Theory of Structures II Force Method of Analysis (Trusses) Reference: Structural Analysis 10th Edition in SI Units by R.C. Hibbeler

1. CED 426
Structural Theory II
Lecture 8
Force Method of Analysis: Trusses
Mary Joanne C. Aniñon
Instructor

2. STATICALLY INDETERMINATE STUCTURES
• When all the on the stable structure can be determined strictly from
the equilibrium equations, the structure is referred to as statically
determinate.
• Structures having more unknown forces than available equilibrium
equations are called statically indeterminate.
• A structure is classified as statically indeterminate when the number
of unknowns exceed the number of equilibrium equations needed to
determine these unknowns.

3. METHOD OF ANALYSIS
• When analyzing any statically indeterminate structure, it is necessary
to satisfy the equilibrium, compatibility, and force-displacement
requirements for the structure.

4. METHOD OF ANALYSIS
• Equilibrium is satisfied when the reactive forces hold the structure at
rest.
• Compatibility is satisfied when the various segments of the structure
fit together without intentional breaks or overlaps.
• The force-displacement requirements depend upon the wat the
structure’s material responds to load. In this book (lecture), we have
assumed this to be a linear elastic response.

5. METHOD OF ANALYSIS
• In general, there are 2 different ways to satisfy these three
requirements.
• Force or Flexibility Method
• Displacement or Stiffness Method

6. METHOD OF ANALYSIS
• FORCE METHOD
• Originally developed by James Clerk Maxwell in 1864 and later refined by
Otto Mohr and Heinrich Muller-Breslau
• Since this method was based on compatibility forms, it has sometimes been
referred to as the compatibility method or the method of consistent
displacement.
• It consists of first writing equations that satisfy the compatibility and force-
displacement requirements for the structure in order to determine the
redundant forces.
• Then once these forces have been determined, the remaining reactive
forces on the structure are determined by satisfying the equilibrium
requirements.

7. PROCEDURE FOR ANALYSIS
• The following procedure provides a general method for determining the reactions of statically
indeterminate structures using the force or flexibility method of analysis.
1.a. Determine the unknown redundant forces.
1.b. Determine the equilibrium equations
1.c. Determine the number of degrees to which the
structure is indeterminate.
1.d. Specify the redundant forces/moments that
must be removed from the structure in order to
make it statically determinate and stable.
1.e. Draw the statically indeterminate structure and
show it equal to a series of corresponding statically
determinate structures.
1.f. Sketch the elastic curve on each structure and
indicate symbolically the displacement or rotation
at the point of each redundant force or moment.

8. PROCEDURE FOR ANALYSIS
• The following procedure provides a general method for determining the reactions of statically
indeterminate structures using the force or flexibility method of analysis.
2.a. Write the compatibility equation for the
displacement or rotation.
2.b. Determine all the deflections and flexibility
coefficients.
2.c. Substitute these results into the compatibility
equations.
2.d. Solve for the unknown redundant.

9. PROCEDURE FOR ANALYSIS
• The following procedure provides a general method for determining the reactions of statically
indeterminate structures using the force or flexibility method of analysis.

10. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Problem

11. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Step 1: PRINCIPLE OF SUPERPOSITION
• Therefore, the truss is indeterminate to the first degree.
• Since the force in member AC is to be determined, member AC will be chosen as the redundant.
• This requires “cutting” this member so that it cannot sustain a force, thereby making the truss
statically determinate and stable.
b - number of bars
r - the total number of external
support reactions
j - number of joints
6 + 3 > 2(4)
9 > 8 ∆′𝐴𝐶= 𝐹𝐴𝐶𝑓𝐴𝐶𝐴𝐶

12. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Step 2: COMPATIBILITY EQUATION
Caused by real load 2 kN Caused by the redundant
force acting alone

13. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Step 2: COMPATIBILITY EQUATION
Here the flexibility coefficient fACAC represents the relative displacement of the
cut ends of member AC caused by a “real” unit load acting
at the cut ends of member AC.
This term, fACAC and ∆AC will be found
using the method of virtual work.

14. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Determine the deflections and flexibility coefficient using
the method of virtual work:
• Remove all the real loads, apply unit load at member AC
• Solve the internal n forces in each truss member
C
FCD
FCB
1
𝐹𝑦 = 0 at point C
−𝐹𝐶𝐵 − 1
3
5
= 0
𝐹𝐶𝐵 = −0.6
𝐹𝑥 = 0 at point C
−𝐹𝐶𝐷 − 1
4
5
= 0
𝐹𝐶𝐷 = −0.8

15. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Determine the deflections and flexibility coefficient using
the method of virtual work:
• Remove all the real loads, apply unit load at member AC
• Solve the internal n forces in each truss member
𝐹𝑦 = 0 at point A
𝐹𝐴𝐷 + 1
3
5
= 0
𝐹𝐶𝐵 = −0.6
𝐹𝑥 = 0 at point A
𝐹𝐴𝐵 + 1
4
5
= 0
𝐹𝐴𝐵 = −0.8
1
FAD
FAB
A

16. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Determine the deflections and flexibility coefficient using
the method of virtual work:
• Remove all the real loads, apply unit load at member AC
• Solve the internal n forces in each truss member
𝐹𝑦 = 0 at point B
𝐹𝐵𝐶 + 𝐹𝐵𝐷
3
5
= 0
𝐹𝐵𝐷 = 1
FBC
FAB
FBD
B
−0.6 + 𝐹𝐵𝐷
3
5
= 0

17. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Determine the deflections and flexibility coefficient using
the method of virtual work:
• Apply the real load and solve the reactions
• Solve the N force in each member due to real load
𝑀𝐴 = 0
−𝑅𝐵𝑦 4 + 2 3 = 0
RBY
RAY
RAX
𝑅𝐵𝑦 = 1.5 𝑘𝑁
𝐹𝑦 = 0
𝑅𝐵𝑦 + 𝑅𝐴𝑦 = 0
𝑅𝐴𝑦 = −1.5 𝑘𝑁
RBX
𝐹𝑥 = 0
−𝑅𝐵𝑥 − 𝑅𝐴𝑥 + 2 = 0
𝑅𝐴𝑥 + 𝑅𝐵𝑥 = 2 (𝑒𝑞. 1)

18. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
RBY
RAY
RAX RBX
𝐹𝑦 = 0 at point C
𝐹𝐶𝐵 = 0
𝐹𝑥 = 0 at point C
−𝐹𝐶𝐷 + 2 = 0
𝐹𝐶𝐷 = 2
C
FCD
FCB
2
0
D FCD
FDA FDB
𝐹𝑥 = 0 at point D
𝐹𝐶𝐷 + 𝐹𝐷𝐵(
4
5
) = 0
𝐹𝑦 = 0 at point D
−𝐹𝐷𝐴 − 𝐹𝐷𝐵(
3
5
) = 0
𝐹𝐷𝐴 = 1.5
𝐹𝐷𝐵 = −2.5
2 + 𝐹𝐷𝐵(
4
5
) = 0
−𝐹𝐷𝐴 − (−2.5) (
3
5
) = 0
3m
4m
53.13
5m
𝐹𝐶𝐷 + 𝐹𝐷𝐵sin(53.13) = 0
2 + 𝐹𝐷𝐵sin(53.13) = 0

19. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
RBY
RAY
RAX RBX
𝐹𝑥 = 0 at point A
𝐹𝐴𝐵 − 𝑅𝐴𝑥 = 0
0
FAD
FAB
A
RAX
RAY
𝐹𝐴𝐵 = 𝑅𝐴𝑥 = 2
𝐹𝑥 = 0 at point B
−𝐹𝐵𝐴 − 𝐹𝐵𝐷
4
5
+ 𝑅𝐵𝑥 = 0
FBC
FAB
FBD
B
RBX
RBY
−𝑅𝐴𝑥 − (−2.5)
4
5
+ 𝑅𝐵𝑥 = 0
−𝑅𝐴𝑥 + 𝑅𝐵𝑥 = −2 (𝑒𝑞. 2)
𝑅𝐴𝑥 + 𝑅𝐵𝑥 = 2 (𝑒𝑞. 1)
−𝑅𝐴𝑥 + 𝑅𝐵𝑥 = −2 (𝑒𝑞. 2)
𝑅𝐴𝑥 = 2
𝑅𝐵𝑥 = 0
Two equations 2 unknowns:

20. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
• Apply Virtual-Work Equation
• arrange the data in a tabular form
Member n (kN) N (kN) L (m)
nNL
(kN2.m)
nnL
(kN2.m)
AB -0.8 2 4 -6.4 2.56
AD -0.6 1.5 3 -2.7 1.08
AC 1 0 5 0 5
BC -0.6 0 3 0 1.08
BD 1 -2.5 5 -12.5 5
DC -0.8 2 4 -6.4 2.56
∑=-28 ∑=17.28

21. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
• Apply Virtual-Work Equation
∆𝐴𝐶= −
28
𝐴𝐸
𝑓𝐴𝐶𝐴𝐶 =
17.28
𝐴𝐸

22. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
• Apply Virtual-Work Equation

23. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
• Substitute the results to the compatibility equation:

24. FORCE METHOD OF ANALYSIS: TRUSS
EXAMPLE 1
Solution:
Step 3: EQUILIBRIUM EQUATION
RBY
RAY
RAX
+1.62
RBX