16.21 Techniques of Structural Analysis and Design Spring 2003 Section 2 - Energy and Variational Principles Unit #7 - Concepts of work and energy Work Figure 1: Work of a force on a moving particle 1 ? ? ? ? Work done by a force: dW = f · du = f i u i = ?f ??du? cos( ? fu) (1) ? B ? B W AB = dW = f · du (2) A A ? Work done by a moment: dW = M · dθ = M i θ i (3) ? B ? B W AB = dW = M · dθ (4) A A ? Extend de?nition to material bodies: total work is the addition of the work done on all particles: – by forces distributed over the volume: W = f · udV V – by forces distributed over the surface: W = t · udS S – by concentrated forces: n W = f i · u(x i ) i=1 Another classi?cation: ? Work done by external forces: we will assume that external forces don’t change during the motion or deformation, i.e., they are independent of the displacements. This will lead to the potential character of the external work and to the de?nition of the potential of the external forces as the negative of the work done by the external forces. ? Work done by internal forces: the internal forces do depend on the deformation. In general, the work done by external forces and the work done by internal forces don’t match (we saw that part of the work changes the kinetic energy of the material). 2 Figure 2: Spring loaded with a constant force Example: Consider the following spring loaded with a constant force: W E = Fδ, F doesn’t change when u goes from 0 to δ (5) = mgδ (6) W I = ? δ 0 F s (u)du, F S : force on spring (7) = ? δ 0 kudu = 1 2 kδ 2 (8) ? W E ?= W I (9) Remarks: 3 ˙ ? ? ? W E = W I would imply δ = 2 mg , which contradicts equilibrium: δ = k mg k , ? before the ?nal displacement δ is reached the system is not in equilib- rium. How can you explain this? Strain energy and strain energy density Figure 3: Strain energy density Strain energy and strain energy density (see also unit on ?rst law of thermodynamics): U = UdV (10) V From ?rst law: ? ? U = σ ij ? ij ?t 4 ˙ ˙ ? ? ? ? ? ? ? ? U ? ij ? ?? ? ?? ij = σ ij ? ij ???? ? ? U σ ij = ?? ij ? ? ij 0 U = σ ij d? ij , not necessarily linear elastic Linear case: U = ? ij 0 C ijkl ? kl d? ij = 1 C ijkl ? kl ? ij = 2 1 σ ij ? ij (11) 2 Complementary strain energy and complementary strain energy density Figure 4: Complementary strain energy density V U c dV (12)U c = ? σ ij 0 U c = ? ij dσ ij (13) 5 ? ? ? ? ? Linear case: ? ij = S ijkl σ kl , where S ijkl = C ?1 ijkl σ ij 1 U c = S ijkl σ kl dσ ij = S ijkl σ kl σ ij = 0 2 1 ? ij σ ij (14) 2 ? U c = U for a linear elastic material (15) Note: In the literature, you will ?nd the following notation used indis- tinctively: U 0 , ? U : to represent strain energy density U 0 ? , ? U c : to represent complementary strain energy density U : to represent strain energy density U ? ,U c : to represent complementary strain energy density Example: Compute the strain energy density, strain energy, and their complementary counterparts for the linear elastic bar loaded axially shown in the ?gure: ? ? 0 ? ?ν? 0 U = σ 11 d? 11 + σ 22 d? 22 + ... 0 ? ? 0 0 1 = E? 11 d? 11 = E? 2 2 0 0 6 ? ? ? ? P From equilibrium we know: σ 0 = A . From the constitutive law: ? 0 = σ 0 = P E AE ? 1 P 2 ? U = ? 2 EA 2 ? ALP 2 P 2 L U = UdV = = V 2EA 2 2EA ? σ 0 ? 0 U c = ? 11 dσ 11 + 0 0 ? σ 0 σ 11 1 σ 2 ? ? σ 0 = 0 E dσ 11 = 2E 11 ? 0 = ? ALP 2 U c = U c dV = ? 22 dσ 22 + ... σ 2 P 2 0 ? = = U !! 2E 2EA 2 P 2 L = = U !! V 2EA 2 2EA Potential Energy Capacity of the system (material body + external forces) to return work Π = U + V ,V : potential of external loads (16) V = ? t ˉ i u i dS ? f ˉ i u i dV (17) S V Π = ? V 1 2 σ ij ? ij dV ? ? S ˉ t i u i dS ? ? V ˉ f i u i dV (18) This expression applies to linear elastic materials (why?). 7