An Example Deflection of a Simply Supported Beam
A simply supported beam (a beam supported at the ends) is bent downwards by the applied load, consisting of the weight of the beam itself plus any other loads.
Figure 111. Diagram of a simply supported beam.
The simply supported steel beam shown in Figure 111 supports a uniformly distributed load of 2000 lb/ft. The length L of the span is 30 feet. The deflection (downward bending displacement) y of the beam as a function of distance x along the span of the beam is given by the secondorder differential equation 111, known as the general equation of the elastic curve of a deflected beam.
M, the bending moment at distance x, is given by equation 112
where L is the length of the beam and w is the weight of the beam per unit length. E is the modulus of elasticity of the beam material; for carbon steel, E = 2.9 x 107 psi, and I is the moment of inertia of the cross section of the beam, given by equation 113.
where b is the width and h the height of the beam cross section. In this example, for a beam 6 in wide x 16 in deep, /= 2048 in4.
Equation 111 can be transformed into the two equations
ax dz M
dx EI
where z is the slope of the beam.
We want to calculate the amount of deflection of the beam at the center of the span. Since the deflection is known to be zero at either end of the beam (y = 0 at x  0 and y = 0 at x — 30), this is a boundary value problem. We will solve it by using the shooting method. We set up the problem as though it were an initialvalue problem, with two "knowns" given at the same boundary, x = 0 in this example. The two known values are the value of y at x = 0 and a trial value of z at x = 0.
The spreadsheet used to solve the problem is shown in Figure 112. To ensure consistency in units, all dimensions have been converted to inches. The values of y along the beam were calculated at increments of 2 inches (rows 13182 are hidden). For simplicity, the values of deflection y and slope z in rows 6 through 185 were calculated by using Euler's method; the formulas in cells B6 and C6 are, respectively,
1 
A 
B 
c 
D 
E 
Beam Deflection Calculated hy Using the Shooting Method  
2 
(Calculations Performed by Using Etilei Method)  
3 
(all quantities must be in inches)  
Distance 
Deflection  
M 
y 
Siope 
Bending  
4 
(in) 
(in) 
z=dy/dx 
moment M 
dz/dx=M.'EI 
5 
0 
0.0000 
0.00000 
o 
0 
6 
2 
0.0000 
0,00000 
59667 
1.00E06 
7 
4 
O.DOOO 
0.00000 
118667 
2.00E06 
8 
6 
0.0000 
0.00001 
177000 
2.98E06 
9 
8 
o.cooo 
0.00001 
234667 
3.95E06 
10 
10 
o.cooo 
0.00002 
291667 
4.91 E06 
11 
12 
0,0001 
0.00003 
348000 
5.86E06 
12 
14 
0.0001 
0,00004 
403667 
6.80E06 
183 
356 
1.6984 
0.01090 
118667 
2.00E06 
184 
358 
1,9202 
0,01091 
59667 
1.00E06 
185 
360 
1.9420 
0.01091 
0 
o : 
Figure 112. Simulation of beam deflection by the shooting method. The boundary values of the deflection and the initial trial value of the slope are in bold. Note that the rows between 12 and 183 have been hidden, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
Figure 112. Simulation of beam deflection by the shooting method. The boundary values of the deflection and the initial trial value of the slope are in bold. Note that the rows between 12 and 183 have been hidden, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
G 
JL 
______H__L 
I J  
5 
Trial 
z 
y  
rr 
1 
0 
1.9420198  
JL 
21 
0.1 
34.05798  
8 

3: 
0.0053945 
Toi 
Figure 113. Calculating the boundary condition by linear interpolation, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
Figure 113. Calculating the boundary condition by linear interpolation, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
With a trial value of z = 0, the value of y calculated at x = 360 is not zero, but 1.9420. We will now proceed to vary z in order to make y = 0. One method that can be used to find the correct value of z is to calculate two values of y at the upper boundary (x = 360), using two trial values of z at the lower boundary (x = 0), and then calculate an improved value of z by using linear interpolation to find the value that makes y — 0. Here, the trial values of z (the slope of the beam) that were used were zero and 0.1. These values of z were entered in cell C5; the resulting values of y that were obtained at x = 360 (in cell B185) are shown in Figure 113.
A 
8 
c 
L D 
E  
1 
Beam Deflection Calculated by Using the Shooting Method  
2 
(Calculations Performed by Using Enter Method)  
3 
(all quantities must be in inches)  
Distance 
Deflection  
X 
y 
Slope 
Sending  
4 
(in) 
m 
z=dy.'dx 
moment M 
dz/dx=WEi 
5 
0 
0.0000 
0.00539 
0 
0 
6 
2 
0 0108 
0.00539 
59667 
1.00E06 
7 
4 
0.0216 
0.00539 
118667 
2.00E06 
8 
6 
0,0324 
0.00539 
177000 
2.93E06 
9 
8 
0.0431 
0.00538 
234667 
3.95E06 
10 
10 
0.0539 
0.00537 
291667 
4,91 E06 
11 
12 
0.0647 
0.00536 
348000 
5.86E06 
12 
14 
0.0754 
0.00535 
403667 
6.80E06 
183 
356 
0.0220 
0.00551 
118667 
2.00E06 
184 
358 
0,0110 
0.00551 
59667 
1.00E06 ; 
185 
360 
0.0000 
0.00552 
0 
0 ; 
Figure 114. Simulation of beam deflection by the shooting method. The final boundary values and the final value of the slope are shown in bold, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
Figure 114. Simulation of beam deflection by the shooting method. The final boundary values and the final value of the slope are shown in bold, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
The calculated value of z for the required boundary value is shown in the third row of the table. The formula in cell H8 is
If the problem is linear, the interpolated value of z obtained in this way will be the desired solution. The spreadsheet with final values is shown in Figure 114. A similar spreadsheet in which the y values were calculated using the Runge custom function can be seen on the CDROM.
This "shooting" procedure was performed manually—that is, successive trial values were entered into the spreadsheet, and the resulting values copied and pasted into the cells shown in Figure 113, in order to use interpolation to find the final value. You can obtain the same final result essentially in one step by using Goal Seek. After entering a trial value, z = 0, in cell C6, use Goal Seek to change cell C6 to make the target cell, B185, attain a value of zero.
The final results are shown in Figures 114 and 115. The maximum deflection, at the midpoint of the beam, is 0.6138 in, within the allowable deflection limit of 1/360 of the span. For comparison, the analytical expression for the deflection at the midpoint of the span, 5wi4/3842?7, yields 0.6137 in.
Figure 115. Beam deflection calculated by the shooting method, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
0 60 120 180 240 300 360 Distance, in
Figure 115. Beam deflection calculated by the shooting method, (folder 'Chapter 11 Examples', workbook 'ODEBVP', worksheet 'Beam deflection (Euler)')
Post a comment