Calculation of continuous beams by the force method. Main system. Lecture 11

1. Lecture 11 Calculation of continuous beams by the force method. Main system

2. Canonical equations of the force method. Mohr integrals

x1 11 x 2 12 x3 13 ......... x n 1n 1P 0
x1 21 x 2 22 x3 23 ........ x n 2n 2 P 0
x1 31 x 2 32 x3 33 ........ x n 3n 3P 0
..................................................................
..................................................................
x1 n1 x 2 n 2 x3 n3 ........ x n nn nP 0
M i M k dx
ik
,
EI
i,k = 1,2,3……..n
ip
M i M p dx
EI

3. Final plots of bending moments and shear forces.

n
M Mi Хi M p ,
i 1
n
Q Qi Х i Q p ,
i 1

4. Example. Construction of single diagrams

5.

6. Final plots of bending moments and shear forces

7. Calculation of statically indeterminate trusses by the force method.

n=C+C0 –2U,
on the right side C is the number of rods that make up
the truss, C0 is the number of support
connections, Y is the number of truss nodes.
To further simplify the calculation, we use the following
assumptions:
1. Connections in knots represent ideal hinges;
2. External load - nodal;
3. The lengths of the rods before and after deformation
are the same.

8. Force method for the calculation of a statically indeterminate truss

9. Main system

10. Canonical equations

x1 11 x 2 12 x3 13 ......... x n 1n 1P 0
x1 21 x 2 22 x3 23 ........ x n 2 n 2 P 0
x1 31 x 2 32 x3 33 ........ x n 3n 3 P 0
..................................................................
..................................................................
x1 n1 x 2 n 2 x3 n3 ........ x n nn nP 0

11. Final plot of longitudinal forces

N N dx
i
k
,
ik
EF
N N dx
i p
ip
EF
i,k = 1,2,3……..n
n
N Ni N p
i 1
N N1 X1 N 2 X 2 N3 X 3 N p

12. Statically indeterminate arches

Statistically indeterminate arches, depending on
the structure of the supports, are divided into
three types:

13. Double-hinged arch

A two-hinged arch - once statically indeterminate
(a), the main system of the force method is
obtained from the given one by discarding an
extra connection (b).
Canonical equation:

14. Определение коэффициента и свободного члена канонического уравнения. Построение эпюр M,Q,N.

15. Single hinged arch

Given single-hinged arch (a), main system
(b)

16. Symmetrical hingeless arch

Given single-hinged arch (a), main system (b)
Canonical equations