERGY. P H. R H. =P/A. P. Linear elastic material. =l/l. l. External work. =. Internal work. =. For a prismatic bar:. S. V. Law of conservation of energy (first law of thermodynamics):. Increment of:. Heat. External work. Adiabatic processes. Static processes.

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ELASTIC ENERGY

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  1. ELASTIC ENERGY

  2. PH RH =P/A P Linear elastic material =l/l l External work = Internal work = For a prismatic bar:

  3. S V Law of conservation of energy (first law of thermodynamics): Increment of: Heat External work Adiabatic processes Static processes Potential energy Kinematical energy Power = work done in a given time Rate of potential energy

  4. For Hooke materials: Specific volumetric energy Specific distortion energy

  5. Specific energy is a potential energy A general form of specific energy for beams: F – cross-sectional force S – beam stiffness κ– shape coefficient

  6. Components of elastic energy formula

  7. Maxwell-Mohr formula

  8. dt External work: function of loading and displacement Corresponding elastic energy Definitions of generalised force and generalized displacement: Generalized force is any external loading in the form of point force, point moment, distributed loading etc. Generalized displacement corresponding to a given generalized force is any displacement for which work of this force can be performed The dimension of generalized displacement has to follow the rules of dimensional analysis taking into account that the dimension of work is [Nm].

  9. P2 P1 u1 u2 M1 M2  Generalized displacement Generalized force dimension Displacement dimension Generalized force [N] u P [m] M [Nm] [1] du/dx q [N/m] [m2] udx But also: Corresponding generalized displacement is the sum of displacements u1+u2 Corresponding generalized displacement is the sum of rotation angles of neighbouring cross-sections

  10. P l For linear elasticity the principle of superposition obeys: or where αij βijare influence coefficients for which Bettiprinciple holds:αij = αjiandi βij =βji The work of external forces (generalized) Piperformedon displacements (generalized) ui is: After expansion of the first term we have:

  11. taking into account that: which after expansion reads: , …

  12. Therefore, for any displacement we have: and since To find an arbitrary generalized displacement of any point of the structure one has to apply corresponding generalized force in this point, and calculate internal energy associated with all loadings (real and generalized), take derivative of this energy with respect to generalized force, and finally set its true value equal to 0: Where Fi is cross-sectional force for each case of internal forces reduction (normal force, shear force, bending moment, torsion moment)

  13. denotes here function of x Making use of superposition principle we have: or where: With general formula for potential energy: we have: where index i has been added for different reduction cases

  14. This is Maxwell-Mohr formula for any generalized displacement . Summation has to be taken over all structural members j and over all internal cross-sectional forces i=4

  15. stop

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