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Statically Indeterminate Problems under Axial Loading

Solve statically indeterminate axial bars using equilibrium plus deformation compatibility, axial stiffness EA/L, stepped members, temperature, and assembly effects.

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A system is statically indeterminate when the number of unknown reactions or internal forces exceeds the number of independent static-equilibrium equations. To determine all unknowns, equilibrium equations must be supplemented by deformation-compatibility equations describing the geometric constraints of the system.

Why equilibrium is not enough

In a statically determinate axial member, reactions can be obtained from equilibrium alone. Additional restraints or interacting members introduce redundant unknowns, so the force distribution also depends on member stiffnesses $EA$ and geometry.

Deformation equations

For a straight prismatic bar under centric tension or compression, when $N$, $E$, and $A$ are constant over the segment and the material behaves linearly elastically:

$$\Delta l=\frac{NL}{EA}.$$

  • $\Delta l$ — change in bar length;
  • $N$ — axial force;
  • $L$ — initial segment length;
  • $E$ — Young's modulus;
  • $A$ — cross-sectional area.

The axial strain is $\varepsilon=\Delta l/L$. For a stepped bar, the total change in length is the sum of segment deformations: $\Delta l=\sum_i N_iL_i/(E_iA_i)$.

For several segments, the total displacement is obtained by algebraically summing their elongations and shortenings. If temperature changes are present, thermal deformation is added to the mechanical deformation.

For a uniform bar subjected to a uniform temperature change and free thermal expansion:

$$\varepsilon_T=\alpha\Delta T,\qquad \Delta l_T=\alpha\Delta T\,L.$$

  • $\varepsilon_T$ — thermal strain;
  • $\alpha$ — coefficient of linear thermal expansion, 1/°C or 1/K;
  • $\Delta T$ — temperature change;
  • $L$ — initial length;
  • $\Delta l_T$ — free thermal change in length.

For heating, $\Delta T>0$ and the bar tends to elongate; for cooling, $\Delta T<0$ and it tends to contract. If free movement is permitted, thermal strain alone does not create mechanical stress.

Compatibility condition

The compatibility condition follows from the actual geometry and restraints. If both ends of a bar are fixed and the distance between the supports does not change, the total change in length is zero: $\sum\Delta l_i=0$. In other systems, selected nodal displacements may be equal or related by a geometric constraint.

  1. Draw the calculation model, identify unknown reactions or internal forces, and determine the degree of static indeterminacy.
  2. Write all independent equations of static equilibrium.
  3. Formulate the geometric displacement-compatibility condition imposed by the supports and connections.
  4. Express segment deformations through internal forces. For a linearly elastic axial member use $\Delta l_i=N_iL_i/(E_iA_i)$; for temperature effects add $\alpha_i\Delta T_iL_i$.
  5. Substitute the deformation expressions into the compatibility equations and solve them together with equilibrium equations for the unknown forces and reactions.
  6. Calculate stresses and perform the required strength and, if necessary, stiffness checks.
  7. Verify signs, units, equilibrium, and compatibility.

Simple example

Consider a uniform bar fixed between two immovable supports and loaded axially at an intermediate point. There are two support reactions but only one independent axial equilibrium equation, so the system is statically indeterminate to the first degree. The second equation follows from the requirement that the total change in distance between the supports is zero. The left and right segment deformations are expressed through their internal forces and axial stiffnesses $EA$.

Effect of stiffness

In statically indeterminate systems, forces are distributed among members according to their stiffness. For the same kinematic condition, a stiffer member generally carries a larger share of the load. Therefore, changing $A$, $E$, or $L$ can change reactions even if the external load remains unchanged.

Assembly and temperature effects

Manufacturing errors, initial gaps, forced assembly, or temperature changes may create internal forces even without an ordinary external mechanical load. These problems are solved by the same principle: equilibrium plus compatibility of total deformations.

Verification

After finding the reactions, verify equilibrium, the geometric compatibility condition, deformation signs, and units. Then calculate stresses in each segment and perform the required strength check.

About this topic

When static equilibrium equations are insufficient to determine support reactions or internal forces, a structure is statically indeterminate. Solving these problems requires additional deformation compatibility equations based on geometric constraints. This section covers degrees of indeterminacy, calculation algorithms for stepped bars and pin-jointed truss systems under axial load, and thermal/assembly stress analysis.