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Lecture 12 deflection in beams
This document discusses stresses in beams and beam deflection. It covers several methods for analyzing bending stresses and deflection in beams, including: [1] the engineering beam theory relating moment, curvature, and stress; [2] double integration and moment area methods for calculating slope and deflection; and [3] Macaulay's method, which simplifies calculations for beams with eccentric loads. Formulas are provided relating bending moment, shear force, curvature, slope, and deflection. Moment-area theorems are also described for relating bending moment to slope and deflection.
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Introduction to Unit 2 covering shear force, bending moment diagrams, stresses and deflection in beams.
Discussion of beam deflection, engineering beam theory, moment-curvature relationship, and bending moment equation.
Presentation of methods for calculating slope and deflection including double integration and Macaulay’s method.
Introduction of Moment-Area Theorems to evaluate deflections using the area under the bending moment diagram.
Elaboration on the Moment Area Method for finding shear and deflection in beams.
An exercise comparing the moment of inertia of beams and their deflection characteristics.
Lecture 12 deflection in beams
- 2. Unit 2- Stressesin Beams Topics Covered Lecture -1 – Review of shear force and bending moment diagram Lecture -2 – Bending stresses in beams Lecture -3 – Shear stresses in beams Lecture -4- Deflection in beams Lecture -5 – Torsion in solid and hollow shafts.
- 3. Beam Deflection Recall: THEENGINEERING BEAM THEORY σ M E = = y I R Moment-Curvature Equation v (Deflection) y € NA A B x A’ B’ If deformation is small (i.e. slope is “flat”):
- 4. 1 dθ ∴ ≈ R dx R δy and δθ ≈ (slope is “flat”) δx B’ δy 1 d2y € A’ ⇒ ≈ 2 R dx € € Alternatively: from Newton’s Curvature Equation y € ⎛ d 2 y ⎞ ⎛ dy ⎞ 2 R ⎜ 2 ⎟ if ⎜ ⎟ <<<< 1 1 ⎝ dx ⎠ ⎝ dx ⎠ = 3 y = f (x) R ⎛ 2 ⎞ 2 ⎛ dy ⎞ ⎜1+ ⎜ ⎟ ⎟ 1 d2y x ⎝ ⎝ dx ⎠ ⎠ ⇒ ≈ 2 € R dx € €
- 5. From the EngineeringBeam Theory: M E 1 M d2y = = = 2 I R R EI dx d2y ⇒ ( EI ) 2 = M dx €Flexural € € Bending Stiffness Moment Curvature €
- 6. Relationship A C B Deflection = y dy Slope = dx d2y A C B Bending moment = EI 2 y dx d3y Shearing force = EI 3 dx d4 y Rate of loading = EI 4 dx €
- 7. Methods to findslope and deflection Double integration method Moment area method Macaulay’s method
- 8. Double integration method d 2 y ⎛ 1 ⎞ ⎟ M Curvature Since, 2 = ⎜ dx ⎝ EI ⎠ dy ⎛ 1 ⎞ ⇒ = ⎜ ⎟ ∫ M ⋅ dx + C1 Slope dx ⎝ EI ⎠ € ⎛ 1 ⎞ ⇒ y = ⎜ ⎟ ∫ ∫ M ⋅ dx⋅ dx + ∫ C ⋅ dx + C 1 2 Deflection ⎝ EI ⎠ € Where C1 and C2 are found using the boundary conditions. € Curvature Slope Deflection y R dy dx
- 9. Double integration method Simplesupported W Slope Deflection L/2 L/2 dy A C B Slope = Deflection = y c dx yc 2 WL3 WL =− = θA = θB = − 48EI 16EI L Uniform distributed load x € Slope € Deflection w/Unit length A C dy B Slope = Deflection = y c dx yc 2 5 WL3 WL =− = θA = θB = − 384 EI 24 EI L € €
- 10. Macaulay’s method Theprocedure of finding slope and deflection for simply supported beam with an eccentric load is very laborious. Macaulay’s method helps to simplify the calculations to find the deflection of beams subjected to point loads.
- 11. Moment-Area Theorems • Consider a beam subjected to arbitrary loading, dθ d 2 y M = = dx dx 2 EI θD xD M ∫ dθ = ∫ EI dx θC xC xD M θ D − θC = ∫ EI dx xC dx CD = Rdθ = dx € € R dθ • First Moment-Area Theorem: € area under BM diagram between € C and D. 9 - 11
- 12. Moment-Area Theorems • Tangents to the elastic curve at P and P’ intercept a segment of length dt on the vertical through C. M dt = xdθ = x dx EI xD x M 1 D 1 − tC D = ∫ x EI dx = ∫ xMdx = EI A x EI xC xC − A= total area of BM diagram between C & D € x = Distance of CG of BM diagram from C • Second Moment-Area Theorem: The tangential deviation of C with respect to € D is equal to the first moment with respect to a vertical axis through C of the area under the BM diagram between C and D. 9 - 12
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- 15. An Exercise- Momentof Inertia – Comparison 1 Load Maximum distance of 4 inch to the centroid I2 2 x 8 beam Load 2 I1 Maximum distance of 1 inch to 2 x 8 beam the centroid I2 > I1 , orientation 2 deflects less Ken Youssefi Engineering 10, SJSU 15