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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 

Image 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

 

 

Table 8.5, Case 5: Externally created concentrated angular displacement 

Notation: W=load (force) etc... 

Image 

Transverse Shear: Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Bending Moment: Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Slope:                 Typesetting:-mrow(Typesetting:-mi( 

Deflection:          Typesetting:-mrow(Typesetting:-mi( 

 

Case 5d:  

Left end fixed. Right end free 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

Section Modulus 

Loading in the y direction, therefore use Ix: Typesetting:-mrow(Typesetting:-mi( 

 

 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi( 

where: Typesetting:-mrow(Typesetting:-mi( 

Note: if Typesetting:-mrow(Typesetting:-mi(", 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 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Typesetting:-mrow(Typesetting:-mi( = 3.648266667 

Typesetting:-mrow(Typesetting:-mi( = 0.2380e-1 

Young's Modulus: Typesetting:-mrow(Typesetting:-mi( 

Section Modulus: Typesetting:-mrow(Typesetting:-mi( = 76248773.34

Beam Data 

Foundation Modulus: Typesetting:-mrow(Typesetting:-mi( 

Beam Length: Typesetting:-mrow(Typesetting:-mi( 

Beam width: Typesetting:-mrow(Typesetting:-mi( 

 

Distance from left edge to displacement: Typesetting:-mrow(Typesetting:-mi( 

 

Initial angular displacement: Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mi( = 5.454660534Note: if Typesetting:-mrow(Typesetting:-mi(", mathvariant = "normal", fence = "false", separator = "false", stretchy = "false", symmetric ..." align="center" border="0">use Table 8.6 instead. 

 

Results 

Typesetting:-mrow(Typesetting:-mi( = 0.11e-1Typesetting:-mrow(Typesetting:-mi( = -0.5208796198e-2Typesetting:-mrow(Typesetting:-mi( = -10227.93106Typesetting:-mrow(Typesetting:-mi( = 0.1145695164e-2

 

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

 

Transverse Shear 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

 

Slope 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

 

Bending Moment 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

 

Deflection 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

 

Maximum Bending Moment 

 

Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

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: Typesetting:-mrow(Typesetting:-mi( = 5.963356970 

Maximum Bending Moment: Typesetting:-mrow(Typesetting:-mi( = -430807.0526 

 

Maximum Shear 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

Typesetting:-mrow(Typesetting:-mi( = 4.518490457Typesetting:-mrow(Typesetting:-mi( = 7.428666803Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

 

Typesetting:-mrow(Typesetting:-mi( 

Maximum Transverse shear: Typesetting:-mrow(Typesetting:-mi( = 166200.1237 

 

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:
 

Typesetting:-mrow(Typesetting:-mi( = Float(undefined) 

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. 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( 

Plot_2d
 

Typesetting:-mrow(Typesetting:-mi( = -0.9383600211e-2

Back-Solving 

 

Example: What applied angular deflection, Typesetting:-mrow(Typesetting:-mi(would give us a maximum deflection of 5 mm? 

 

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( = 0.1118973499e-1 

 

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