Crushing damage of web girders under localized static loads

ELSEVIER 0 1 4 3 - 9 7 4 X ( 9 4 ) 0 0 0 0 3 - 4

J. Construct. Steel Research 33 (1995) 199-235 Elsevier Science Limited

Printed in Malta. 0143-974X/95/$9-50

Crushing Damage of Web Girders Under Localized Static Loads

Tomasz Wierzbicki & Jennifer Culbertson Driscoll

Department of Ocean Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

A B S T R A C T

This paper examines the crushing characteristics of web girders subjected to local in-plane crushing loads. First, a simplified model of the post-buckling deformation of the web girder is developed. This model is then used to derive an approximate solution for the plastic behavior of the deforming web girder under localized in-plane loading. Finally, the results of small-scale crushing tests of extruded aluminum double-hollow forms are presented to verify the assumptions made concerning the deformation mode and to demonstrate the accuracy of the solution. Although the present analysis focuses on damage of longitudinal web girders in unidirectionally-stiffened double-bull (USDH) ships, this analysis can be applied to many web girders commonly encountered in civil engineering applications.

N O M E N C L A T U R E

Hull parameters H B tw tp

L H* B* t*

L*

Depth of web Longitudinal girder spacing Thickness of web Thickness of upper and lower hull plating flanges Distance between transverse bulkheads H/tw B/tp tp/tw L/H

199

200 T. Wierzbicki, J. Culbertson Driscoll

Material parameters ao Flow stress trust Ultimate tensile yield stress eult Ultimate tensile strain Mo Fully plastic bending moment for the web Mo Fully plastic bending moment for the flanges M~ (aut2)/4 No Fully plastic membrane force for the web -~o Fully plastic membrane force for the flanges

Geometric parameters A A*

Oo Oi li 0 ¢

Uf

Ux

~f

~x

~avg PB Psh

Crushing depth A/H Local crushing depth for global bending mode Vertical extent of damage for local denting mode (/H Angle between first and second hinge lines Angle between second and third hinge lines Angle of indentation due to crushing; Angle between regions connected to the ith hinge line; Length of ith hinge line Angle between the first and third hinge lines Angle between center line and third hinge line Angle of depression in global bending mode Stretching of flange Maximum stretching of web Average strain for flange Maximum strain for web Average strain for structural unit Crushing force Shear force

1 I N T R O D U C T I O N

Plastic crushing failure of web girders in plated structures occurs fre- quently in many industrial applications. For example, decks of offshore exploration and production platforms which are supported by a system of orthogonal stiffeners may be subjected to large localized forces, such as those caused by the impact of massive objects. The decks and bulkheads of a ship involved in a collision may suffer substantial damage due to large compressive in-plane loads. Finally, transverse frames in either

Crushing damage of web girders 201

single- or double-hull ships may be crushed during grounding on a hard surface.

The problem of ultimate strength of web girders has been extensively studied for civil engineering and naval architecture applications. 1-3 The post-failure behavior of web girders, however, is not yet well understood. Langseth 4 presented a thorough theoretical-experimental study of impact damage of offshore platform decks. The main failure mode considered was local denting and possible perforation of the steel deck plating caused by accidental impact of a drill collar. Little attention was paid to the damage of stiffeners themselves.

Most ship damage models have dealt with conventionally-framed ships which utilize a gridwork system of large transverse web frames intersec- ted by smaller longitudinal stiffeners. In general, for both collisions and groundings, these transverse web frames are subjected to large compressive in-plane loads.

McDermott et al. 5 developed a mathematical model of the various stages involved in a minor collision of longitudinally-framed oil tankers. This model was applied to both single-hull and double-hull tankers. McDermott assumed that the resistance offered by a buckled web frame remained constant during deformation of the web frame in the transverse direction. Chang, 6 in his discussion of McDermott's paper, recommended that the post-buckling resistance of web frames be further investigated because many structural members offer decreased resistance after an ultimate strength has been reached.

In the naval architecture field, interest in the post-failure resistance of web girders has been renewed with more recent studies of grounding damage of ships. Wierzbicki et al. 7 investigated the extent of damages involved in high energy groundings of tankers for a study conducted for the Joint M.I.T.-Industry Program on Safe Tankers. For this study, a simplified model was developed to predict the energy dissipated by the crushing of transverse web frames.

This model utilized the assumption that the deformation of the transverse web frame and the hull plating, which acts as a flange, could be separated into two mechanisms. It was assumed that the web detaches from the flange and rolls into two cylinders of equal radii, one on either side of the point of load application. It was assumed that the hull plating deforms into a trough shape. Geometric compatibility of the web-flange connection, however, was not maintained.

Another naval architecture interest which motivates study of the post- failure response of web girders is the recent interest in novel double- hullforms, which may be environmentally safer than conventional hull-


forms. In 1985, The Hitachi Zosen Corporation of Japan presented the concept of an advanced double-hull design, which utilizes longitudinal girders connecting inner and outer hulls to form a cellular structure that wraps around the girth of a hull. 8 Among other advantages, this unidirectionally-stiffened double-hull (USDH) structure, shown in Fig. 1, offers the potential of increased protection against local damage resulting from accidental loading. This potential has contributed to recent interest in the USDH structure for both commercial and naval ship designs. 9

The crushing of web girders presents a problem similar to that of crushing tubular members. Several studies have been conducted which have explored the large deflection plastic behavior of thin-walled tubular members. Reference 10 contains a review of several of these studies.

The crushing of longitudinal girders in the USDH structure is similar to the crushing of tubes subjected to lateral loads rather than axial loads. De Oliveira e t al.1 ~ presented an analysis of crushing of rectangular tubes due to lateral loads. This class of problem is difficult to analyze, due to a lack of symmetry and due to the fact that a tube's cross-sectional shape changes during deformation. If the crushing load is localized rather than distributed, the distortion of the cross-sectional shape occurs locally and propagates outward as loading increases. The work on rectangular tubes was later extended by Wierzbicki and Suh 12 to include lateral crushing of circular tubes.

I CL

Fig. 1. The USDH hullform.

Crushin9 damage of web 9irders 203

The problem of damage to web girders in the USDH structure re- sembles crushing of a multiple box column-type structure subjected to a localized lateral load. Based on the successful application of large-deflection plasticity theory to lateral crushing of tubes, this same approach is used in the current study to develop the load-deformation characteristics for longitudinal web girders subjected to local in-plane crushing loads.

First, models for two modes of deformation of a web girder--local denting and global bending--are developed based on some photographic evidence of the deformation of fully-clamped plates that were subjected to in-plane impact loading. Next, a load-deflection relationship for the deforming web girders is derived for each mode. Also, a simplified load-deflection relationship for shear failure is developed. Finally, these theoretical predictions are compared to the results from a limited numb- er of small-scale crushing experiments.

2 P R O B L E M D E F I N I T I O N

Three modes of deformation are postulated for this study. The first mode involves local indentation of the web girder. The second mode involves the development of a three-hinge mechanism to induce global bending of the web girder between transverse bulkheads. The third mode is pure shear failure. These postulated deformation mechanisms are described in Sections 3, 4 and 5. Assumptions concerning the geometry and boundary conditions of the deforming body, the material properties of the body and the applied external loads for this solution are described in the following sections.

2.1 Structural unit

Figure 2 shows a section of the USDH structure consisting of rectangular cells. This cellular structure runs between transverse bulkheads and wraps around the girth of a hull. Due to symmetry, it is sufficient to define a structural unit which consists of one longitudinal web girder

J

J

Fig. 2. USDH section.


,bT-B "1

Fig. 3. Structural unit.

with the attached inner and outer hull plating. The structural unit, as shown in Fig. 3, can actually represent any web girder encountered in civil engineering applications.

The depth of the web girder is denoted by H and the thickness of the web girder is denoted by tw. Because the inner and outer hull plating act as fully effective flanges, the breadth of the attached plating is equal to the girder spacing, B. The thickness of the hull plating flanges is denoted by tp. The length of the web girder, L, is determined by the distance between transverse bulkheads. This length is much greater than the girder depth and spacing.

Due to symmetry, the flanges are constrained from in-plane movement in the X-direction along the longitudinal edges (X = _ B/2). Both ends of the structural unit are considered to be fully-clamped at the transverse bulkheads (Y= _ L/2).

2.2 Loading condition

The web girder is considered to be subjected to a local crushing load, such as that delivered by a knife-edge indenter or a rigid punch normal to the longitudinal axis of symmetry. This loading is quasi-static, therefore dynamic effects are ignored. The applied load has a magnitude of PB. This load causes the flange to deflect a distance A relative to the undeformed region of the flange. The rate of deflection is denoted by A.

2.3 Equilibrium

The equilibrium of the structural unit is expressed by the principle of virtual work:

~s N PB/k = N~a~p dSo + ~ MoOili (i) o i=1

where ~,p are components of the in-plane membrane strain rates and Nap are the corresponding components of the membrane force tensor. The

Crushin9 damaoe of web 9irders 205

integration is performed over the plasticly deforming region of the structural unit.

Equation (1) incorporates the assumption that plastic bending resistance of structural components is decoupled from the membrane resistance. Furthermore, the bending deformations are concentrated in the plastic hinge lines. Accordingly, the last term in the right-hand side of eqn (1) represents the rate of plastic energy dissipation which is a product of the fully plastic bending moment, Mo, the rate of rotation of' the ith hinge, 0i, and the length of the ith hinge, li.

Based on the assumptions implicit in eqn (1), the solution is reduced to finding a kinematically admissible field of strain rates, ~,~; displacement rate, A; and rotation rates 0i. Such a field will be constructed for both the local denting mode, the global bending mode and the shear failure mode.

2.4 Material properties

The material is assumed to be rigid/plastic, isotropic, time-independent, characterized by the flow stress, a0. Strain hardening effects may be included by adjusting the value of the flow stress based on the average plastic strains of the deforming regions during the deformation process. The average plastic strain may be obtained by integrating the strain over the area of the deforming region:

~avg = S ~ dS (2)

3 L O C A L D E N T I N G M O D E

3.1 Simplified model

Figure 4 shows a photograph of a fully-clamped plate which was deformed by a crushing load applied by a rigid punch to the upper edge of the plate. This photograph is from full scale experiments conducted by S. Shimizu of Shimushu University in Japan. 13 Based on this photograph and observations of failure modes in the scale model tests conducted at M.I.T., the collapse mode for the local-denting deformation mechanism shown in Fig. 5 was developed.

The out-of-plane deformation of the web involves both bending and stretching. The bending of the web is caused by the formation of plastic hinge lines, which are shown in Fig. 5. The web deforms symmetrically


Fig. 4. Deformation of clamped plate,

H

I

Fig. 5. Local denting model.


about the point of application of the load. Six stationary plastic hinge lines define boundaries of rigid triangular plates that rotate with respect to each other. Due to symmetry, only the three hinge lines on one side of the vertical plane of symmetry will be described. The first hinge line forms at the junction between the web and the upper flange. The second hinge line forms at an angle ~ below the first hinge line. The third hinge line forms at an angle fl below the second hinge line. The angle between the first and third hinge line is taken as a process parameter, ~.

The photographs examined in developing this model showed the length of the deformed region in the longitudinal direction to be approximately twice the length in the vertical direction. This characteristic was included in the model by fixing the angle between the upper edge of the undeformed web and the third hinge line at rt/4. The vertical length of the deformed zone is taken as a parameter (.

3.2 Deformation sequence

Figure 6 depicts the geometry of the deformed web. The applied force, PB, causes the upper edge of the web to indent a distance, A. The angle of indentation is 0o. As A increases, the second hinge moves out of plane. The angle between the first and third hinges, ~, decreases. The angle between the vertical plane of symmetry and the third hinge line, ~b, remains constant at ~/4.

The shaded area in Fig. 6 illustrates the stretching in the deforming web. The attached flange deforms with the web, as shown in Fig. 7. Again, the shaded area illustrates the stretching in the deforming flange.

H

L ~ Ip b

Fig. 6. Deforming web.


Fig. 7. Deformed flange.

3.3 Bending energy rate

In the local denting mode, the deformation is simplified by modeling all plastic hinges as stationary hinges. The bending energy rate term simplifies to

/~a = 2Mo{[/)1]/1 + ['02"]/2 + [03]/3} (3)

where Mo is the fully plastic bending moment:

ao t2 M ° = 4 (4)

The assumption is made that all hinge lengths remain constant during the deformation process. The hinge lengths can be determined from the geometry of the model shown in Fig. 6.

12 = (/cos(~t) 13 = ~/-2~

(5)

The rotation rates, 0i, for each hinge line can be related to the angle between the first and third hinge lines. For each hinge line, a section normal to the hinge line is used to determine the rotation rate. These derivations are included in Appendix A.

3.4 Membrane energy rate

Both the attached upper plating (or flange) and the web stretch while deforming, therefore

/~M total =/~M flange ~ EM web (6)

In each of the two plate elements of the web which rotate with respect to each other, a local convective coordinate system (~, r/) which rotates with


the respective plate section is chosen. According to the present model, plate elements are inextensible in the r/-direction and undergo stretching in the l-direction. Furthermore, the shear strain rate component is assumed to be zero, 4,¢ =0. The strain rate vector, therefore, takes the following form:

(7)

Substituting the above result into eqn (1), the rate of membrane energy expression becomes:

;s ; /~M = ~¢,NodS=2 No~--~d~dt/=2N0 ti(~/)d~/ (8) i

0 0 0

where No = aot is the fully plastic membrane force per unit length and 2ti is the velocity with which material points on either side of the centerline of the shaded regions shown in Figs 6 and 7 move apart relative to each other.

3.5 Load-deflection relationship

The rate of work done in deforming the structural unit is

I'~'e~, = PsA (91

where A is related to the process parameter • by

I)V¢,, = - pao((2 + A 2) (10)

Applying the principle of virtual work, eqn (1), the rate of external work is equated to the rate of internal energy dissipation. For approximately a~<~z/12, the linear assumption discussed in Appendix A is valid. The load-deflection relationship in dimensionless terms is:

P B

Mo 2[(K1 - K 2 ) - ( K 2 + 2)~b]

COS 0~(1 - - I ~ ) ~ / ( K 3 - - K4)71 - ( K 4 - - 2)~k


+ 2&*H*wG - K6) - &$I cos’ a - K4$

+ $[45*H* tan a + 8B*t*2]

where the dimensionless parameters are

and

(11)

(12)

(13)

The constants ICI--K7 are functions of a and j? which are listed in Appendix A.

3.6 Physical limit

For the local denting mode, a is modeled as a parameter. The choice of a affects not only the magnitude of the crushing force for a specified crushing depth, but also the range of freedom for the model. The degree of freedom for the denting mode is exhausted when locking occurs-when the upper flange makes contact with the ‘upper flap’ of the web, the region between the first and second hinge lines. When this occurs, 13~ =x/2 and

A* 0 cos2a+2cosp cos2a-cos2fi - i* limit= 2 ~0s’ p- cos’ a

(14)

3.7 Initial phase envelope

For the local denting mode, it is postulated that the longitudinal and vertical extent of deformation, [, is initially zero but spreads during the deformation process until the deformed region extends the depth of the web ([ = H).

Figures 8 and 9 show the local denting solution for several values of c* ranging from [* = 0.1 to [* = 1.0. The parameters of the structural unit are


0

800

700

60O

500

40O

300

200

100

Local denting model (Alpha = Pill2)

(H* = 100; B* = 75; T* = 1)

~* = 0.2 [* = 0.4

~* =0.1 I I I

0.05 0.10 0.15

[*=0.6 [*=0.8 [ * = 1 . 0

I I I

0.20 0.25 0.30 0.35

Delta*

Fig. 8. Local dent ing m o d e for g=Tt/12.

8OO

700

600

500

0

3011

2OO

1011

Local denting model (Alpha = PI/8)

(H* = 100; B* = 75; T* = 1)

[* = 1.0

I I

0 o12 04 016 o , 10

Delta*

Fig. 9. Local denting mode for e = re/8.

H * = 100; B * = 75; t * = 1.0. Figure 8 depicts the load-def lect ion curve for ~=7r/12. The load-def lect ion curve for ~=7r/8 is shown in Fig. 9.

Both figures show envelopes for the initial phase of deformation. Due to the complexi ty of the load-def lec t ion relationship, eqn (11), it is difficult to analytically determine this envelope.


3.8 Strain and strain hardening

F r o m the geometry of the model, the strain averaged over the entire deforming area can be shown to be

where

x / ~ - cos • - sin ef (cos ~ + sin ~) (16)

and

(,,/2 cos ~ -- cos fl) sin (I) + cos fl cos (I) - cos :c ex - [-cos fl(sin (I)-- cos (I)) + cos ~]

(17)

Strain-hardening effects can be accounted for by adjusting the value of Oo based on the average plastic strain. For example, if the stress-strain relationship for a given material can be described by a power-law relationship similar to the R a m b e r g - O s g o o d formula:

(eavg/" (18) O" 0 ~-- O'ul t - - \ S u i t /

w h e r e Ou l t , eult and n are material constants and eavg is evaluated from eqn (15).

4 G L O B A L B E N D I N G M O D E

4.1 Simplified model

This second mode of failure involves generation of three hinge beam- bending mechanisms, shown in Fig. 10. The crushing load is applied at mid-span and the length between the hinge mechanisms is equal. This model could be adjusted to account for unequal lengths between hinge mechanisms.

The hinges are modeled as shown in Fig. 11. This model is similar to the local indenta t ion model, except that there is no axial stretching in the


Fig. 10. Global bending model.

PB

/_

PB

Fig. 11. Hinge mechanism.

flange. The vertical extent of deformation, ~, is equal to the depth of the web; therefore, ( = H. Also, a fourth hinge line forms along the bottom flange at the centerline of each hinge mechanism.

4.2 Deformation sequence

Figures 10 and 11 show the web deforming in the global mode. The applied force, PB, causes the 'legs' of the deforming girder to rotate an angle ~,; this causes the angle ~ for each hinge mechanism to decrease. The second hinge line moves out of plane and the angle between the first and third hinge lines, O, decreases; this results in a local deflection, 6, at the central hinge mechanism. This local deflection, 5, is part of the total deflection, A, as shown in Fig. 10. The shaded area in Fig. 11 illustrates the stretching of the web involved in deforming the hinge mechanism. Note that there is no extensional deformation in either the upper or lower flange in the global bending mode.

The range of A* is limited to A* ~<0.5. The extension of the two 'legs' of the deforming girder, therefore, is neglected. The hinge mechanisms are separated by a distance (L/2--H). The central hinge is designated the primary mechanism. The two outer hinges are designated secondary mechanisms. The subscripts 'p' and 's' are used to designate the primary and secondary mechanisms, respectively. As the central mechanism


deflects, the two 'legs' of the overall structure rotate an angle 7. From Figs 10 and 1 t, the angles ~bp and ~b~ can be related to 7:

7Z ~,bp=~-7 (19)

and

rc 7 (20) q~s-4 2

In Appendix B, the internal rate of energy dissipation for the primary mechanism is formulated as a function of 7 and ~, fp(7, ~). In the global bending mode, energy is dissipated not only by the primary hinge mechanism, but also by the two secondary hinge mechanisms. The total energy dissipated, therefore, is

Lot., = f .(7, + 2L(7, (21)

where fp and fs are the functions formulated in Appendix B for the total rate of internal energy dissipation for the primary and secondary hinge mechanisms, respectively. The functions fp and fs are related by eqns (19) and (20) so that

The deformation of the web girder in the global bending mode, like that in the local denting mode, involves both bending and stretching. The components of the internal rate of energy dissipation are discussed below.

4.3 Bending energy rate

Like the local denting model, the bending model consists of six stationary hinge lines in the web. In addition, one hinge line forms across the bottom flange, as shown in Fig. 10. The bending energy rate term is

/~B,, = 2Mo {ll [Olp] +/2 [02p] +/3 [Oap]} + )~ro/4[04p] (23)

where Mo is the fully plastic bending moment for the flange:

)~,1o = tr4 t2 (24)


............. I .............................................. I

PB

........... ................ I [ ........

Fig. 12. Set-up for denting experiment.

The assumption is made that the lengths of the hinge lines remain constant. From the geometry of the model, shown in Fig. 11, the hinge line lengths can be determined:

Ii = H

12 = H/cos

la = x /2H

14 = B

(25)

Details of the calculations of the rotation rates, 0i, and the bending energy rate,/~B, are given in Appendix B.

4.4 Membrane energy rate

The rate of membrane energy is defined by eqn (8). For the web, the maximum displacement, Ux, is shown in Fig. 11. The average displacement, Uw, avg for the shaded region in Fig. 12 is

1 Uweb,avg = ~ Ux (26)

Details of the calculation of the membrane energy rate, /~M, are given in Appendix B.

The internal rate of energy dissipation for the secondary mechanisms follows the same function as that for the primary mechanism evaluated at 7/2 rather than at 7. Both the bending energy rate term and the membrane energy rate term for the secondary mechanisms are listed in Appendix B.

4.5 Rate of external work

The rate of external work performed in deforming the web girder is

l?V'ext = PBA (27)


The local deflection, 3, at the central mechanism can be described as a function of 7 by considering the geometry of the model:

~ = x f H 2 - 2 H 2 sin 2 ~ b p = H x ~ 27 (28)

The total deflection, A, is the sum of the local deflection and the deflection due to the rotation of the two legs of the entire mechanism:

A = ~ + ( L - - H ) tan~ (29)

Therefore

~ ( L - H ) see27 (30) A=~+

4.6 Physical limit

As in the local denting mode, ~ is modeled as a parameter for the global bending mode also. The choice of ~ affects not only the magnitude of the crushing force for a specified crushing depth, but also the range of freedom for the model. The degree of freedom for the bending mode is exhausted

Combined denting-bending model

(H* = 100; B* = 75; T* = 1; L* = 9; Alpha = PI/8) 1000

o 600

400 ~ ~ ~ Bending model

i~..... -

I I I I

0 0.2 0.4 0.6 0.8 1.0

Delta*

Fig. 13. Comparison of failure modes.


when the connection of the web to the flange moves out of the Z - Y plane; this occurs when

(1 n sin2 fl) x/2 7limit = ~ -- arctan sin2 (31)

4.7 Combined bending-denting model

Figure 13 shows the combined local denting and global bending load- deflection curves for a structural unit representing a typical 40 000 DWT USDH tanker. The solution was constructed for ~ = re/8. Transition from the local denting mode to the global bending mode is indicated where the two solutions cross.

5 S H E A R F A I L U R E M O D E

The problem of ultimate strength of web girders has received a great deal of attention in civil engineering applications. For web girders characterized by a large height to thickness ratio, H/tw, the dominant stresses are shear stresses. Most of the existing analyses are concerned with elastic shear buckling. 14 Approximate engineering solutions have been derived using the concept of a diagonal tension. 3

The post-failure range of interest in ship groundings, however, has received little attention to date. Recently, a comprehensive theoretical experimental study on plastic shear failure was conducted by Price. 15 A picture of a plate deformed by the application of shear at a shear angle of 4,=20 ° is shown in Fig. 14(a).

Initially, the shear buckling starts with the theoretical elastic buckling wavelength:

2el = 0.7H (32)

As the shear angle displacement increases, the buckling wavelength, 2pl, shortens. For ~ = 2 0 °, it reaches approximately one-half the elastic wavelength:

1 2pl =~2el = 0"35H (33)

The analysis presented in Ref. [16] shows that a good estimate of the peak shear resistance force is obtained by considering a pure plastic in-plane shear of the web, even though the plate actually deforms in the


P S H

r . . . .

Fig. 14. Shear failure of a web girder. (a) Photograph of deformed specimen. (b) Simple computational model.

out-of-plane mode. Referring to Fig. 14(b), the total resistance of the structural unit in the shear model can be calculated from the principle of virtual velocities:

Psh A =/~h +/~M +/~B (34)

where/~sh represents the rate of plastic work in the web and/~u and/~B are the rates of membrane and bending work in the upper and lower flanges. These fractional contributions are calculated below:

/~h = 2tw I k~y dSw = 2tw~ k i l l= 2kl~twH JSw

(35)


where k = ao/3 is the yield stress in pure shear and Sw = Hl is the area of the web;

/~M=4 I AA jsf No£x dSy = 4N0 T f B l (36)

where Sf = Bl is the area of the flange;

A E8 = 8MoOB = 8Mo f B (37)

Substituting eqns (35-37) into eqn (34), the shear resistance of the structural unit becomes:

{,. =8 1 H + f + Mo tp tp tp

(38)

The last term in eqn (38) describes the bending resistance of the flanges and is quite small compared to the remaining terms for deflections greater than the plate thickness, A > tf.

As an example, consider a structural unit characterized by the following geometrical parameters:

tw=tp=t I=9H B=0"75H

H - -=100 t

(39)

Substituting the above values into eqn (38) leads to

Psh = 460 + 133A* (40)

Mo

where A* =A/H is the dimensionless central deflection. The shear resistance function has a constant component, due to the plastic shear in the web, and a linearly increasing component, due to tension developed in the flanges.

A comparison of all three failure modes for a structural unit with the geometric parameters listed in eqn (39) is shown in Fig. 13. The line


corresponding to shear failure lies above the curves representing local denting mode and global bending mode. The same tendency persists for other combinations of geometric parameters.

It can be concluded that the web girder will not fail in the plastic shear buckling mode because other plastic collapse modes give lower resistance. Because of its simplicity, however, eqn (38) can be used to estimate the order of magnitude of the maximum resisting force in shear webs subjected to localized loads.

6 C R U S H I N G E X P E R I M E N T S

6.1 Background

Four extruded aluminum double-hollow sections were crushed for this study. Two experiments were set-up to initiate the local denting mode. The other two experiments were designed to simulate the global bending mode.

The depth-to-thickness ratio for these specimens was H/tw = 20.3, which is at the lower end of the thin-wall member spectrum. Also, the material exhibited significant strain-hardening properties.

The extruded aluminum specimens were provided by Reynolds Alumin- ium, Holland B.V. This size of extruded shape was used in order to maintain a small scale which could easily be handled with standard laboratory machines. The specimens were 300 mm long, 127 mm wide and 50.8 mm deep. The issue of structural discontinuity and inhomogeneity due to welds was avoided by using extruded forms.

The double-hollow profile was chosen in order to provide some restraint against motion in the transverse direction. A multiple cell section, such as that shown in Fig. 2 would have been an ideal choice. The double-hollow form, however, was selected due to availability.

Figures 12 and 15 show the experimental set-up for the local denting and global bending experiments. Photographs of the deformed specimens from these experiments are shown in Figs 16 and 17.

6.2 Local denting

Due to the significant strain hardening effects, it was difficult to see the propagation of the deformation during the crushing process. The photograph in Fig. 16 shows the permanent deformation of the first dented specimen with one of the outer webs cut away to expose the interior web. The pattern of deformation is very similar to that postulated


PB

w ~ II

Fig. 15. Set-up for bending experiment.

Fig. 16. Photograph of dented specimen.

Fig. 17. Photograph of bent specimen.


for the simplified model. The angle ~ between the flange-web connection and the first hinge line is approximately 15 ° . The shape of the lower hinge line is parabolic rather than straight. The longitudinal length of the deformed region, however, is approximately twice the depth of the web.

In both denting experiments, fracture occurred at the juncture of the outer two webs and the upper flange. During the crushing process, the location of fracture was obscured by the rigid punch. Therefore, the crushing depth, A*, at which fracture began was not recorded.

Both denting specimens showed initial signs of fracture along the second hinge line; this appears to be a result of the fact that a structure cannot assume infinitely small radii of curvature. Because the local denting model employs stationary hinges and is formulated for only a discontinuous deformation field, this feature is not included in the present theory.

The ends of the specimens for both denting experiments were clamped to the work surface only at the lower flange. A fully-clamped condition was not achieved. A small degree of'pull-in' in the transverse direction was observed during the crushing process. The local denting mode solution was developed with the assumption of fully-clamped ends. The amount of energy dissipated by stretching in the experiments was substantially less than that predicted by the theoretical model.

6.3 Load-deflection curves

The load-deflection curves labeled 'THEORY' in Figs 18 and 19 were developed from the theory presented in Section 3. The curves shown are the envelopes of the family of curves as shown in Figs 8 and 9. Each specimen was modeled as three structural units with the following geometric parameters:

~=rt/12 H* =20"3

B*=16"9 t* =1"0

Although the two outer structural units are channel type sections rather than I-beam type sections, it was assumed that the two outer webs deform in the same manner as the inner web during the crushing process.

Strain-hardening effects were included as discussed in Section 3.8. The stress-strain curve for the specimens was approximated by a power law with the exponent n = 0.165.

As shown in Figs 18 and 19, the local denting model over-predicts the energy absorbed in the crushing process. It also does not capture the load


250

201)

150

100

50

250

200

150

I00

50

Denting No. 1 (H* = 20.3; B* = 16.9; T* = 1; Alpha = Pl/12)

I I 0.115 12 0.05 0.10 0. 0

Delta*

Fig. 18. F i r s t d e n t i n g expe r imen t .

Denting No. 2

(H* = 20.3; B* = 16.9; T* = 1; Alpha = PIll2)

Experiment

T h e o r y ~

0.25

Experiment

I I I I 0 0 . 0 5 0 . 1 0 0 .15 0 . 2 0 0 .25

Delta*

Fig. 19. S e c o n d d e n t i n g expe r imen t .

reduction which occurs at approximately A*= 0-10. The theoretical model shows an almost linear increase in load after the initial phase. Some of the load reduction may be due to fracture and the lack of fully-clamped end conditions.

From the present theory, it can be shown that the membrane energy involved in stretching the flange is the most significant factor in this region


of the load-deflection curve. The relative magnitude of the crushing load for the model and the general character of the curves, however, show good agreement with the experimental results.

6.4 Global bending mode

Figures 20 and 21 show the load-deflection curves for the two bending experiments. The photograph in Fig. 17 shows the permanent deformation viewed after experiments were completed. The deformation is similar to that postulated for the bending model. The angle ~ is approximately 22 °. The parabolic shapes of the hinge lines are more pronounced in these specimens than in those of the denting experiments. In the first bending experiment, fracture occurred at the juncture of the two outer webs with the upper flange. The second bending experiment was halted at a smaller value of A*. The initiation of fracture is evident at the same locations for this specimen.

6.5 Load-deflection curves

The load-deflection curves for the bending experiments were compared with the theoretical solution in Figs 20 and 21.

250

200

150

100

50

Bending No. 1

(H* = 20.3; B* = 16.9; T* = 1; L* = 4.4; Alpha = PI/8)

E x p e r i m e n t ~ ' ~ - - - _ _ _ ~ Bending mode l

i 01! 0.2

Delta*

Fig. 20. F i r s t b e n d i n g expe r imen t .

013 0.4


a ~

250

200

1 5 0

100

50

0 0 . 4

Bending No. 2

(H* = 20.3; B* = 16.9; T* = 1; L* = 4.4; Alpha = PI/8)

Experiment j ~ . ~ Bending model

I I I 0.I 0.2 0.3

Delta*

Fig. 21. S e c o n d b e n d i n g expe r imen t .

The specimens were modeled as three structural units with the following geometric parameters:

~ = re/8 H* =20"3

B*=16"9 t * = l ' 0

L* =4"4

The global bending mode was treated as bending of a simply supported beam.

The theoretical models for the bending and shear modes show very close agreement with the experimental results, both in the shape and the magnitude of the load-deflection curves. The theoretical denting model, however, over-predicts the required force during the initial phase prior to transition to overall collapse.

7 C O N C L U S I O N S A N D R E C O M M E N D A T I O N S

The objective of this study was to investigate the crushing characteristics of web girders for a USDH structure. The crushing experiments conducted provided some insight into the shape of the deformation that a web girder


undergoes when subjected to a concentrated lateral collapse load. Based on these experiments, a simple computational model was developed and the load-displacement relations were derived for the local denting and global bending deformation modes.

The local denting model over-predicted the amount of energy absorbed for the denting experiments. The theoretical load-deflection curves for the global bending model closely followed the experimental results for a simply-supported structural unit. The shape of the load-deflection curve for the global bending model is very similar to the shape of experimental results presented by de Oliveira 11 and Paik 16 for a horizon- tally-free circular tube subjected to a concentrated lateral crushing load.

The membrane energy involved in the extension of the two 'legs' of the bending mechanism was neglected for the theoretical model; this may be a significant factor for a fully-clamped structural unit. Crushing experiments for fully-clamped conditions would be worthwhile.

For this study, several assumptions were made in order to keep the analysis of the crushing characteristics relatively simple and math- ematically tractable. These assumptions may require further investigation.

In particular, the assumption that there is no load-interaction between bending moments and membrane forces was made to uncouple the energy dissipation terms. The inclusion of the interaction surface for the different forces would make the problem significantly more difficult, as pointed out by de Oliveira. ~

A further refinement of the method of including strain-hardening effects could be made. Strain-hardening was accounted for in a relatively crude manner which utilized the average strain over the entire structure. The strain for each deforming region could be used to adjust the flow stress for each region.

Another factor which requires further investigation is fracture. Not only may fracture affect the deformation mode and the load-carrying capacity of a web girder unit, it may also intitiate flooding in a ship's double-hull. For the models, an estimate of initial fracture could be made by using a simple critical strain to rupture criterion for the regions of maximum strain or some more advanced criteria.

Finally, the present analysis which was developed for grounding damage of unidirectionally-stiffened double-hulls could be directly applied to damage assessments of any web girder subjected to accidental loads.


R E F E R E N C E S

1. Jones, N. & Mansour, A., Ultimate strength of web girder frames. J. Ship Res., 1978.

2. Hughes, O. F. Ship Structural Design, John Wiley & Sons, New York, 1983. 3. Dubas, P., Gehri, E. (eds), Behavior and design of steel plated structures. In

Applied Statics and Steel Structures. ETH-Honggerberg, Zurich. 4. Langseth, M., Dropped objects: plugging capacity of steel plates, Ph.D.

Thesis, Norwegian Institute of Technology, 1988. 5. McDermott, J. F., et al., Tanker structural analysis for minor collisions.

SNAME Trans., 89 (1974) 382-407. 6. Chang, P. Y., Review of Tanker structural analysis for minor collisions, by J.

F. McDermott et al. SNAME Trans., 89 (1974) 408. 7. Wierzbicki, T., et al., Damage estimates in high energy groundings. Report

No. I: Joint M.I.T.-lndustry Program on Safe Tankers, pp. 51-56. 8. Okamoto, T., et al., Strength evaluation of novel unidirectional girder

system product oil carrier by reliability analysis. SNAME Trans., 93 (1985) 55 77.

9. Beach, J. E., Advanced surface ship hull technology--cluster B. Naval Engrs J., 103 (6) (Nov. 91), 28-31.

10. Johnson, W. & Reid, S. R., Metallic energy dissipating systems. Appl. Mech. Rev., 31 (3) (1978) 277-288.

11. Oliviera, de, Wierzbicki, T. & Abramowicz, W., Plastic behavior of tubular members under lateral concentrated loading. Det Norske Veritas Technical Report No. 82-0708 (Progress Report 30, July 1982), pp. 1--4.

12. Wierzbicki, T. & Suh, M. S., Indentation of tubes under combined loading. Int. J. Mech. Sci,, 30 (1987) 229-248.

13. Shimizu, S., Private communications. 14. Southwell, R. V. & Skau, S. W., On the stability under shearing forces of a

fiat elastic strip. Proc. Royal Soc. A, 105 (1924) 582-607. 15. Price, S. R., Plastic shear buckling of ship hull plating induced by ground-

ing, M.Sc. Thesis, M.I.T., May 1992, (Report No. 9: Joint M.I.T.-Industry Program on Tanker Safety.)

16. Paik, J. K., Damage estimates for offshore tubular members under quasi- static loading. J. Soc. Naval Architects of Korea, 26 (4) (1989) 81-93.

A P P E N D I X A

A.1 Bending energy rate

From the geometry of the model, shown in Fig. 5, the following can be determined:

cos 01 _ c o s f l - c o s ~ cos • (AI'I sin ct sin •


cos ~ - cos 3 cos cos 02 = (A2)

sin fl sin

- cos a c o s / / + cos cos 03 = (A3)

sin 0t sin fl

Using the relationship

d - t i dt [arccos (u)] - x / ~ - u 2) (A4)

with eqns (A1)-(A3), the rotation rates can be shown to be

O~ = - ¢ ( c o s . - cos 3 cos O) sin @x/C1 + C2 cos @ - c o s 2

(A5)

sin • 02 = (A6)

4 C 1 -~- C 2 cos ( I ) - cos 2 (I)

03= -+(cos 3-cos .cos ~) sin Ox/C1 + C2 cos @ - c o s 2

(A7)

where

C1 = (sin 2 a - cos 2 fl)

C2 = (2 cos ot cos fl) (A8)

By noting that the sum of the depression angle, 0o, and • remains constant and equal to re/4, cos @, sin • and @ can be related to A and A.

cos re+A) 2 x / ~ + A 2

(A9)

sin @_ x /~ (~ -A) (AIO) 2 %//~ + A 2

~ _ -A~ ((2 +A 2) (A11)


Substituting eqns (A5)-(A7), into eqn (2) yields

/~a = - 2~Mod~ (C3 - C4 cos • - cos z ~)

cos e sin q~x/C1 + Cz cos ~ - c o s 2 (A12)

where

C 3 = ( c o s 2 ~-1- N / 2 c o s (x c o s fl-]- l )

C 4 = ( N / ~ c o s 2 o~ + c o s o~ c o s fl)

(A13)

A.2 Membrane energy rate

First, consider the stretching involved in deforming the flange. Figure 7 shows the trough shape of deformation that the flange assumes. Although the stretching of the flange occurs across the entire deformed region, the shaded area in Fig. 7 is used to approximate the average strain involved. The amount of stretching is constant in the !-direction, normal to the longitudinal girder.

From the geometry of the model, it follows that

u f - - - ( - ( (A14) cos 0o (cos • + sin O)

Differentiating eqn (A14) and substituting the result into eqn (8) yields:

I B

EM, flange = 2No fi dr/ dO

2 /~N BrdP (cos ~- -s in ~) o

(A15)

where N0 is the fully plastic membrane force for the flange:

No=aotp (A16)

Now, consider the stretching involved in deforming the web. Again, the shaded region in Fig. 6 can be used to approximate the average strain tor the web. The amount of stretching varies along the depth of the web. The displacement, uf, at the web flange connection and the maximum displacement, Ux, at the intersection of the second hinge lines are used to determine

230 T, Wierzbicki, J. Culbertson Oriscoll

the average displacement

1 /'/web, avg = 2 Ux + { tan ~t / f (A17)

where ux is:

ux = ( [(x/~ cos ~ - cos fl) sin ~ + cos fl cos ~ - cos ~] [cos fl(sin (I)-cos O)+ cos a] (A18)

Substituting eqns (A14) and (A18) into eqn (A17) and then differentiating this and substituting the result into eqn (8) yields

J~M w e b = - - / / 9 ~ ? ' 2 A T { (COS OC COS fl - - COS 20~COS(I)) , X./ ~ ' ~ ~ ' O / ' r - - ~.-'xz~--- . . . . . . . . . 2 \Lcos/~(sin ~- -cos ~ )+ cos ~]

sin or(cos (I)- sin ~) "] + (A19) cos a(1 + 2 cos (b sin ~)J

A.3 L O A D - D E F L E C T I O N R E L A T I O N S H I P

Equating the external rate of work to the internal rate of energy dissipation gives

l)ge xt =/~tota, =/~B +/~M, nan~e +/~M, web (A20)

Substituting eqns (A12), (A15) and (A19) into eqn (A20) and making the result dimensionless yields

PB ~ . 2 { 2(C3 - - C 4. c o s (I) - - c o s 2 (I)) = *2 *2

Mo (( +A ) cos~zsin~v/C~+C2cos~_cos2~

, , (cos ~ c o s / ~ - cos 2 0~ cos ~ ) + 4x/~( H . . . . . . . . . . .

(cos fl(sin O - cos O) + cos o0 2

sin c~ (cos (I)- sin (I)) -]

+ 8,f 8,t, 2 (cos e - s i n e) (1T~ ~os O----~--n (I)i/ ( A 2 1 )

Crushino dama#e of web 9irders 231

where

c o s , "fi + A*) 2 x/(* 2 + A .2

sin q) = x / ~ ( (*-A*) 2 ~ / ( , 2 + A , 2

(A22}

The dimensionless parameters variables are

A* A ( , ~ H* H H ' H ' tw'

, B tp B = - - " t * = - -

tp tw

(A23)

Substituting eqn (A23) into eqn (A22) yields:

2[Kx(I+~2)_2~b_Kz(I+d/)(I+~kz)x/2] PB 1 cos

2 x / ~ * H * ([Ks(1 -[- ~2)-- g6(1%¢)(1 ~_. ~2)1/2]~ + 1 ~ - ~ \ [-~v~-~-K4~0(l +~0z) ~/z +cos~] /

+ ((4(*H* tan ~ + 8B*t*2)'~ ~ - 2 )~k(1 + ~kz) '/2 (A24}

K1 =2 cos 2 ~ + 2x//-2 cos ~ cos fl+ 1

K2 = 2 cos 2 a + ~/2 cos a cos fl

g 3 = 1-2(cos 2 ~ - c o s 2 fl)

K4 = 2x/~ cos a cos fl

K5 = 2 cos a cos fl

K 6 = ~/2 COS

K7 =2 COS 2 fl+COS 2 0~

(A25}

A* (A26}


If ~, < 1, then ff2,~ 1. If this case holds, then eqn (A24) can be linearized by neglecting ~k 2. Equation (A24) becomes

2[(K1-K2)-(K2+2)~] ) PB = cos ~(1 - ff)~/(K3 - K4) + (K, - 2)~b Mo

_.[_(2N/~*H*[(Ks- K6)- K61//]~ \ cos ]

(A27)

+ (~k[4~*H* tan ~ + 8B*t*2])

A P P E N D I X B

B.1 Bending energy rate

First, consider the fourth hinge line for the primary hinge mechanism. The rotation rate of the fourth hinge line is simply

04p = 2~ (B1)

Now consider the other hinge lines for the primary hinge mechanism. The rotation rates, [0~], for the first, second and third hinge lines are related to the angle • between the first and third hinge lines by the same relationships used for the local denting model--eqns (A5)-(A7); this is due to the similar geometry of the two models. The relationship of • to A, however, differs from that used for the local denting model.

To relate Op to 7, the following geometric relationships are used

cos 00 = x/~ sin ~bp (B3)

sin 0o = x/cos 2 ~bp- sin 2 ~bp (B4)

cos ~p =-~-- {(1 - s i n 2V) + (cos V + sin V)w/s~n 27} (ss)

sin ~p =--~-x/~ {cos 2V- (cos y - s i n V) w/s-~n 27} (B6)

Crushin9 damaoe of web 9irders

- 2~,{sin 27x/sin 2 7 - sin 2 27 +cos 27(½- sin 27)}

233

x/sin 27 -- sin 2 2? X/1 - sin 2? + 2 sin 2 27 + 2 cos 2? x / s in 2? - sin 2 2?

(B7)

Substitution of eqns (B5)-(B7) into eqns (A5)-(A7) gives expressions for [01], [02], and [03] in terms of ?. Substitution of these expressions and eqns (24) and (B1) into eqns (22) yields

/

/~Bp = 2~MoHZp ~cos

+ 2B~to~

C 3 - C 4 c o s (I)p - c o s 2 (I)p

s in O p x / C 1 -~- C 2 c o s O p - c o s 2 0 p / (B8)

where C1 and C 2 a r e given by eqn (A8); C 3 and C4 are given by eqn (A13); cos Op and sin ~p are given by eqns (B5) and (B6); and

Z p -

2{sin 2?x/sin 27 - sin 2 2? + cos 27(½- sin 27)} m

{sin 2 7 - sin 2 27}1/2{1 - s i n 2y+2 sin 2 27+2 cos 2?x/sin 2 y - sin 2 27} 1/2

(B9)

B.2 Membrane energy rate

For the global bending model, the flange remains the same length and therefore does not undergo stretching. The web, however, does stretch. The maximum displacement, Ux, occurs at the intersection of the second hinge lines. The average displacement, Uweb, avg, is

Uweb,avg ~--- ½U x (BIO)

where Ux is defined by eqn (A18) with ( = H . Differentiation of eqn (B10) and substitution of the result into eqn (7)

yields

cos ~ cos f l - cos 2 ~ cos *p /~Mp = -- s e / ~ p N o H2 {COS fl(sin ~ p - cos Op) + cos ~} 2 (Bll)


B.3 S e c o n d a r y hinge m e c h a n i s m s

The bending energy rate terms and the membrane energy rate terms for the secondary mechanisms are equal to those for the primary mechanism evaluated at 7/2 rather than y. The bending energy rate term is:

EB=2~MoHZs(. C 3 - - C 4 c ° s O s - C ° S 2 (I) s )

\cos e sin *sx/C~ + C2 cos O s - cos e *s (B12)

+BMo~

where

Z s -

2{sin 7x/sin ~,- sin 2 7 +cos ~(½- sin 7)} 7----

{sin 7 - sine Y} ~/2{ 1 - sin y + 2 sin / 7 + 2 cos 7 x/sin ~,- sin 2 7} 1/2 (B13)

cos Os =-~-~{(1 - s i n y) + (sin 7+sin 2 7) :/2} (B14)

sin Os = - - ~ {cos y - ( s in y - s i n e 7) t/2 } (B15)

The membrane energy rate term for the secondary mechanisms is

COS CX COS fl -- COS 2 ~X COS tI) s E m = -- x / ~ ¢ s N o H 2 (cos fl(sin Cs-- cos Os) + cos cz} 2 (B16)

B.4 L o a d - d e f l e c t i o n relat ionship

The total internal rate of energy dissipation consists of the energy rate terms for the one primary mechanism and the energy rate terms for the two secondary mechanisms. Combining these terms, equating the sum to the external rate of work and making dimensionless yields

Pa _-- 4Zp ( C3 - - C 4 c o s (I)p - - c o s 2 (I)p

Zv \cos ~ sin Opx/C1 + C2 cos tI)p-COS 20p M0


2w/2H*(cos ~ cos f l - c o s / a cos (I)~)'~

+ (cosf l ( s in~p_cOS~p)+C~s~)~ ,]

~_8Z s / C 3 - C4. c o s (I) s - c o s 2 (I) s . . . . . I-

v k, cos ~ sin *~x/C1 + C2 cos *~--cos 2 *s

2x/~H*(cos ~ cos f l - c o s 2 ~ cOS*s)'~ 4B't*3 + (-~sos fl-~ ~s-~C-~-s O~-+~os ~-~ )+ Z,rH* (B17)

where

The dimensionless parameters are listed in Appendix A (A23) and a new dimensionless parameter is introduced:

L L* = - - (B19)

H

Crushing damage of web girders under localized static loads

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