This document discusses the concept of statically indeterminate structures and the virtual work method for analyzing them. It defines statically indeterminate structures as those where the number of unknown reaction forces is greater than the number of equilibrium equations. The virtual work method makes indeterminate structures determinate by releasing supports, then calculates displacements due to applied loads to determine reactions. An example problem demonstrates applying this method to analyze the bending moments in an indeterminate frame. In conclusion, virtual work provides a powerful analysis technique for trusses, beams and frames that does not require disassembling the system of equations.

1. Kingdom of Saudi Arabia
Ministry of education
Northern Border University
Engineering College
Civil Engineering Department
1401-303, Structural Analysis II -
Spring 2020
STATICALLY INDETERMINATE BY VIRTUAL WORK
PREPARED BY:
BEDAIWI ISMAIL BEDAIWI ALRUWAILI
IBRAHIM ALASMAR FAYYADH ALANAZI
KHALED HAMDAN SALEM ALINZE
RAKAN JAZAA AQEEL ALMIJLAD
SAUD SELFIG SAUD ALSHLIKHI
TARIQ SALEH AJAJ ALENZY SUPERVISED BY:
Dr. Osama Salem Hussien

2. Contents:
• INTRODDUCTION.
• EXAMPLE ON FRAMES.
• EXAMPLE ON TRUSSES.
• VIRTUAL WORK METHOD
• EXAMPLE ON V.M.W
• CONCLUSION
• REFERENCES

3. Introduction:
What is the meaning of Statically Indeterminate?
Statically indeterminate structures are defined as structures that can't be statically
analyzed using only equilibrium equations (statics).
Meaning that: ∑Fx = 0 , ∑Fy= 0 and ∑M=0
Indeterminacy Degrees:
The difference between the number of unknown force reaction forces and the
number of equations of equilibrium is called the degree of indeterminacy.
Meaning that: Un-Eq = Indeterminacy Degree

4. Examples on frames:
Figure1 Solution
Un=3+3=6
Eq=3+2=5
Un>Eq
Indeterminate
Un-Eq=6-5=1
Once statically indeterminate

5. Examples on trusses:
Figure2 Solution
Un=5+8=13
Eq=2*5=10
Un>Eq
Indeterminate & stable
Un-Eq=13-10=3
3 times statically indeterminate.

6. Virtual Work Method:
By this method we can calculate any distance or slope at any point by
this low:
𝑆 =
𝑀0 ∗ 𝑀1
𝐸𝐼
+
𝑄0 ∗ 𝑄1
𝐸𝐼
+
𝑁0𝑁1
𝐸𝐼
.
But in beams & frames we neglect the effect of shear & normal force. In
truss we calculate the displacement due to normal force only.
-How can we benefit from V.W.M to analyze the indeterminate structures?
This method depends on making the indeterminate structures, determinate.
Then calculate the displacement due to releasing supports by V.W.M which is
equal (S10), then calculate the displacement due to the load 1ton in the same
direction of the releasing support which is equal (S11).
The sum. of displacements must equal zero :
S10+XS11=0
After that, calculate the factor X . Then the final moment equal:
Mf=M0+XM1
** Application of this principal, gives a very powerful method to

7. Example:
Determine the B.M.D by using virtual work method for the following
frame:
Figure3

8. Solution:
1)It’s indeterminate frame. So, we must change the support to
make it determinate.
Figure4

9. Solution:
2) Calculate & draw the moment (M0) and displacement due to
releasing supports by V.W.M which is equal (S10).
Figure5

10. Solution:
3) Calculate & draw the moment (M1) due to the load 1ton in the same
direction of the releasing support which is equal (S11).
Figure6

11. Solution:
4)Draw the M1:
figure7

12. Solution:
5) Now, integrate & calculate the (S10) and (S11)
by using the values of (M0) and (M1).
After that, calculate the factor X by using:
S10+XS11=0
Finally calculate & draw the final moment using
the following low:
Mf=M0+XM1

13. Conclusion:
Virtual work is perhaps the most useful and widely
applicable method of other methods. Unlike the moment
area method and the conjugate beam method, it can be
used to find the deflections of trusses, beams or frames (or,
in fact any mechanical system) because it is not necessary to
dismember the systems in order to establish relations
between the active forces. Relations between active forces
can be determined directly without reference to the reactive
forces.

14. References:
1. https://study.com/academy/lesson/statically-indeterminate-
definition-calculation-examples.html
2. http://people.duke.edu/~hpgavin/cee421/virtual-work.pdf
3. https://www.slideshare.net/scemd3/principle-of-virtual-work-in-
structural-analysis
4. https://yasserelleathy.com/eng-akram-mostafa/
5. C Russell c. Hibbeler , Structural Analysis , Pearson Prentice Hall
2012 , ISBN 10: 0 13 257053 X .
6. https://structural-analyser.com

15. The End
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