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A tensile structure is a construction of elements carrying only tension and

no compression or bending. The term tensile should not be confused with tensegrity, 

which is a structural form with both tension and compression elements.

Most tensile structures are supported by some form of compression or bendingelements, such as masts (as in The O2, formerly the Millennium Dome), compression

rings or beams.

Tensile membrane structures are most often used as roofs as they can economically

and attractively span large distances.

This form of construction has only becomemore rigorously analyzed and widespread in large structures in the latter part of the

twentieth century. Tensile structures have long been used intents, where the guy

ropes provide pre-tension to the fabric and allow it to withstand loads.

Russian engineer Vladimir Shukhov was one of the first to develop practical

calculations of stresses and deformations of tensile structures, shells and

membranes. Shukhov designed eight tensile structures and thin-shell

structures exhibition pavilions for the Nizhny Novgorod Fair of 1896, covering the

area of 27,000 square meters. A more recent large-scale use of a membrane-covered tensile structure is the Sidney Myer Music Bowl, constructed in 1958.

Antonio Gaudi used the concept in reverse to create a compression-only structure

for the Colonia Guell Church. He created a hanging tensile model of the church to

calculate the compression forces and to experimentally determine the column and

vault geometries.

The concept was later championed by German architect and engineer Frei Otto, 

whose first use of the idea was in the construction of the German pavilion at Expo

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67 in Montreal. Otto next used the idea for the roof of the Olympic Stadium for

the 1972 Summer Olympics in Munich. 

Since the 1960s, tensile structures have been promoted

by designers and engineers such as Ove Arup, Buro Happold, Walter Bird of Birdair,Inc., Frei Otto, Eero Saarinen, Horst Berger,Matthew Nowicki, Jorg Schlaich, the duo

of Nicholas Goldsmith & Todd Dalland at FTL Design & Engineering

Studio and David Geiger. 

Steady technological progress has increased the popularity of fabric-roofed

structures. The low weight of the materials makes construction easier and cheaper

than standard designs, especially when vast open spaces have to be covered.

The horizontal and vertical reactions :

By geometry: 

The length of the cable:

The tension in the cable:

By substitution:

The tension is also equal to:

T = wR 

The extension of the cable upon being loaded is (from Hooke's

Law, where the axial stiffness, k, is equal to ):

where E  is the Young's modulus of the cable and A is its

cross-sectional area. 

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If an initial pretension, T 0 is added to the cable, the

extension becomes:

Combining the above equations gives:

By plotting the left hand side of this equation

against T, and plotting the right hand side on the

same axes, also against T, the intersection will give

the actual equilibrium tension in the cable for a

given loading w and a given pretension T 0.

[edit]Cable with central point load

A similar solution to that above can be derived

where:

By equilibrium: 

By geometry: 

This gives the following relationship:

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As before, plotting the left hand side and right hand side of the equation against the

tension, T, will give the equilibrium tension for a given pretension, T 0 and load, W .

Tensioned cable oscillations

The fundamental natural frequency, f 1 of tensioned cables is given by:

where: T = tension in newtons, m = mass in kilograms and L =

span length.

Torsion Theory

Shear Strain:  s = r / L 

Shear Stress: s = T r /J 

Angle of Twist: = TL/(JG) 

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s = Shear strain (radians)

s = Shear Stress (MPa)

= Angle of twist (radians)

L = Gauge length (mm)

r = Radius (mm)

G = Shear Modulus (MPa)

T = Torque (Nm)

J = Polar Moment of Inertia (mm4)

Where J is the Polar Second Moment of Area, which is used when under

torsion. For a cylinder... 

For a pipe, calculate J for the outside diameter, then SUBTRACT the J forthe inside diameter (the hole).

We do not have torsion equations for square shapes - these equations only work for circularshapes (round rods and pipes). Anything else will deform under torsion and will not obey

these formulas. Open shapes like a slit pipe can be almost as strong in bending as a closed

pipe, but hundreds of times weaker in torsion.

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General Background

If a beam is supported at two points, and a load is applied anywhere on the beam,

deformation will occur. When these loads are applied either longitudinally outside or inside

of the supports, this elastic bending can be mathematically predicted based on materialproperties and geometry.

Determination of Curvature

Curvature at any point on the beam is calculated from the moment of loading (M), the

stiffness of the material (E), and the first moment of inertia (I.) The following expression

defines the curvature in these parameters as 1/ρ, where ρ is the radius of curvature. 

 I  E 

 M 

  

1

 

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Equation 1

Equation 1 does not account for shearing stresses.

Curvature can also be found using calculus. Defining y as the deflection and  x as the position

along the longitudinal axis, the expression becomes

23

2

2

2

1

1

 

  

 

dx

dy

dx

 yd 

   

Equation 2

Central Loading

Central loading on a beam can be thought of as a simple beam with two supports as shown

below.

Figure 1

Applying equilibrium to the free body equivalent of Figure 1, several expressions can be

derived to mathematically explain central loading.

20

220

0

P R RP RF 

P R L R

 LP M 

 RF 

aycay y

C C  A

ax x

 

Equation 3, 4, and 5

Figure 2 and 3 act as free body diagrams for the section between AB and BC respectively.

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Figure 2

Figure 3

Solving the reactions between AB and BC, equation 1 can be expressed as

 L x L LP xP

dx

 yd  I  E 

 L x

 xP

dx

 yd  I  E 

222

20

2

2

2

2

2

 

Equation 6, 7

Integrating twice, Equation 6 becomes

 L x L

C  xC  x LP xP

 y I  E 

 L xC  xC 

 xP y I  E 

2412

20

12

43

23

21

3

 

Equation 8, 9

To determine the constants, conditions at certain positions on the beam can be applied.

Knowing the deflection at each of the supports, as well as the slope at the top of the curve is

zero, the constants can be derived to

4816

30

16

3

4

2

32

2

1

 LPC  LPC C 

 LPC   

Equation 10, 11, 12, and 13

Combining Equations 8 and 9 with 10 through 13, the expressions for deflection can be

expressed as

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 L x L LP x LP x LP xP

 y I  E 

 L x

 x LP xP y I  E 

24816

3

412

20

16123223

23

 

Equation 14, 15

Overhanging Loads

Overhanging loading on a beam is similar to that of central loading. In overhanging loading,

a simple beam is supported with two supports and two loads as shown below.

Figure 4

Using similar methods used previously for central loading, the equation for determination of 

deflection as a function of position, load, length, stiffness, and geometry can be derived as

L x xba LP xaP

ba L

 xP y I  E  02

626

23

 

Equation 16

DIRECT SHEAR TEST

Purpose:This test is performed to determine the consolidated-drained shear strengthof a sandy to silty soil. The shear strength is one of the most important engineeringproperties of a soil, because it is required whenever a structure is dependent on theAnalysis:

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1) Calculate the density of the soil sample from the mass of soil and volume of the shear

box.

2) Convert the dial readings to the appropriate length and load units and enter

the values on the data sheet in the correct locations. Compute the sample area A, and the

vertical (Normal) stress, sv.

sv=Nv /A

Where: Nv = normal vertical force, and sv = normal vertical stress

3) Calculate shear stress (t) using t =Fh/A

Where Fh= shear stress (measured with shear load gage)

4) Plot the horizontal shear stress (t) versus horizontal (lateral) displacement

?H.

(5) Calculate the maximum shear stress for each test.

6) Plot the value of the maximum shear stress versus the corresponding

vertical stress for each test, and determine the angle of internal friction (f)

from the slope of the approximated Mohr-Coulomb failure envelope.

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