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# Roark and Young Calculation Sheet: Transverse Shear, Slope, Bending Moment and Deflection along a Beam

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Roark and Young Calculation Sheet: Transverse Shear, Slope, Bending Moment and Deflection along a Beam

? Maplesoft,  2006

Problem Definition: Table 8.5, Case 5

 > restart; mf := (x, a, n) -> (x-a)^n*Heaviside(x-a): calc:=(expr,val)-> evalf(eval(expr,val)):

Section Properties: Appendix A.1, Case 6: Wide-flange beam with equal flanges

 Area and distances from centroid to extremities Moments and Products of inertia and radii of gyration about central axes

Table 8.5, Case 5: Externally created concentrated angular displacement

 Transverse Shear:   Bending Moment:   Slope:                   Deflection: Case 5d:   Left end fixed. Right end free Section Modulus Loading in the y direction, therefore use Ix:

where:

Note: if ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric ..." align="center" border="0">use Table 8.6 instead.

Solution

Enter numeric design parameters

 Section Properties      =   =   Young's Modulus:   Section Modulus: = Beam Data  Foundation Modulus:   Beam Length:   Beam width:     Distance from left edge to displacement:     Initial angular displacement:     = Note: if ", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric ..." align="center" border="0">use Table 8.6 instead. Results  = = = =

Plots of Transverse Shear, Slope, Bending Moment and Deflection along the beam

Transverse Shear

Slope

Bending Moment

Deflection

Maximum Bending Moment

The Maximum Bending moment occurs at the load point (x=a), which cannot be solved algebraically (inspection of the expression for diffM shows there is a Dirac function at that point which may account for this. Also, inspection of the deflection plot shows a discontinuity at this point). Therefore, we will need to solve numerically using the fsolve() command.

Distance for Maximum Bending Moment: =

Maximum Bending Moment: =

Maximum Shear

= =

Maximum Transverse shear: =

Maximum Deflection

By inspection, the maximum deflection as at the point of applied displacement, a. So we should be able to calculate the maximum displacement where x=a:

=

So, what happened? Things become clear when we plot the derivative of the displacement. There is a discontinuity at x=a that is mathematically undefined, so we need to evaluate numerically. In fact, if we overplot the Bending Moment, we see that the maximum bending moment is also at x=a, so we can use the distance, xMmax, that was found using fsolve() earlier.

=

Back-Solving

Example: What applied angular deflection, would give us a maximum deflection of 5 mm?

=

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