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# Matrix form of the displacement method. Stiffness matrix. Lecture 15

## 1. Lecture 15. Matrix form of the displacement method. Stiffness matrix.

Lr Z R p 0L r - is a matrix consisting of unit coefficients,

Z

Rp

- a vector whose components are unknown

displacements in the introduced links,

is a vector whose components are the free terms of the

canonical equations.

## 2. Matrix Actions

r11 r12 r13Lr r21 r22 r23 ,

r r r

31 32 33

Lr L1 BL1 ,

1 p

Rp 2p ,

3p

C L1 B,

X1

Z X2

X

3

0

R p CL p ,

## 3. Determination of the coefficients and free terms of the canonical equations of the displacement method

M i M k dxrik

,

EI

Rip

0

M i M p dx

EI

•i,k = 1,2,3……..n

M 0p

represents a diagram of bending moments in any statically

determinate system, obtained from a given one, from an

external influence p.

## 4.

L1 - a matrix whose elements are the ordinatesof unit diagrams of bending moments at fixed

points of the main system of the displacement

method

M i , i 1,2,......n

L1

- transposed matrix, which is obtained from

a direct matrix by replacing rows with columns,

0

Lp

- a matrix, the elements of which are the

ordinates of the load diagram М0р in any

statically determinate system obtained from the

given one at fixed points, i.e. ordinates of the

diagram of bending moments in any basic

system of the method of forces from external

influence R.

## 5. Compliance matrix B

B10

0

0

B

0

.

.

0

0

B2

0

0

0

.

.

0

0

0

B3

0

0

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0

0

0

0

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0

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. 0

. 0

. 0

. 0

.

. 0

. .

. .

. B s

## 6. Construction of the final diagrams of internal forces

1Z Lr R

M L1 Z L p ,

dM

Q

K M

dx

1

K

L

1

L

## 7. Matrix K

1L1

0

K

.

.

,

1

L1

0

0

.

.

1 1

L2 L2

.

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,

,

0

,

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0

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1 1

Ls Ls

.

0

## 8. Final diagrams of internal forces

nM M i Zi M p ,

i 1

n

Q Qi Z i Q p ,

i 1

n

N Ni Zi N p

i 1

## 9. Static and kinematic checks of M, Q, N diagrams

• Static checksF 0, F 0

x

y

• Kinematic checks

M i Mdx

0

EI

M s Mdx

EI 0,

i 1,2,...n

M s M 1 M 2 ... M n