Say I want to determine the vertical displacement of the truss at Node B

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How do I tell whether the vertical displacement is upwards or downwards just by looking the real forces? Where do I draw the virtual unit load? Upwards or downwards?


1 Answer 1


In the image below, the left side shows the applied (horizontal) load and the reaction forces. The right side shows the virtual load in the direction of the displacement to be determined. The principal of virtual work states, that the deflection can be calculated as follows: $$ w=\sum \frac{N_i \cdot \bar{N_i}}{EA}L $$ The direction of the virtual load ist not really relevant. If the calculated deflection is positive, the deflection is in the direction of the virtual load. For a negative results it's the other way around.

Now, from looking at the system on the right, one can see that the only member forces are in the leftmost column (bold), i.e. those are the only two members with a non-zero virtual force. As they are in compression (virtual), while the real part (left) are in tension, you will get a negative value for $w$, which means the deflection will be upwards (opposite direction of the virtual load).

Truss with reaction forces for applied load (left) and virtual load (right).

  • $\begingroup$ Hi Andrew thanks for the answer, So I can just pick a direction to apply the vertical virtual unit load, and proceed with all the calculations involving reactions and method of joints and only in the end when I apply the formula and depending on the solution, I can just reverse the direction of the arrow. $\endgroup$
    – CountDOOKU
    Commented Apr 21, 2022 at 5:09
  • $\begingroup$ The leftmost column isn't entirely in compression is it? ie when you consider equilibrium equations, the sum of forces wouldn't be zero? $\endgroup$
    – CountDOOKU
    Commented Apr 21, 2022 at 5:43
  • $\begingroup$ Also I have done some hand calculations, I don't understand why every other members except the left column is a zero-force member for the real system (I agree with the virtual system), I found internal forces on those? What software are you using? Many thanks again. $\endgroup$
    – CountDOOKU
    Commented Apr 21, 2022 at 5:44
  • $\begingroup$ @CountDOOKU: It is only in the system on the right, where all member forces, except in the leftmost column are zero. This is not true for the system on the left. However, to determine $w$ you need to multiply every real member force $N$ with its virtual counterpart $\bar{N}$. Thus, only non-zero forces (be it real or virtual) contribute to the result of $w$. $\endgroup$
    – Andrew
    Commented Apr 21, 2022 at 6:01
  • $\begingroup$ If you want to check it, there's a free(ish) software: skyciv.com/de/free-truss-calculator $\endgroup$
    – Andrew
    Commented Apr 21, 2022 at 6:04

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