Preliminary analytical studies on web crippling

Preliminary analytical studies on web crippling

Citation for published version (APA):

Peköz, T. (1986). Preliminary analytical studies on web crippling. (EUT report. B, Dept. of Architecture Building and Planning; Vol. 86-B-02). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1986

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PRELIMINARY ANALYTICAL STUDIES ON

WEB CRIPPLING

by

Teoman Pekoz

The objective of this report is to explore simple analytical procedures to estimate web crippling of thin walled steel members.

For this purpose various cases of loading will be considered. At this phase only single unreinforced webs will be studied.

INTERIOR TIm FLANGE LOADING

The first model tried for t~is case is illu~trat~rl in Fiq.l. In this ~odel the web is assumed to be a beam column of width b and length L which is equal to the web depth d. The load is applied with an excentricity e. The failure is defined by the interaction equation:

Pe - 1

P

-(1 - p::-) Mo cr P +

~

eq. 1

where P is the failure load when only the axial load is present. P shall be defiRed to different ways. One is to define it ignoring column bu8kling namely

Po

=

cry bt eq. 2

Pnshall also be taken as the concentrically loaded column buckling load.

Mo

is the plastic moment, namely: bt 2

Mo ::::

--r-

cry eq. 3

Per is the elastic column buckling load for the idealized section.

It can be shown that the miminum root Jf I::q.l is

where P

=

- B -

V

B 2 - 4AC \ zA

A

=

Mo / Pcr B

= -

(Mo + Po e + ~1l 0 / P cr ) C

=

PoMo eq. 4 eq. 5 eq. 6 eq. 7 It can be shown that provided P > P ,the value of P determined from

Eq. 4 approaches P _cr as e approRchescO.

This analytical formulation of failure shall be checked against the test rtj!sultsrepr.;0rtedin Ref. 1. The appearance of the failed specimen

2

-The parameters of the test specimens of Ref. 1 and the results of the calculations with the approach described above are summarized in Table 1

and 2. The failure loads given in table 1 are the averages for identical sections.

In Table 2 the failure loads calculated using different values of load eccentricity and idealized plate width are tabulated. It is assumed that the loaded edges of the plates are simply supported. It is seen that to get a failure load equal to the failure load observed experimentally a plate length larger than the specimen length is necessary in each case even if the eccentricity is taken as zero.

A

possible explanation would be that the angles used as bracing at the ends of the specimens as seen in Fig. 45a of Ref. 1 restrained the flanges and hence the webs.

Tables 3 and 4 give an idea about the restraints on the webs. In Table 3 the width of plate tabulated is that necessary to have a calculated

concentrically loaded column buckling load equal to that observed in tests.

All the webs fell into the elastic buckling mode according to the AISI Specifi-cations (Ref. 2).

Namely the slendersness ratios KL/r were larger than C (with K taken as 1.0) L is the web depth and C_c is defined in Ref.2 as c

and

r

=

t

=

-2

V3'

For this case, the buckling load is

1f2Ebtr2

=

(KL)2 24263 bt3

=

(KL)2

and the required plate width to get the observed test load is

2 eq. 8 eq. 9 eq. 10 eq. 11 eq. 12 eq. 13 eq. 14 P test (KL) b

=

eq. 15 24263 t3

-3-In table 3 b is calculated taking K

=

1 in Eq. 15. In Table 4 the value of effective length coefficient K is calculated taking b

=

SL.

K

=yl2~263

bt ' eq. 16

L _Ptest

It is seen in Table 4 that a significant amount of restraint of the webs was present.

The preceeding calculations were based on assuming an idealized section of width b as shown in Fig. 1. The idealized section was treated as a column or a beam-column. Yamaki presents in Ref. 3 buckling coefficients for rectangular plates compressed by two opposite concentrated loads. This results are presented in Fig. 2. The buckling load can be calculated by the equation.

P _ krrD

cr -

n

eq. 17

or the buckling coefficient can be calculated from the test load as

=

hP test

K - eo. 18

test n.D

The plate buckling coefficients calculated by Eq. 18 are tabulated in Table 5. It is seen that the values of Ktest are reasonably close to that given by Yamaki for the case long edges built-in. It is also possible that the fact that the load is distributed over some distance might have led to loads on the high side. The ratios Rs and Rfix give an idea about the ratio of the test load to that calculatea by the charts of Yamaki. As seen the load calculated by assuming long edges built-in is reasonably close

to the observed ul timate loads. .

CONCLUSIONS:

The ultimate loads of the test specimens can be calculated by the two following approaches:

1. By a column formula taking the full length of the specimen as the width of the idealized concentrically loaded web and an effective length coefficient of .85. The values obtained using E q. 14 are given in Table 6.

2. By the plate buckling formula of Eq. 17.

Since the results appear to be very sensitive to the length of the specimen, there is question whether the test specimens represent the conditions in an actual member.

END TWO FLANGE LOADING

In Table 8 the results of Ref. 1 end two flange loading tests are evaluated. The evaluation approach is similar to that used for interior one flange loading case. The values of effective length factor calculated using Eq. 16 with b

=

SL/2 are tabulated. Also tabulated are the values of b according

4

-CONCLUSIONS:

Though not fully studied here it appears possible to develop approaches similar to those for interior two flange loading for determining the failure loads in the test specimens.

INTERIOR ONE FLANGE LOADING

The results of tests on interior one flange loading are given in Table 5a on pages 131, 132 and 133 of Ref. 1. Reference 4 presents a relevant approach for this case. In this approach a buckling load is determined using the equation

P _cr

=

_NPcr Eq. 19

and

K 1f 2D

Pcr

=

_L2

Eq. 20

where N is the bearing length and D is the flexural regidity of the plate and L is the length of the plate. The plate buckling coefficient K is determined from Fig. 3. An evaluation of the test results using Eq. 19 is shown in Table 9. It is seen that the approach gives very conservative results. The calculated results are so far from those observed that the approach seems to be inapplicable.

Another possible approach is that of a failure mechanism as described in Ref. 5. A computer program given in the Appendix was prepared on the basis of this reference. The results are given in the Appendix and summarized in Table 10. The correlation with the test results does not seem to be good.

CONCLUSIONS:

Other than the empirical formulations given in various specifications, there appears to be no satisfactory analytical solution for this case. Efforts are currently underway to develop a satisfactory analytical approach.

Eindhoven, 1985, January 18th.

REFERENCES

1. Hetrakul, N. and Yu, W.W., UStructural Behavior of Beam Webs Subjected to Web Crippling and a Combination of Web Crippling and Bendingll, Civil Engineering Study 78-4, Structural Series, Dept. of Civil Engineering, Univ. of Missouri-Rolla, Rolla, Mo., June 1978.

2. "Specification for Design of Cold-Formed Steel Structural Members", American Iron and Steel Institute, ~Iashington, D.C. 1980 Edition. 3. Yamaki, N., "Buckling of a Rectangular Plate under Locally

Distributed Force Applied on two Opposite Edges", 1st and 2nd Report, The Institute of High Speed Mechanics, Tohuku University, Japan, vol. 3, 1953.

4. Zetlin, Lev., "Elastic Instability of Flat Plates Subjected to Partial Edge Loadingsll, Journal of the Structural Division, ASCE Proceedings, Vol. 81, September 1955.

5. Roberts, T.M. and Rockey, K.C., "A Mechanism Solution for Predicting the Collapse of Slender Plate Girders when subjected to in-plane Patch loading", Proc. Instn. of Civil Engineers, part 2, 1979, Mav. pp. 155-175.

(f)/

1 /

l'

L~/t·

l;

Idealized member Fig. 1. Interior two flanges loading case

\

i

Actual member

Fig. 1. Interior two flanges loading case

T

I , 1 1

L

b

...-i

/1 , !

p

6*

p 16

~Q

\

_LU

\

~: "-12 K

I~

""-

s b fCJF / 5

~

"'~

m:1 / 8

sr:!Js

_""-

m=l _"<..

K

--

!---s 4 . , 0.6 1.0 1.4 1.8 L/h

(a) Values of K for Long Edges. Simply Supported (Yamaki) p K 16

~O

\

~

m=2 v

_{l :}

J

m:i'

v _~ 14

1\

'9""

if"

\

~

mol

"-

r--v 12

SOs"

'"

m=1

~

F' _V ~: _t---

""-

t--10 ) 0.6 1.0 1.4 1.8 L/h

(b) Values of K for Long Edges Built-in

(Yamaki)

Fig. 2 Variation of Buckling Coefficient K Versus the Aspect Ratio of the Plates Compressed by Two Opposite Concentrated loads

I I

\tfH'

I

rrr-

NXP:P

600

hI

H

\

N L

l

" , --

h/L

=c

.2~

\

~

I

F}2

.172

\

,\

400

\

\

f - -

hl

=2.'"

rl-~

, K

200

\ \

1\

\

[><

r -I -

ft :

II.U

~

"-I~

V

"

t'-.

r--"

I'-.

"'"

I'-.

_~ ... r---t'--

~

-...

1-I-r--- --

---

t-

r"--;_ .~

J-.GtJ-v

0.01

0.05 0.5

Figure 6. Buckling Coefficient K versus NIL ratio for Simply Supported Rectangular P;ates

I

-TABLE 1

INTERIOR TWO FLANGE LOADING

I I

SPECIMEN C I _h

cry P test hit SL

NO~ (i n) (; n) (ksi) (k) {approx} (i n) -SU - 1 1 9.7 43.82 .78 200 21 3 II II _.81 "

"

SU - 2 1 12.09 II _{.61 250 24} 3 II " . 61 _"

..

SU - 4 1 4.90 47.12 1.72 95 15 2 _" II _1.95 II II 3 II " 2.26 II II SU - 5 1 6.10 II _{1.52 120 15} 2 " II 1.63 II II 3 II II _1.62 " II SU - 6 1 7.30 01 _{1.35 150 15} II " 1.25 01

..

l. II II , , ,

TABLE 2

INTERIOR TWO FLANGE LOADING

CALCULATED OBSERVED

SPEC.

NO~ e

=

.18 in e

=

0

b P_ult P_ult SL _{P ult} C _{P ult}

(i n (k) (k) (i n) K (in) (k) (I) (2) (3 ) (4) ---

₁₌₌₌₌₌

---

_{1======= ======== ========= 1========= ============:}

--- ---SU - 1 6 .14 .17 21 .60 1 .78 9 .21 .26 3 .81 18 .43 .51 36 .85 1.02 SU - 2 18 .19 .33 24 .45 1 .61 36 .58 .66 3 .61 40 .65 .73 SU - 4 18 1.14 1.94 15 1.68 1 1.72 30 1.90 3.23

_I

2 1.95 36 2.28 3.87

I

3 2.26 SU - 5 18 .87 1.28

I

15 1.08 1 1.52 ! I 30 1.45 2 1.63 36 1.74 2.56 3 1.62

,

SU - 6 18 .68 .90 15 .76 1 1.35 36 1.35 1.80 2 1.25 3 1.29 (1) calculated taking P _o

=

a _y bt

(2) calculated taking Po as the concentric column buckling load of the idealized member (fig. 1).

(3) SL - specimen length (fig. 1).

TABLE 3

INTERIOR TWO FLANGE LOADING

!

SPECIMEN P_test b ca1c SL

I

NO. ( k) (i n) (i n)

t===========

------

_{============== ===========}

~SU

-

1 .80 28.05 21 SU

-

2 .60 32.68 24 SU

-

4 1.98 17.72 15 SU - 5 1.59 22.05 15 SU

-

6 1.30 25.82 15 TABLE 4

INTERIOR TWO FLANGE LOADING

SPECIMEN _{P test} k nit

NO. _(k) ---

_{=========== f'============ :=:::=====::=:}

---SU

-

1 .80 .87 200 SU

-

2 .60 .86 250 SU

-

4 .98 .92 95 SU

-

5 1.59 .82 120 SU

-

6 1.30 .76 150

-average .85

I

-TABLE 5

INTERIOR TWO FLANGE LOADING

,

!

SPECIMEN _{Ktes t} Kca 1 c Kca 1 c Rss Rfi x

!

NO. s . s . fix ! i (1) (2) (3) (4)

1=========== ===::======r==========

F========F======== ~========== . J SU - 1 8.2 I 4.1 9.1 2.0 .9

su

-

2 7.9 If \ SU

-

4 10.4 II i i SU

-

_{5 10.4}

..

SU

-

6 10.1 II

(1) For long edges simply supported

(2) For long edges built-in

(3) Rss :: Ktest / Kca1c S.s. (4) R_fix :: K_test / K ca1c fix. II _{1.9 .9} II 2.5 1.1 II 2.5 1.1 It _2.5 ! 1.1

TABLE 6

INTERIOR TWO FLANGE LOADING

I ; SPECIMEN NO. Ptest _{( k)} Peale (k) (1 )

,=========== =====================F===============

I . i SU - 1 .80 .83 .96 i ; SU - 2 .60 .61 .98 I I \ I i SU - 4 1.98 2.32 .85 ! ; SU - 5 ! 1. 59 1.50 1.06 \ SU - 6 1.30 1.05 1.24 , I _1.02

I

Avg. i St. Dev. .14

(1) calculated using eq. 14 with k

=

.85 and b = SL

TABLE 7

INTERIOR TWO FLANGE LOADING

, . SPECIMEN

I

P

r

test NO.

I

(k)

l.'=;:=:=~====t====~;~==

~

, su -

2 . .60

. su -

4 1.98 SU - 5 , SU - 6

I

Avg.

St.Dev. 1.59 1.30 Peale Ptest/Pcalc (k) (1) ========== =============== = .80 .90 .71 .86 1.75 1.14 1.40 1.14 1.17 1.11 1:'04 , .15

...1---.---, ... ).

TABLE 8

END TWO FLANGE LOADING

SPECIMEN (0) or (L) SL P k b NO. (i n) (i n) (k) (1) (i n) =========;;:::;::

============:

---_{--- ========} ====;::==

1=========

SU

-

1 9.70 21 .34 .94 11.92 SU

-

2 12.09 24 .29 .87 15.80 SU

-

4 4.90 15 .73 1.07 6.53 SU

-

5 6.10 15 .64 .92 8.88 SU

-

61 _{7.30 15 .61 .79 12.11}

(1) k calculated by eq. 16 taking b ::: SL/2 (2) b calculated by eq. 15 taking k 1.

TABLE 9

INTERIOR ONE FLANGE LOADING

SPEC. N SL H h/SL NIL k _Pcalc P_{test P test/P ca 1 c}

NO. (in) (in) (i n) _{(1) (k) (k)}

(2)

===== f::==== ======

--- ---_{--- ---} --- ------

---

--- ------

_{===============}

1 1 42 9.70 .23 .02 380 .64 1. 22 1.91 3 _" 11 " .07 170 .85 1.42 1.67 ~ 1 48 12.09 .25 .02 380 .49 1.23 2.51 3 II II " .06 170 .65 1.42 2.18 5 1 30 6.10 .20 .03 260 .85 1.44 1.69 2 _" " " .07 130 .85 1. 79 2.11 3 II 11 11 _{.10 100 .98 1.96 2.00} 6' 1 30 7.30 .24 .03 260 .85 1.53 1.80 2 II II II _{.07 130 .85 1.85 2.18} 3 " " II _{.10 100 .98} 1.99 2.03

(I) From fig. 3.

TABLE 10

INTERIOR ONE FLANGE LOADING

SPECIMEN N _{Pcale P test P eal/ P test}

NO. (i n) _{(k) ( k)} (1) ---

_{F====:::=====}

---

_F===========

=================== --- ---1 1 .90 1.22 .74 3 II 1.42 .63 2 1 .92 1.23 .75 3

..

1.42 .65 5 1 .99 1.44 .69 2 II 1. 79 .55 3

..

1.96 .51 61 1 .96 1.53 .63 2 _" 1.85 .52 3 II 1.99 .48

APPENDIX

An implementation of the formulation of ref. 5.

100 :L10 120 :1.30 :lAO 150 :1.60 170 180 :1.90 ;;'~OO 210 ;;'~~~O 230 240 :~50 2[,0 270 280 290 300 310 320

:no

340 350 360 37() ~m() 390 400 410 420 43() 440 430 460 470 480 4('~0 ;;;; + 04t::: L ",:: ~::<;o () READ TN,S,B,C,ALFA PRINT "TN,S,B,C,ALFA" PRINT TN,S,B,C,ALFA PRINT "TN

=

";TN DATA 11,43.82,1.5,1,1.62 DATA 13,43.82,1.5,3,1.62 DATA 21,43.82,1.5,1,2.02 DATA 23, 43.82,1.5,3,2.02 DATA 41,47.12,2.2,1,0.82 DATA 42,47.12,2.2,2,0.82 DATA 43,47.12,2.2,3,0.82 DATA 51,47.12,2.7,1,1.02 DATA 52,47.12,2.7,2,1.02 DATA 53,47.12,2.7,3,1.02 DATA 61,47.12,3.2,1,1.22 DATA 62,47.12,3.2,2,1.22 DATA 63,47.12,3.2,3,1.22 CB :::: 4

*

E

*

.048 I (S

*

B): PRINT ·CB :::: ·;CB TX :::: 89 IT :::: TX TT

=

TT

*

3.14159 I 180 THETA :::: I I FF:::: COS (TH) I (1 - SIN (TT» - CD IF FF

<

0 THEN GOTO 370 TX :::: TX - • O~) GOTO 300 MW :::: S

*

T

*

T I 4

BETA:::: SQR (ALFA

*

COS (THETA» Cl :::: C:C2 :::: C

IF C

>

2

*

(BETA) THEN Cl :::: 2

*

BETA D :::: I.,

*

AL.FA

IF C

>

.2

*

D THEN C2 :::: .2

*

D CS ::: Cl

IF CS

>

C2 THEN CS ::: C2

ETA:::: (4

*

BETA

+

2

*

CS)

*

MW I «CMW I ALFA)

*

COS (THETA)

+

S PI :::: 4

*

MW I BETA

+

(4

*

BETA + 2

*

CS - 2

*

ETA)

*

MW I (AL.FA

*

PFHNT-"pr-~", '-~F'f ~.- ETA "" .; ETA BETA:::: SQR (4

*

MW I (S

*

T»

P2 - 4

*

MW I BETA

+

S

*

T

*

(BETA

+

C) 500 PRINT ·P2

=

";P2

510 PU :::: PI: IF P2

<

PU THEN PU :::: P2 520 PRINT ·PU

=

";PU

530 PRINT: PRINT 540 GOTO 110 '1

*

T)

*

ALFA

*

COS (THETA» COS (THETA»

RUN TN,S,B,C,ALFA 1.1 43.82 1.5 1 1.62 TN :::: ₁₁

eB

:::: _86.1706984 P1 :::: _{.900555674 ETA} :::: _.4912928:1. F' ':) A ..

-

3.02500618 PU ::: _.900555674 Hh S, Po, C,ALFA 1.3 43.82 1.5 3 1.b2 TN ::: ₁₃ CII .... 86.1706984 Pl ,,- _{• 90055~i674} ETA ::: _.491292Bl P'"> G,.

--

7.23172618 PU ::: _.90055~:j674 TN,S,B,C,ALFA ;! 1. 43.82 1.5 1 2.02 TN :::: ₂₁ CB

_-

86.1706984 Pi ₌₌ .921686336 ETA ₌₌ .439984359 Ff2 "" 3.02500618 PU ::: _.921606336 TN,S,B,C,ALFA 23 43. B;') l • ~i 3 2.02 TN ::: ₂₃ CB ::: _8tl.1706984 Pl :::: _{.921686336 ETA}

--

_.439984359 F'2 ,- _7.231.726l.8 PU ::: _.921686336 nhs,B,C~ALFA 41 47.12 '") r) ~+ ... :1. .82 TN ::: _4:1. ell :::: _54.6380615 P:I, ::: _{.87896897 ETA}

--

_.553508203 F'''

..

, ::: 3.25281358 ,PU

--

.87B96897 TN,S,B,C,ALFA 4':> _{47.1.2 :~. 2} P) ,'" .B2 TN :::: 4') ~. ell .- 54.6:~806l.5 P1

--

.87896897 ETA

_--

.553508203 F'2 :::: _5.51457358 PU ::: _.87896897

TN,S,B,C,ALFA 43 47.12 2.2 3 .82 TN :::: ₄₃ CEl

_.

54.6380615 Pi

-

.87896897 ETA

_-

.5!:i3508203 F'2

-

7.7763335B PU _. .87896(-197 TN,S,B,C,ALFA 51. 47. :1.2 2.7 1 :1. .0;;.) TN :::: ₅₁ CEl

_.

44.5199019 P1. .... .990820603 ETA _ . .446()4B836 F":l A ..

-

;3. 25281.3!;;i8 f'U " M .990820603 TN,S,l),C;,ALFA 52 47.1.2 "'l _{.... + ,"} "1 "'l _1.0::.' "-TN .... c' "'l "I,;.. cn ::;: _44.~i1.99019 Pi

_.

.990820603 ETA ::;: _.44604BE136 P''l

...

-

5. ~:j14~;7358 PU

_.

.990820603 TN,S,B,C,Al..FA 53 47.l~! 2.7 3 1 .• 02 TN

-

~j3 CB :::: _{44.51 <?9019} Pl. .... .9(;'0820603 ETA - .44b048836 P') A ..

_.

7.7763335B PU

-

.990820.603 TN,S,B,C,ALFA 61 47.12 3

....

P) ₁ 1 ")") + ",,',. "'-TN :::: ₆₁ CEI ... ~~7 • 5636672

P:I. :::: _{.96:i403538 ETA .376593473}

P2

-

3.2~':;2B135B

TN,S,B,C,ALr~1 62 47.1.2 3.2 2 1 ~)") " iI.: •• A-TN :::: ₆₂ CD :::: _37.5636672 Pi - • 96340:~538 ETA

_.

• 37659347:~ P"'") .:. :::: 5.51457358 PU :::: _.963403538 TN,S,B,C;,ALFA 63 47.12 3. ::,~ 3 1. ~ 2;! TN :::: ₆₃ CB -- 37.5636672 Pi -- .963403538 ETA -. • 376~j9:'5473 P':) A_ :;:; 7.77633358 PU :::: _{• 9634()3538}