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Mechanics of Materials CIVL 3322 / MECH 3322 Deflection of Beams
The Elastic Curve
¢ The deflection of a beam must often be limited in order to provide integrity and stability of a structure or machine, or ¢ To prevent any attached brittle materials from cracking
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The Elastic Curve
¢ Deflections at specific points on a beam must be determined in order to analyze a statically indeterminate system.
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The Elastic Curve
¢ The curve that is formed by the plotting the position of the centroid of the beam along the longitudal axis is known as the elastic curve.
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The Elastic Curve
¢ At different types of supports, information that is used in developing the elastic curve are provided l Supports which resist a force, such as a pin, restrict displacement l Supports which resist a moment, such as a fixed end support, resist displacement and rotation or slope
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The Elastic Curve
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The Elastic Curve
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The Elastic Curve
¢ We can derive an expression for the curvature of the elastic curve at any point where ρ is the radius of curvature of the elastic curve at the point in question 1 M = ρ EI
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The Elastic Curve
¢ If you make the assumption to deflections are very small and that the slope of the elastic curve at any point is very small, the curvature can be approximated at any point by
2 v is the deflection of d v M the elastic curve = dx2 EI
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The Elastic Curve
¢ We can rearrange terms
d 2v EI = M dx2
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The Elastic Curve
¢ Differentiate both sides with respect to x
d ⎛ d 2v⎞ dM EI = = V(x) dx ⎝⎜ dx2 ⎠⎟ dx
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The Elastic Curve
¢ And again differentiate both sides with respect to x
d 2 ⎛ d 2v⎞ dV EI = = w(x) dx2 ⎝⎜ dx2 ⎠⎟ dx
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The Elastic Curve d 4v ¢ So there are EI 4 = w(x) three paths to dx finding v d 3v EI = V(x) dx3 d 2v EI = M (x) dx2 13 Beam Deflection by Integration
Boundary Conditions
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Boundary Conditions
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Boundary Conditions
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Boundary Conditions
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Boundary Conditions
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Boundary Conditions
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Combining Load Conditions
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Cantilever Example
¢ Given a cantilevered beam with a fixed end support at the right end and a load P applied at the left end of the beam. ¢ The beam has a length of L.
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Cantilever Example
¢ If we define x as the distance to the right from the applied load P, then the moment function at any distance x is given as
M (x) = −Px
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Cantilever Example
¢ Since we have a function for M along the beam we can use the expression relating the moment and the deflection M (x) = −Px d 2v EI = M(x) dx2 d 2v EI 2 = −Px 23 Beam Deflection by Integration dx
Cantilever Example
¢ Isolating the variables and integrating
⎛ d 2v⎞ EI = −Px ⎝⎜ dx2 ⎠⎟ ⎛ dv⎞ Px2 EI = − + C ⎝⎜ dx ⎠⎟ 2 1
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Cantilever Example
¢ Integrating again
⎛ dv⎞ Px2 EI = − + C ⎝⎜ dx⎠⎟ 2 1 Px3 EIv = − + C x + C 6 1 2
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Cantilever Example
¢ To be able to define the function for v, we
need to evaluate C1 and C2 ⎛ dv⎞ Px2 EI = − + C ⎝⎜ dx⎠⎟ 2 1 Px3 EIv = − + C x + C 6 1 2
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Cantilever Example
¢ The right end of the beam is supported by a fixed end support therefore the slope of the
deflection curve is 0 and the deflection is 0 ⎛ dv⎞ Px2 EI = − + C ⎝⎜ dx⎠⎟ 2 1 Px3 EIv = − + C x + C 6 1 2
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Cantilever Example
¢ In terms of boundary conditions this means
⎛ dv⎞ Px2 EI = − + C dv ⎝⎜ dx⎠⎟ 2 1 x = L : = 0 dx Px3 EIv = − + C x + C x = L :v = 0 6 1 2
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Cantilever Example
¢ Evaluating the expressions at the boundary
conditions ⎛ dv⎞ Px2 dv EI ⎜ ⎟ = − + C1 x = L : = 0 ⎝ dx ⎠ 2 dx 2 2 PL PL x = L :v = 0 EI (0) = − + C1 ⇒ C1 = 2 2 Px3 PL2 EIv = − + x + C 6 2 2 PL3 PL2 PL3 EI (0) = − + L + C2 ⇒ C2 = − 6 2 3 29 Beam Deflection by Integration
Cantilever Example
¢ So the expression for the slope (θ) and the
deflection (v) are given by 1 ⎛ Px2 PL2 ⎞ dv θ = ⎜ − + ⎟ x = L : = 0 EI ⎝ 2 2 ⎠ dx P θ = L2 − x2 x = L :v = 0 2EI ( ) 1 ⎛ Px3 PL2 PL3 ⎞ v = − + x − EI ⎝⎜ 6 2 3 ⎠⎟ P v = −x3 + 3xL2 − 2L3 6EI ( ) 30 Beam Deflection by Integration
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Combining Load Conditions
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Example 1
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Example 1
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Homework
¢ Show a plot of the shear, bending moment, slope, and deflection curves identifying the maximum, minimum, and zero points for each curve. Use separate plots for each function. ¢ Show the mathematical expression(s) for each function. ¢ Problem P10.4 ¢ Problem P10.8 ¢ Problem P10.11 34 Beam Deflection by Integration
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