Indeterminate Trusses
A truss is statically indeterminate when it has more unknown member forces or reactions than available static equilibrium equations. This can occur due to:
- Excess support reactions beyond 3 $(in\ planar\ trusses)$
- Additional members beyond a statically determinate configuration
Steps Using the Superposition $(Consistent\ Deformation)$ Method:
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Determine the degree of indeterminacy. Select a suitable redundant force (denoted as R) to remove. This may be a support reaction or a member force.
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Primary Truss 1 $(Real\ System)$:
- Remove the redundant R.
- Apply the actual external loads.
- Compute internal member forces F.
- Calculate the deflection at the location of the removed redundant in the direction of R. Denote this as $\Delta$.
-
Primary Truss 2 $(Virtual\ System)$:
- Apply a unit load at the location and direction of the redundant R.
- Compute internal member forces due to this unit load. Denote them as Fv.
- Calculate the flexibility coefficient y.
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Apply the Principle of Virtual Work:
The total displacement at the redundant’s location must be zero. Thus:
$$ \Delta + y \cdot R = 0 \quad \Rightarrow \quad R = -\frac{\Delta}{y} $$
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Substitute the value of R back into the structure and solve for final support reactions and member forces using static equilibrium.
Virtual Work Table of Values:
The following table summarizes the required values per member for computing $\Delta$ and $y$:
| Member |
Length $L\ (m)$ |
Area $A\ (mm²)$ |
Force $F\ (kN)$ |
Virtual Force $F_v\ (kN)$ |
$F_vFL$ |
$F_v^2L$ |
| Member 1 |
$L_1$ |
$A_1$ |
$F_1$ |
$F_{v1}$ |
$F_{v1}F_1L_1$ |
$F_{v1}^2L_1$ |
| Member 2 |
$L_2$ |
$A_2$ |
$F_2$ |
$F_{v2}$ |
$F_{v2}F_2L_2$ |
$F_{v2}^2L_2$ |
| ⋮ |
⋮ |
⋮ |
⋮ |
⋮ |
⋮ |
⋮ |
| Sum |
|
$\sum F_vFL$ |
$\sum F_v^2L$ |
Final General Formulas:
$$ \Delta_n = \sum \left( \frac{F_v \cdot F \cdot L}{AE} \right), \quad y_n = \sum \left( \frac{F_v^2 \cdot L}{AE} \cdot R \right) $$
$$ \Delta_n + y_n R = 0 \quad \Rightarrow \quad R = -\frac{\Delta_n}{y_n} $$
Indeterminate Truss by Redundant Bar Member
In some trusses, an extra member may be present that makes the system statically indeterminate. These additional members—called
redundant bars—introduce an unknown internal force. We can solve these cases using the Superposition Method,
specifically through Virtual Work.
General Procedure:
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Identify the Redundant Bar: Choose the redundant bar (e.g., member AC) and remove it to create a statically determinate Primary Truss 1.
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Analyze Primary Truss 1 $(Real\ System)$: Solve for internal axial forces in each member due to actual loads. Denote these forces as $F$.
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Analyze Primary Truss 2 $(Virtual\ System)$: Apply a unit load in the direction of the redundant force $(i.e.,\ axial\ in\ the\ removed\ bar)$.
Solve for internal virtual axial forces in all members, denoted as $F_v$.
-
Use the Virtual Work Table: Fill up the same virtual work table as previously shown. It helps compute both the
real deformation $\Delta$ and the flexibility coefficient $y$ using:
$$ \Delta = \sum \frac{F_v F L}{AE}, \quad y = \sum \frac{F_v^2 L}{AE} $$
-
Apply the Compatibility Equation:
The total relative deformation along the removed member must be zero:
$$ \Delta_{\text{redundant}} + y_{\text{redundant}} = 0 $$
Or simply:
$$ \Delta + y \cdot R = 0 \quad \Rightarrow \quad R = -\frac{\Delta}{y} $$
-
Substitute $R$ Back: Once the force in the redundant bar is known, reanalyze the complete structure by adding it back and checking equilibrium.
This approach is especially useful when a single bar is added for stiffness or bracing, but its internal force must still be determined consistently with compatibility and equilibrium.
