Esercitazione 1 - Elementi finiti per travi

C

0 γ

ε  

 

C E E

1 0 0 2 (1 +

γ −ν −ν ν)

Where is the so-called flexibility matrix. In the hypothesis of linear elasticity, the relationship

C

can be also inverted: = = (1.10)

−1 Dεε

σ ε

C

Where is the stiffness matrix, with value:

D 1 1 0 0

   

− ν ν ν

0

E

D 1 0 1 0

= (1.11)

= =

ε −

ν ν ν G

D

D D

0 γ

ε (1 + (1 2ν)    

D −

ν) 1 0 0 1

γ −

ν ν ν

The above constitutive laws are valid if applied to the elastic strain. In fact, it may happen that

the total strain in a body has also a thermal strain contribute due to temperature changes:

T

1 1 1 0 0 0

= (1.12)

ε α∆T

0

Where is the thermal coefficient of the material. In order to get the elastic strain it is sufficient

α

to subtract this constribute: = =⇒ = (εε ) (1.13)

e − −

ε ε ε σ ε

D

0 0

1.1.1 Beam hypothesis

Beams are bodies which satisfy the geometrical condition of having one dimension that is much

greater than the other two directions. Typically the two "small" dimensions are the width and the

height of the beam cross section, while the other dimension is the length of the beam. Given these

conditions, following Saint Venant theory of beams, the only components of stress and strain are

, and , while = = = 0. Therefore, the constitutive law becomes:

σ τ τ σ σ τ

x xy xz y z zy 0 0

     

σ E ε

x x

0 0

= =⇒ = (1.14)

Dεε τ G γ

σ xy xy

     

0 0

τ G γ

xz xz

The vector of thermal strain becomes: T

1 0 0

= (1.15)

ε 0

In general, beams are analyzed in terms of generalized stress and strain components that, for a

beam in the 2D space, are equal to:

= = (1.16)

ε γ ϑ N T M

ε σ

a 0

Where is the axial force, the shear force and the bending moment. With reference to

N T M

Figure 1.3, the generalized components of stress are equal to the stress resultants:

Z Z Z

= dA = dA = (1.17)

− −

N σ T τ M σ ydA

x xy x

A A A

Computational Structural Analysis 3

1.2. Principle of virtual work 1. Introduction

The generalized components of strain are equal to:

du dv dφ

= = = (1.18)

−

ε γ ϑ

a 0

dx dx dx

The Bernoulli beam hypothesis, which is the hypothesis commonly adopted, reads that in slender

beams the shear deformability is negligible and therefore the sections stay plane, i.e. orthogonal

to the beam axis. Therefore we can write:

(y) = = (1.19)

− −γ

ε ε ϑy γ

x a xy 0

Using the constitutive laws we can write:

Z Z N (1.20)

= dA = dA = =⇒ =

N Eε Eε Eε A ε

x a a a EA

A A 1

Z Z Z T

A

= dA = dA = = =⇒ = (1.21)

dA =

− − · · ·

T Gγ τ̄ Gγ Gγ γ χ

xy 0 0 0

χ χ GA

A A A

Z Z Z M

= = dx = dA = =⇒ = (1.22)

2 2

−

M Eε ydx Eθy Eϑ y EθI ϑ

x EI

A A A

Where is the correction shear factor, used to compute the effective shear area =

χ A A/χ.

ef f

Figure 1.3: Generalized stress and strain components of a beam

1.2 Principle of virtual work

The cornerstone of structural mechanics is the principle of virtual work which, thanks to its flexibil-

ity, can be applied to a great variety of problems to get results in terms either of displacements and

strains or forces and stresses. To demonstrate the principle of virtual work, let us consider a generic

body, in the hypothesis of homogeneous, isotropic, linear elastic material and small displacements.

The body is of constant cross section and of unitary thickness = 1. On the boundary the

A t S

normal vector is defined as: T T

sin cos

= = (1.23)

−

n n α α

n x y

Let us also define two system, that are independent one from each other. The first system is

equilibrated while the second is compatible. The related quantities are:

T

=

′ ′ ′

F F

F x y T

=

′′ ′′ ′′

u v

u

T (1.24)

=

′ ′ ′

f f

f x y T

=

′′ ′′ ′′ ′′

ε ε γ

ε x y xy

T

=

′ ′ ′ ′

σ σ τ

σ x y xy

4 Computational Structural Analysis

1. Introduction 1.2. Principle of virtual work

The equilibrium equations for system 1 are:

′

′ ∂τ

∂σ xy

+ + = 0

′

x F + =

′ ′ ′

x σ l τ m f

∂x ∂y x xy x

in in (1.25)

A S

′ ′ + =

′ ′ ′

∂τ ∂σ τ l σ m f

yx y

+ + = 0

′ yx y y

F y

∂x ∂y

Now we multiply the equilibrium equations in by the corresponding displacement function in

A

system 2, sum up and then integrate over A.

′ ′ ′

′ ∂τ ∂τ ∂σ

Z ∂σ xy yx y

+ + + +

+ dA = 0 (1.26)

′ ′′ ′ ′′

x F F

u v

x y

∂x ∂y ∂x ∂y

A

Noting that for each term: (σ )

′ ′′ ′′

′ ∂ u ∂u

∂σ = (1.27)

′

x

x − σ x

∂x ∂x ∂x

We can rewrite in:

(" #

′ ′′ ′ ′′ ′ ′′

(σ )

′ ′′ ∂ τ u ∂ τ v ∂ σ v

Z ∂ u xy yx y

+ + + +

x

∂x ∂y ∂x ∂y

A (1.28)

′′ ′′ ′′ ′′

∂u ∂u ∂v ∂v

+ + + + + dA = 0

′ ′ ′ ′ ′ ′′ ′ ′′

− σ τ τ σ F u F v

x xy yx y x y

∂x ∂y ∂x ∂y

The first part of the integral can be rewritten using the Green’s identity:

)

(σ ′′

′ Z

Z u

∂ dA = (1.29)

′ ′′

x σ u ldS

x

∂x S

A

At the end the equation becomes:

Z + + dA =

′ ′′ ′ ′′ ′ ′′

σ ε τ γ σ ε

x x xy xy y y

A Z Z

= + + + dS + + dA =

′ ′ ′′ ′ ′ ′′ ′ ′′ ′ ′′ (1.30)

σ l τ m u τ l σ m v F u F v

x yx xy y x y

S A

Z Z

= + dS + + dA

′ ′′ ′ ′′ ′ ′′ ′ ′′

f u f v F u F v

x y x y

S A

We obtained the principle of virtual work, which reads:

"The ′ ′′

σ ε

internal work done by the stresses for the strains equals the work done by the set of

".

′ ′ ′′

F f u

volume forces and surface forces for the displacements

In matrix form: Z Z Z

T T T

dA = dA + dS (1.31)

′′ ′ ′′ ′ ′′ ′

ε σ u F u f

A A S

It can be observed that:

• The only conditions needed for the validity of the Principle of Virtual Work are the equi-

librium of system 1 of stresses and forces, and compatibility of system 2 of strains and

displacements;

• The equilibrated system 1 and the compatible system 2 are arbitrary and not necessarily

related to the actual stresses and strains of the body;

• The Principle of Virtual Work does not in any way depend on the constitutive laws;

• Depending on the purpose, the virtual work equation can be specialized in the form of virtual

displacements or virtual forces.

Computational Structural Analysis 5

1.2. Principle of virtual work 1. Introduction

1.2.1 Principle of virtual displacements

The principle of virtual displacements is obtained choosing for the compatible system 2 an in-

finitesimal variation of the actual kinematic field, while selecting the actual forces and stresses for

system 1. = in

′′ δu A

u = (1.32)

′′ δεε

ε

= = 0 in

′′ δ S

u ū u

The principle of virtual work becomes:

Z Z Z Z

dA = + dS + (1.33)

T T T T

δεε σ δu δu δ

FdA f ū rdS

A A S S

σ u

Since is prescribed, its variation is null. It is interesting to notice that the only unknowns in this

ū

case are the displacements, in fact:

Z Z Z

(Bu ) dA = + dS (1.34)

T T T T

−

δu ε δu δu

B D FdA f

0

A A S

σ

Where is the internal compatibility matrix. With reference to Figure 1.4, for beam bodies, the

B

equation is written as:

l l

Z Z X

(δε + + ) dx = (x) (x) dx + (1.35)

N δγ T δθM δv p δv F

a i i

0

0 0 i

1.2.2 Principle of virtual forces

In this case strains and displacements of system 2 are the actual one, while stresses and forces of

system 1 are an infinitesimal variation of the actual ones.

= = 0 in

′ δF A

F = = 0 in = (1.36)

′ ′ σ

δf S

f σ δσ

σ

= in

′ δr S

f u

The principle of virtual work becomes:

Z Z Z Z

dA = + + (1.37)

T T T T

σ

δσ ε δF δf δr

udA udS ūdS

A A S S

σ u

It is interesting to notice that the only unknowns in this case are the stresses, in fact:

Z Z

[Cσ + ] dA = (1.38)

T T

σ σ

δσ ε δr ūdS

0

A S

u

Always with reference to Figure 1.4, for beam bodies,