A simplified method for the impact test of beams using a pseudo-dynamic (PSD) process

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MECHANICSRESEARCH COMMUNICATIONS

Mechanics Research Communications 33 (2006) 190–205

A simplified method for the impact test of beams usinga pseudo-dynamic (PSD) process

Francisco J.Q. Melo a,*, Joaquim A.O. Carneiro b, Cassilda L. Tavares c,Pedro P. Camanho c, Paulo T. de Castro c

a Department of Mechanical Engineering, University of Aveiro, Portugalb Department of Physics Engineering, University of Minho, Guimaraes, Portugal

c Group of Structural Design and Experimental Validation IDMEC-Pole FEUP, Institute of Mechanical Engineering,

Faculty of Engineering of University of Porto, Portugal

Available online 16 June 2005

Abstract

The impact test in structural parts for dynamic applications is an essential procedure for their certification in thepresence of time dependent loads. In the case of beam elements, either built with one material or as an assembly ofdifferent material members joined with recent developed bonding techniques, an impact test is of leading importance,once the dynamic resistance of the joints involved in the beam fabrication is assessed and evaluated. The pseudo-dynamic method is an alternative to dynamic analysis of structures, here offering to the researcher the possibility ofexamining with detail the specimen in test for the initiation and progress of eventual damage mechanisms arising inthe beam element joints whenever included in the design.� 2005 Elsevier Ltd. All rights reserved.

Keywords: PSD; Pseudo-dynamic techniques; Impact analysis; Pultruded beams; Composite beam structures; FRP; Fibre-reinforcedplastics

1. Introduction

1.1. General procedures in dynamics of structures

The behaviour of a structure under the impact of a moving body consists in converting kinetic energyinto deformation energy, where the structure may undergo elastic or plastic deformations. This subject

0093-6413/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechrescom.2005.05.003

* Corresponding author.E-mail addresses: [email protected] (F.J.Q. Melo), [email protected] (J.A.O. Carneiro).

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F.J.Q. Melo et al. / Mechanics Research Communications 33 (2006) 190–205 191

is a particular field of the extensive area of structural dynamics with participation of efficient numerical orexperimental tools for the assessment of the structure behaviour under impact or generalized dynamicloads. Such techniques lead to accurate results, provided that realistic modelling of operating conditionsin test specimens is observed. In the finite element method the dynamic equilibrium of the structure submit-ted to time variable loads and inertial effects is governed by the equation:

Fig. 1.horizo

½M �€Uþ ½C� _Uþ ½K�U ¼ FðtÞ ð1Þ
where [M], [C] and [K] are, respectively, the mass, the damping and the stiffness matrices; €U; _U and U arethe structure acceleration, velocity and displacement vectors. The solution of (1) may be obtained either bymodal superposition or using direct time integration techniques. For fast rate dynamics and non-linearstructure behaviour, the modal superposition gives place to the more efficient direct time integrationmethods.
Procedures in experimental dynamics usually include essential accessories as shakers, excitation tables,accelerometers and eventually high-speed data recording equipment. The task is performed as in an open

loop routine, as the operator analyses the results only at the final step. The initiation of a damage config-uration can hardly be detected with inexpensive methods for thoroughly investigation.

1.2. Pseudo-dynamic (PSD) techniques

This is a less conventional tool in dynamic analysis operating in a closed loop. At each time-step, a struc-ture parameter status is recorded at an experimental task and fed back to a computer program running adirect time integration algorithm for the structure dynamic analysis. A new structure configuration is com-puted in a next time step, which is prescribed to the structure in test via the test rig actuators. The operatingprinciples of pseudo-dynamic techniques impose some limitations to the type of degrees of freedom pre-scribed to the structure in the test rig.

Fig. 1 exemplifies a structure for dynamic analysis submitted to a base horizontal motion. The displace-ment field consists mainly in horizontal relative motions between each storey and the base (Clough andPenzien, 1992).

Horizontal inertial forces are assigned at each storey level as in Fig. 1(b). Stiffness structure values areassigned to the columns with horizontal slabs rigid. The column pair of each storey cell is displaced in par-allel, allowing a dynamic model in Fig. 1(c). This is compatible with the operating principles of pseudo-dynamic techniques, where the test-rigs are equipped with linear displacement actuators.

Applications of pseudo-dynamic techniques have major incidence in seismic analysis, where the timedependent structure configuration is mainly defined along the horizontal direction (Fig. 1). This configura-tion can be prescribed in a test rig at each storey level. A set of linear displacement actuators impose thedisplacement vector while a load cell reads the structure internal reaction force vector at each degree of free-dom level. The method deals now with a modified version of Eq. (1):

U1

U2

F2

F1

m2

m1

k1

k2

2k1

2k2

m2

m1

U0

a b c

Multi-storey structure under seismic loads: (a) basic structure; (b) and (c) simplified dynamic models with lumped masses andntal displacements.

192 F.J.Q. Melo et al. / Mechanics Research Communications 33 (2006) 190–205

½M �€Uþ ½C� _Uþ Rst ¼ FðtÞ ð1aÞ
where vector Rst is the structure internal restoring force vector read from load cells at the degrees of freedom.For structures where the displacement vector is defined along a common direction, the stiffness matrix in-cludes efforts associated to linear displacements, which is typical of multi-storey structures under seismicloads. Fig. 2 shows schematically the set-up for the method described with the operation sequence diagram.A set of linear actuators at the level of each degree of freedom prescribes a displacement vector. The loadcells at each actuator axis read the internal reaction force vector.
PSD techniques have reached success without major modifications since the early experiments (Mago-nette et al., 1997; Molina et al., 2000). Innovation fields were attempted, not only in the design of test rigsbut also in applications, as piping engineering (de Melo et al., 2001; Carneiro et al., 2001), where detail wasfocused on signal processor equipment and test-rig design to meet requirements assigned to geometry com-plexity and test specimens displacement field. However, as this method is performed quasi-statically, thecontribution of viscous damping forces in the dynamic behaviour of the structure practically is not de-tected. An upgrading of the method has consisted on operating the cycle of tasks depicted in Fig. 2 in acontinuous process. This is known as continuous pseudo-dynamics and allows that contributions of dampingforces arising from internal friction in the dynamically loaded structural members are included in the inter-nal restoring force vector Rst (Eq. (1a)).

In this work it is discussed the application of a pseudo-dynamic analysis to the investigation of the struc-tural behaviour of a beam subjected to an impact load. The evaluation of the displacement of the structureat the contact point with a rigid striker is carried out as described next. This is an important step for theassessment of the structure performance, once it leads to the evaluation of the energy absorption capacityfor the structure under impulsive forces.

The displacement field of a vibrating structure can be decomposed in a set of characteristic vibratingmodes. The fundamental or first vibrating mode corresponds to the minimum deformation energy associ-ated to each of the superimposed modes defining the global structure motion; alternatively, it correspondsto a dominant flexibility of the structure. Following this, in a dynamically loaded structure where it is ex-pected that the first vibrating mode is dominant, the approached analysis here proposed can produce resultswith a precision correspondent to the structure discretization into a single degree of freedom (SDOF). Tominimize the errors arising from such a coarse finite element structure division, this analysis considers a

U(t) and Rst

Displacementtrasducerrecord

Actuatordrive unit

Servo -actuators with load cells

Computer

Displacement transducers

U(t+∆t)prescribe r ead

Reaction wall

Test-specimen

Fig. 2. Closed-loop diagram for the operating principle of pseudo-dynamic techniques.


unique external force acting in a mode to induce mainly a fundamental vibration mode as described in theforegoing analysis.

2. Beam structures submitted to impact actions

2.1. Basic assumptions and validity of pseudo-dynamic techniques

From previous discussion, it can be drawn that the exploitation of the potentialities of pseudo-dynamictechniques is limited to structural models where the finite element discretization refers to displacements ofthe same type of the actuators fitted in the rig. Their number must be restricted, otherwise, the rig is expen-sive and its operation is complicated.

An objective of the present work refers to the evaluation of the energy absorbing capacity of beam struc-tures under impact loads. Truss or beam elements present simple damage mechanisms when they reach acollapse configuration. In the case of beam structures under multiple external loads, the collapse configura-tion can result from the combination of elemental damage mechanisms (Horne, 1979) as presented in Fig. 3.

In the static analysis of framed structures a collapse configuration results from the combination of someelemental mechanisms as the represented in previous figure. If the dynamic rate of the external loads is nothigh, the expected collapse configuration is similar to the one statically generated. However, for fast ratedynamic loads it is possible that the previous configurations present an evolution similar to the presentedin Fig. 4, these resulting from the combined action of the external loads and the inertial forces, which playan important role when high accelerations are present.

2.2. Set-up of a simple collision model

It is understood that physically the collision between two bodies is a deformation process involving twofields, where these refer to a local surface distortion (resulting from contact forces) and a propagated defor-mation field inducing bending or shear effects, as is the present case in discussion.

a

Plastic hinges

FF

b c

Fig. 3. Example of a portal frame and associated damage mechanisms for a single concentrated load: (a) basic design; (b) swaymechanism and (c) beam mechanism.

Plastic hinges

FF

a b c

Fig. 4. Example of a portal frame and the possible collapse mechanisms in a fast rate dynamic load: (a) basic design; (b) swaymechanism (plastic hinges in axial wave propagation) and (c) beam mechanism (plastic hinges in flexural wave propagation).


As mentioned previously, it is adopted in this work a dynamic model with a single degree of freedom,assuming that it can practically condense the global dynamic behaviour of the structure at the structure/striker contact point. This holds for deformation cases as the presented in Fig. 3, where the structure dis-placement at any point can be defined via simple expressions based on the deformed structure geometry anddepending mainly on the displacement at the impacted section.

In this section, the effect of the impact force of a striker against a beam structure is analysed under twoaspects:

• The extended effect of the impact force in the beam structure, where this load is the cause for bendingand shear deformation in a single element or for its propagation to adjacent elements integrating thestructure.

• The local effect of the contact forces, which can be materialized for elastic or plastic surface deformationseither in the striker or the still structure. The structure displacement field from contact forces is muchsmaller than the one resulting from bending deformations; however the consequences of this localizedeffect have important repercussion on the time dependent structure displacement at the contact point,as it will be analysed next.

The structure in analysis has a lumped mass distribution at the loaded point and remaining nodes(Fig. 5). The collapse mechanisms may involve sway or beam type, according to the external forceorientation.

From Figs. 4 and 5, it is observed that the external force F and the respective displacement are enough todefine an equivalent dynamic model, where the structure stiffness Ks is calculated from the ratio of appliedforce F(t) and the consequent displacement at the contact point. The equivalent structure mass for this sin-gle degree of freedom discretization is calculated as resumed in Fig. 5. The equivalent stiffness value Ks maybe defined either in the elastic or plastic field. Cases of plastic behaviour in pseudo-dynamic methods areexperimentally evaluated at each load increment as referred in the foregoing analysis.

To set-up the dynamic model of an impact problem using a pseudo-dynamic approach, a simple exampleis analysed next. The striker colliding with the structure is a virtual element only playing role in the algo-rithm in the pseudo-dynamic procedure as resumed in this sequel. Its contact force with the structure cor-responds to the external load F(t) in Eqs. (1) or (1a).

The following problem refers to a simple system composed by a rigid striker and a deformable structureas the presented in Fig. 6 where no damping effects are included.

The dynamics of the striker/structure pair has two possible physical states:

v

ccmc/2 mc/2

mv/2 mv/2

F

F mv+ mc

Ks

(a) - Basic system; (b) mass lumping (c) equivalent dynamic system

(a)

v

ccmc/2 mc/2

FF 2×mv/4

Ks

(b) (c)

mv/4 mv/4

Fig. 5. Dynamic system equivalent to a structure under a single load: (a) basic system; (b) mass lumping and (c) equivalent dynamicsystem.

Striker at initialspeed V0

System

M

Ms

K0

Fig. 6. Dynamic system for collision analysis in a single degree of freedom system.


• The striker keeps in contact with system, at least during one time increment Dt and looses quantity ofmotion, impelling the system with the contact force, so increasing its potential deformation energy.

• In a non-contact situation, the striker and system form independent dynamic models, where the first onetravels at constant speed (eventually after a previous collision) as a rigid body and the second vibratesfreely.

These kinetic states are bound at each time step by the first striker/system collision instant until the com-plete stop of both entities. The last state occurs when the striker kinetic energy is totally converted in thestructure deformation energy. The mathematical expressions defining the previous kinetic states are re-sumed in the statements:

(i) The contact/impact model in a linear-elastic structure deformation. The first of previous assumptions isnumerically defined from the dynamic equilibrium equation written at instant t + Dt as follows:

• Structure or system behaviour:

M €XtþDt þ KXtþDt ¼ FtþDt ð2aÞ
• Striker behaviour (from theorem of impulse)
FtþDt � Dt ¼ M s � ð _XtþDt

s � _Xt

sÞ ð2bÞ
where M, K, Xt + Dt, and €XtþDt are, respectively, the system mass and stiffness matrices and displace-ment and acceleration vectors at time step t + Dt, while Ms, X
tþDts and _X

tþDt

s are, respectively, the stri-ker mass, the displacement and the speed at time step t + Dt. In Eq. (2a) Ft+Dt stands for the systemdeforming force, coming from the impact of the striker having mass Ms. This, for its turn, reduces thespeed according to the theorem of impulse, as by Eq. (2b).
From the definition of constant acceleration and mean velocity between two consecutive time steps inthe Newmark method, the following expressions are useful for the complete algorithm:
_XtþDt ¼ _Xt þ 0.5Dt � ð€XtþDt þ €XtÞ ð2cÞXtþDt ¼ X t þ 0.5Dt � ð _XtþDt þ _XtÞ ð2dÞ

Substituting previous expressions in (2a), it is possible to get updated values of the structure acceler-ation at time t + Dt

ðM þ 0.25Dt2KÞ€XtþDt ¼ FtþDt � KðX t þ Dt _Xt þ Dt2 €XtÞ ð3Þ


The external force vector Ft+Dt in Eq. (3) results form collision with the striker. Two dynamic statesmay occur:(a) Striker and system are both in contact (at least during a time interval Dt). Solving (2b) for Ft+Dt and

eliminating it in (3), an equilibrium expression for contact conditions results:

Fig. 7.structu

M€XtþDt ¼ �M s

Dtð _Xt � _X

t

sÞ � 0.5M s€Xt � FtþDt

with : M ¼ M þ 0.25Dt2 � K þ 0.5M s

and : FtþDt ¼ KðX t þ Dt _Xt þ 0.25Dt2 €XtÞ

ð4Þ

Ms and _Xt

s are the striker mass and speed at time t. The updated value of striker speed at instantt + Dt is the same as the system, _XtþDt, having assumed a total contact system/striker during Dt.

(b) Striker and system are independent dynamic models. Here, the striker travels at constant speed_XsttþDt ¼ _Xstt , while the system vibrates freely until next collision:

M€XtþDt ¼ �KðXt þ Dt _Xt þ 0.25Dt2 €XtÞwhere : M ¼ M þ 0.25Dt2 � K

ð5Þ

(ii) The contact/impact model in a generalized structure deformation. Modifications are necessary in pre-vious analysis, now replacing the product KXt on the right-hand side of (4) or (5) by Rs, the internalreaction force vector. This vector corresponds to the product KXt only in a linear-elastic behaviour,while in a generalized non-linear behaviour, the internal reaction force vector Rs, is evaluated at eachtime step from experimental measurement. In Eqs. (4) or (5) other terms also contain the contribu-tion of the stiffness matrix K. It is usual to delete them in seismic analysis, once the structure velocityand acceleration vectors are not large; furthermore, such terms are multiplied by quantities Dt andDt2 which are generally small. However, in a higher-rate dynamic analysis, velocity and accelerationare not negligible; such terms are multiplying the stiffness matrix K(t + Dt) in (4) or (5) at time step t.As the stiffness matrix may change during the deformation, a linear approach to the calculation ofthe structure stiffness results from a next internal reaction force vector at time-steps t + Dt; thisallows the evaluation of a tangent stiffness matrix at time t:

Kðt þ DtÞ ¼ RtþDts � Rt

s

XtþDt � Xtð6Þ

where internal structure reaction forces Rts and RtþDt

s are read from load cells at the level of structuredegrees of freedom.

It was considered that the structure deformation resulted practically from bending or the combination ofbending and shear effects, according with the deformation model in the numerical approach. The contact

∆F ∆F

∆l

F limit

∆l

F limit

≈

Local deformation on the structure from the effect of contact loads of a rigid striker. The constitutive relation associated to there response is approximated as rigid/perfectly plastic.


force of structure/striker pair was calculated after the theorem of impulse having assumed an elastic shock.The consequent displacement at the contact point was considered much smaller when compared with otherstructure displacements; therefore its contribution is neglected. To investigate the influence of an eventualnon-elastic shock in the global displacement of the deformed structure, it is now considered that the contactforce can reach a limit value set from a constitutive law. Such relation can be set either experimentally orby a numerical approach via finite element techniques. In the present analysis, a simple approach consistedin a deformation model for the contact effects where it was adopted a rigid-perfectly plastic constitutiverelation as sketched in Fig. 7. It is considered that the striker is rigid, while only the structure can undergoplastic deformations. The influence of a rigid-plastic shock in the global structure deformation is analysednext.

3. Pseudo-dynamic procedures in a structure under an impact load

At each time step, the Newmark algorithm calculates an updated structure displacement vector. In theoperation mode of the pseudo-dynamic method, this displacement develops is prescribed to the structure intest, giving rise to the internal reaction force vector. This vector is fed back to the Newmark algorithmallowing for a new displacement vector, as described. The computer program using algorithms (4) or (5)calculates updated displacement, velocity and acceleration vectors for system and striker. In next time step,the system displacement vector is prescribed to the structure via precision actuators with load cells at thelevel of the structure degrees of freedom. The procedure is resumed on test examples reported in the fore-going analysis.

Particular details are important in the performance of the method in discussion:

• Though the Newmark constant acceleration method is unconditionally stable, accurate values resultfrom a time step smaller than a critical Dtc (Bathe, 1992):(a) For single degree of freedom

Dtc ¼2p10

ffiffiffiffiffiMK

rð7Þ

This is the natural oscillation period of the system(b) For multiple degrees of freedom, Dtc = 1/10 of the least period associated to highest system

eigenfrequency.
• The striker was considered perfectly rigid . Effects of surface distortion from local plastic deformation incontact are not considered.
4. Numerical examples: cantilever beam with concentrated mass under transverse impact

Next examples have as objective some insight in the numerical approach to the contact-impact between amoving rigid body and a deformable structure. The problems were solved with a numerical approach viathe Newmark algorithm. For sake of simplicity, the structure was assumed to undergo linear elasticdeformations.

Example 1. The assessment of previous algorithm is performed with a simple example, where results arecompared with data from solid dynamics theory. Fig. 8 shows the model used in the analysis, involving asingle degree of freedom system and striker.

The critical time step by (7) is Dtc = 0.0347 s. Smaller intervals lead to precision increased solutions,having been adopted Dt = 0.01 s. Fig. 9 shows the results for the displacement progression of the elastic

Fig. 8. Dynamic model of an SDOF system under transverse shock load.

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

Syst

em d

ispl

acem

ent (

m)

Time (s) 0.00 0.05 0.10 0.15 0.20 0.25 0.30

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Strik

er s

peed

(m/s

)

Time (s) a b

Max system displacement 0.157 m

Fig. 9. (a) System displacement and (b) striker speed drop.


system and the successive striker speed drop as consequence of the of the shatter between the striker andsystem mass. The striker speed drop is evident in graphic (b), matching with time steps at displacementirregularities in graphic (a). The conservation of the total energy predicts that in a linear-elastic problem,the kinetic energy in the striker is converted in elastic deformation energy:

1

2M st

_X2

st0 ¼1

2KX 2

tmax ð8Þ

This gives a maximum displacement Xmax = 0.1747 m for the system, while present result Xmax is0.157 m, as shown in Fig. 9. This discrepancy can be analysed from Eq. (4), where the equivalent mass sys-tem is

M ¼ M þ 0.25Dt2 � K þ 0.5M s ð4aÞ


The equivalent mass (4a) increases with mass M of the deformable system, so determining in the next time-step smaller values of acceleration vector (4a) with incidence on the velocity and displacement in (2c,d).Physically this is understood from the fact of a moving body hitting elastically another one in a still posi-tion, a larger speed drop occurs with a larger mass of the still body. This influence is inspected in the nextcase.

Example 2. The striker mass is maintained at 10 kg but the elastic system mass is reduced to 0.25 kg(remaining parameters unchanged). The maximum displacement for the system is about 0.17 m(Fig. 10), closer to the theoretical value of 0.175 m from Eq. (8), still exhibiting some shatter from stri-ker/system contact, as occurring in the ideally elastic contact between the striker and the structure. How-ever, this is not realistic in real impact tests, where it is likely to occur a local smoothing effect from someplastic deformation at the contact surface.

Next examples refer to the impact simulation of a beam with irrelevant end support design, as only anequivalent stiffness value definition at the contact point interests. The beam is of an ideal material. The firstimpact analysis considers that the contact force does not exceed a limit value, while in the second test thatforce could exceed it, leading to a local plastic deformation from the assumption of a rigid-plastic criterion.The problem data is as follows:

Striker mass rms = 10 kgs.Initial striker velocity Vst = 1 m/s.Concentrated equivalent system mass rm = 1 kg.Equivalent structure stiffness at contact point K = 300 N/m.Structure initial displacement = 0.Structure initial velocity = 0.Maximum contact force prior plastic deformation (rigid-plastic model).

• First problem Fclimit = 5000 N.
• Second problem Fclimit = 1500 N.
Fig. 11 refers to the test where the contact behaviour of the structure–striker pair is perfectly elastic orexhibits a local damage.

0.00 0.05 0.10 0.15 0.20 0.25 0.300.00

0.05

0.10

0.15

0.20

Syst

em d

ispl

acem

ent (

m)

Time (s)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Strik

er s

peed

(m/s

)

Time (s)

Max. displacement: 0.17 m

Fig. 10. System displacement and striker speed drop for a system with an equivalent mass of 0.25 kg.

0.00 0.05 0.10 0.15 0.20 0.25 0.300.00

0.05

0.10

0.15

0.20

Dis

plac

emen

t (m

)

time (s)0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

Dis

plac

emen

t (m

)

Time (s) a b

Fig. 11. (a) Evolution of the transverse displacement in a beam hit by a striker: striker mass = 10 kg; equivalent system massrm = 1 kg; system stiffness K = 300 N/m and limit contact force: 5000 N. In this problem the peak contact force was about 2500 N, notexceeding the limit contact force. (b) Evolution of the transverse displacement in a beam hit by a striker: striker mass = 10 kg;equivalent system mass m = 1 kg; system stiffness K = 300 N/m and limit contact force: 1500 N. In this problem the peak force wasexceeded (if the contact was perfectly elastic, the contact force reached about 2500 N). As the contact force is now limited to aprescribed maximum value, the maximum gap between the structure and the striker is now much smaller than in a rigid collision, whichcontributes for a smoothed aspect of the displacement graphic.


The structure displacement exhibits a graphic also obtained in previous examples, where the typical as-pect arises from the rebound between the beam and the striker as consequence of the interaction betweenthe elasticity and equivalent mass of the system at the contact point. Alternatively, when the limit value forthe contact force is exceeded, the structure displacement graphic at the contact point appears as inFig. 11(b), now much smoother as a result of a locally lower shock restitution factor compared with thefirst case.

On analysing the present results it is possible to note that, in spite of the different behaviour of both dy-namic entities according to the contact characteristics, both the maximum structure displacement and thetotal elapsed time from first impact until the complete striker stop did not experiment a significant change.This result only applies naturally whenever the displacement at the contact zonal is much smaller than theone from the bending effect. If the objective of the experience is the evaluation of the maximum bendingdisplacement that a beam under impact can withstand, it is possible to ignore whether there is a non-rigidcontact in the impact process, once both the maximum displacement and the time for that value come prac-tically unchanged.

5. Pseudo-dynamic simulation of the collision of a rigid striker against a simply supported beam

Next experiment refers to the transverse impact of a rigid striker against a glass/polyester compositebeam produced by pultrusion process with remarkable performance in the mechanical properties, this alliedto an attractive aspect. Automotive and aeronautic industries are increasing the application of this type ofelements as a partial alternative to lightweight pressed sheet metal hollow section elements.

The aim of the next experiment is the evaluation of the shock energy absorption capacity for a pultrudedbeam element. The test was carried out in a multi-axial frame INSTRON machine as shown in Fig. 15. Asingle actuator set-up was used in the experiment. The test specimen has end joints bolted to the machinecrosshead beams working as partially built-in edges (Fig. 12).

Pull-downLoad anchor

Crosshead beams

INSTROM Servo-cylinder

with internal LVDTlinear displacement

transducer

INSTROM Load cell

Pultruded beam(test specimen) with

bolted ends

Digital displacement gauge

computer

Line feedback for displacement vector

(optional)

Line feedback for internal restoring force vector

INSTRON Control unit

Next time-step displacement vector

Fig. 12. Schematic representation of servo-cylinder drive control in the INSTRON machine for pseudo-dynamic three-point pull-down test bending of a pultruded beam.


The beam end bolts support essentially shears forces, here assumed as equivalent to a single resultantreaction force and a bending moment. Dimensions and test-specimen properties are included in Fig. 13.

Fig. 14 shows the design of bolted joint end supports with shear inserts fastening a composite beam to ametallic bracket. The adopted joint design has eight 4 mm-diameter holes drilled at each end of the com-posite beam supporting the material located at the exit of the drill to avoid damaging the composite. Fol-lowing the procedure proposed by Camanhoe and Matthews (2000), metallic inserts with 4 mm and 6 mminternal and external diameters, respectively, were bonded into the holes using an epoxy structural adhesive.The use of the cylindrical bonded inserts leads to a redistribution of contact stresses in the composite,reducing the bearing stress concentration factor (Camanhoe and Matthews, 2000). The end distance andthe adopted one between holes follow EUROCOMP�s (1996) recommendations. Considering e as the edgeand p the distances between the hole centres, ratios e/d = 3 and p/d = 6 were used (Fig. 14).

70 mm

30 mm

Length L=1.4 m

Transversesection

Thickness 3mm

Fig. 13. Overall dimensions of a beam made by pultrusion process.

Fig. 14. Beam test specimen end support with a bolted joint including shear inserts.


A load was applied at the beam mid span section using the INSTRON machine actuator for evaluationof an initial stiffness value to be added to the previous structure data in Fig. 13. In this experiment the limitcontact force was supposed high enough to induce an elastic contact. The pseudo-dynamic process per-forms the following tasks during each time-step:

1. Prescription of a small displacement at the load point and reading the internal reaction force to calculatean initial SDOF stiffness K0 = F0(load cell)/Dl in a three-point beam bending. The evaluation of the struc-ture stiffness was achieved defining a static force–displacement diagram on operating very slowly themachine in a set of prescribed displacement increments, pausing at each increment to read the structurereaction. This procedure practically suppresses the contribution of internal friction forces and allows astructure adaptation prior a next load increment.

2. Calculation of the equivalent structure (or system) mass at the contact/impact structure section,m � qAL/2, where q is the specific mass, A is the transverse area and L is the beam length.

3. Calculation of the critical time step Dt (from (7)).4. Use of Eqs. (4) and (2a,b) for contact striker/system during increment Dt.5. Analysis of gap between striker and structure (contact or non-contact).6. Choice of Eqs. (4) or (5), depending on the contact status.7. Reading from load cell of internal reaction force form test specimen and feedback it to Eqs. (4) or (5).8. Calculation of next system acceleration, speed and displacement for system and striker; send data to

actuator control unit and repeat procedure from step 5.

Fig. 15 shows the experimental set-up used in the experiment. The INSTRON machine has an actuatorwith a built-in internal LVDT (linear voltage displacement transducer) along the hollow cylinder ram.

Fig 16 shows the displacement at midspan contact section past successive time steps until the strikerstop. It is noted the rebound between the test specimen and the striker as a consequence of the elastic con-tact predicted by the numerical model.

The maximum transverse displacement experimentally measured at the contact section is about0.00795 m as depicted in previous graphical results. This value is in good agreement with a theoretical resultfrom (8) where the kinetic energy of the striker is 10 J. The elastic energy absorbed by the beam is:Ue = 300,000 · (0.00795)2/2 � 9.48 J. The kinetic energy in the striker is Uk = 10 J.

This discrepancy, though small, may be due to a discontinuous system kinetic state, with successive stri-ker–system rebound until complete stop. In fact, the experiment cannot be included in a linear analysis,once its geometry and inertial parameters change with time.

It is noted here that the time scale in graphical results of Fig. 16 refers to virtual values. The proceduredealt with a modified (non-automated) equipment to cope with the operating method of pseudo-dynamictechniques, each time step carried out in the numerical/experimental tasks of the technique have in factincluded delay pauses, this making the method largely dissimilar of the continuous pseudo-dynamic

Fig. 15. Test specimen installation in the INSTRON machine for three-point bending.

0,00E+001,00E-032,00E-033,00E-034,00E-035,00E-036,00E-037,00E-038,00E-039,00E-03

0,00E+00 5,00E-03 1,00E-02 1,50E-02 -0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Transverse displacement at contact point (m)

Time (s)

Striker speed drop (m/s)

Time (s)

Fig. 16. Displacement at impact point and striker velocity drop.


techniques. The real time of the test took about one minute; that is to say, a value in excess about 6000times of the predicted time for the complete striker stop in the collision with the structure in test. This factnaturally excludes the contribution of internal friction forces in the deformed structure acting as a struc-tural damping component.

6. Conclusions and discussion for technical applications

The problem was formulated with a single degree of freedom, where concepts of dynamics of rigid ordeformable bodies played role in the algorithm of the structure equilibrium. The method is proposed as analternate procedure in the evaluation of the shock absorbing energy of structures under impulsive loads, asfor example, test pendulum used in the certification of separate parts for specific application structures, astransport vehicle bodies (Dagson, 2001). The effect of an impact load from an external striker on simple struc-tures as cantilever or simply supported beams can be simulated with acceptable accuracy with the presentmethod, even when a contact surface damage can occur. This last case was approached having used a quitesimple model, where the local structure behaviour was assumed a rigid-perfectly plastic constitutive model.

The tool is an economic alternative for a real impact analysis with the attribute of being performed in aset of virtual time steps. The process can be interrupted when desired for test-specimen inspection, seekingfor collapse mechanism initiations in bolted, welded or bonded joints if used in the design.


This method used has, however, important drawbacks resumed as follows:

• The simple formulation of the method, with only one degree of freedom, masks the presence of moredeformation modes inherent to bending wave propagation when high impact speeds are involved. Theperformed experiments suggest that this approached analysis can be applied to truss or portal framesstructures whenever the displacement field can be defined with reasonable accuracy from a unique sec-tion, precisely the one at the contact zone with the striker. Essentially, the displacement field of the struc-ture in test is resumed to the first (or fundamental) bending eigenmode, the one involving the minimumdeformation energy. This constitutes a limitation for the technique when the structure displacement fieldis too complex, making the method inapplicable in the characterization of the damage models as analternative to true impact tests.

• The procedure was carried out in a sequence of prescribed displacements in each time increment, buteffectively it corresponded to a long real time delay during each step of the process. As the internalrestoring forces, resulting from the imposition of the updated displacement at each time step, were readafter the structure in test was still, the contribution of internal friction forces in the structure deforma-tion process was neglected. This suppressed an important contribution for a realistic dynamic behaviour,but for elastic deformations (which was the case of the composite beam analysed in this work), this omis-sion may not be very detrimental in the precision of the results.

• The case of large deformations with non-linear material structure behaviour has a more relevant impor-tance in the prediction of the dynamic behaviour of structural elements subjected to impact loads. Theprocedure here proposed needs the inclusion of an additional force vector resulting from damping effectsin Eq. (1) of this work. As an accurate and complete characterization of the damping forces in dynam-ically stressed structures is not possible, an alternative to this consists on evaluating an equivalent vis-cous damping factor as described by Clough and Penzien (1992). With the equipment used in thiswork, a procedure to estimate an equivalent viscous damping factor in presence of large plastic defor-mations is proposed as follows:(a) Define a structure stiffness K as described previously (with small, slowly imposed and paused dis-

placements to read the internal structure load). The procedure applies even with plastic deforma-tions, where a tangent value for K can be calculated.

(b) Change the structure deformation rate in the INSTRON machine, running the actuator ram at con-stant and increasing velocities to plot a set of displacement/force results read from the LVDT and theload cell. The evaluation of an equivalent viscous damping factor to be used in Eq. (1) of thedynamic equilibrium of the structure appears after subtracting in the structure internal forces readfrom the load cell the component resulting from the product of the stiffness K with the correspondingdisplacement. It is noted here that, the maximum deformation velocity available in a INSTRONmachine remains well lower than values expected in impact tests. The process here proposed is nat-urally an approach with a predictable accuracy highly dependent on the material properties andstructural constitution of the test-specimen, Nevertheless, it appears as a simple process to avoidthe set-up of complicated algorithms in the method.

References

Bathe, K.-J., 1992. Finite Element Procedures, second ed. Prentice-Hall, Englewood Cliffs, NJ.Camanhoe, P.P., Matthews, F.L., 2000. Bonded metallic inserts for bolted joints in composite laminates. Journal of Materials Design

and Applications: Proceedings of the Institution of Mechanical Engineers Part L 214, 33–40.Carneiro, J.A.O., Pereira, J.P.T., Rocha, A.J., de Melo, F.Q., 2001. Structural validation of motor vehicle seats by pseudo-dynamic

analysis. Paper 2001-01-3462, ATT/SAE, Automotive and Transport Technology Congress and Exhibition, Barcelona, October2001.


Clough, R., Penzien, J., 1992. Dynamics of Structures, second ed. McGraw-Hill, New York.Dagson, N., 2001. Influence of the forming process on the crash response of a roof rail component. Master Thesis, Department of Solid

Mechanics, Linkoping University, Sweden (Reg. No. LiTH-IKP-EX-1836), March 2001.de Melo, F.J.Q., Carneiro, J.A.O., Lopes, H.R., Rodrigues, J.F.D., 2001. The dynamic analysis of piping systems using pseudo-

dynamic techniques. Journal of Strain Analysis ImechE 36 (5), 441–451.EUROCOMP, 1996. In: Clarke, J.L. (Ed.), Eurocomp Design Code and Handbook. E&FN Spon.Horne, 1979. Plastic Analysis Design. Pergamon Press, Oxford.Magonette, G., Molina, F.J., Taucer, F., Verzeletti, G., Renda, V., Tognoli, P., 1997. Contribution of the JRC ISPRA to the inter-

comparison of analysis methods for seismically isolated nuclear structures. In: International Post-SMIRT Conference Seminar onSeismic Isolation, Passive Energy Dissipation and Active Control of Vibrations of Structures, Proceedings InternationalSymposium, Taormina, 25–27 August 1997, Italy.

Molina, F.J., Verzeletti, G., Magonette, G., Taucer, F., 2000. Dynamic and pseudodynamic responses in a two-storey buildingretrofitted with rate-sensitive rubber dissipaters. In: Proceedings of the 12th World Conference on Earthquake Engineering, NewZealand.

A simplified method for the impact test of beams using a pseudo-dynamic (PSD) process

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