Interactive Roark’s Formulas: Five Problems and Solutions

Let's load a case from the Interactive Roark's Formulas application and solve a problem. Along the way, you will learn about many of the software's features.

 

Problem 4
Let's switch from beams to plates and learn more about how Interactive Roark's Formulas on TK Solver works. The next problem involves a flat annular plate with uniform annular line load, fixed on the outside and free in the center.

Using the Applications Menu in TK Solver, select Interactive Roark's Formulas on TK Solver and load Table 11.2, Case 1e.

 

 
Enter the following inputs into the Variable Sheet:
 

uniform annular line load (w)

30 lbf/in

radius to annular line load (r0)

6 inches

material number (matnum)

20

outer radius of the plate (a)

18 inches

inner radius (b)

1 inch

plate thickness (t)

.25 inch

radius (r)

6 inches

 
Solve.

The Variable Sheet will appear as shown:

 

Variable Sheet

St

Input

Name

Output

Unit        

Comment

 

 

 

 

 

Case 1 - Annular plate with a uniform

 

 

 

 

 

annular line load

 

 

 

 

 

 

 

30

w

 

lbf/in

Uniform annular line load

 

6

r0

 

in

Radius to annular line load

 

 

 

 

 

 

 

 

 

 

 

Table 11.2: Roark's Formulas

 

 

 

 

 

Formulas for flat circular plates of

 

 

 

 

 

constant thickness

 

 

case

'Case_1e

 

Reference number

 

 

caution

'_

 

Caution Message

 

 

caution

'_

 

 

 

 

 

 

 

 

 

20

matnum

 

 

Material Number (See Material Table)

 

 

matl

“Steel - s

 

Material name

 

 

plot

'y

 

Generate plots? 'n=no (Default=yes)

 

 

 

 

 

 

 

 

E

3E7

psi

Young's Modulus

 

 

nu

.285

 

Poisson's ratio

 

 

 

 

 

 

 

18

a

 

in

Outer Radius

 

1

b

 

in

Inner Radius

 

.25

t

 

in

Plate Thickness

 

 

D

42515.85

lbf-in

Plate Constant

 

 

 

 

 

 

 

 

 

 

 

AT RADIUS:

 

6

r

 

in

 Radius

 

 

y

-.0872459

in

 Deflection

 

 

th

8.67535E-3

rad

 Radial Slope Angle

 

 

Mr

73.147

lbf-in/in

 Radial Bending Moment

 

 

Mt

77.327

lbf-in/in

 Tangential Bending Moment

 

 

Q

0

lbf/in

 Shear Force

 

 

sigma_r

7022.136

psi

 Radial Bending Stress

 

 

sigma_t

7423.401

psi

 Tangential Bending Stress

 

 

 

 

 

 

 

 

 

 

 

AT OUTER EDGE:

 

 

ya

0

in

 Deflection

 

 

tha

0

rad

 Radial Slope Angle

 

 

Mra

-80.65

lbf-in/in

 Radial Bending Moment

 

 

Qa

-10

lbf/in

 Shear Force

 

 

 

 

 

 

 

 

 

 

 

AT INNER EDGE:

 

 

yb

-.1157804

in

 Deflection

 

 

thb

3.85214E-3

rad

 Radial Slope Angle

 

 

Mrb

0

lbf-in/in

 Radial Bending Moment

 

 

Qb

0

lbf/in

 Shear Force

 
Open the Plot Sheet and view the graphical output by highlighting each of the plot names and pressing F7. You may want to close the plots after viewing them to avoid cluttering the screen.
 

 
The main design concerns appear to be at the center of the plate. This is where the deflection and tangential stress are maximized. In fact, there may be too much deflection. You can investigate how changes in the plate thickness will affect the deflection in the center. Specifically, you might like to generate a plot of the center deflection (yb) versus the plate thickness (t) as t goes from .25 to 1 in steps of .05.
 
Use the List Solving Wizard in TK Solver for this. Start the wizard by picking it from the Wizards menu in the Menu bar. Follow the instructions in the opening screen, then click Next.
 
The next screen prompts you for independent (input) and dependent variables. We are plotting the relationship of t, which is the input variable, and yb. Type t in the input variable field and select yb from the list of output variables, as shown below.
 

 
Click Next. The next screen brings up the List Fill dialog box shown below. Enter the values shown to create a list for t.
 

 
Click Next. With the next screen, shown below, TK Solver asks whether you want to build a plot and-or a table, which displays the two lists in tabular form. Let's do both. Click the check boxes, and name the table and the plot ybcheck, as shown.
 

 
Click Next. TK Solver repeatedly solves the equations for each of the t values and stores the resulting yb values in a list. The resulting plot and table are shown below.
 

 

 
As you have seen, list solving can be used to study the relationships between any of the variables in a model and the results can be displayed in plots or tables. This capability is in addition to the default plots that are automatically generated by the software.

Before proceeding to the next problem, close all the sheets except the Variable Sheet.