StiffnessCoefficients.pdf

September 18, 2002

Ahmed Elgamal

Stiffness Coefficients for a Flexural Element Ahmed Elgamal

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September 18, 2002

Ahmed Elgamal

Stiffness coefficients for a flexural element (neglecting axial deformations), Appendix 1, Ch. 1 Dynamics of  Structures by Chopra. u2

u4

u3

u1

Positive directions k 21

4 degrees of freedom

k 41

k 11 k 31

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September 18, 2002

Ahmed Elgamal

Stiffness coefficients for a flexural element (neglecting axial deformations), Appendix 1, Ch. 1 Dynamics of  Structures by Chopra. u2

u4

u3

u1

Positive directions k 21

4 degrees of freedom

k 41

k 11 k 31

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Ahmed Elgamal

To obtain k coefficients in 1 st column of stiffness matrix, move u1 = 1, u2 = u3 = u4 = 0, and find forces and moments needed to maintain this shape.

u1 1.0 L

3

12EI

September 18, 2002

Ahmed Elgamal

3

L

These are (see structures textbook) 6EI 6EI

L2

2

L

12EI

 Note that that Σ Forces = 0 Σ Moments = 0

L3

ΣM= i.e. remember 

12EI 3

L

, and you

can find other forces & moments

12EI 3

L

-

12EI 3

L

= 0

Positive directions k 21 k 41

k 11 k 31 4

September 18, 2002

⎡ k 11 ⎢k  21 k = ⎢ ⎢k 31 ⎢ ⎣k 41

Ahmed Elgamal

k 12

k 13

k 22

k 23

k 32

k 33

k 42

k 43

⎡ 12 ⎢ EI ⎢ − 12 k = 3 L ⎢− 6L ⎢ ⎣− 6L

k 14 ⎤

⎥ ⎥ k 34 ⎥ ⎥ k 44 ⎦ k 24

k ij = k , where i is row number  and j is column number 

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 5

September 18, 2002

Ahmed Elgamal

u2 = 1, u1 = u3 = u4 = 0

1.0

u2

L

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Ahmed Elgamal

12EI L3

6EI L2 6EI

u2 = 1

L2 12EI

Σ M = 0, Σ F = 0 ⎡ ⎢ EI ⎢ k = 3 L ⎢ ⎢ ⎣

- 12 12 6L 6L

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

L3

Positive directions k 22 k 42

k 12 k 32 7

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Ahmed Elgamal

u3 = 1, u1 = u2 = u4 = 0

1.0

u3 = 1

L

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⎡ ⎢ ⎢ ⎢ k = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Ahmed Elgamal

2EI

4EI

L

L

6EI

6EI

L2

L2

- 6EI L2 6EI 2

L 4EI L 2EI L

Σ M = 0, Σ F = 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Positive directions k 23 k 43

k 13 k 33 9

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Ahmed Elgamal

u4 = 1, u1 = u2 = u3 = 0

u4 = 1

1.0

L

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Ahmed Elgamal

4EI

2EI

L

⎡ ⎢ ⎢ ⎢ k = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L

6EI

6EI

L2

L2

Σ M = 0, Σ F = 0 - 6EI ⎤ L2 ⎥ 6EI ⎥ 2

L 2EI L 4EI L

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Positive directions k 24 k 44

k 14 k 34 11

September 18, 2002

Ahmed Elgamal

Example: Water Tank 

u1

m

m is lumped at a point & does not contribute in rotation

u2 L

ug u2 above was u3 in the earlier section of these notes

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Ahmed Elgamal

Example: Water Tank (continued) u1

k11

m

u2

k12

k21

k22

L

ug

⎡ 12EI ⎡m ⎤ ⎡ &u&1 ⎤ ⎢ L3 ⎢ ⎥ ⎢&u& ⎥ + ⎢ − 6EI 0 ⎣ ⎦⎣ 2 ⎦ ⎢ ⎣ L2

− 6EI ⎤ L2 ⎥ ⎡ u1 ⎤ = − ⎡m⎤ &u& ⎢0⎥ g 4EI ⎥ ⎢⎣u 2 ⎥⎦ ⎣ ⎦ ⎥ L ⎦

“Note Symmetry”

Rotational (used to be u 3) 13

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Ahmed Elgamal

Example: Water Tank (continued) Static Condensation: Way to solve a smaller system of equations by eliminating degrees of freedom with zero mass. e.g., in the above, the 2nd equation gives

− 6EI 2

L

u1 +

4EI L

u2 = 0

or  u2 =

6EI L 2

L 4EI

u1 =

6 4L

u1 =

3 2L

u1

----- *

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September 18, 2002

Ahmed Elgamal

Example: Water Tank (continued) Substitute * into Equation 1

⎛ 12EI 6EI 3  ⎞ m&u&1 + ⎜ 3 − 2 ⎟u 1 = − m&u& g L 2L ⎠ ⎝  L or, ⎛ 24EI − 18EI ⎞ ⎟u 1 = − m&u& g 3 2L ⎝   ⎠

m&u&1 + ⎜

or,

⎛ 3EI ⎞ u = −m&u& g 3 ⎟ 1 ⎝  L  ⎠

m&u&1 + ⎜

 Now, solve for u1 and u2 can be evaluated from Equation * above. Static condensation can be applied to large MDOF systems of  equations, the same way as shown above.

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Ahmed Elgamal

Example: Water Tank (continued)

⎛ 3EI ⎞ u = −m&u& g 3 ⎟ 1 ⎝  L  ⎠

m&u&1 + ⎜

k of water tank as we were given earlier. or of column

if EI b = 0 I b

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Mandatory Reading Example 9.4 page 362-364

 bending  beam

Example 9.8 page 368-369 Sample Exercises: 9.5, 9.8, & 9.9

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Example (a)

(b)

Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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Ahmed Elgamal

(c)

(d)

Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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Ahmed Elgamal

(e)

(f)

Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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Ahmed Elgamal

(g)

(h)

Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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Ahmed Elgamal

(i)

(j)

Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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Ahmed Elgamal

therefore,

⎡ 12 − 12 − 6L − 6L⎤ ⎢ ⎥ 6L 0 EI ⎢ − 12 24 ⎥ k = 3 2 2 4L 2L ⎥ L ⎢− 6L 6L ⎢ 2 2 ⎥ 6L 0 2L 8L − ⎣ ⎦

,

⎡ m1 ⎢ M=⎢ ⎢ ⎢ ⎣

m2

⎤ ⎥ ⎥ ⎥ 0 ⎥ 0⎦

Sample Exercise: For the above cantilever system, write equation of  motion and perform static condensation to obtain a 2 DOF system. Reference: Dynamics of Structures, Anil K. Chopra, Prentice Hall, New Jersey, ISBN 0-13-086973-2

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September 18, 2002

Ahmed Elgamal

Column Stiffness (lateral vibration) m  bending  beam

L

k =

3EI 3

L

,

ω

=

k  m

 beam with EI beam = 0

⇒

⎛ 3EI ⎞ 3 ⎟ ⎝  L  ⎠

k = 2⎜

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September 18, 2002

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Case of  EI b = (rigid roof)

∞

u

L

k  =

⎛ 12EI ⎞ 3 ⎟ L ⎝   ⎠

12EI

k = 2⎜

L3

k  =

h1

h2

12E1 I1 h

3 1

+

12E 2 I 2 h 32

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September 18, 2002

Ahmed Elgamal

(See example 1.1 in Dynamics of Structures  by Chopra)

h

L = 2h

k =

96EI c 7h

3

if  EI b = EI c  beam

24EIc 12ρ + 1 , k = h 3 12ρ + 4

ρ

=

column

I b 4I c

& E c = E b = E 26

September 18, 2002

Ahmed Elgamal

Obtained by “static condensation” of 3x3 system u1

θ2

force

θ1

f s

call it u2 call it u3 ⎡ ⎢ ⎢ ⎢⎣

⎤ ⎡ u 1 ⎤ ⎡f S ⎤ ⎥ ⎢u ⎥ = ⎢ 0 ⎥ ⎥⎢ 2 ⎥ ⎢ ⎥ ⎥⎦ ⎢⎣ u 3 ⎥⎦ ⎢⎣ 0 ⎥⎦

and get f s = ku1

Use to represent u2 and u3 in terms of  u1 & plug back into Technique can also be used for large systems of equations

(See example 1.1 in Dynamics of Structures by Chopra)

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Ahmed Elgamal

Draft Example  Neglect axial deformation u3

u2 u1

I b Ic

Ic

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September 18, 2002

Ahmed Elgamal

u1 = 1 u2 = u3 = 0 6EI c

6EI c

L2

L2

(2)⎛ ⎜

12EI c  ⎞

⎟

3 ⎝  L  ⎠

L

L

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September 18, 2002

Ahmed Elgamal

u2 = 1 u1 = u3 = 0 2EI b

4EI b

4EI c

L

L

L 6EI c L2

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September 18, 2002

Ahmed Elgamal

u3 = 1 u1 = u2 = 0 4EI b

2EI b

L

L

6EI c L2

4EI c L

6I c L ⎡ 24I c E ⎢ k = 3 6I c L 4(I b + I c )L2 L ⎢ ⎢⎣6I c L 2I b L2

⎤ ⎥ 2I b L2 ⎥ 2 4(I b + I c )L ⎥⎦ 6I c L

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September 18, 2002

Ahmed Elgamal

If frame is subjected to lateral force f s Then (for simplicity, let Ic = I b = I)

⎡ 24 6L EI ⎢ 2 6L 8L L3 ⎢ ⎢⎣6L 2L2

6L ⎤ ⎡ u 1 ⎤

⎡f s ⎤ ⎥⎢ ⎥ ⎢ ⎥ 2L2 u 2 = 0 ⎥⎢ ⎥ ⎢ ⎥ 2 8L ⎥⎦ ⎢⎣ u 3 ⎥⎦ ⎢⎣ 0 ⎥⎦

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September 18, 2002

Ahmed Elgamal

Static condensation: From 2nd and 3rd equations, ⎡8L = − ⎢ 2 ⎢u ⎥ ⎣ 3⎦ ⎣2L ⎡u 2 ⎤

2

−1

2L ⎤ ⎡6L⎤ 2

⎥ ⎢

 Note matrix inverse:

⎥ u1

8L2 ⎦ ⎣6L⎦

−1 = 64L4 − 4L4

⎡ 8L ⎢ 2 ⎣- 2L 2

−1

- 2L ⎤ ⎡6L⎤ 2

⎡a  b ⎤ ⎢c d⎥ ⎣ ⎦

−1

⎡ d − b ⎤ = ⎢ ⎥ ad − cb ⎣− c a ⎦ 1

⎥ ⎢ ⎥ u1 8L ⎦ ⎣6L⎦ 2

− 6 ⎡1⎤ = ⎢ ⎥ u1 10L ⎣1⎦

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September 18, 2002

Ahmed Elgamal

Substitute into 1st equation EI ⎡

36

36 ⎤

168EI

− ⎥ u 1 = f s = 24 − u1 3 ⎢ 10 10 ⎦ L ⎣ 10L 3

or  k =

168EI 3

10L

(check this result)

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Draft Example 2

u3

u2 u1

L

2L

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September 18, 2002

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u1 = 1 u2 = u3 = 0 6EI c

6EI c

L2

L2

(2 )⎛ ⎜

12EI c  ⎞

⎟ 3 ⎝  L  ⎠

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Ahmed Elgamal

u2 = 1 u1 = u3 = 0 2EI b

4EI b

4EI c

2L

2L

L 6EI c L2

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September 18, 2002

Ahmed Elgamal

u3 = 1 u1 = u2 = 0

6EI c L2

4EI b

2EI b

2L

2L 4EI c L

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September 18, 2002

Ahmed Elgamal

⎡ ⎢ 24I 6I c L ⎢ c E ⎛ I  ⎞ k = 3 ⎢6I c L 4⎜  b + I c ⎟L2 L ⎢ ⎝  2  ⎠ ⎢ ⎢6I c L I b L2 ⎢⎣

⎤ ⎥ 6I c L ⎥ ⎥ I b L2 ⎥ ⎥ ⎛ I b  ⎞ 2 ⎥ 4⎜ + I c ⎟L ⎝  2  ⎠ ⎥⎦

For simplicity, let I b = Ic ⎡ 24 6L EI ⎢ 6L 6L2 3 ⎢ L ⎢⎣6L L2

6L ⎤ ⎡ u 1 ⎤

⎡f s ⎤ ⎥⎢ ⎥ ⎢ ⎥ L2 u 2 = 0 ⎥⎢ ⎥ ⎢ ⎥ 2 6L ⎥⎦ ⎢⎣ u 3 ⎥⎦ ⎢⎣ 0 ⎥⎦

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September 18, 2002

Ahmed Elgamal

Static condensation:

⎡6L2 ⎢u ⎥ = −⎢ 2 ⎣ 3⎦ ⎣L ⎡u 2 ⎤

=

−1

L ⎤ ⎡6L⎤ u1 ⎥ 2⎥ ⎢ 6L ⎦ ⎣6L⎦

−1 36L4 − L4

2

⎡6L ⎢ 2 ⎣- L

2

−1

- L ⎤ ⎡6L⎤ 2

⎥ ⎢ ⎥ u1 6L ⎦ ⎣6L⎦ 2

− 30 ⎡1⎤ − 6 ⎡1⎤ u1 = u1 = ⎢ ⎥ ⎢ ⎥ 35L ⎣1⎦ 7L ⎣1⎦

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September 18, 2002

Ahmed Elgamal

Substitute in 1st Equation EI ⎡

36

36 ⎤

24 − − ⎥ u 1 = f s ⎢ 7 7⎦ L ⎣ 3

or,

f s =

96 EI

or,

k =

96EI

3

7 L

3

7L

u1

Same as in Example 1.1, Dynamics of Structures by Chopra

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September 18, 2002

Ahmed Elgamal

Sample Exercise 1) 1.1 Derive stiffness matrix k  for  EI b h1

EIc1 EIc2

h2

L 1.2 For the special case of Ic1 = Ic2 = I b, h1 = h2 = h and L = 2h, find lateral stiffness k of the frame. 42

September 18, 2002

Ahmed Elgamal

Sample Exercise 2) Derive equation of motion for: m 2

EI b Flexural rigidity of beams and columns

h

EIc

m

EIc

E = 29,000 ksi,

EI b use 600 lb/ft

h

EIc

EIc 2h

Columns W8x24 sections with Ic = 82.4 in4 h = 12 ft I b = ½ Ic

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