Bending Deflection – Statically Indeterminate Beams

1 Bending Deflection . Statically Indeterminate Beams AE1108-II: Aerospace Mechanics of Materials Dr. Calvin Rans Dr. Sofia Teixeira De Freitas Aerospace Structures & Materials Faculty of Aerospace Engineering Recap Procedure for Statically Indeterminate Problems I. Free Body Diagram II. Equilibrium of Forces (and Moments). III. Displacement Compatibility IV. Force-Displacement (Stress-Strain) Relations Solve when number of equations = number of unknowns V. Answer the Question! Typically calculate desired internal stresses, relevant displacements, or failure criteria For Bending , Force-Displacement relationships come from Moment-Curvature relationship (ie: use Method of Integration or Method of Superposition). Statically Indeterminate Beams Many more redundancies are possible for Beams : - Draw FBD and count number of redundancies - Each redundancy gives rise to the need for a compatibility equation P P.

2 MA. HA. A B. VA VB. - 4 reactions 4 3=1 1st degree Statically Indeterminate - 3 equilibrium equations Statically Indeterminate Beams Many more redundancies are possible for Beams : - Draw FBD and count number of redundancies - Each redundancy gives rise to the need for a compatibility equation P P MB. MA. HA HB. A B. VA VB. - 6 reactions 6 3=3 3rd degree Statically Indeterminate - 3 equilibrium equations Solving Statically Indeterminate Beams using method of integration What is the difference between a support and a force? F. Displacement Compatibility (support places constraint on deformation). Method of Integration P. A B. What if we remove all redundancies and replace with reaction forces? P VB. If we treat VB. as known, we can solve! A B. Formulate expression for d 2v Can integrate M(z). M EI 2. dz to find v Method of Integration (cont). P VB. At A, v = 0, = 0. A B. 2 A. d v EI 2. M z , VB.

3 Dz . EIv M z , VB dz C1. Determine constants of integration from Boundary Conditions EIv M ( z , VB ) dz C2. v( z ,VB ) How to determine VB? Method of Integration (cont). P VB P. =. A B A B. v( z , VB ) How to determine VB? VB2. P VB changes displacement! VB1 Must be compatible! Compatibility BC: At B, v = 0 Solve for VB. Compatibility equations for Beams are simply the boundary conditions at redundant supports Example 1 q Problem Statement Determine Deflection equation for the beam using method of A B. integration: Solution 2) Equilibrium: 1) FBD: q MA. F H A 0 HA.. A B. F . VA VB qL. 2 2. VA VB. qL qL. M A M A LVB 2.. 2. LVA. Treat reaction forces as knowns! Aerospace Mechanics of Materials (AE1108-II) Example Problem 11. Example 1. 4) Determine moment equation: z q z MA. M ccw . z M M A VA z qz . 2. M. q 2 A. M A VA z z V. 2 VA. Can also use step function approach always on always off M M A z 0 VA z 0 VB z L.

4 0 q 2. z 0. 2. Aerospace Mechanics of Materials (AE1108-II) Example Problem 12. Example 1. 5) Integrate Moment equation to get z q v' and v q 2. EIv M A VA z z = M(z) A B. 2.. VA 2 q 3. EIv M A z z z C1 = -EI (z). 2 6. M A 2 VA 3 q 4. EIv z z z C1 z C2 = -EIv(z). 2 6 24. We now have expressions for v and v', but need to determine constants of integration and unknown reactions Aerospace Mechanics of Materials (AE1108-II) Example Problem 13. Example 1. 5a) Solve for Constants of Integration z q using BC's: 0. VA 2 q 3. EIv M A z z z C1 = -EI (z) A B. 2 6. Fixed support 0 = 0, v = 0. M A 2 VA 3 q 4. EIv z z z C1 z C2 = -EIv(z). 2 6 24. Boundary Conditions: q 3 VA. EI 0 0 0 M A 0 C1. 2. At z = 0, = 0. 6 2. C1 0. Aerospace Mechanics of Materials (AE1108-II) Example Problem 14. Example 1. 5a) Solve for Constants of Integration z q using BC's: 0. VA 2 q 3. EIv M A z z z C1 = -EI (z) A B. 2 6. Fixed support 0 0 = 0, v = 0.

5 M A 2 VA 3 q 4. EIv z z z C1 z C2 = -EIv(z). 2 6 24. Boundary Conditions: q VA MA. EI 0 0 0 0 C2. 4 3 2. At z = 0, v = 0. 24 6 2. C2 0. Aerospace Mechanics of Materials (AE1108-II) Example Problem 15. Example 1. 5b) Solve for Reaction Forces using z q BC's (imposed by redundant support): VA 2 q 3. EIv M A z z z = -EI (z) A B. 2 6. roller support M A 2 VA 3 q 4 v=0. EIv z z z = -EIv(z). 2 6 24. Boundary Conditions: q VA MA. EI 0 L L L . 4 3 2. At z = L, v = 0. 24 6 2. Recall from equilibrium: qL 3M A qL2. VA MA LVA VB qL VA. 4 L 2. Aerospace Mechanics of Materials (AE1108-II) Example Problem 16. z q Example 1. 5b) Solve for Reaction Forces using A B. BC's (imposed by redundant support): qL 3M A qL2. VA , MA LVA , VB qL VA. 4 L 2. 5. VA qL. 8 MA. qL2. MA A B. 8. 3 VA VB. VB qL. 8. Aerospace Mechanics of Materials (AE1108-II) Example Problem 17. z q Example 1. We were asked to determine Deflection A B.

6 Equation: M A 2 VA 3 q 4 0. EIv z z z 2 6 24. qz 2.. v 3 L2. 5 Lz 2 z 2. 48 EI. Max Displacement: =0 0 1L. v . q 48 EI. 6 L2 z 15Lz 2 8 z 3 0 z L. qL2. v( L) EI. Aerospace Mechanics of Materials (AE1108-II) Example Problem 18. Example 1. Now that the reactions are known: z q q 2 A B. M ( z ) EIv M A VA z z L. 2. 4. qL2 5qLz qz 2 M. qL2. 8 8 2 . d 8. dz 5qL 5. V ( z ) EIv qz 8. qL. 8 V. 5 3. L qL. 8 8. Aerospace Mechanics of Materials (AE1108-II) Example Problem 19. Solving Statically Indeterminate Beams using superposition Method of Superposition Determine reaction forces: P. 2L L 1) FBD: P. MA. HA. A B. VA VB. 2) Equilibrium: . - 4 reactions F H A 0 - 3 equilibrium equations . F . VA VB P 1st degree Statically Indeterminate M A 3 LVB 2 LP M A. Method of Superposition (cont). How do we get compatibility equation? P. A B. Split into two Statically determinate problems VB. P. B2. A B B1 A B. Be careful!

7 3) Compatibility: B1 = B2 vB1 vB 2 0 +ve v . Method of Superposition (cont). How do we get Force-Displacement relations? We have been doing this in the previous lectures Integrate Moment Curvature Relation Standard Case Solutions P. d 2v Can integrate A B. M EI 2 to find v dz L. + +. PL2 PL3. B vB . 2 EI 3EI. Method of Superposition (cont). From the standard case: P PL2. B . 2 EI. A B PL3. 4) Force-Displacement: vB . L 3EI. 2L P L. P 2L . 2. 2 PL2. P . B1 2 EI EI. A B. vB1 vP P L. straight P 2 L 2 PL2. 3. L. 3EI EI. 14 PL3.. 3EI. Method of Superposition (cont). From the standard case: P PL2. B . 2 EI. A B PL3. 4) Force-Displacement: vB . L 3EI. VB. B2 VB 3L . 3. 9VB L3. vB 2 . A B 3EI EI. Compatibility: 14 PL3 9VB L3 14. vB1 vB 2 0 VB P. 3EI EI 27. Additional remarks about Bending deflections Remarks about beam Deflections Recall Shear deformation V. +. V. +. Moment deformation V(x). M M. M(x) +.

8 Negligible (for long Beams ). Bending Deformation = Shear Deformation + Moment Deformation Remarks about beam Deflections P P MB. HB. MB. A B A. P B. VB HB. VB. B. C P. Axial deformation P of AB due to HB. B C. A. P. Axial deformation of BC due to VB. C. Axial deformation << Bending deformation! Remark about beam Deflections For Bending deformation problems negligible Deformation = Axial Deformation + Shear Deformation + Moment Deformation BUT! P MB. If moment deformation is not HB. present, deformation is not negligible A B. VB. q Example 2. B C. Calculate reaction forces at A and D: L, EI. First, can we see any simplifications? A D. Symmetry! Symmetry implies: q - Reactions at A = reactions at D HF. - Slope at symmetry plane = 0 MF. B. - Shear force at symmetry plane = 0. We will solve using superposition and standard cases A. Aerospace Mechanics of Materials (AE1108-II) Example Problem 30. q HF.

9 Example 2 B. MF. F. Calculate reaction forces: 1) FBD. 2) Equilibrium A. HA. VA. F H A H F. MA. qL. F . VA . 2. qL2. M ccw . A MA . 8. HF L M F. Split problem into two: beam AB and beam BF. Aerospace Mechanics of Materials (AE1108-II) Example Problem 31. Example 2. Calculate reaction forces: 3) Compatibility MF - qL2/8. q HF qL/2 HF MF - qL2/8 q MF B. B HF MF. F HF. + B F. =. qL/2. A A. F 0, v B 0. (axial deformation negligible). Aerospace Mechanics of Materials (AE1108-II) Example Problem 32. MF - qL2/8. Example 2 qL/2 HF. B. 4a) Force-Displacement for beam AB. MF - qL2/8. HF qL/2. B B. B. A. + +. A A A. H F L3 M F L2 qL4. vB vB 0 vB . 3EI 2 EI 16 EI. H F L2 M F L qL3. cw . B B 0 cw . B . 2 EI EI 8 EI. Aerospace Mechanics of Materials (AE1108-II) Example Problem 33. Example 2. 4a) Force-Displacement for beam AB. MF - qL2/8. qL/2 HF. H F L3 M F L2 qL4. vB 0. B 3EI 2 EI 16 EI. H F L2 M F L qL3. Bcw.

10 2 EI EI 8 EI. A. Recall compatibility: Still need F F 0, v B 0. Aerospace Mechanics of Materials (AE1108-II) Example Problem 34. MF + qL2/8 q Example 2 HF HF MF. 4b) Force-Displacement for beam BF B F. qL/2 L/2. q HF MF. + +. B F B F B F. q L 2 . 3 MFL. cw . F 0 Fcw . F. 6 EI EI. qL3 MFL. cw . F Wait! Not entirely correct! 48 EI EI. Aerospace Mechanics of Materials (AE1108-II) Example Problem 35. Example 2. Previous angle relative to fixed support B. B. q HF MF. B F. qL3 M L. cw . F F B. 48 EI EI. 7 qL3 2 M F L H F L2. cw . F 0. 48 EI EI 2 EI. Aerospace Mechanics of Materials (AE1108-II) Example Problem 36. Recall equilibrium: Example 2 . F H A H F. qL. Solve: F . VA . 2. Recall compatibility: qL2. F 0, v B 0. M ccw . A MA . 8. HF L M F. 7 qL3 2 M F L H F L2. cw . F 0 qL. 48 EI EI 2 EI HF HA . qL. 8 8. qL2 2qL2. H F L3 M F L2 qL4 MF MA . vB 0 24 3. 3EI 2 EI 16 EI. Aerospace Mechanics of Materials (AE1108-II) Example Problem 37.