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 5
Continue with the annular plate from problem 4. To optimize the design, you would like to specify that the maximum deflection in the center be at most 10% of the plate thickness. You could use the plot generated in Problem 4 to estimate the thickness at which this occurs. There's another, more precise method that TK Solver  allows. You can simply add another equation.
 
Open the Rule Sheet and enter the following equation on an open row:

yb = -t/10

 
The Rule Sheet now looks like the following:
 

 
Make the Variable Sheet active again. If you try to solve at this point, TK Solver reports an inconsistency. This is because you added a new constraint without adding an unknown. Let's make t a guess variable by placing a G in its Status field. Solve.

After a few iterations, success! If you make the plate at least .37 inches thick, the deflection will be ok.

 

Variable Sheet

St

Input

Name

Output

Unit        

Comment

 

 

 

 

 

Case 1 - Annular plate with a uniform

 

 

 

 

 

annular line load - Page 400

 

 

 

 

 

 

 

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 -

 

Material name

 

'n

plot

 

 

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

L

 

t

3.6674E-1

in

Plate Thickness

 

 

D

134221.559

lbf-in

Plate Constant

 

 

 

 

 

 

 

 

 

 

 

AT RADIUS:

 

6

r

 

in

 Radius

 

 

y

-.0276359

in

 Deflection

 

 

th

2.74799E-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

3263.034

psi

 Radial Bending Stress

 

 

sigma_t

3449.494

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:

L

 

yb

-0.0366745

in

 Deflection

 

 

thb

.0012202

rad

 Radial Slope Angle

 

 

Mrb

0

lbf-in/in

 Radial Bending Moment

 

 

Qb

0

lbf/in

 Shear Force

 

Technical Time Out

You probably noticed that the Rule Sheet for Problem 5 includes more than equations. There are conditional and logical rules along with function calls. Take a few minutes to investigate these.
 

 
Rule 1 checks several of the dimensions to make sure the problem is feasible as described.

Rule 2 checks the thickness of the plate in comparison with the radius. Taken to an extreme, as a plate gets thicker and thicker, it becomes a column and the formulas are no longer accurate.

Rule 3 computes the plate constant.

Rule 4 is a call to the function get_tab, with four variables involved. This function performs the material table look-ups. You can open the get_tab subsheet directly from the Rule Sheet by placing the highlight over the rule and clicking the right mouse button.

 

 
These rules assign given values of mat1,E, and nu if they exist. If not, they are taken from the lists in the property table. Close the subsheet.

Rule 5 calls the case function, which includes the specific case equations from the book. Open the subsheet and the book and compare.

 

 
The case formulas include the C and L factors from the book. These are placed in functions in TK Solver called cal_LG and cal_C. Open these subsheets and compare the functions with those in the book. Close them again when you are finished. Also, close the case function and return to the Rule Sheet.
 

 

 
Rule 6 calls the load function, which includes the general Case 1 load formulas from the book. Again, these formulas reference factors from the book.
 

 
Rule 7 and Rule 8 computes the radial and tangential stress.

Rules 9-11 generate the lists used in the various plots.

TK Solver's unique ability to maintain and merge functions helped us avoid repetitive entry and storage of formulas that in turn has helped us in maintaining the software as periodic corrections to the book are made.