Bernoulli Beams & Trusses

Basically, bars oriented in two dimensional Cartesian system.

Trusses support compressive and tensile forces only, as in bars.

Translate the local element matrices into the structural (global) coordinate system.

2D TRUSSES

CONSIDER A TYPICAL 2D TRUSS IN GLOBAL X-Y PLANE

Local system:

𝑢′=[𝑢 ′ 1𝑢 ′ 2]Global system:

𝑢=[𝑢1𝑢2𝑢3𝑢4 ]

𝑢′=[𝑢 ′1=𝑢1∗ cos𝜃+𝑢2∗ sin 𝜃  𝑢′ 2=𝑢3∗cos𝜃+𝑢4∗ sin𝜃 ]=[cos𝜃 sin𝜃 0000cos𝜃 sin 𝜃 ]∗ [𝑢1𝑢2𝑢3𝑢4]

(𝑥1 , 𝑦1)

(𝑥2, 𝑦 2)

𝜃

=m=

cos𝜃=   l  =𝑥2− 𝑥1𝑙𝑒

𝑢′=[𝑙𝑚000 0 𝑙𝑚]∗[𝑢1𝑢2𝑢3𝑢4 ] 𝑢′=𝐿∗𝑢

𝑙𝑒=√(𝑥2−𝑥1)2+(𝑦2−𝑦1)

2

STIFFNESS MATRIX

Strain Energy:

𝑑𝑈=12∗𝜎 𝑥∗𝜀𝑥∗ 𝐴∗𝑑𝑥

𝑈=𝑢 ′ 𝑡∗𝐾 ′∗𝑢 ′Energy for the local system:

𝑢′=𝐿∗𝑢

)

𝑈=𝑢  𝑡∗(𝐿𝑡∗𝐾 ′∗𝐿)∗𝑢

K

𝐾=𝐸∗𝐴𝑙𝑒 [ 𝑙0𝑚00000 ]∗[ 1 −1

−1 1 ]∗[𝑙𝑚0000 𝑙𝑚]

Stiffness matrix for the local system:

𝐾 ′=𝐸∗𝐴𝑙𝑒

∗[ 1 −1−1 1 ]

…

𝐾=𝐸∗𝐴𝑙𝑒 [ 𝑙−𝑙

𝑚−𝑚−𝑙 𝑙−𝑚𝑚 ]∗[𝑙𝑚0000 𝑙𝑚]

𝐾=𝐸∗𝐴𝑙𝑒

∗ [ 𝑙2 𝑙∗𝑚 − 𝑙2− 𝑙∗𝑚𝑚∗𝑙 𝑚2 −𝑚∗𝑙−𝑚2

−𝑙2− 𝑙∗𝑚

−𝑚∗𝑙−𝑚2

𝑙2 𝑙∗𝑚𝑙∗𝑚𝑚2 ]

Stiffness matrix for the global system

STRESSES AT THE ELEMENT

𝜎=𝐸∗𝜀 𝜎=𝐸∗𝑢 ′ 2−𝑢 ′1

𝑙𝑒𝜎=

𝐸𝑙𝑒∗ [−1 1 ]∗[𝑢 ′ 1𝑢 ′ 2]

𝑢′=𝐿∗𝑢

𝜎=𝐸𝑙𝑒∗ [−𝑙−𝑚𝑙𝑚 ]∗ [𝑢1𝑢2𝑢3𝑢4 ]

Local system:

Global system:

BERNOULLI BEAMS• Beams are subject to transverse loading. including

transverse forces and moments that result in transverse deformation.

• They are deflection in the y direction (w), and rotation in the x-y plane with respect to the z axis.

• Each two-noded mean element has total of four degrees of freedrom(DOFs)

INTRODUCTION• The Euler-Bernoulli beam

theory assumes that undeformed plane sections remain plane under deformation.

w= deflection• Strain are defined as:

STRAIN ENERGY

Taking :

Inertia:Then strain energy:

SHAPE FUNCTION CONSTRUCTION

• As there are four DOFs for a beam element, there should be four shape functions.

Shape functions:

For N1:

SHAPE FUNCTION CONSTRUCTION For N2:

For N3:

SHAPE FUNCTION CONSTRUCTIONFor N4:

The shape functions defined as:

The transverse displacement is interpolated by Hermite shape functions as:

Taking :

Then:

The strain energy is obtained as:

We know that:

STIFFNESS MATRIX

Deriving shape functions:

Each element of the matrix is integrated between [-1,1]:

The stiffness matrix is: