65002�

d

ELASTIC STRESSES AND DEFLECTIONS FOR FLAT CIRCULAR PLATES WITH UNDER UNIFORM PRESSURE

1. NOTATION

Both SI and British units are quoted but any coherent system of units may be used

diameter of plate m in

Young’s modulus N/m2 lbf/in2

von Mises-Hencky equivalent stress

=

N/m2 lbf/in2

, , principal stresses N/m2 lbf/in2

maximum total stress due to bending and membrane tension effects at centre

N/m2 lbf/in2

maximum total radial stress due to bending and membrane tension effects at edge

N/m2 lbf/in2

maximum total tangential stress due to bending and membrane tension effects at edge

N/m2 lbf/in2

ratio of equivalent stress of thick plate to equivalent stress of thin plate

, ,

,

ratios of stresses in plate edge elastically restrained against rotation to stresses in plate with edge fixed against rotation

ratio of deflection of plate with edge elastically restrained against rotation to deflection of plate with edge fixed against rotation

ratio of deflection of thick plate to deflection of thin plate

pressure on plate N/m2 lbf/in2

thickness of plate m in

maximum deflection of plate m in

rotational restraint provided by edge support N m/m rad lbf in/in ra

Convention: Tensile stresses are positive, compressive stresses arenegative. A negative sign in the figures denotes acompressive stress (for example, – .

D/t 4≥

D

E

fe

12--- f1 f2–( )2

f2 f3–( )2f3 f1–( )2

+ +

1/2

f1 f2 f3

fT

fTr

fTθ

kE

ke kT

kTr kTθ

kδ

k∆

p

t

δ

λ

fTθ )Issued September 1965

1

65002�

circular

its may

d

tes

ned from

nle in

2. INTRODUCTION

Curves are given that enable the maximum stresses and deflection to be determined for initially flat plates of uniform thickness under uniform pressure, with various edge conditions.

The curves apply only to stresses within the elastic range of the material.

For thin plates , (f/E)(D/t)2 and are plotted against (p/E)(D/t)4 on Figures 1 and 2, stresses,deflection and pressure being reduced to non-dimensional forms. Any consistent system of untherefore be used.

For thick plates , factors and kE are plotted against t/D on Figures 4 and 5 respectively.

For plates with edges elastically restrained against rotation, factors k on stresses and deflection are plotteagainst on Figure 3.

Table 2.1 indicates which figures should be used for a given edge condition.

3. USE OF THE CURVES

For thin plates , Figures 1 and 2 give values of maximum total stresses and deflection for plawith fixed and simply-supported edges respectively.

When the edge is elastically restrained against rotation, the values of stresses and deflection obtaiFigure 1 must be multiplied by factors ke , kT , kTr , and k obtained from 3 for the given value of

.

(For example, fT(elastic restraint) = fT(fixed) × kT.)

The curves of Figure 3 may be used for values of (p/E)(D/t)4 up to 5 when = 0 increasing to 30 whe. The derivation of the curves in Figure 3 assumes that the membrane stresses are negligib

comparison to the bending stresses. Beyond the limiting values of (p/E)(D/t)4 this is no longer true.Reference should then be made to Item No. 65003 for larger deflections of thin plates.

TABLE 2.1

Edge condition Number of figure giving

Rotation Translation fe fT fTr

fixed fixed or free 1 1 1 1 1 and 4

free fixed or free 2 2 – 2 2 and 4

elastic fixed or free1, 3

and 51 and 3 1 and 3 1 and 3

1, 3 and 4

D/t 20≥( ) δ/t

4 D/t 20<≤( ) k∆

λD/Et3

fTθ δ

D/t 20≥( )

kTθ δλD/Et

3

λλ ∞=

2

65002�ust bee

int, the

te.

tion is

.

For thick plates values of stress and deflection obtained by the method given mmultiplied by factors dependent upon t/D. Only in the cases of and fe do these factors exceed about fivper cent. Figure 4 gives the factor so that

(thick plate) = (thin plate) × .

Figure 5 gives the factor kE for so that

fe(thick plate) = fe(thin plate) × kE .

For , the factor kE = 1.0.

When plastic deformation sets in at the edge of plates with fixed or elastic rotational edge restrarestraint conditions rapidly approach those for a plate with its edge free to rotate.

4. LOCATION OF MAXIMUM TOTAL STRESSES

fT is a tensile stress on the unloaded face of the plate. fTr is a tensile stress on the loaded face of the plaTable 4.1 gives the location of ·

5. DERIVATION

6. EXAMPLE

A flat circular plate with its edge elastically restrained against rotation and fixed against translasubjected to a uniform pressure of 2000 lbf/in2 .

TABLE 4.1

Edge condition

Rotation Translation Plate face Stress

fixed fixed loaded tension

fixed free unloaded compression

free fixed unloaded tension

free free loaded compression

1. TIMOSHENKO, S. Theory of elasticity. Second edition. Chapter 13. McGraw-Hill, NewYork, 1957.

2. TIMOSHENKO, S. Theory of plates and shells. Second edition. Chapters 3 and 13McGraw-Hill, New York, 1959.

4 D/t 20<≤( )δ

k∆

δ δ k∆

λD/Et3 1.4≥

λD/Et3 1.4<

fTθ

fTθ

3

65002�

It is required to find the maximum deflection and total stresses in the plate if D = 2.0 in, t = 0.25 in. Therotational restraint along the edge is = 350 000 lbf in/in rad. E for the plate material is 30 × 106 lbf/in2 .

.

Figures 1 and 3 are used to find the total stresses.

From Figure 1, for a fully fixed edge and

From Figure 3, at

Hence,

.

Similarly,

Hence,

Hence,

λ

PE---

Dt----

4

0.273= , E tD----

2

469 000 lbf/in2,=

λE---

D

t3---- 1.49,=

tD---- 0.125=

pE---

Dt----

4

0.273,=

fTE----

Dt----

2

0.034.=

λE---

D

t3---- 1.49,=

kT 1.20.=

fT 0.034 1.20 469 000××=

19 130 lbf/in2

=

fTr

E-------

Dt----

2

0.052= and kTr 0.86.=

fTr kTr× 0.052 0.86 469 000××=

21 000 lbf/in2,=

fTθE

------- Dt----

2

0.016= and kTθ 0.54.=

fTθ kTθ× 0.016 0.54 469 000××=

4050 lbf/in2.=

4

65002�

Figures 1, 3 and 5 are used to find the von Mises-Hencky equivalent stress.

From Figure 1, for a fully fixed edge and

From Figure 3, at

From Figure 5, at

Hence,

This occurs at the edge of the loaded face.

Figures 1, 3 and 4 are used to find the deflection.

From Figure 1, for a fully fixed edge and

From Figure 3, at

From Figure 4, at

Hence,

pE---

Dt----

4

0.273,=

feE----

Dt----

2

0.046.=

λE---

D

t3---- 1.49,=

ke 0.90.=

t /D 0.125,=

kE 1.06.=

fe ke kE×× 0.046 0.90 1.06 469 000×××=

20 600 lbf/in2.=

pE---

Dt----

4

0.273,=

δ / t 0.0029.=

λE---

D

t3---- 1.49,=

kδ 1.42.=

t /D 0.125= andλE--- D

t3---- 1.49,=

k∆ 1.24.=

δ kδ k∆×× 0.0029 1.42 1.24 0.25×××=

0.00128 in.=

5

65002�

FIGURE 1 EDGE FIXED AGAINST ROTATION

p D 4

E t

10−1 100 101

δ/t

10−4

10−3

10−2

10−1

100

δ/t

( )

fTr D 2

E t

102

101

100

10−1

10−2

Edge fixed against translation

Edge free in translation

( ) f D 2

E t( )fT D 2

E t( )

fe D 2

E t( )

− fTθ D 2

E t( )

fTθ D 2

E t( )

6

65002�

FIGURE 2 EDGE FREE IN ROTATION

10−1 100 101

δ/t

10−4

10−3

10−2

10−1

100

Edge fixed against translation

Edge free in translation

δ / t

102

101

100

10−1

10−2

f D 2

E t( )

=fT D 2

E t( ) fe D 2

E t( )− fTθ D 2

E t( )

fTθ D 2

E t( )

p D 4

E t( )

7

8

65002�

101

Loaded face at edge

FIGURE 3

λD

Et3

10−3 10−2 10−1 100

k

−2.00

−1.00

0.00

1.00

2.00

3.00

4.00

5.00

Unloaded face at centre

kδ

kTθ

kTr

ke

kT

65002�

FIGURE 4 THICKNESS CORRECTION FACTOR FOR DEFLECTION

t/D

0.00 0.05 0.10 0.15 0.20 0.25

k∆

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

∞

5.0

0

0.1

0.2

0.5

1.0

2.0

λD

Et3

δ

9

65002�

FIGURE 5 THICKNESS CORRECTION FACTOR FOR EQUIVALENT STRESS

t/D

0.00 0.05 0.10 0.15 0.20 0.25

kE

1.0

1.1

1.2

1.3

f e

λD

Et3

--------- 1.4≥

10

65002�

gth of

mmittee

THE PREPARATION OF THIS DATA ITEM

The work on this particular Item was monitored and guided by the Stress Analysis and StrenComponents Committee which has the following constitution:

The Item was accepted for inclusion in the Structures Sub-series by the Aerospace Structures Cowhich has the following constitution:

The members of staff of the Engineering Sciences Data Unit concerned were:

ChairmanMr H.L. Cox – National Physical Laboratory

MembersProf. J.M. Alexander – Imperial College of Science and TechnologyMr C.E. Day – National Coal BoardMr J.R. Dixon – National Engineering LaboratoryDr H. Fessler – University of NottinghamMr N.E. Frost – National Engineering LaboratoryDr R.B. Heywood – A. Macklow-Smith LtdMr M.J. Kemper – A.P.V. Company LtdMr M.J.M. Raymond – Imperial Chemical IndustriesDr R.T. Rose – John Thompson LtdMr G.P. Smedley – Lloyd’s Register of ShippingMr J. Spence – Babcock and Wilcox Ltd.

ChairmanMr F. Tyson – Handley Page Ltd

MembersMr H.L. Cox – National Physical LaboratoryMr K.H. Griffin – College of AeronauticsProf. W.S. Hemp – Oxford UniversityMr H.B. Howard – IndependentDr E.H. Mansfield – Royal Aircraft EstablishmentMr P.J. McKenzie – Hawker Siddeley Aviation LtdMr I.C. Taig – British Aircraft Corporation LtdMr A.W. Torry – Hawker Siddeley Aviation LtdMr A.J. Troughton – Hawker Siddeley Aviation Ltd.

Mr E.R. Welbourne – Head of Solid Body Mechanics GroupMr R.L. Penning – Solid Body Mechanics Group.

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