Comments concerning the notion of prestress.
Effect of prestress:
Applying prestress in a tensioned structure leads to increase the stiffness of the
element, which is not the case for the more conventional materials such as concrete or
steel. The effects of prestress cannot be compared in both cases.
In the following examples, the effect of prestress is presented in the cases of two
colinear cables, two orthogonal cables (bi-cable), an inflatable beam, and a rectangular
tensioned membrane.
1. Effect of prestress on a system of two colinear cables:
Source; Article "Techniques de l'ingénieur, C2580, Metal constructions, cable
structures", author: Jean-Pierre LAUTE
This example is voluntarily the simplest imaginable system, consisting of two identical
aligned cables 𝐺2 𝐾 and 𝐾𝐺3 (with the same length 𝐿, the same Young modulus 𝐸,
and the same section 𝑆). It is attached to the supports 𝐺2 and 𝐺3 . Two cases are
studied:
Figure 1: model of the colinear cables
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1.1 Without initial prestress: The initial tension is nil (figure a), and a load 𝑃1 is
applied at the point 𝐾, following the direction 𝐺2 𝐺3 .
This leads to (Fig. b):
 a support action at the point 𝐺2 : 𝑃1
 the value of the tension in 𝐺2 𝐾: 𝐹 = 𝑃1
 the tension in 𝐾𝐺3 is nil, which means that it is a totally relaxed state section (if
such an element would be part of a framework, it would be likely to float or
flap under the effect of a transverse stress, which is not conceivable).
The stress in 𝐺2 𝐾 is then 𝑛 =
𝑃1
, and the elongation of 𝐺2 𝐾 (following Hooke's law):
𝑃1 𝐿
𝑃1
∆𝐿1 =
=
𝐸𝑆
𝐸𝑆
𝐿
𝐸𝑆
where = 𝐾𝑛 represents what may be called the normal stiffness of the 𝐺2 𝐾 element.
𝑆
𝐿
1.2 With initial prestress: 𝐺2 𝐺3 is subjected to an initial tension 𝐹0 (Fig. c), and then
the same force 𝑃1 is applied at point 𝐾 in the 𝐺2 𝐺3 direction.
This leads to (Fig. d):
 two new support actions 𝐹2 and 𝐹3 ;
 an elongation ∆𝐿2 of 𝐺2 𝐾;
 a shortening of the same value 𝐺2 𝐾 of 𝐾𝐺3
𝐺2 𝐾 and 𝐾𝐺3 are initially identical (this is valid after the application
of F0 and before the application of 𝑃1 ). The increase of the tension in 𝐺2 𝐾 equals in
absolute value the reduction of the tension in 𝐾𝐺3 (Hooke's law), then:
𝐹2 − 𝐹0 = 𝐹0 − 𝐹3 or 𝐹2 + 𝐹3 = 2𝐹0
The static equilibrium leads in absolute value to 𝐹2 − 𝐹3 = 𝑃1 . It gives
𝑃1
𝑃1
𝐹2 = 𝐹0 +
𝐹3 = 𝐹0 −
2
2
So, to ensure that 𝐾𝐺3 remains always in tension (𝐹3 ≥ 0), despite its shortening ∆𝐿2 ,
it is sufficient. to have: 𝐹0 ≥
𝑃1
2
.
If this strictly sufficient value is adopted for 𝐹0 , 𝐹0 =
𝑃1
2
, one can obtain:
𝐹2 = 𝑃1 (idem of case 1)
𝐹3 = 0 (idem of case 1)
Conclusion: The implementation of a judicious pretension in 𝐺2 𝐺3 in order to make it
able to withstand the applied force 𝑃1 at the point 𝐾, in the middle of 𝐺2 𝐺3 , following
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the 𝐺2 𝐺3 direction, and avoiding the slackening of 𝐾𝐺3 therefore leads to, under the
effect of 𝑃1 :
 the same reactions at 𝐺2 and 𝐺3 than without pretension
 the elongation in 𝐾 is reduced by half, because the increase of the tension in
𝐺2 𝐾(𝑃1 −
𝑃1
2
=
𝑃1
2
) gives:
𝑃1 𝐿
1
= ∆𝐿1
2 𝐸𝑆 2
Note: the pretension, which leads to a better use of the material than when lacking,
gives the system an increased stiffness, and therefore, in principle, does not
penalize dimensioning of the supports.
∆𝐿2 =
2. Effect of prestress on a bi-cable modelling a membrane
Source; Article "Techniques de l'ingénieur, c2580, Structures textiles", authors
Marc MALINOWSKY, Christian LYONNET
To illustrate the various aspects of the pretension, it is possible, with a simplified
model, to study the equilibrium of a point on the surface. In the case of the negative
double curvature, the model consists of two cables AB and CD respectively fixed in
two "high" and two "low" points. The horizontal projection of the four points gives a
quadrangle (see Figure 2).
For simplicity, let pose AB = CD = 2L. The length of the cables in their unstressed
state is 2ℓ0 . The distance between the lines AB and CD is chosen greater than 2ℎ0 ,
where ℎ0 is the distance from the point M to the chord AB (or CD) corresponding to a
zero tension in the cable AMB (or CMD). This same distance is ℎ0 + 𝑧 in the
prestressed state, where z is the displacement of the point M.
Figure 2: model of the bi-cable
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In this symmetric case, the preload consist to align the two points M of the cables
halfway between the two horizontal planes: the corresponding deformation of the two
cables is similar to applying a force 𝑇2 to the upper cable and a force 𝑇1 to the lower
cable.
In both cases, it is possible to write the relation between the force applied to the node
and the corresponding displacement relative to the unstressed state, which corresponds
to the diagram Figure 3:
√𝐿2 + (ℎ0 + 𝑧) − ℓ0
𝑇 = 2𝐸𝐴 sin 𝛼
ℓ0
where 𝐸 is the Young's modulus, 𝐴 is the section of the cable, and 𝛼 is the angle
between the cable and the horizontal.
This relationship introduces the terms of the second order with respect to the
displacement, thus taking into account large displacements while remaining in small
strains and linear elasticity.
This relationship between 𝑇 and 𝑧 is valid for both cables. The origin corresponds to
the original undeformed geometry, where z = 0.
When no external action is applied, the static equilibrium leads of course to: 𝑇1 = 𝑇2 .
In this case, the displacement is 𝑧1 , which corresponds to the prestressed state ( fig. 3).
When a vertically force F is applied downwardly, the point M common to both cables
undergoes a displacement in the same direction. When the static equilibrium is reached
again, the displacement of M being 𝑧2 , we have:
𝑇1′ + 𝐹 = 𝑇2′
The displacement 𝑧2 of the equilibrium point is such that the length 𝐾𝐶 on the diagram
in Figure 4 is equal to F.
From point M representing the prestressed state(𝑧 = 𝑧1 ), the released lower cable is
represented by a symmetrical curve to the one of the upper tensioned cable relatively
to a vertical line passing through M.
Figure 3: operating diagram of the bi-cable
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As a result:
 The equilibrium when the system is loaded shows an increased force applied to
the upper cable, but the value 𝑇2′ is lower than the sum 𝑇2′ + 𝐹 because 𝑇1′ is
smaller than 𝑇1 (= 𝑇2 ). There is therefore no addition of the effects of the
pretensioned and applied loads, but combination;
 The displacement of the crossing point M of the two cables is measured by the
difference 𝑧2 − 𝑧1 , which is less than in the abscissa 𝑧1 when there is no
pretension. The pretension therefore has a stiffening action on the membrane;
 It is necessary in any case to ensure that the lower cable does not "Float", which
corresponds to a tension equal to zero, so that 𝑧2 is smaller than 𝑧𝑚𝑎𝑥 , 𝑧𝑚𝑎𝑥
being the abscissae of the intersection point of the curve with the z axis
Regarding the level of pretension, a first analysis could lead to choose high values so
that the displacement of the membrane is limited, regardless the climate actions
characterized in our example by the force 𝐹 .
But applying such a state of prestress to the membrane would stress the fabric
excessively and therefore cause fatigue and premature aging of the fabric, and this
only to support extreme values of actions that are seldom reached, and for a short
period. It follows that the choice of the values of pretension is always a compromise
between the displacements of the membrane considered as eligible, based on its shape
and its use, and the fatigue imposed to the fabric. These values depend mainly on the
climate action that may be applied to the membrane, but in practice values between
180 and 350 𝑑𝑎𝑁/𝑚 are considered.
4. Effect of the prestress on inflatable beams:
Source:
Q.T. Nguyen, J.C. Thomas, and A. Le van. Some new theoretical results for the orthotropic inflatable
beams.
11th biennal conference on Engineering Systems Design and Analysis ESDA, Nantes, France, 2012.
A. Le van and C. Wielgosz. Bending and buckling of inflatable beams : some new theoretical results.
Thin-Walled Structures, 43 :1166–1187, 2005.
The point of departure to is the total Lagrangian formulation written in order to take
into account the internal pressure which induces follower forces (they follow
orthogonally the membrane, which is different to dead loads like the weight, which
keeps the same direction). This means that the geometrical non linearity is taken into
account.
Timoshenko's kinematic has to be used in the case of thin-walled structures.
After a final linearization, one obtains a set of linear equations which allows analytical
formulations for the deflection.
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For an cantilever inflatable tubular beam:
𝐹 is the force applied, ℓ0 is the length, Eℓ is the Young modulus in the longitudinal
direction of the beam, 𝐼0 is the second moment of area, 𝑆0 is the surface of the section,
𝐺ℓt is the shear coefficient, 𝑃 = 𝑝𝜋𝑟02 is the effect of the pressure 𝑝 on the end surface
of the beam (radius r0 ).
𝑣(𝑥) =
𝐹
𝑃
(𝐸ℓ + 𝑆 ) 𝐼0
(
ℓ0 𝑋 2 𝑋 2
𝐹𝑋
− )+
(𝑃 + 𝑘𝐺ℓ𝑡 𝑆0 )
3
6
0
In this formula, one can see that:
 the relation between the load and the displacements is linear
 the relation between the presstress effect and the displacement is non-linear
 the effect 𝑃 = 𝑝𝜋𝑟02 of the inside pressure 𝑝 of the beam reinforces explicitly
𝑃
the bending stiffness(𝐸ℓ + ) 𝐼0 and the shear stiffness 𝑃 + 𝑘𝐺ℓ𝑡 𝑆0
𝑆0
 Another interpretation is that the pressure p increases the material coefficients:
𝑝r
𝑝r
𝐸ℓ is replaced by 𝐸ℓ + 0 , and 𝐺ℓ𝑡 is replaced by 𝐺ℓ𝑡 + 0 = 𝐺ℓ𝑡 + 𝑝r0
2
2k
because 𝑘 = 0,5 in the case of a circular thin wall beam.
5. Vibration of a rectangular tensioned membrane:
Source http://www.mcise.uri.edu/sadd/mce565/Ch6.pdf
In the case of a rectangular isotropic membrane, considering a uniform prestress, the
analytical formulation for the eigenfrequencies are:
1
𝑇
𝑚2
𝑛2
𝑓 = 2 √𝜌 √ 𝑎 2 + 𝑏 2 .
where 𝑇 is the linear tension, 𝜌 is the masse/area, 𝑎 and 𝑏 are the width and length of
the membrane, 𝑚 et 𝑛 are the numbering of the frequencies
2
𝜔
𝑖
𝐸𝐼
In comparison, the eigenfrequencies of a beam is: 𝑓 = 2𝜋
= 𝜋√ 4 .
2
𝜌𝑆𝐿
Here also, the tension 𝑇 has the same effect than the stiffness.
6. Conclusions:
 prestress is a state of pretension before other external loads are applied,
 it is combined with the external loads,
 it reinforces the stiffnesses of the structures.
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In regard of these facts, it seems inappropriate to consider the prestress in the same
way as an external action.
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