Exact Analysis of Indeterminate Structures: Force Methods

Using compatibility of displacements to solve for redundant forces.
Unlike determinate structures, statically indeterminate structures cannot be solved using equilibrium equations alone because they have more unknown forces than available equations. To analyze them, we must consider the deformations of the structure.
There are two primary categories of exact analysis methods: Force Methods (also called Flexibility Methods) and Displacement Methods (also called Stiffness Methods). This module focuses on the Force Methods.

Core Principle of Force Methods

Redundant forces and compatibility equations.
In Force Methods, the unknown redundant forces (or moments) are chosen as the primary variables. The method relies on the principle of geometric compatibility, which states that the deformed shape of the structure must satisfy the physical constraints at its supports.

The General Procedure

  1. Identify the Degree of Indeterminacy (ii): Determine how many redundant forces exist.
  2. Choose the Primary Structure: Mentally remove a number of "redundant" supports equal to ii to make the structure statically determinate and stable. This is called the "primary structure."
  3. Calculate Deformations (Primary): Apply the actual given loads to this primary structure and calculate the displacement (or rotation) at the points where the supports were removed.
  4. Calculate Deformations (Redundant): Remove the given loads. Apply an unknown redundant force (represented by a variable, e.g., X1X_1) at each removed support location. Calculate the displacement at those points due only to the redundant forces. (Usually done by applying a unit force and scaling it by X1X_1).
  5. Enforce Compatibility: Write an equation stating that the sum of the displacements from step 3 and step 4 must equal the actual known displacement at that support (usually zero). Solve these equations for the redundant forces X1,X2X_1, X_2, etc.

Force Method: Propped Cantilever Simulation

Observe how the method of consistent deformations solves for the redundant reaction $R_B$. The primary structure (a simple cantilever) deflects downwards due to the uniform load. The redundant force $R_B$ must push upwards exactly enough to bring the net deflection at support B back to zero.

Calculations

  • Length ($L$): 10 m
  • Flexural Rigidity ($EI$): 10000 kN·m²
  • Primary Deflection at B (DeltaB0\\Delta_{B0}): 1250.00 mm (down)
  • Flexibility Coefficient (fBBf_{BB}): 33.33 mm/kN
  • Redundant Reaction ($R_B$): 37.50 kN (up)
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Key Takeaways
  • The Force Method (Method of Consistent Deformations) solves for redundant forces by enforcing geometric compatibility (displacements must match boundary conditions).
  • The primary structure must be statically determinate and geometrically stable.

The Method of Consistent Deformations

Analyzing structures with redundant reactions.
Let's break down the procedure for a structure that is indeterminate to the first degree (i=1i = 1).

Flexibility Coefficient (fijf_{ij})

A flexibility coefficient fijf_{ij} is defined as the displacement at point ii caused by a unit load applied at point jj. It quantifies the "flexibility" of the primary determinate structure.

Procedure

  1. Identify the Redundant: Choose one support reaction to be the redundant force (X1X_1). Remove this constraint to create the statically determinate "primary structure." Example: For a propped cantilever (fixed at A, roller at B), choose the roller reaction RBR_B as the redundant. The primary structure is a simple cantilever fixed at A.
  2. Calculate Primary Deflection (Δ10\Delta_{10}): Apply the external loads to the primary structure. Calculate the deflection at the point and in the direction of the removed redundant force. Let this be Δ10\Delta_{10} (deflection at point 1 due to load 0).
  3. Calculate Flexibility Coefficient (f11f_{11}): Remove the external loads from the primary structure. Apply a unit force (magnitude = 1.0) at the point and in the direction of the redundant. Calculate the resulting deflection at that same point. Let this be f11f_{11} (deflection at point 1 due to a unit force at point 1).
  4. Write Compatibility Equation: The total deflection at the support must equal the known settlement value (usually zero).
  5. Solve for Redundant: Calculate X1X_1 by dividing the primary deflection by the flexibility coefficient (X1=−Δ10f11X_1 = -\frac{\Delta_{10}}{f_{11}}). A negative result means the assumed direction of the redundant force was incorrect.
  6. Final Analysis: Now that X1X_1 is known, apply it along with the external loads to the primary structure (or the original structure) and use statics to find the remaining reactions, shear, and moment.

Compatibility Equation (1 Degree of Indeterminacy)

Equates total deflection to support settlement (usually zero).

$$ \Delta_{10} + f_{11} X_1 = 0 $$

Multiple Degrees of Indeterminacy

Extending the method to more complex structures.
If a structure is indeterminate to the nn-th degree, you must choose nn redundant forces (X1,X2,…,XnX_1, X_2, \dots, X_n) and write nn compatibility equations.

Procedure

  1. Choose nn redundants to form a stable, determinate primary structure.
  2. Calculate primary deflections Δ10,Δ20,…,Δn0\Delta_{10}, \Delta_{20}, \dots, \Delta_{n0} due to applied loads at the locations of the redundants.
  3. Apply a unit force at location 1 and calculate flexibility coefficients f11,f21,…,fn1f_{11}, f_{21}, \dots, f_{n1} at all redundant locations.
  4. Repeat step 3 by applying unit forces at locations 2 through nn to find all fijf_{ij} coefficients (remembering fij=fjif_{ij} = f_{ji}).
  5. Set up a system of nn linear equations, which can be represented in matrix form as [Δ0]+[F][X]=0[\Delta_0] + [F][X] = 0:
[Δ10Δ20⋮Δn0]+[f11f12…f1nf21f22…f2n⋮⋮⋱⋮fn1fn2…fnn][X1X2⋮Xn]=[00⋮0]\begin{bmatrix} \Delta_{10} \\ \Delta_{20} \\ \vdots \\ \Delta_{n0} \end{bmatrix} + \begin{bmatrix} f_{11} & f_{12} & \dots & f_{1n} \\ f_{21} & f_{22} & \dots & f_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ f_{n1} & f_{n2} & \dots & f_{nn} \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ \vdots \\ X_n \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} 6. Solve the system of equations simultaneously ([X]=−[F]−1[Δ0][X] = -[F]^{-1} [\Delta_0]) for the unknown redundants.

Other Effects: Settlements, Temperature, and Errors

The compatibility equation can be easily expanded to account for internal forces induced by non-load effects in indeterminate structures:
  • Support Settlements: Set the right side of the compatibility equation to the known settlement value instead of zero. (e.g., Δ10+f11X1=Δsettlement\Delta_{10} + f_{11}X_1 = \Delta_{settlement}).
  • Temperature Changes: Calculate the thermal deformation of the primary structure using ΔT=αΔTL\Delta_T = \alpha \Delta T L (for axial elongation) and add it to the primary load deflection. (e.g., Δ10+ΔT+f11X1=0\Delta_{10} + \Delta_{T} + f_{11}X_1 = 0).
  • Fabrication Errors: If a truss member is manufactured too long or too short, calculate its effect on the primary structure's deflection and include it in the compatibility equation.
Key Takeaways
  • The compatibility equation relates the deflection of the primary structure due to applied loads (Δi0\Delta_{i0}) to the deflection caused by the unknown redundant forces (fii⋅Xif_{ii} \cdot X_i).
  • The flexibility coefficient fijf_{ij} represents the deflection at ii due to a unit load at jj.
  • Non-load effects like settlements and temperature changes can be directly incorporated into the compatibility equations.
  • For multiple degrees of indeterminacy, a system of linear equations is set up to solve for all redundant forces.

Maxwell-Betti Reciprocal Theorem

A fundamental theorem relating deflections caused by different forces.
The Force Method relies heavily on flexibility coefficients (fijf_{ij}). The Maxwell-Betti Reciprocal Theorem simplifies calculations when dealing with multiple redundants.

Maxwell-Betti Theorem

For any linear elastic structure subjected to two independent force systems (System A and System B):

The work done by the forces of System A moving through the displacements caused by System B is equal to the work done by the forces of System B moving through the displacements caused by System A.

A direct consequence of this theorem is the symmetry of flexibility coefficients:
$$ f_{ij} = f_{ji} $$
Meaning: The deflection at point ii caused by a unit force at point jj is exactly equal to the deflection at point jj caused by a unit force at point ii. This cuts the number of required deflection calculations in half when forming the flexibility matrix for highly indeterminate structures.
Key Takeaways
  • The Maxwell-Betti theorem proves that fij=fjif_{ij} = f_{ji}, simplifying calculations for multiple redundants.

The Three-Moment Equation

An application of force methods specifically for continuous beams.
The Three-Moment Equation (often called Clapeyron's theorem of three moments) is a specialized formulation of the Force Method used for the analysis of continuous beams. It establishes a relationship between the bending moments at three consecutive supports of a continuous beam, based on the compatibility of rotations (slopes) at the middle support.

Three-Moment Equation Formula

For two adjacent spans, L1L_1 and L2L_2, supported at points A, B, and C with internal moments MAM_A, MBM_B, and MCM_C, the general equation is:

$$ M_A L_1 + 2M_B (L_1 + L_2) + M_C L_2 = -6 \\left[ \\frac{A_1 \\bar{x}_1}{L_1} + \\frac{A_2 \\bar{x}_2}{L_2} \\right] $$

Three-Moment Equation Interactive Lab

Adjust the span lengths and uniform loads for a two-span continuous beam. Exterior supports are simple pins/rollers ($M_A=0, M_C=0$). Observe how the internal moment at the middle support ($M_B$) changes.

Calculation

$M_A L_1 + 2M_B(L_1 + L_2) + M_C L_2 = dots$

$dots - rac{w_1 L_1^3}{4} - rac{w_2 L_2^3}{4}$

$0(5) + 2M_B(5 + 5) + 0(5) = dots$

$dots - rac{(10)(5)^3}{4} - rac{(10)(5)^3}{4}$

$20 M_B = -625.00$

Internal Moment $M_B$-31.25 kNm
Key Takeaways
  • The Three-Moment Equation connects moments at three consecutive supports, reducing the need to find every flexibility coefficient.

Three-Moment Equation Formula

For a continuous beam with two adjacent spans (L1L_1 and L2L_2), supported at points 1, 2, and 3, the general Three-Moment Equation relating the internal moments at the supports is:
M1L1+2M2(L1+L2)+M3L2=−6A1a1L1−6A2b2L2 M_1 L_1 + 2M_2(L_1 + L_2) + M_3 L_2 = - \frac{6A_1 a_1}{L_1} - \frac{6A_2 b_2}{L_2}
Where AA represents the area of the bending moment diagram for the applied loads on a simple span, and aa and bb represent the centroidal distances from the supports.

Symmetry and Anti-Symmetry

Exploiting structural properties to simplify analysis.

Simplifying Indeterminate Structures

If a structure is symmetric in geometry, material properties, and support conditions, the analysis can be drastically simplified:
  • Symmetric Loading: The internal forces and deformations will be symmetric. The structure can be cut in half at the axis of symmetry, with specific boundary conditions applied at the cut (e.g., zero shear, zero rotation), halving the degree of indeterminacy.
  • Anti-Symmetric Loading: The internal forces will be anti-symmetric. The structure can be cut at the axis of symmetry with different boundary conditions applied (e.g., zero moment, zero axial force).
  • Any general loading on a symmetric structure can be decomposed into a superposition of symmetric and anti-symmetric loading cases.
Key Takeaways
  • Utilizing symmetry and anti-symmetry can significantly reduce the computational effort required in force methods.
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