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## 1. Lecture 14. Construction of the final diagrams of internal forces in a statically indeterminate system by the displacement

methodThe principle of independence of the action of forces

(Principle of superposition)

The result from the action of a combination of forces is

equal to the sum of the results from each force separately.

This principle is valid for small loads and small

deformations of an elastic system.

On the basis of this principle, the final forces in a given

statically indeterminate system are decomposed into the

load forces and the reactions of the discarded bonds

separately.

## 2.

After calculating all the coefficients and free terms, thecanonical equations are solved and the unknown

angular and linear displacements Z1 , Z2, Z3 ……Zn are

determined.

The final internal forces are determined using the

principle of independence of the action of forces

(principle of superposition), by the following

summation (1):

n

M M i Zi M p ,

i 1

n

Q Qi Z i Q p ,

i 1

n

N Ni Zi N p

i 1

## 3.

On the left side of (1), respectively, M, Q, N are thefinal forces: bending moment, transverse and

longitudinal forces,

M ,Q, N

on the right side with a dash

internal forces in the main system from single

unknown displacements, the last terms represent

M p ,Qp , N p

internal forces in the main system of the

displacement method from a given external load.

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

1. Static checksF 0, F 0

x

y

2. Kinematic checks (superscript 0 means that unit

diagrams are built in the main system of the force

method)

## 5. The use of symmetry. A symmetric system refers to a system in which there is an axis of symmetry, i.e. with respect to which

there is a mirror image. In the presented system, the degree ofkinematic freedom is equal to 4. Let us decompose the

constraints in the main system of the displacement method into

symmetrical and inversely symmetrical, i.e. We use the grouping

of unknowns.

Z1 , Z 3, Z 4 -reversely symmetric unknowns;

Z2 - symmetrical angular displacement

## 6. Canonical equations of the displacement method

The canonical equations fall into two systems: The firstsystem refers to symmetric unknowns ( Z1 , Z 3 , Z 4 ) ;

the second - to the inversely symmetrical unknown .

Reactions in symmetric bonds from unit

Z 2 displacements

of inversely symmetrical bonds are equal to zero and

vice versa.

Z1r11 Z3r13 Z 4 r14 R1p 0

Z1r31 Z3r33 Z 4 r34 R 3p 0

Z1r41 Z343 Z 4 r44 R 4p 0

Z 2 r22 R1p 0

## 7. Formulas for final forces in a symmetrical system

After calculating all the coefficients and free terms, thecanonical equations are solved and the unknown

angular and linear displacements Z1, Z2, Z3, Z4 are

determined.

The final internal forces are determined using the

principle of independence of the action of forces

(principle of superposition), by the following summation

(1):

4

M M i Zi M p ,

i 1

4

Q Qi Zi Q p ,

i 1

4

N N i Zi N p

i 1

## 8. MD algorithm

1. From a given statically indeterminate system, they pass to themain system of the displacement method - kinematically

determinable, which is obtained from the given one by

introducing constraints that prevent linear and angular

displacements of the system nodes.

2. The unknowns are the angular and linear displacements in

superimposed constraints.

3. Static equations are compiled that negate reactions in

superimposed constraints (Canonical equations).

4. Using the tabular values of the reactions, unit and load

diagrams are built, from which the coefficients and free terms of

the canonical equations are determined.

4. By solving these equations and determining the unknown

displacements in superimposed constraints, plots of internal

forces are built using the principle of independence of forces.

## 9. Matrix form of the displacement method. Stiffness matrix.

Lr Z R p 0• Lr

- is a matrix consisting of unit coefficients,

Z

- a vector whose components are unknown displacements in

the introduced links,

• Rp

is a vector whose components are the free terms of the

canonical equations.

## 10. 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 ,

## 11. 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.

## 12.

L1 - a matrix, the elements of which are theordinates of unit diagrams of bending moments at

fixed points of the main system of the displacement

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

method

L1 - transposed matrix, which is obtained from a

direct matrix by replacing rows with columns,

L0p - 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.

## 13. Compliance matrix B

B10

0

0

B

0

.

.

0

0

B2

0

0

0

.

.

0

0

0

B3

0

0

.

.

0

0

0

0

.

0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

. 0

. 0

. 0

.

. 0

. .

. .

. B s

## 14. 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

## 15. Matrix K

1L1

0

K

.

.

,

1

L1

0

0

.

.

1 1

L2 L2

.

.

.

.

,

,

0

,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

,

,

,

,

,

.

0

.

.

.

.

1 1

Ls Ls

.

0

## 16. 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

## 17. 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