🧱Structural Analysis Unit 9 – Force Method for Indeterminate Structures

Key Concepts

• Indeterminate structures have more unknown forces or moments than available equilibrium equations
• Statical indeterminacy arises when the number of reactions exceeds the number of equilibrium equations
• Kinematic indeterminacy occurs when a structure has more unknown displacements than available compatibility equations
• Degree of indeterminacy represents the number of additional equations needed to solve for unknown forces or displacements
• Force method relies on the principle of superposition to analyze indeterminate structures by considering the effects of redundant forces separately
• Compatibility equations ensure that the deformations at the points where redundant forces are applied are consistent with the support conditions
• Flexibility coefficients relate the applied forces to the resulting displacements in the structure
• Redundant forces are additional support reactions that make a structure statically indeterminate

Fundamental Principles

• Force method is based on the principle of superposition which states that the total response of a linear elastic structure can be obtained by summing the individual responses caused by each load case
• The method involves selecting redundant forces or moments to remove from the structure, rendering it statically determinate
• The primary structure is analyzed under the action of external loads and redundant forces treated as unknown loads
• Compatibility equations are formulated to ensure that the deformations at the locations of redundant forces are consistent with the support conditions
• The redundant forces are determined by solving the compatibility equations
• Once the redundant forces are known, the total response of the structure can be obtained by superimposing the effects of external loads and redundant forces
• The force method is particularly useful for analyzing structures with a small degree of indeterminacy

Types of Indeterminate Structures

• Beams with more than two supports (continuous beams) are statically indeterminate
• Frames with more members than necessary to maintain equilibrium under applied loads are indeterminate
• Trusses with more members than required to ensure stability are internally indeterminate
• Arches with fixed supports at both ends are statically indeterminate
• Closed frames, such as portal frames with fixed supports, are indeterminate
• Structures with redundant supports, such as beams with fixed ends or propped cantilevers, are indeterminate
• Cable-stayed bridges with multiple cables anchored at different points along the deck are indeterminate
• Continuous beams with settlement of one or more supports become indeterminate due to additional unknown displacements

Force Method Steps

1. Identify the degree of indeterminacy (n) by comparing the number of unknown forces or moments to the available equilibrium equations
2. Select the redundant forces or moments (n in number) to be removed from the structure, rendering it statically determinate (primary structure)
3. Analyze the primary structure under the action of external loads (P) and redundant forces (X) treated as unknown loads
   • Determine the reactions and internal forces in the primary structure due to external loads (P) alone
   • Determine the reactions and internal forces in the primary structure due to each redundant force (X) acting separately
4. Formulate the compatibility equations by equating the deformations at the locations of redundant forces to the known support conditions
   • Express the deformations in terms of flexibility coefficients and unknown redundant forces (X)
5. Solve the compatibility equations to determine the values of redundant forces (X)
6. Superimpose the effects of external loads (P) and redundant forces (X) to obtain the total response of the structure
   • Calculate the final reactions, internal forces, and displacements by adding the contributions from external loads and redundant forces
7. Verify the results by checking the equilibrium and compatibility conditions of the original indeterminate structure

Compatibility Equations

• Compatibility equations ensure that the deformations at the locations of redundant forces are consistent with the support conditions
• The number of compatibility equations is equal to the degree of indeterminacy (n)
• Each compatibility equation represents a condition that must be satisfied for the structure to remain continuous and properly supported
• For a redundant force (X) acting at a specific location, the compatibility equation states that the displacement at that location due to external loads (P) and redundant forces (X) must be equal to the known support condition
• The displacement at the location of a redundant force can be expressed as a linear combination of the flexibility coefficients and the unknown redundant forces
  • $\delta_{P} + \sum_{i=1}^{n} f_{ii} X_{i} = 0$
  • where $\delta_{P}$ is the displacement due to external loads (P), $f_{ii}$ are the flexibility coefficients, and $X_{i}$ are the unknown redundant forces
• Flexibility coefficients ($f_{ii}$) represent the displacement at the location of redundant force $i$ due to a unit load applied at the same location, with all other redundant forces set to zero
• The flexibility coefficients can be determined using methods such as the moment-area method, conjugate beam method, or virtual work method
• The compatibility equations form a system of linear equations that can be solved simultaneously to determine the values of the redundant forces (X)

Redundant Forces and Moments

• Redundant forces and moments are additional support reactions that make a structure statically indeterminate
• The number of redundant forces or moments is equal to the degree of indeterminacy (n)
• Redundant forces can be horizontal or vertical reactions at supports, while redundant moments are typically moments at fixed or partially fixed supports
• The selection of redundant forces or moments is not unique, and different choices may lead to different primary structures
• When selecting redundant forces or moments, it is generally preferable to choose those that result in a primary structure that is easier to analyze
• Redundant forces and moments are treated as unknown loads acting on the primary structure during the analysis process
• The values of redundant forces and moments are determined by solving the compatibility equations
• Once the redundant forces and moments are known, they are superimposed with the effects of external loads to obtain the total response of the structure
• The presence of redundant forces and moments provides additional load paths and enhances the overall stability and safety of the structure

Worked Examples

1. Analyze a propped cantilever beam with a concentrated load at the free end
   • Identify the degree of indeterminacy (n = 1)
   • Select the redundant force (X) as the vertical reaction at the propped end
   • Analyze the primary structure (cantilever beam) under the action of external load (P) and redundant force (X)
   • Formulate the compatibility equation by equating the displacement at the propped end to zero
   • Solve the compatibility equation to determine the value of the redundant force (X)
   • Superimpose the effects of external load (P) and redundant force (X) to obtain the total response of the beam
2. Analyze a two-span continuous beam with a uniform load on one span
   • Identify the degree of indeterminacy (n = 1)
   • Select the redundant moment (X) as the moment at the intermediate support
   • Analyze the primary structure (simply supported beams) under the action of external load (w) and redundant moment (X)
   • Formulate the compatibility equation by equating the slope at the intermediate support to zero
   • Solve the compatibility equation to determine the value of the redundant moment (X)
   • Superimpose the effects of external load (w) and redundant moment (X) to obtain the total response of the continuous beam
3. Analyze a portal frame with fixed supports and a horizontal load at the top
   • Identify the degree of indeterminacy (n = 3)
   • Select the redundant forces and moments (X1, X2, X3) as the horizontal reaction, vertical reaction, and moment at one of the supports
   • Analyze the primary structure (pinned-pinned frame) under the action of external load (H) and redundant forces and moments (X1, X2, X3)
   • Formulate the compatibility equations by equating the displacements and rotation at the fixed support to zero
   • Solve the compatibility equations to determine the values of the redundant forces and moments (X1, X2, X3)
   • Superimpose the effects of external load (H) and redundant forces and moments (X1, X2, X3) to obtain the total response of the portal frame

Common Pitfalls and Tips

• Ensure that the selected redundant forces or moments are independent and do not violate the equilibrium conditions of the primary structure
• Be consistent with the sign convention used for forces, moments, and displacements throughout the analysis
• Double-check the formulation of compatibility equations to avoid errors in the coefficients or constants
• When solving the compatibility equations, make sure to use the correct values of flexibility coefficients and external load effects
• Verify that the results obtained from the force method satisfy the equilibrium and compatibility conditions of the original indeterminate structure
• If the primary structure is unstable or has large deformations, consider using alternative methods such as the displacement method or approximate methods
• When dealing with symmetric structures or loading conditions, exploit the symmetry to simplify the analysis and reduce the number of unknowns
• Use sketches and diagrams to visualize the primary structure, redundant forces, and deformations at various stages of the analysis
• Cross-check the results obtained from the force method with other methods, such as the moment distribution method or slope-deflection method, to ensure accuracy