Solving indeterminate structures requires advanced structural analysis techniques that go beyond what simple static equilibrium equations can provide. These methods account for the additional stiffness and redundancy present, ensuring a complete and accurate analysis.
Indeterminate structures are those where the number of unknown reactions or internal forces exceeds the number of available independent static equilibrium equations. This "excess" of unknowns means that basic equations of statics (sum of forces and moments equal to zero) are insufficient to fully determine all forces and moments within the structure. To solve them, additional equations derived from the compatibility of deformations and the material's constitutive laws are required.
There are two primary categories of methods used to analyze indeterminate structures:
- The Force Method (also known as the Compatibility Method or Method of Consistent Deformations)
- The Displacement Method (also known as the Stiffness Method)
Each method offers a systematic approach to solving these complex structural problems, with the choice often depending on the specific structure's characteristics and the analyst's preference.
The Force Method (Method of Consistent Deformations)
The Force Method focuses on satisfying the compatibility conditions of the structure. It involves identifying and treating redundant forces (or moments) as the primary unknowns. By temporarily removing these redundants, the structure is transformed into a statically determinate "primary structure," which can then be analyzed using standard static equilibrium equations. The deformations caused by external loads and the unknown redundant forces are then combined to satisfy the original structure's compatibility requirements.
Here are the step-by-step procedures for applying the Force Method:
Step 1: Determine the Degree of Indeterminacy ($D_i$)
The first crucial step is to ascertain how many "extra" unknowns exist. The degree of indeterminacy represents the number of redundant reactions or internal forces that cannot be determined solely by static equilibrium equations.
- How to calculate: For plane structures, this is generally calculated as $D_i = (\text{Number of unknown reactions} + \text{Number of internal redundants}) - (\text{Number of equilibrium equations available})$. For example, a fixed-end beam has three reaction components at each support (vertical, horizontal, moment), totaling six unknowns. With three equilibrium equations available, the degree of indeterminacy is $6 - 3 = 3$.
- Practical Insight: A higher degree of indeterminacy indicates more complex calculations with the Force Method, as it will lead to a larger system of simultaneous equations.
Step 2: Choose the Redundant Reactions
Next, you must strategically select the redundant forces or moments whose removal will render the structure statically determinate and stable. These are the primary unknowns you will solve for.
- Selection Criteria:
- Choose redundants that simplify the primary structure's analysis (e.g., turning a continuous beam into a series of simply supported beams or cantilevers).
- Ensure that removing the chosen redundants results in a statically determinate and stable primary structure. An unstable primary structure cannot be used.
Step 3: Obtain the Primary (Determinate) Structure
Once the redundant reactions are chosen, physically (or conceptually) remove them from the original indeterminate structure.
- Process: If a support reaction is chosen as redundant, replace it with a freedom of movement in its direction. If an internal force (like a moment in a continuous beam) is chosen, introduce a hinge or cut at that location.
- Result: This process yields a statically determinate structure that can be readily analyzed using the basic equations of static equilibrium.
Step 4: Formulate the Compatibility Equations
This is the core of the Force Method. For each redundant reaction removed, an equation must be written to ensure that the deformations in the primary structure are compatible with the original indeterminate structure's constraints.
- Principle: The total deflection or rotation at the point where a redundant was removed, and in the direction of that redundant, must be consistent with the original support conditions.
- If a support reaction (e.g., vertical support) was removed, the total deflection at that point in the direction of the redundant must be zero.
- If an internal moment was removed (by introducing a hinge), the relative rotation at that hinge must be zero.
- Mathematical Expression: For each redundant $Ri$, the compatibility equation typically takes the form:
$\Delta{i,total} = \Delta{i,0} + \sum{j=1}^{Di} f{ij} R_j = 0$ (or a known displacement).- $\Delta_{i,0}$ is the deflection (or rotation) at point $i$ in the primary structure due to external loads.
- $f_{ij}$ are flexibility coefficients.
- $R_j$ are the unknown redundant forces/moments.
Step 5: Compute the Flexibility Coefficients
Flexibility coefficients ($f_{ij}$) are essential components of the compatibility equations. They quantify the deformation at a specific point ($i$) due to a unit force or moment applied at another point ($j$) on the primary structure.
- What they are: $f_{ij}$ represents the displacement at point $i$ due to a unit load applied at point $j$.
- Calculation: These coefficients are typically computed using methods like the principle of virtual work, Castigliano's theorem, or direct integration of the elastic curve, applied solely to the primary (determinate) structure.
- Matrix Form: For multiple redundants, these coefficients form a flexibility matrix that relates the redundant forces to the displacements.
Step 6: Solve for Redundant Forces
After formulating all compatibility equations and computing the flexibility coefficients, you will have a system of simultaneous linear equations (equal in number to the degree of indeterminacy).
- Process: Solve this system of equations to determine the numerical values of the unknown redundant forces or moments ($R_1, R2, ..., R{D_i}$).
Step 7: Analyze the Original Indeterminate Structure
Once the magnitudes and directions of all redundant forces are known, apply them back to the original indeterminate structure along with the external loads.
- Final Step: With all reactions determined (or at least enough to make the structure statically determinate), you can now use the basic equations of static equilibrium to calculate all internal forces (axial forces, shear forces, bending moments) and reactions throughout the entire structure. This allows for a complete analysis, including drawing shear and moment diagrams and calculating deflections.
The Displacement Method (Stiffness Method)
In contrast to the Force Method, the Displacement Method focuses on satisfying the equilibrium conditions of the structure. It treats unknown joint displacements (rotations and translations) as the primary unknowns. This method is particularly well-suited for computer-aided analysis due to its systematic matrix formulation. Key techniques under this umbrella include:
- Slope-Deflection Method: For beams and frames, it relates end moments to joint rotations and chord rotations.
- Moment Distribution Method: An iterative technique for solving continuous beams and rigid frames.
- Matrix Stiffness Method: The most general form, forming the basis for most structural analysis software, directly relates forces to displacements via a stiffness matrix.
Choosing the Right Method
- Force Method: Often more efficient when the degree of indeterminacy is low (e.g., 1 to 3 redundants), as it involves solving a smaller system of equations. It offers a deeper conceptual understanding of force flow.
- Displacement Method: Generally more efficient when the number of unknown joint displacements (degrees of freedom) is low. It is the preferred method for complex structures and is almost universally employed in commercial structural analysis software (e.g., SAP2000, ETABS, STAAD.Pro) due to its algorithmic nature.
Both methods, when applied correctly, will yield the same accurate results. The choice often comes down to the analyst's familiarity, the structure's complexity, and the tools available. A fundamental understanding of structural behavior, statics, and mechanics of materials is essential for success with either approach.
