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1.050 Engineering Mechanics I Lecture 25: Beam elasticity – problem solving technique and examples Handout 1 1.050 – Content overview I. Dimensional analysis 1. On monsters, mice and mushrooms 2. Similarity relations: Important engineering tools II. Stresses and strength 3. Stresses and equilibrium 4. Strength models (how to design structures, foundations.. against mechanical failure) III. Deformation and strain 5. How strain gages work? 6. How to measure deformation in a 3D structure/material? IV. Elasticity 7. Elasticity model – link stresses and deformation 8. Variational methods in elasticity V. How things fail – and how to avoid it 9. Elastic instabilities 10. Plasticity (permanent deformation) 11. Fracture mechanics Lectures 1-3 Sept. Lectures 4-15 Sept./Oct. Lectures 16-19 Oct. Lectures 20-31 Oct./Nov. Lectures 32-37 Dec. 2 11.050 – Content overview I. Dimensional analysis II. Stresses and strength III. Deformation and strain IV. Elasticity Lecture 20: Introduction to elasticity (thermodynamics) Lecture 21: Generalization to 3D continuum elasticity Lecture 22: Special case: isotropic elasticity Lecture 23: Applications and examples Lecture 24: Beam elasticity Lecture 25: Applications and examples (beam elasticity) Lecture 26: … cont’d and closure … V. How things fail – and how to avoid it 3 Beam bending elasticity Governed by this differential equation: d 4ξ fz z= dx4 EI Integration provides solution for displacement Solve integration constants by applying BCs Note: E = material parameter (Young’s modulus) I = geometry parameter (property of cross-section) fz = distributed shear force (force per unit length) fz = pb0 where p0=pressure, b=thickness of beam in y-direction 4 24-step procedure to solve beam elasticity problems • Step 1: Write down BCs (stress BCs and displacement BCs), analyze the problem to be solved (read carefully!) • Step 2: Write governing equations for ξz ,ξx ... • Step 3: Solve governing equations (e.g. by integration), results in expression with unknown integration constants • Step 4: Apply BCs (determine integration constants) Note: Very similar procedure as for 3D isotropic elasticity problems 5 Difference in governing equations (simpler for beams) Physical meaning of derivatives of ξz d 4ξz = fz d 4ξz EI = f Shear force density dx4 EI dx4 z ddx 3ξ 3 z = − QEIz − d 3ξ 3 z EI = Qz Shear force dx d 2ξz = − My − d 2ξz EI = M Bending moment dx2 EI dx2 y dξz = −ω y − dξz =ω y Rotation (angle) dx dx ξ ξ Displacement z z 6 38Step-by-step example z p = force/length x l length Step 1: BCs ξz (0) = 0 EI x = 0 ω (0) = 0y ξz (l) = 0 x = l My (0) = 0 7 Step 2: Governing equation d 4ξzfz d 4ξzp= = −dx4 EI dx4 EI p applied in negative z-direction Step 3: Integration ξ '''= − px + Cz 1 EI 2 ξz '' = − p x + C1x + C2 EI 2 ξ '''' = − p 3 2 z EI ξz ' = − p x + C1 x + C2 x + C3 EI 6 2 4 3 2 ξz = − p x + C1 x + C2 x + C3x + C4 EI 24 6 2 4Step 4: Apply BCs ξz '''= − px + C1 = − Qz EI EI zξ 2 ξz '' = − p x + C1x + C2 EI 2 3 2 ξz ' = − p x + C1 x + C2 x + C3EI 6 2 EI M y y −= −= ω 4 3 2 = − p x + C1 x + C2 x + C3x + C4EI 24 6 2 Known quantities are marked 9 Step 4: Apply BCs (cont’d) ξz (0) = 0 → C4 = 0 ωy (0) = 0 → C3 = 0 ξz (l) = 0 → − p l4 + C1 l3 + C2 l2 = 0EI 24 6 2 My (0) = 0 → − p l2 + C1l +C2 = 0EI 2 ⎛ l3 l2 ⎞ ⎛⎜ l4 ⎞⎟ C1 = p 5 l ⎜ ⎟⎛ C1 ⎟⎟⎞ = p ⎜ 24 ⎟ EI 8 ⎜⎜6 l 12 ⎟⎟⎜⎜⎝C2 ⎠ EI ⎜ l2 ⎟ p 12 ⎝ ⎠ ⎜ ⎟C = − l ⎝ 2 ⎠ 2 EI 8 10 5Solution: Qz (x) = p⎜⎛ x − 5 l ⎟⎞ ⎝ 8 ⎠ My (x) = p⎜⎜⎛ 1 l2 + x2 − 5 lx⎟⎟⎞ ⎝ 8 2 8 ⎠ ω y(x) = EIp ⎜⎜⎛⎝ 18 l2 x + x 63 − 165 lx2 ⎟⎟⎞⎠ ξz (x) = − p ⎜⎜⎛ 1 l2 x2 + x4 − 5 lx3 ⎟⎟⎞ EI ⎝16 24 48 ⎠ 11


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