CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND …

CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES

1

INTRODUCTION

? We learned Direct Stiffness Method in Chapter 2

? Limited to simple elements such as 1D bars

? we will learn Energy Method to build beam finite element

? Structure is in equilibrium when the potential energy is minimum

? Potential energy: Sum of strain energy and potential of

applied loads ? Interpolation scheme:

3UV

Potential of applied loads

v(x)

??N(x)?? {q}

Strain energy

Beam Interpolation Nodal deflection function DOF

2

BEAM THEORY

? Euler-Bernoulli Beam Theory

? can carry the transverse load ? slope can change along the span (x-axis) ? Cross-section is symmetric w.r.t. xy-plane ? The y-axis passes through the centroid ? Loads are applied in xy-plane (plane of loading)

y Neutral axis

L

xz F

y Plane of loading

A F

3

BEAM THEORY cont.

? Euler-Bernoulli Beam Theory cont.

? Plane sections normal to the beam axis remain plane and normal to the axis after deformation (no shear stress)

? Transverse deflection (deflection curve) is function of x only: v(x) ? Displacement in x-dir is function of x and y: u(x, y)

u(x, y)

u0 (x)

y dv dx

y

Neutral axis

H xx

w u wx

du0 dx

y

d 2v dx2

y(dv/dx)

T dv dx

x

y

T = dv/dx

L

F

v(x)

4

BEAM THEORY cont.

? Euler-Bernoulli Beam Theory cont. ? Strain along the beam axis: H0 du0 / dx

H xx

w u wx

du0 dx

y

d 2v dx2

? Strain Hxx varies linearly w.r.t. y; Strain Hyy = 0

? Curvature: d 2v / dx2

? Can assume plane stress in z-dir

basically uniaxial status

V xx

EH xx

EH 0

Ey

d 2v dx2

? Axial force resultant and bending moment

? ? ? P

V xxdA

A

EH0 dA

A

E

d 2v dx2

A

ydA

M

? yV xxdA

A

? ? EH0 ydA A

E

d 2v dx2

A

y 2dA

Moment of inertia I(x)

P EAH0

M

EI

d 2v dx2

EA: axial rigidity EI: flexural rigidity

5

BEAM THEORY cont.

? Beam constitutive relation

? We assume P = 0 (We will consider non-zero P in the frame element) ? Moment-curvature relation:

M

EI

d 2v dx 2

Moment and curvature is linearly dependent

? Sign convention

+M +P

y

+Vy +M

x

+P

+Vy

? Positive directions for applied loads y p(x)

x

C1

C2

C3

F1

F2

F3

6

GOVERNING EQUATIONS

? Beam equilibrium equations

? fy

0 p(x)dx

? ?Vy

?

dVy dx

dx

? ??

Vy

0

dVy p(x) dx

M

? ??

M

dM dx

dx

? ??

pdx dx 2

Vydx

0

Vy

dM dx

? ?

Combining three equations together: Fourth-order differential equation

EI

d 4v dx 4

p(x)

M Vy

p

Vy

dVy dx dx

M dM dx dx

dx

7

STRESS AND STRAIN

? Bending stress V xx

Ey

d 2v dx2

V xx (x, y)

M

EI

d 2v dx 2

M (x)y I

Bending stress

? This is only non-zero stress component for Euler-Bernoulli beam

? Transverse shear strain

J xy

wu wy

wv wx

wv wv 0 wx wx

u(x, y)

u0 (x)

y dv dx

? Euler beam predicts zero shear strain (approximation)

? Traditional beam theory says the transverse shear stress is W xy

VQ Ib

? However, this shear stress is in general small compared to

the bending stress

8

POTENTIAL ENERGY

? Potential energy 3 U V

? Strain energy

? Strain energy density

U0

1 2

V

xxH

xx

1 2

E(H

xx

)2

1 2

E

? ? ?

y

d 2v dx2

?2 ? ?

1 2

Ey

2

? ? ?

d 2v dx2

? 2 ? ?

? Strain energy per unit length

UL (x) U L ( x)

?U0(x, y, z) dA

A

1 2

EI

? ? ?

d 2v dx2

? 2 ? ?

?A

1 2

Ey

2

? ? ?

d 2v dx2

?2 ? ?

dA

? 1

2

E

? ? ?

d 2v dx2

? 2 ? ?

A

y 2dA

Moment of

inertia

? Strain energy

?L

U 0 U L (x) dx

?1

2

L 0

EI

? ??

d 2v dx2

? 2 ??

dx

9

POTENTIAL ENERGY cont.

? Potential energy of applied loads

V

? ? ? L

p(x)v(x) dx

0

NF

Fiv(xi )

i1

NC i1

Ci

dv(xi ) dx

? Potential energy

? ? ? ? 3 U V

1 2

L 0

EI

? ? ?

d 2v dx2

?2 ? ?

dx

L

p(x)v(x) dx

0

NF

Fiv(xi )

i1

NC i1

Ci

dv(xi ) dx

? Potential energy is a function of v(x) and slope ? The beam is in equilibrium when 3 has its minimum value

3

w3 0 wv

v*

v

10

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