Library / Civil / CIV-33
Civil
Symmetric two-bar truss member force
Axial force in each of two identical inclined bars that join an apex to supports at the same lower level when a downward load P acts at the apex.
- Ref
- CIV-33
- Rev
- v1
- Field
- Civil
- Result
- N [N]
At the apex of a symmetric two-bar truss the vertical load P is balanced by the vertical components of the two bar forces, 2N sin θ = −P, so N = −P/(2 sin θ). The bars are in compression (negative) for a downward load. As the bars flatten (θ → 0) the force grows without limit, so shallow trusses are inefficient.
Axial force in each of two identical inclined bars that join an apex to supports at the same lower level when a downward load P acts at the apex. θ is each bar’s angle to the horizontal in radians, 0 < θ < π/2. A negative result is compression. The force grows without limit as the bars flatten.
Calculator
- Horizontal thrust at each support, P/(2 tan θ)25,980 N
- Bar angle in degrees30 °
- Force relative to P, |N|/P1
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Each bar carries 30 kN of compression, 1 times the applied load, and each support has a horizontal thrust of 25.98 kN.
- At θ = 30° (0.5236 rad) the bar force equals P exactly; flatter bars carry more, steeper ones less.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| P Apex load | 100 to 1,000,000 N | −1,000,000 | −100 | 999,900 | +10% | +10 % |
| θ Bar angle to the horizontal | 0.1 to 1.5 rad | −150,300 | −15,040 | 135,200 | +10% | −8.2 % |
Sweep of Bar angle to the horizontal
| Bar angle to the horizontal (rad) | Member force (N) |
|---|---|
| 0.1 | −150,300 |
| 0.275 | −55,240 |
| 0.45 | −34,490 |
| 0.625 | −25,640 |
| 0.8 | −20,910 |
| 0.975 | −18,120 |
| 1.15 | −16,430 |
| 1.325 | −15,460 |
| 1.5 | −15,040 |
Sweep one variable
Variables and how each one contributes
Member force
What the equation gives you. Everything else feeds it.
Apex load
The downward load at the apex, in N. Scales the member force in direct proportion. Raising it by 10 % (to 33,000) moves the result by +10 %, from −30,000 to −33,000 N.
Bar angle to the horizontal
The angle of each bar to the horizontal, in radians. A flatter bar (smaller θ) carries much more force, since sin θ is small. Raising it by 10 % (to 0.576) moves the result by −8.2 %, from −30,000 to −27,540 N.
Worked example with the values above
- Vertical equilibrium at the apex: 2N sin θ = −P with sin θ = 0.5.
- 2 sin θ = 1.
- N = −30,000 / 1 = −30,000 N (compression).
Assumptions and limits
- Pin joints, loads at the joints, small displacements and linear elastic members.
- Tension positive in both the real and virtual force systems.
Common mistakes
- Using θ measured from the vertical instead of the horizontal.
- Forgetting that the result is compression and needs buckling checks.
- Using the formula for unsymmetrical bars or off-centre loads.
Statistics and uncertainty
Truss displacements are exact for pin-jointed bars with axial deformation only. Real joints are slightly flexible and members have length tolerances, so measured displacements are usually 5 % to 20 % higher than the pin-jointed prediction.
Where it is used
- Deflection and stiffness checks for trusses, braces and frames.
- Estimating thermal movement and strain energy of members.
Maths you need
- Virtual work: external virtual work equals internal virtual work
- Axial deformation: ΔL = NL/EA
- Strain energy: the work stored by elastic deformation