Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 15363799

Viscoelastic analyses of a prestressed Howe Bridge

D.A. Gasparini & 1. Bruckner Department ofCivil Engineering, Case Western Reserve University, Cleveland, Ohio, US

F. da Porto Department of Strue/ural and Transpor/a/ion Eng. , University of Padova, Padova, l/ai)'

ABSTRACT: Many wooden trusses built in the 19th Century were designed to be prestresseel, in order to have both diagonaIs active without tensile connections. However, the viscous behavior of wood can cause temporal losses in prestress. In the present contribution, a standard-linear-solid model for wood is used to carry out viscoelastic analyses. Simple prestressing experiments, to quantify the model parameters, are proposed. Results ofviscoelastic analyses ofthe Pine BluffHowe Bridge in Indiana are presented.

fNTRODUCTION

Many ofthe well-known American truss bridges, built of wood or iron or using a combination of wood and iron in the first half of the 19th century, were prestressed. The advantages of prestressing wooden bridges, in particular, were that connections were simplified, some wood tensile connections were elim­inated anel, if ali elements did not loosen, the stiffness of a bridge was increased.

The concept of prestressing was initially applied to the construction of wooden covered bridges by Stephen Harriman Long (Wooel, 1966; Gasparini and Provost 1989; Gasparini and Simmons, 1997). In his first truss form , patented in March 1830, the prestress­ing was achieved by driving wood wedges between the diagonaIs and the chords, to induce compressive forces in the diagonaIs and tensile forces in the verticaIs and chords (Long, 1836). In his second patent, November 1839, he detailed tension connections for the diagonaIs and drove wedges between the verticaIs and the top chorei, thus obtaining the opposite state ofprestressing.

Long's 1830 patent became outdated when the William Howe patent, shown in Figure I , was granted in August 1840. Howe replaced the vertical wooden

Figure I. Howe truss elevation, from his August 3, 1840 U.S. patent.

posts with wrought iron threaded rods, and prestressed the bridge by tightening the nuts, thus proposing a much easier prestressing processo

The prestress state in Howe 's truss was exactly the same as that of Long's 1830 truss (Gasparini & Simmons, 1997). Other prestressed truss forms , which reproduced the prestress state of Long's 1839 truss, were patented in the following years (Thomas and Caleb Pratt, 1844; Nathaniel Rider, 1845 ; Stephen Moulton in Englanel, 1847).

For the purpose of understanding the structural behaviour of such wooden trusses, in order to improve their maintenance and rehabilitation , it is necessary to quantify the effects oftime-dependent stress- strain properties of wood and to analyse their influence on the temporal loss of prestress. This paper describes studies ofa wooden Howe truss, the Pine BluffBridge in Indiana. Simple experimental and analytical stud­ies are first presented in order to understand the behaviour ofthe Howe truss formo A linear viscoelas­tic formulation using a standard-linear-solid model for wood is then presented and a method for estimat­ing the model parameters from prestressing experi­ments is described. Finally, the results of analyses for time-dependent effects on the Pine Bluff Bridge are presented.

2 LINEAR ELASTIC STUDlES OF THE PINE BLUFF BRlDGE

2.1 Description of lhe bridge

The Pine Bluff Bridge over Big Walnut Creek in Putnam County, Indiana, was built by Joseph Albert

983

Figure 2. Internai view af the bridge.

Britton in 1886. The bridge consists of two simply­supported spans approximately 30 m in length. Each span has ten panels. The bridge is a classic Howe form, with diagonais framing within only one panel , threaded vertical wrought iron rods and cast iron nodes. Figure 2 shows an internai view of the bridge.

In its present condition, practically ali the counter­braces are loose. The bridge truss therefore essentially behaves as a statically determinate truss. lt is only the action of the dead load that prevents the diago­nais near midspan from becoming slack when a heavy live load traverses the span. The present condition of the bridge indicates that current maintenance practices do not recognize the original technology ofthe classic Howe truss.

2.2 Analysis under dead and tive load

Two linear elastic plane frame models, one with ali diagonais active and another with the counter­diagonais inactive, were used for the linear elastic analyses of the Pine Bluff Bridge. The wood species used for the construction of the bridge is white pine, which is assumed to have a modulus of elasticity of 8.27 GPa. The vertical iron rods are assumed to have a modulus of elasticity of 200 GPa. The total weight of one span is approximately 530 kN, that is 26.5 kN per each panel of the two-dimensional frame model. Length and cross sectional area of the truss elements are reported in Table 1. Figure 3 shows the overall geometry of one span.

Analyses of the non-prestressed model were per­formed for the action of the actual dead load and for a unit live load. Maximum dead load stresses in the chords at midspan were approx imately 1.5 MPa. The maximum dead load vertical displacement at midspan was approximately I cm for the model with ali diagonais active and 1.4 cm for the model with counterdiagonals inactive.

From an analysis under concentrated unit gravity load applied on each no de along the length ofthe truss,

Table I. Length and cross sectianal area af the truss elements.

Length Area Elemenl Type (m) (m2)

LOLl, L I L2, L2L3, lower chord 3.048 0.1503 L3L4, L4L5 UOU I, U IU2, U2U3, upper chord 3.048 0.1306 U3U4,U4U5 LlUO counter 5.8 17 0.0232 L2UI , L3U2 counler 5.817 0.0310 L4U3 counler 5.8 17 0.036 1 L5U4 counter 5.81 7 0.04 13 LOU I diagonal 5.817 0.0929 L IU2 diagonal 5.8 17 0.0826 L2U3, L3U4 diagonal 5.817 0.0723 L4U5 diagonal 5.817 0.0619 LO UO, L5U5 vertical rod 4.953 0.0010 LlUI vertical rod 4.953 0.0023 L2U2 vertica l rod 4.953 0.0020 L3U3 vertical rod 4.953 0.00 16 L4U4 vertical rod 4.953 0.00 13

Figure 3. Geometry of the Pine Blulf Bridge and element numbering of the plane model.

it was found that the element which is subjected to the highest axial tension force is the mid-span diago­nal. The live load that, in combination with the dead load, produces a null total axial force in the midspan diagonal is found by superposition. Such a load would produce a kinematic mechanism in the bridge if ali joints were perfectly pinned. Because of the conti­nuity of the chords, the bridge does not develop a mechanism, but nonetheless such a load represents a limit state at which a change in the behaviour of the bridge occurs (da Porto, 2002). To prevent loose ele­ments, the diagonais ofHowe trusses were prestressed in compression by tightening the vertical rods.

2.3 Prestressing analysis and physical experiments

Since the actual pretension forces achieved in the rods immediately afier prestressing were unknown, unit effective nodal forces from prestressing were used to study effects of prestressing. Analyses revealed that the prestress action obtained by tightening a rod, is effective primarily in the elements ofthe two adjacent

984

Figure 4. Element axial forces from the prestressing action.

Figure 5. Element axial forces from the combination ofthe prestressing action and the change in temperature .

panels. The upward vertical displacements ofthe truss, caused by prestressing, are very smal!. For realistic initial prestress forces, the corresponding vertical dis­placements are almost surely not sufficient to lift a bridge and relieve the dead load forces on the false­work, thus making the full dead load active (da Porto 2002). '

One of the physical experiments performed on the Pine BluffBridge was to loosen two rods and manually retlghten them, with one person using a wrench with an approximate moment arm ofO.6 m. The axial strains in the bars were measured with electrical resistance strain gauges. The average prestressing force induced in the rods was approximately 30.5 kN per bar. At present, the actual size ofwrench and method used by the original builders to tighten the nuts is unknown. Nevertheless, almost certainly a greater torque was used to tighten rods in the nineteenth century, thus achieving greater prestressing forces.

The prestress forces in the truss members may change wlth a variation in temperature. The coef­ficients of thermal expansion for wood in the lon­gitudinal direction and for steel are different and respectively equal to about 3.6 x 10- 6 LlUC and I J.7 x 10- 6 LlLoC. Therefore, additionallinear elas­tic analyses of the Pine Bluff Bridge were done for an assumed prestress condition and an increase in tem per­ature. Specifically, effective nodal prestressing forces of 250 kN, ul11form along the span, were assumed and effective nodal forces from a t. T = 25°C were computed.

Figure 4 shows element forces from the prestressing actlOn only and Figure 5 shows element axial forces from the combination ofthe prestressing action and the change in temperature. Prestress losses for the applied 1I1crease oftemperature vary from 17% to about 40% . A good practice, in order to minimize the decreases of

prestress forces , may be that of prestressing the truss at warm temperatures. A decrease in temperature will in fact increase prestress forces.

3 TIME DEPENDENT BEHAVIOUR OF WOOD AND LABORATORY TESTS

3.1 Effect of shrinkage and creep

Shrinkage and creep consist in deformations related respectively to the change in moisture content and to the behaviour of wood under load. Both phenomena cause displacements and, possibly, decreases in the initial prestressing forces in the truss members. Both shrinkage and creep are time-dependent phenomena.

In particular, shrinkage is a property of wood that depends on moisture and temperature conditions and is affected by wood anisotropy. It consists in temporal 1I1crease 111 strain while the wood is losing mois­ture. Although it is higher in the radial and tangential direction, it is likely that prestress states in Howe trusses are primarily affected by longitudinal shrink­age strains. Maximum longitudinal shrinkage strain is 111 the range of 0.00 I to 0.003 (Forest Products Labo­ratory, 1999). A simplified analysis of the Pine Bluff Bridge under shrinkage revealed that a mean shrink­age strain equal to 0.002 can reduce prestress forces to zero. The shrinkage action was modelled by applying nodalloads in the direction ofthe elements, assuming ali nodes are fixed and shrinkage is hindered, and by superposing the effect of fixed end forces in the truss members (da Porto, 2002). Therefore, properly sea­soned wood must be used, or periodical re-prestressing to counteract the effect oflongitudinal wood shrinkage is required.

Creep consists in an increase in strain given a con­stant stress. A property related to creep is relaxation which is a temporal decrease in stress given a con~ stant strain . The time-dependent properties of wood under stress have been studied extensively (Wood, 1951 ; Kingston, 1962; Senft and Suddarth 1971· Barrett and Foschi , 1978; Bodig, 1982 ; Mad~en anct Johns, 1982; Hoyle, Griffith and Itani , 1985; Tissaoui , 1996 and others).

A self-equilibrated state ofpre-stress in a wood-iron truss is neither a creep nor a relaxation condition in the sense that both stress and strain vary with time. To obtain data about the prestress state, loss-of-prestress experiments were performed at Case Western Reserve University. A linear viscoelastic model of the time­dependent behaviour of wood was then adapted to prestressed members, in order to carry out viscoelastic studies on the Howe truss.

3.2 Prestress fests on wood specimens

The loss-of-prestress experiments were performed on a seasoned and a green timber specimen. The

985

Figure 6. Prestress test set-up.

6.0

5.5 .. ~ 5.0

:; 4.5 o ~ 40

.: 3.5

ª 3.0 Ui 2.5

~ 2.0

q 1.5

1.0

:''''-""-

I'"

- - r- -

'''v 1

~

....... '" '"-f-

- -

DOL g l. ~ I

teF ~ -Wh

o 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 Time (days)

Figure 7. Loss of prestress with time for the two wood specimens.

two speeimens were initially prestressed by hydrauli ­eally tensioning and anehoring two 32 mm Dywidag bars. A 18 em x 18 em x 3 m Douglas fir timber, eut and fi rst used about 75 years ago, was eompressed with a 4.14 MPa initial prestress. A reeently-eut 18 em x 18em x 3 m Eastem white pine timber was eompressed with a 3.69 MPa initial prestress. Figure 6 shows the speeimens and the prestressing test set-up.

Figure 7 shows the loss of prestress with time. The Douglas f ir speeimen lost 23% of prestress after 130 days at whieh time the loss rate is very smal!. On the other hane!, the reeently-eut white pine spee­imen lost about 60% of its prestress after 130 days. Loss-of-prestress on actual Howe trusses will depend not only on the wood properties, but also on atmo­spherie conditions, on the behaviour of the joints and on the re lative ax ial stiffness ofthe timber member and the metall ie rods.

4 LINEAR VISCOELASTIC MODELLING OF TIME-DEPENDENT BEHAVIOUR

4. 1 Assumed viscoelaslic model

Time-dependent stress- strain properties of wood have often been modelled using linear viseoelastie eonstitutive equations (Flügge, 1975; Christensen, 1982; Aklonis and MacKnight, 1983; Lakes, 1999).

(2) A2, E2, E'2, J-l2, L

b~ K2= A2E2

L

a (1) AI , EI, L K I= AlEI L

Figure 8. Model for prestressing experiment.

The Burger linear viscoelastic model is a simple and reliable model to deseribe the creep behaviour of wood (Bodig, 1982). The Burger model is eom­posed by a Kelvin body, whieh in tum eonsists of a linear elastie component in parallel with a linear vis­eous component, inserted between the linear elastie and the linear viseous eomponents of a Maxwell body. Fridley (1 992) applies a Burger linear viseoelastie model (Flügge 1975) to prediet creep in Douglas fir. Fridley notes that most of the primary ereep oeeurs in a few months, followed by long-term creep at a mueh smaller, eonstant rate. If this long-term creep rate is negleeted, then a simpler, three-parameter, "standard linear solid" (Flügge, 1975) viscoelastie model may be used. This model may be represented by a linear elastic component in series with a Kelvin body. The model is expressed mathematieally in one dimension by the fo llowing three equations:

S(l) = se (t) + Si (t)

u(t) = Eee (t)

u(t) = E' li; (t) + f1&; (t)

(1)

(2)

(3)

in which Bel) is the total strain and is expressed as a sum ofan elastie se(t) plus an inelastic strain, Si(t), and t(l) is the time rate ofthe total strain. The three model parameters are the eonventional modu lus of elastieity, E, another modulus, E' , and a viseosity parameter, /.L.

4.2 Modelling of lhe preslress lests

The three-parameter standard-linear-solid model assumed fo r wood may be eombined with a linear elastie model for steel to predict the behaviour of the simple prestressing experiment shown in Figure 6. The system may be represented as shown in Figure 8, with the linear elastie steel element labelled as I and the wood element labelled as 2.

For one-dimensional axial elements of length L, and constant eross-seetional area A, the model may be directly expressed in terms ofelement ax ial forces, N(t), and element axia l deformations, e(t), taking into account that N(t) = Acr(t) and e(t) = Ls (t). The force-deformation equation for the linear elastic steel element is:

(4)

986

in which elO is the prestressing deformation, consid­ered negative if the element must be stretched to fit between end nodes. For the wood element, the force­deformation equations may be derived from eqs.(2) and (3) that respectively become:

N2(t)=E2A2 e2e(t) L2

(5)

(6)

The nodal displacement at node b is equal to the axial deformations in the steel and the wood elements, as expressed by the following eqs.(7) and (8):

el (t) = ele (t) + O = ub (t)

e2(1)= e2e(t)+ e2i(t)= ub (I)

Eqs.(5), (6) and (8) may be combined to give:

E2(Ub -e2i)=Eíe2i(t)+ Jl2e2i(t)

(7)

(8)

(9)

Horizontal equilibrium imposes that the sum ofthe axial forces, N I (t) in lhe steel element and N2(t) in the wood element, is equal to zero. Given eqs.(4) and (7), and eqs.(5) and (8), the equilibrium condition may be written as:

(10)

Eqs.(9) and (10) may be solved for the inter­nai state variable e2i(t) and the nodal displace­ment Ub(t) using the initial condition e2i(0) = O or Ub(O) = KI elO/(K I + K2). The nodal displacement is given by:

(li)

where c and dare respectively equal to:

c=~(E; + __ E":"2 __ )

Jl2 (1 + K 2 1 K I )

(12)

d = (E2 + E;)e1o

Jl2(1 + K2 1 K1)

(13)

With the time histories of e2i(t) and Ub(t) , the fol­lowing relations are found for the ratio of the axial stress in the wood at t = 00 to the original axial stress (eq.(14» and for the initial time rate ofthe axial stress in the wood (eq.( 15»:

(14)

(15)

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Since E2 is simply the conventional modulus of elasticity of wood, eqs.( 14) and (15) may be used for directly estimating the viscoelastic model parameters, E'2 and f.L2, from simple tests such as that shown in Figure 6, by evaluating 0-2(0) and 02(00)102(0) from experimental curves sue h as those shown in Figure 7.

It can be also observed that, in analogy with pre­stressed concrete, the first equation indicates that for any given ratio E2/E' 2, the loss of prestress decreases with KI /K 2, the ratio ofaxial stiffness ofthe steel rod to the axial stiffness of the wood. Therefore, using smaller diameter, high strength rods with large pre­strains, reduces the loss of prestress from the viscous behaviour of the wood.

5 LINEAR VISCOELASTIC ANALYSIS OF THE PINE BLUFF BRIDGE

5.1 Analytical approach

The approach used for the two-element prestressing experiment was applied to the truss of the Pine­Bluff Bridge, in order to predict its time-dependent behaviour. The formulation used to express the truss behaviour is a direct extension of that obtained for the two-element system. Let the subscript I denote ali the linear elastic elements and the subscript 2 denote ali the linear viscoelastic elements, modelled with the standard-linear-solid model. The element axial defor­mation matrix e may be expressed, again, as a sum of elastic deformations and inelastic (null for the lin­ear elastic steel elements) defonnations. The axial force-axial deformation equations for the linear elas­tic elements and for the linear viscoelastic elements become:

NJ(t)=KJe(ele - elO)

N 2 (t)=K 2e e2e

N 2 (t)= K 2e e2i+ fJ2 é1i

(16)

(17)

(18)

in which K le and K 2e are diagonal matrixes of lin­ear elastic axial stiffnesses, EiA/ Li, el O is a vector of prestressing deformations, considered negative if elements are stretched, K'2e is a diagonal matrix of stiffness terms, E'iA/Li, and /L2 is a diagonal matrix of viscosity terms, /LiA/ Li.

The linear geometric equations that relate nodal dis­placements, V(t) , to the element axial deformations may be expressed as:

AV=e=ee+ei

AtU=ele

A2 U = e2e +e2i

(19)

(19a)

(19b)

where A is the matrix that contains direction sines and cosines, ee and ei represent the elastic and the inelas­tic deformations. Eqs.(19a) and (19b) correspond to eq.( 19), partitioned for the linear elastic and the linear viscoelastic elements, respectively. Eqs.(17), (18) and (19b) may be combined to give:

(20)

Eqs.( 16) and (17) may be written in compact form:

Substituting eq.( 19) into eq.(2 1):

(22)

and considering that the nodal equilibrium equations can be written in compact form as:

(23)

in which Q is a vector of applied nodal loads, the equilibrium equation can be rewritten as foll ow:

(24)

whereATKeA is the system stiffnessmatrix, K. Noting that ei and eo have null parts, gives the following form for the equi librium equations:

T T K U = Q + A 2 K2e e2i+ AI K le elO (25)

in which the two matrix products on the right-hand­side are effective nodal loads from inelastic defor­mation and from prestressing. Eqs.(20) and (25) are the basic coupled equations that must be solved for the time histories of internai state variables, eZi(t), and nodal displacements, U(t). Eqs.(20) are first order linear ordinary differential equations with initial conditions e2i(0) = O. To solve the equilibrium equa­tions (25), displacement boundary conditions must be imposed. Time histories of e2i(t) and U(t) may in tum be backsubstituted to find the element forces:

NI (t) = K le (AI U - elO)

N2 (t) = K2e (A2 U - e2i)

(26)

(27)

The basic equations were written and solved using MATLAB to compute viscoelastic time histories ofthe responses of the Pine Bluff Bridge.

=t -· .. - - " -----1' --,

-, 100 '"

~ .. I ·f \ .'00~::;==~\ ==::;;;" ~==;

26

L ~ '200~--==========================i

I '300.Ó-' ~~-':;20:--- , ___ ~~_ --------_ .... ---1-------

~ ~ ~ ~ m Tmeino.y.

Figure 9. Time histories of selected member axial forces.

5.2 Response ofthe Pine BluffBridge

Viscoelastic analyses were first done using the actual member areas of the Pine Bluff Bridge. The values assumed for the analyses were the same as those adopted in the linear elastic analyses. The dead load was 26 .5 kN per panel, the effective nodal forces from prestressing were again assumed to be uniform along the span, equal to 250 kN per node. The white pine used on the Pine BluffBridge was not tested, therefore parameters of the standard-linear-solid viscoelastic model had to be assumed. The modulus of elasticity for the wood was taken equal to 8.27 OPa. The other parameter values, E'= I 1.7 OPa and J.1,= 123.50Pa­day, were taken proportional to those given by Fridley (1992) for Douglas fir.

Figure 9 shows time histories of selected member axial forces (Figure 3 shows locations of the mem­bers). Ali axial stresses in wood members are below 3.5 MPa and ali axial stresses in the rods are well below 140 MPa. At t = O, member axial forces predicted with the linear viscoelastic analysis are equal to those pre­dicted by means of linear elastic analyses. With the standard-linear-solid model and its assumed parame­ters , most of the changes in the member fo rces from viscous behavior occur within about two months.

The axial fo rce in end counterdiagonal6 is predicted to decrease about 44% from its initial (prestressing plus dead load) value. Vertical rod member 2 is pre­dicted to have an 18.7% decrease. The final forces in the end counterdiagonals (most tensioned mem­bers due to a uniformly distributed live load over one half of span) and in both diagonais near midspan (most tensioned members due to a concentrated live load), obtained as a sum of prestressing and dead load plus the effect ofthe time-dependent phenomena, are important because they control the live load that can cause " Iooseness" in the members. The temporal increase in the midspan vertical displacement is given

988

~.008

j ~Ol

i ~

! 4012

i -0.014

.Q0160

Figure 10. midspan.

300

1--. 200

100

o

V ·200

V

t - V~O!!Ip. N.Node15 1

'od""'"

100 120

Time histories of vertical displacement at

23

_2

lO • ,. ~ " 5

20 eo 100 120

Figure 11. Time histories of selected member axial forces with reduced rod areas.

by the curve labeled "original rod areas" in Figure I O. The displacement is predicted to increase by 73%. By comparing Figures 9 and 10, it can be seen that ele­ment forces and nodal displacements do not vary in time with equal rates. Changes in displacements gen­erally follow the rates of change of the inelastic axial deformations ofthe members, eZi(t) = deZi(t) j dt.

Figure 11 shows time histories of element forces computed using rod areas reduced to one-quarter ofthe original values. The initial prestressing axial deforma­tions are increased proportionately in order to obtain the same effective nodal prestress loads. This of course causes a different initial prestress state. The axial force in end counterdiagonal6 is predicted to decrease about 19% from its initial (prestressing plus dead load) value, which in tum is significantly greater than that com­puted using the original rod areas. Vertical rod member 2 is predicted to have a 9.2% decrease. The tempo­ral increase in the midspan vertical displacement is given by the curve labeled "rod areas/4" in Figure I O. Although the initial elastic displacement is slightly greater, the long-term displacement is slightly smaller than that predicted using the original rod areas.

6 CONCLUSIONS

In prestressed wooden or wood and iron trusses, the viscous behavior of wood causes temporal losses of prestress. Linear viscoelastic models and analyses can predict time histories offorces and displacements. The analyses can be easily formulated and solved with­out large, general purpose programs. The parameters needed for a standard-linear-solid viscoelastic model may be directly obtained from simple prestressing tests . Smaller, high-strength rods with larger initial prestrains will produce prestress states that will be less affected by viscous behavior of wood. Adoption of high-strength steel and modem prestressing tech­nologies is useful for rehabilitation and maintenance of Howe trusses, being at the same time respectful of the original design concept.

ACKNOWLEDGEMENTS

The U.S . Federal Highway Administration supported the Historic American Engineering Record study of the Pine Bluff Bridge. The wood species used for the prestressing experiments were identified by Ronald W. Anthony ofFort Collins, CO. The actual timbers used for the experiments were lent to Case Westem Reserve University by Carl Steepleton of Alliance, Ohio. Neil Hamar, Department Engineer, instrumented and mon­itored the long-term prestressing tests.

REFERENCES

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da Porto F. (2002) "Pine Bluff Bridge" HAER No. IN-I03 HAER Collection, Library ofCongress, Washington De.

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Madsen, B. & Johns, K. (1982) "Duration ofLoad Effects in Lumber. Part III : Code Considerations" Cano J. Civ. Eng. Vol. 9, p 526-536.

Senft, J.F. & Suddarth, S.K. (1971) "An Analysis of Creep­inducing Stress in Sitka Spruce" Wood Fiber, Vo l. 2, No. 4, P 32 1- 327.

Tissaoui, 1. ( 1996) "Effects of Long-Term Creep on the Integrity of Modern Wood Structures" Dissertation sub­mitted to the Faculty of the Virginia Polytechnic Institute and State Univers ity in partial ful fi llment of the requ ire­ments fo r the degree of Doctor of Philosophy in Civil Engineering, Blacksburg, VA.

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