Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 220

TABLE 4.3
Transfer Matrix for a Beam Element, Part a:
Definitions: k ∗ )/ λ = k / EI , η = k /( k s GA ) , ζ = ( N − EI Sign Convention 1

e 1 + ζ e 3 − e 2 − e 4 / EI + ( e 2 + ζ e 4 )/ k s GA − e 3 / EI

λ e 4 e 1 − η e 3 e 3 / EI ( e 2 − η e 4 )/ EI

U i =

λ EI ( e 2 + ζ e 4 ) − λ EIe 3 e 1 + ζ e 3 − λ e 4

λ EIe 3 EI ( e 0 − η e 2 ) e 2 e 1 − η e 3

0
b

w

w
V
M

0
b
V b
M b

θ

U i z a + z i z i z b = z = =

p a e 5 + p

− p

w

0
b

= (

e 6 + m a e 4 + m e 5 )/

−

p a ( e 3 + ζ e 5 )/

(

e 4 + ζ e 6 )/ EI k s GA k s GA

M Ta e 3 + M T

− (

e 4 )/

;

EI

[ p a e 4 + p

θ

0
b

=−

e 5 + m a ( e 3 − η e 5 ) + m

(

e 4 − η e 6 )

/ +

[ M Ta ( e 2 − η e 4 ) ] EI

+ M T

(

e 3 − η e 5 ) ] / EI ;

e 4 ) − p e 4 − M T V b =− m a e 5 + m p a ( e 2 + ζ

(

e 3 + ζ e 5 ) + λ( e 6 ) − M Ta λ

λ

e 5 ;

p a e 3 − p M b =− e 4 ) − m e 4 − m a ( e 2 − η

(

e 3 − η e 5 ) + M Ta ( e 1 − 1 − η e 3 )

+ M T

(

e 2 − − η e 4 )

M Tb − M Ta p

=

p b − p a m b − m a

M T m

=

=

b − a b − a b − a

p a is the magnitude of the distributed applied force at x = a (force/length).

m a is the magnitude of the distributed applied moment at x = a (force-length/length).

p

is the gradient of the distributed applied force, linearly varying in the x direction
(force/length 2 ).

m

is the gradient of the distributed applied moment, linearly varying in the x direction
(force-length/length 2 ).

k s is the shear form factor.

is the Winkler (elastic) foundation modulus (force/length 2 ). k

k ∗ is the rotary foundation modulus (force-length/length).

N is a compressive axial force.

E is the modulus of elasticity of the material.

I is the moment of inertia taken about the neutral axis.

G is the shear modulus of elasticity.

A is the cross-sectional area.

M Ta is the magnitude of the distributed thermal moment at x = a .

M T / is the gradient of the thermal moment, linearly varying in the x -direction. The moment

is defined as follows:

M T = E α TzdA

A

where T is the temperature change and α is the coefficient of thermal expansion.