6.6 Membranes

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Tents or similar space-like membranes can be analyzed with special flat elements

by combining the structural behavior of a prestressed membrane with

a rigid cloth.

If it is assumed that the horizontal prestressing force H in the membrane

is uniform in all directions, the deflection of the membrane satisfies the differential

equation

− H (w,xx +w,yy ) =p p= wind pressure . (6.45)

As expected, this is the extension to 2-D problems of the one-dimensional

equation −Hw__ = p of a taut rope. If the model is extended and it is assumed

that the prestressing force in the x-direction Hx is different from the force Hy

in the y-direction, then it seems reasonable to modify the differential equation

as follows

− Hx w,xx −Hy w,yy = p . (6.46)

Green’s first identity for this equation is

G(w, ˆ w) =

_

Ω

(−Hx w,xx −Hy w,yy ) ˆ wdΩ (6.47)

+

_

Γ

(Hx w,x nx + Hy w,y ny) ˆ wds − a(w, ˆ w) = 0, (6.48)

where the strain energy product is

a(w, ˆ w) =

_

Ω

(Hx w,x ˆ w,x +Hy w,y ˆ w,y ) dΩ . (6.49)

To understand how the analysis proceeds, let us consider a bar element.

6.6 Membranes 499

Fig. 6.14. Rotation of a bar element

Originally the stiffness matrix of the bar element is a 2 × 2-matrix, which

is enlarged to a 4 × 4-matrix to account for possible rotations of the element

(see Fig. 6.14):

EA

le

_

1 −1

−1 1

_ _

u1

u2

_

⇒ EA

le

⎡

⎢⎢⎣

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

⎤

⎥⎥⎦

⎡

⎢⎢⎣

u1

u2

u3

u4

⎤

⎥⎥⎦

. (6.50)

Assuming that the bar element is stabilized by a horizontal force H, we obtain

Ku =

⎧⎪⎪⎨

⎪⎪⎩

EA

le

⎡

⎢⎢⎣

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

⎤

⎥⎥⎦

+ H

le

⎡

⎢⎢⎣

0 0 0 0

0 1 0 −1

0 0 0 0

0 −1 0 1

⎤

⎥⎥⎦

⎫⎪⎪⎬

⎪⎪⎭

⎡

⎢⎢⎣

u1

u2

u3

u4

⎤

⎥⎥⎦

= f .

(6.51)

The horizontal force H adds vertical stiffness to the bar, because the force H

tends to pull the bar straight.

The structure of this matrix resembles the stiffness matrix of a beam in

second-order beam theory. The first part is the linear stiffness matrix, and the

second is the so-called geometric stiffness matrix, which is just the stiffness

matrix of the rope,

H

le

_

1 −1

−1 1

__

u1

u2

_

=

_

f1

f2

_

, (6.52)

extended to 4×4. Hence the stiffness matrix of a membrane element consists

of the stiffness matrix KS of the cloth (orthotropic material) and a membrane

matrix KM:

K = KS +KM . (6.53)

An appropriate choice for the membrane part is the rectangular Q4+2 element,

and to these four bilinear shape functions are added the unit deflections

500 6 Shells

Fig. 6.15. Roofing of sand boxes with a prestressed membrane: a) system,

b) deformations due to gravity load + live load + snow

of the four nodes, so that the membrane matrixKM contains the strain energy

kM

ij =

_

Ω

(Hx ϕi,x ϕj ,x +Hy ϕi,y ϕj ,y ) dΩ . (6.54)

Hence, a prestressed membrane is analyzed in two steps: first the shape is

found, then the stresses are calculated.

In the first step only the geometric matrix due to the prestressing forces is

activated. In other words, in this step the extensional stiffness is assumed to be

zero. To prevent the nodes from swimming on the surface of the membrane, as

if on a soap film, the nodes are stabilized in the tangential direction by small

springs. In the second step the stresses within the membrane are calculated.

Figure 6.16 a illustrates finding the shape of a tightly stressed rope. The

deflection of nodes 1 and 4 is given: u1 = 1.0 m, u4 = 1.4 m. The unknowns are

the associated nodal forces f1, f4 and displacements u2, u3 of the free nodes,

so that the system for the unknowns is

found, then the stresses are calculated; see Fig. 6.15.

6.6 Membranes 501

Fig. 6.16. Determining the shape a) of a taut rope, and b) of a membrane

Fig. 6.17. Determining the shape of a membrane supported along its edge:

H

le

⎡

⎢⎢⎣

2 −1 0 0

−1 2−1 0

0 −1 2−1

0 0−1 2

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

1.0

u2

u3

1.4

⎤

⎥⎥⎦

=

⎡

⎢⎢⎣

f100

f4

⎤

⎥⎥⎦

. (6.55)

The starting point for finding the shape of a membrane can either be a

3-D system (see Fig. 6.17) or a plane system. If the analysis starts with a

3-D shape, the individual elements are initially flat. If it starts with a plane

a) starting position, b) final shape

502 6 Shells

Fig. 6.18. Wind load: a) undeformed system, b) deformations due to wind load;

(the maximum deflection at the center of the membrane was about 30 cm)

Fig. 6.19. Wind load a) undeformed system b) deformations under wind load, the

maximum deflection at the center of the membrane was about 30 cm

system as in Fig. 6.16 b, the edge of the membrane is moored at the supports

and pulled upwards.

Once the shape has been found, the stresses resulting from wind loads and

snow must be determined. While snow is a simple load case, the wind load

depends on the height, position and orientation of the individual elements; see

Figs. 6.18 and 6.19. Under the action of high wind pressure, the magnitude of

the tensile stresses from the prestressing forces may not be great enough and

wrinkles will develop in the membrane. Because the full wind load does not

converge in one step, the wind load must be applied in single steps instead.

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