1/141/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy ELASTIC ENERGY

2/142/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy P ll PHPH Linear elastic material External work  =P/A  =  l/l RHRH Internal work = = For a prismatic bar:

3/143/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy V S Law of conservation of energy (first law of thermodynamics): Adiabatic processes Static processes Power = work done in a given time HeatExternal work Rate of potential energy Potential energy Kinematical energy Increment of:

4/144/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy For Hooke materials: Specific volumetric energy Specific distortion energy

5/145/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy Specific energy is a potential energy A general form of specific energy for beams: F – cross-sectional force S – beam stiffness κ – shape coefficient

6/146/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy Components of elastic energy formula Specific case Cross-sectional force F Beam stiffness S Shape coefficient Tension NEA 1 Bending MEJ 1 Shear QGA Torsion MsMs GJ s

7/147/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy Maxwell-Mohr formula

8/148/14 M.Chrzanowski: Strength of Materials SM2-09: 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].  dt Corresponding elastic energy External work: function of loading and displacement

9/149/14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy Generalized displacement Displacement dimension Generalized force dimension [1] P [Nm] q M [N]u [m] [N/m][m 2 ] du/dx  udx But also: Corresponding generalized displacement is the sum of displacements u 1 +u 2 P2P2 P1P1 u1u1 u2u2 M2M2 M1M1  Corresponding generalized displacement is the sum of rotation angles of neighbouring cross-sections Generalized force

10 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy For linear elasticity the principle of superposition obeys: or where α ij β ij are influence coefficients for which Betti principle holds: α ij = α ji and i β ij =β ji The work of external forces (generalized) P i performed on displacements (generalized) u i is: After expansion of the first term we have: P ll

11 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy taking into account that: which after expansion reads: …,

12 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy 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 F i is cross-sectional force for each case of internal forces reduction (normal force, shear force, bending moment, torsion moment)

13 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy Making use of superposition principle we have: where: With general formula for potential energy: or we have: where index i has been added for different reduction cases denotes here function of x

14 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy 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 /14 M.Chrzanowski: Strength of Materials SM2-09: Elastic energy  stop