Analysis of the stresses in the corner zone of a C-frame press

Analysis of the stresses in the corner zone of a C-frame press

Citation for published version (APA):

Singh, U. P., Veenstra, P. C., & Ramaekers, J. A. H. (1979). Analysis of the stresses in the corner zone of a C-frame press. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0447). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979

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WT-rapport Nr. 0447

Analysis of the stresses in the corner zone of a C-frame press.

U.P. Singh, Eindhoven University of Technology/NL; P.C. Veenstra, Eindhoven University of Technology/NL (1); J.A.H.

Ramaekers, Eindhoven University of Technology/NL.

Summary: It is observed in practice that the corner zone of a press-:frame is highly sensitive to rupture-failure. The extant classical methods to evaluate the stressed state of the corner zone appear to be inadequate to predict such a fa i 1 u reo

The appl ication of both the finite element method and the finite difference method proves to be laborious and rather expensive. By means of the application of the theory of the thin walled-curved beam with large curvature it is possible to analyse the stresses in a corner zone with fair reliability.

2

1. I NTRODUCT I ON

Both experimental and theoretical works conducted by several investigators [1, 2, 3J confirm that the corner zone of a press frame is the most heavily stressed section. The actual stresses in this section are considerably higher than those calculated by conventional methods of calculation.

The appl ication of the finite element method using beam elements in analysing the stress-deformed state of a corner zone does not provide results with adequate accuracy to meet the present

demand of modern press designs. Although the appl ication of the finite difference method as well as the finite element method using more sophisticated elements such as rectangular,

tetrahedronal or rectangular-hexahedronal etc. can provide more accurate results [4, 2J, the use of these methods is restricted in general usuage due to cost of labour and computation. Hence simpler approaches for reasonably reT iable estimation of the stresses at corner zones are needed.

In view of the practical appl ication an attempt is made in the present paper to analyse the stresses at the corner zone of 20 x 10

4

kN-C frame press (fig. l.a) by thin walled-curved beam theory and compared with other methods I ike the classical

straight beam (conventional) and the finite beam element. Based on both the theoretical analysis and experiments, it is

shown that this approach viz., the theory of the thin wall-curved beam with large curvalure provides results which agree better with practice.

It will be shown that the influence of deformation of the contour of cross-sectional profile is accounted for if the

actual flange width (2a) is replaced in such a way that the

elementary theory of bending appl led to such a transformed section (equivalent cross section) gives the correct value of maximum bending stress [5J. 2~

____________

~ corner zone l(AI-column - - H - _ o ·i~3=50~--.j.l-..:r

liable

600

Fig. l.a. Schematic view of 20 x 10

4

kN C-frame press.

LO m

N ..c:.

A-A

2 0 2 -z 1 + - - - -201 - - - - . . . ; - - + - 4 _ _ . _

-1. b.

Fig. l.b. Cross-sectional view of the corner zone. 2. ANALYSIS

1.c.

Fig. l.c. Equivalent cross-section of

2.1. Formulation of the problem

For the study of the corner zone the cross-section as mentioned in fig. l.c. and subjected to concentrated load is relevant. Basically the following is assumed:

1. \oIhen circular plates (web and ribs) are subjected to symmetrical loading, they may be analysed as per theory, referred to in [6].

2. Since the condition RZ ~ Rl + _{(H 1} + H2)/2 holds (as shown in fig. 2), the shape of the outside edge has vertually no influence on the stiffness [7J.

3.

Since the thickness of the cyl indrical plate (flange) is much smaller than the overall height of the cross-section, the situation is one of pure bending for analysing the plate in the t-directlon (as shown in fig. 4.b).

Fig. 2. The shape of the outside edge of an angular element according to Kaminskaya.

With the above mentioned assumptions the object of the study would be:

a. to obtain an equivalent cross-section from the equil ibrium conditions of the coupling at the junction between the flanges and the web,

b. to determine the stresses in the flanges from the theory of cy 1 i nd rica 1 she II s ,

c. to determine the stresses in the web from the theory of symmetrical bending of circular plates and

d. finally to apply the results so obtained to analyse the stressed-deformed state of the corner zone.

6

2.2. The model and evaluation

According to Panamarov-Biderman [8J for the case of pure bending, the circular plate (web) is considered to be loaded through the cyl indrical plates (flanges) which in turn are assumed to be cylindrical shells as shown in fig. 3.

x R

=

the radius of curvature. h

=

thickness of the flange. n

=

distance between the central layer of the flange and the neutral axis of the cross-section.

Fig. 3.a. Spatial view of the I-sectional profile.

Fig. 3.b. View of cylindrical plate (flange) element before deformation.

Fig. 3.c~ View of flange element after deformation

Under the action of an external load a local displacement w of a cylindrical plate occurs in radial direction. The strain of acyl indrical plate in tangential direction is written as:

n + z ~de w

€t

=

R + z . ~ - R + z

Since the thickness h of the cyl indrical plate compared to nand R is small, the equation (1) can be written as:

n AdS w (

€t =

R .

~

- R

2)

The strain of the cyl indrical element in axial direction is represented by:

K' L I - KL

£ :::

X KL

The axial and tangential stresses in the cross-section of the

cy lind

rica:

x

e

~

erne;: :::

+:::~r:i

ne: 2

a(

+ z

d2~

+v(; .

_{'adda _}

~

)]

l-v l-v

L

dx

(4)

E ( ) E

[11

fid

e

w

crt ::: - - 2 E:t+v£x

=

- - 2 R'ere - R + \IE: + vz --Y!.. d

2

J

dx2

1-v 1-v 0

(5)

where £0 is the strain in the central layer.

The forces per unit length on the cyl indrical element are found to be: h/2 Eh [ ( n 'd

e

w ) ] T :::

-h~2

cr dz

_{::: 1-}}

_{£0 + \) R .}

_{de -}

_R x x

(6)

h/2 Eh

[11

fide w

J

T t ::: _-h/2

S

crtdz ::: - - 2 _l-v R'

de -

R -v£o (7)

and the corresponding moments per unit length are:

h/2 d2 M ::: ~ cr zdz::: D --Y!.. ' (8) x -h/2 x dx2 h/2 d2 M_t :::

S

crt zdz ::: vD ~ (9) -h/2 dx Eh2 '.

where D ::: 2 IS the flexural rigidity of a flange element.

12(1-v )

E

and v are the modulus of elasticity and the poisson ratio of the material of the element, respectively.

When considering the equilibrium of the element as shown in fig.

4.b.

with respect to forces along x and z-axes and the moments about t-axis, the following is obtained

.: 0 ::: -V

(* .

fia d

e

8 -

*)

(

1 0 ) ( 11 fide

w)

T_{t :::} Eh

R'

'"'Cl'8 -

R

(11) E d2w 12 Mx (1 ::: - - Z :: Z (12) x 1-v2 ' dx2 h3 T t 12 Mt (1t="h+ h₃ z (13) d2Mx 1 dx2 :::

R

T t ( 14 )

8

-+1

L J

.~

-dx

....

dw dw + d2w dx

CJX

Ox

ax

- _ . - - I t - - - 4+ ,

-@

Fig. 4.a. Deformation of the cyl indrical plate element.

Z'

X'

z !~'--_ _ t'

. Substituting the expressions (8) and (11) for Mx and Tt into equation (14) results in:

Eh

3

d

4

w :-.

.1

Eh

(21

tide -

~)

(15) 12(1-v2) • dx4 R R • de R

The expression in equation (15) can be written differential equation which gives the relative cylindrical plate as follows:

in the form of a deformation of the where 4 ~ + 4 k4w

=

4 k4 6de • 't n de dx k =- 4 3(1-}) R2 h2 ( 16) (17) The general solution of the equation (16) is given as [8J:

6de

w

=

n (fIf + C₁

v

₁ (kx) +

e

2V2(kx) + C3V3(kx) +

e

4V4(kx)

( 18) where

e

_l,

e

2, C3 and C4 - constants of integration,

V1' V2' V3 and V

_{4 -}

functions of Krilov which are defined as:

V 1 (kx)

=

ch(kx) cos (kx) ; V 2(kx)

=

t

[ch(kx) sin (kx) + sh(kx) cos (kx)]; V 3(kx) :::

t

sh(kx) sin (kx) ; V₄(kx) :::

*

[ch(kx) sin (kx) - sh(kx) cos (kx)]

2.3.

~~l ication of the model

As a definition for the equivalent flange width it is introduced: 2aeq .

=

K 2a

Where K and 2a are the width reduction factor and the actual flange width respectively.

By setting the origin of the coordinates at the periphery of the inside flange (point 0 in fig. 1.b) the following boundary

conditions are obtained: For x ::: 0:

10 ( 2 ) d

w,

(Mx₁ )x=O = - 2 - ::: 0 dx x=Q ::: 0 (20)

=

0 For x = a_{1 ,2:} (21) 2.3.2. ~Ql!:!!lQ!]

From equations (20) and (21) it follows:

C

31

=

C32

=

C41 ::: C42 ::: 0

Hence the expression (18) for displacements of outside and inside flanges can be written as:

The forces per unit length on the inside respectively are determined as:

(nl t.de W1)

Ttl = Ehl ~.

de -

R, ;.

(n2 lide

W2)

T t2 ::: Eh2 ~. de - ~

and outside flanges

The total forces acting on the inside and outside flanges respectively are:

=

2fll t.de Ft ~ T_t dx

=

E R de n,h,2K,a,

(22)

(23) 1 1 1

=

2132 t.de (24) F_t J_ _{Tt dx::: E R} 2de n2h22K2a2 2 0 2

where 1 sh (2k 1a1) + sin (2k 1a1) K,

=

k,a 1 2+ch(Zk1a,) + cos(2k,a1) 1 sh (2k 2a2) + sin (2k2a2) K2

=

k 2aZ Z+ch(2k2a2) + cos(2k_, 2a2} (25)

where K, and K

_z

are the width reduction factors of the inside flange and outside flange respectively.

The total forces acting on the web and the ribs respectively are determined as:

=

h3

E

~de

b;n,

z dz = h

E

t.de

~b-

In

R2) ;

de J p +z 3 de Pn

R,

-11

,

n (26) -11 "de

f.

Rl)

F~

=

2h_q E

~ddee

~' ~ ~~

= 2h_q ETe\b,-P_n lnlf t

z

-(b 1+11,) n

band h3 are the height and thickness of the web respectively, whereas b, and hq are those for the- rib.

Since the normal force on the cross-section, in case of pure bending, is absent, it holds:

~ lI! ~de

fn,

11 2

Ft,+Ft2+Ft,+Ft2 = E

CRl

LR7

h,2K,a, + ~ h22K2a2 +

+h3 (b-Pn In

:~)

+ 2h4 (bl-Pn In R

i)]

=

0 (2]) The equation (27) allows to determine the radius of curvature of

the neutral axis as follows:

(28)

The bending moment acting on the cross-section is determined as:

lI! where O"t

,

b-n M

₌

F _t

n,

₊ F _{t 112 +} h

₃

(1 _J - , 2 - ~ lI! 0"

t,

Z dz + -11, lI!

S

O"t z dz -(b,+11,) 2 (29)

and O"lI! are the stresses in the web and the rib resp.

12

(30)

For a curved beam with large curvature it holds:

M

=

E 8

a

dee s

\-/here S =

f,

the modified static moment of the cross-section with respect to the neutral axis.

Analogeous to this, for a curved beam with large curvature when taking into account the deformation of cross-section contour it is proposed:

M

=

E 8de seq

de (31)

From equations (30) and

qn

it fol.lowr

seq - h 12K1a 1 ;: +

h22~a2 ;~

+ h3 b [

~

-n₁+Pn

t~

-

~J

+

+ 2h'4b

rp

(Pn -

1) -

~

-

n

J

(32) 1 ~ n b

1 2 1

Equation (32) represents the modified static moment of the equivalent cross-section (fig. l.b) with respect to its neutral axis.

When analysing the stresses in the flanges due to bending the maximum moments Mx and M_t at the points of junction of the flanges and the web are determined as:

The stresses in the middle layer of the inside and outside flanges respectively are: T tl o =

-t,o

hl T t

(34)

Z °t 20

=

~

The stresses due to bending in the inside and outside flanges are: o _{= 6Mt1}

=

v 6(Mx)x=al

t,.b

h~

h~

o

=

_6Mt2 == v 6 (r'1) x=a2

t2 • b hZ h2

Z Z

Substituting the quantity

E 6::

through equation

(3')

into equations

(34)

and

(35)

and solving it through equation

(33),

the total stresses in the outside and inside flanges are found to be: \/here

ttlJx=a1

T t 6(M ) M n ] 1 x x=a, 1

r

= - + v = - - l + v K hl h2 SeqR 0₁ 1 1

~)

Ttz 6(M)x=a _{M nZ}

r ]

(J

= -

+ v 2

=

1 +vK t2x x=a h Z hZ Seq RZ ' "\02 2 Z ch(Zk

1a,) - cos (Zk1a,) 2+ch(2k,a 1) +cos(2k1a1) ch(Zk Za2) - cos (2k2a2) 2+ch(2k 2a2) + cos (2k2a2) (36)

The total maximum tensile stress in the web is composed of the stress in the central layer of the web and the stress due to bending moment Ma at the junction point of the web and flange. The stresses in the central layer of the web are determined as:

~ M (p n - r)

°t.O

=

seq • r

where r is the radius of curvature of the fibre considered. The stresses due to bending moment Ma are determined as:

14

(40)

The net maximum stress in a cross-section of the corner zone is determined as the sum of the stresses due to bending and due to tension:

Ci

=

max (41)

where M.

=

F (a + p . cos ~.) and N.

=

F cos W.,

I C CI I I I

Ai is the cross-sectional area of the i-th section of the corner zone and M., n., s~q, K . and N. correspond to that cross-section.·

I I I Ci I I P

ci is the radius of curvature of the i-th centroidal line of the cross-section.

3.

RESULTS AND CONCLUSIONS

,

,

1

5 3 2 A - Measurement 1 -Conventional method 2 -F.EM,

3 -Method using the theory of thin

walled curved beam

IPloegr",,)

Fig. 5.

Maximum stress

(01

at mid-point C (refer fig. l.b) of the inside flange of the corner zone as a function of angle cp.

1. Fig. 5 shows the relationship between the net maximum

stress at the midpoint C (refer fig. l.b) by different methods viz., conventional (method 1), the finite beam element (method 2) and thin wall-curved beam as a function of the angle cp. The angle cp describes the position of the cross-section of the corner zone with respect to plane 1-1. It can be seen that the method 3 agrees better with measurements than the methods 1 and 2. This is due to the fact that the method 3 takes into account the deformation of the contour of the

cross-sectional profile, whereas methods 1 and 2 neglect this effect.

2 @-for ~ =Q93 15 30 X-06 _{a - .} x-corrected in x-direction 45 from fig. 7 A-measured -method 3 60 75 90 Fi g. 6. Maximum stress (0) for the rela-tive distance

(x/a=0.6)

from the midpoint C (refer fig. l.b) as a function of angle fj). - - -... 'Ptdegreel

2. Fig. 6 shows the ditribution of the net maximum stresses for points lying along the entire length of the corner zone at a distance of

77

mm from the midpoint of the inside flange

(x/a

=

0.6). In view of the fact that the measuring points for cp

=

00 and ql

=

7.5

0 are located at 120 mm from the

midpoint C (refer fig. l.b) and for fj)

=

85

0 at a distance of

77

mm, the measured stress values corresponding to ql

=

00 and

to

0.6 16 1::. _ _ _ 1::. Measured 2 _ _ method 3 0.2 0 0.2 _{0.6 x 10}

--

a Fig. 7. Maximum stress a as a function of the flange width at the cross-section A-A

(refer fig. 1.a).

3.

Fig.

7

shows the distribution of the net maximum stress along the width of the inside flange for the cross-section A-A (refer fig. l.b) as obtained by method

3.

The maximum

tensile stress attains the highest value at the midpoint of the inside flange and gradually decreases in the outward direction. Hence it may be concluded that in designing the corner zone, the application of a beam with a wide flange should be avoided as a considerable portion of the beam does not effictively contribute to the bending stiffness.

6 Omox 5 10' Nm2 4

1

3 2 0 -1 ·2 -3 -4 ·5 -6 6 measured - method 3 04 0..5 0.8 -~-Fig.

8.

Variation of the maximum stress (a)' along the height (H) of the cross-section (A-A).

4. Fig. 8 sho\,/S the distribution of maximum stress (a) along the central axis (Z-Z) over the entire height of the cross-section A-A (refer fig. l.b).

1.0...---~18""O-;;::3---,2.0

K

I....:.-__

Ko,

1

1 . 8 Ko 1.5

t

The width reduct ion factor Kl Fig. 9 • and the stress concentration factor

Ka

1 as a function of 10 geomet rica 1 factor •

.6

. 2

2 4 6 8 100

01 2

-"R1fil

5.

Fig.

9

shows the relationship between the width reduction factor Kl and dimensionless factor aT/Rthl and similarly

a2

between the stress concentration factor K and ----R h

a l l

From this it may be concluded that

a. with r~spect to strength it is not effictive to choose a₁

Rl1

> 1.8

1 1

b. however, with respect to bending stiffness a virtual optimum is achieved for

a2

Rl1

~ 6

1 1

In the latter case eq. (25) reduces to

REFERENCES

(42)

1. Nudelman, L.G. et al.: "Eksperimentalnoe issledovanye prochnosty stanyn gYdravlicheskykh pressove dl ia

plast-massovikh izdelii (Experimental investigation into the strength of hydraulic press-frames for manufacturing plastic products)". Sb Nauka i proizvodstvo vip 2, Orenburg, 1965.

2. Greiger, M.: "Beitrag zur rechnerunterstUtzten Auslegung von pressengestel1en". Verlag W. Giradet, Essen, 1974.

3. Dipt. Ing. K.H. ButtsHidt und Dipt. Ing. Theimert Darmstadt: "Der Einsatz der Finite-Element-Methode im Pressenbau". Werkstatt und Betrieb, october 1974.

18

4. Nudelman, L.G. et al.: IIJhestkost stanin odnostoechnikh gydropressov (Stiffness of a single column hydraulic press-frame)lI. «Kuznechnoe stampovochnoe proizvodstvo» No.2, 1974.

5.

Timoshenko, S.P. and Goodier, J.N.: "Theory of Elasticity". McGraw-Hill, Third Edition.

6. Timoshenko, S.P. and ~Joinowsky-Krieger, S.: "Theory of plates and shells". McGraw-Hill, Second Edition.

7. Kam i nskaya, V. V. et a 1.: "Stan I ny I korpusn ic deta 1 I metal1rejhushikh stankov (Frames and housing elements of metal cutting lathes}lI. Mashgiz, 1960.

8. Panmarov, S.D., Biderman, V.L. et a1.: "Raschety na prochnost v mashinostroeni i (Cakulation of strength in machine design)", Tom 1, Mashgiz, 1956.