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A uniform horizontal beam of length, L. units, having a simple support at its left end and a fixed support at its right end will be distorted, due to its own weight, into a curve: y=f(x) as shown in the figure. 0 L y=f(x) This curve is called the deflection curve of the beam and it satisfies the differential equation: E-Idy w along with the four boundary conditions: y(0) = 0, y(L) = 0, y" (0) = 0 y' (L) = 0 Problem: Substitute each boundary condition into the appropriate equation: (2), (3) or (4) from page 17.7 of the article: Topic 17: The Deflection Equation of a Uniform Beam and then solve for the coefficients, C. i=1,2,3,4, to verify that the beam's equation is: y= [2x*-3Lx³ + L³x] 4RFI

A uniform horizontal beam of length, L. units, having a simple support at its left end and a fixed support at its right end will be distorted, due to its own weight, into a curve: y=f(x) as shown in the figure. 0 L y=f(x) This curve is called the deflection curve of the beam and it satisfies the differential equation: E-Idy w along with the four boundary conditions: y(0) = 0, y(L) = 0, y" (0) = 0 y' (L) = 0 Problem: Substitute each boundary condition into the appropriate equation: (2), (3) or (4) from page 17.7 of the article: Topic 17: The Deflection Equation of a Uniform Beam and then solve for the coefficients, C. i=1,2,3,4, to verify that the beam's equation is: y= [2x*-3Lx³ + L³x] 4RFI

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 54E: Plant Growth Researchers have found that the probability P that a plant will grow to radius R can be...
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Integrating a second time, we find that SO > SE.I. E.I. Integrate a third time to obtain: E.I. E.I E.I. Integrate a fourth time to get: E I E.I. E·I· y = d³y dx³ dy dx dy dx = w dx dx = W dx = " dx 6 W 24 d³ y dx.³ = d'y dx² - dx = √ w-x dx + [ c, dx W 2 W 6 = w.x + C₁ x² + C₁₂ 6 + fx [x dx + G₂₁₂ • ¹ a 1 dx W ·•ƒ x² dx + G₂ • Sx x² dx + ·x² + G·x + C₂ 2 x dx + .f x² dx +²³+ +ƒ •x² + C₂ ·x + C₂ C₂ dx c₂.f -x² + C₁-x + C₂ (3) x dx + C₂ dx (4) (1) (2) The values of the coefficients: C₁, C₂, C3, C4 are dependent upon the types of supports that are provided to the beam at its ends. 17.3 17.4
Transcribed Image Text:Integrating a second time, we find that SO > SE.I. E.I. Integrate a third time to obtain: E.I. E.I E.I. Integrate a fourth time to get: E I E.I. E·I· y = d³y dx³ dy dx dy dx = w dx dx = W dx = " dx 6 W 24 d³ y dx.³ = d'y dx² - dx = √ w-x dx + [ c, dx W 2 W 6 = w.x + C₁ x² + C₁₂ 6 + fx [x dx + G₂₁₂ • ¹ a 1 dx W ·•ƒ x² dx + G₂ • Sx x² dx + ·x² + G·x + C₂ 2 x dx + .f x² dx +²³+ +ƒ •x² + C₂ ·x + C₂ C₂ dx c₂.f -x² + C₁-x + C₂ (3) x dx + C₂ dx (4) (1) (2) The values of the coefficients: C₁, C₂, C3, C4 are dependent upon the types of supports that are provided to the beam at its ends. 17.3 17.4
A uniform horizontal beam of length, L units, having a simple support at its left end and a fixed support at its right end will be distorted, due to its own weight, into a curve: y=f(x) as shown in the figure. L y = f(x) This curve is called the deflection curve of the beam and it satisfies the differential equation: E- I dy = w along with the four boundary conditions: dx y(0) = 0, y(L) = 0, y" (0) = 0 y'(L) = 0 W 4RFI X Problem: Substitute each boundary condition into the appropriate equation: (2), (3) or (4) from page 17.7 of the article: Topic 17: The Deflection Equation of a Uniform Beam and then solve for the coefficients, C₁, i = 1,2,3,4, to verify that the beam's equation is: y= [2x-3Lx³ + L³x]
Transcribed Image Text:A uniform horizontal beam of length, L units, having a simple support at its left end and a fixed support at its right end will be distorted, due to its own weight, into a curve: y=f(x) as shown in the figure. L y = f(x) This curve is called the deflection curve of the beam and it satisfies the differential equation: E- I dy = w along with the four boundary conditions: dx y(0) = 0, y(L) = 0, y" (0) = 0 y'(L) = 0 W 4RFI X Problem: Substitute each boundary condition into the appropriate equation: (2), (3) or (4) from page 17.7 of the article: Topic 17: The Deflection Equation of a Uniform Beam and then solve for the coefficients, C₁, i = 1,2,3,4, to verify that the beam's equation is: y= [2x-3Lx³ + L³x]
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