Approximate Analysis of Statically Indeterminate Structures

Practical tools for preliminary design and checking.
Exact analysis of highly indeterminate structures (like multi-story building frames) is mathematically rigorous and often requires structural analysis software. However, before the advent of computers, engineers developed simplified, "approximate" methods. Even today, these methods remain valuable for:
  1. Preliminary Design: Quickly sizing members before a detailed analysis.
  2. Verifying Computer Output: Providing a "sanity check" to ensure computer models don't have gross errors.
  3. Understanding Structural Behavior: Developing a qualitative feel for how forces flow through a building.

Assumptions and Limitations

The core principle of approximate analysis is to make the structure statically determinate by introducing assumptions. For a structure indeterminate to the nn-th degree, you must make nn independent, reasonable assumptions about its behavior.

Limitations of Approximate Methods

  • Regularity Required: These methods are primarily valid for relatively regular building frames (uniform story heights, similar bay widths).
  • Significant Errors: Applying these methods to highly irregular structures (e.g., frames with large transfer girders, drastically changing stiffnesses, or complex geometries) can lead to significant errors (often >20%> 20\%) compared to exact analysis.

Assumptions for Vertical Loads

Approximating the behavior of beams in a rigid frame.
When analyzing a continuous beam or a rigid frame under vertical (gravity) loads, a common assumption is to estimate the locations of the points of inflection (points where the bending moment is zero).

Points of Inflection

In a rigid frame subjected to uniformly distributed gravity loads, the beams bend in double curvature. The points where the curvature changes from positive (sagging) to negative (hogging) are the points of inflection (M=0M=0).

  • Assumption: For a typical interior span of a continuous beam or rigid frame under uniform load, the points of inflection are approximately 0.1L0.1L from each support.
  • By assuming an internal hinge at each point of inflection (M=0M=0), a highly indeterminate continuous beam can be broken down into a series of determinate simply supported and cantilevered segments.

Substitute Frame Method

For large multi-story buildings, it is often impractical to analyze the entire frame for vertical loads simultaneously. The Substitute Frame Method assumes that the vertical load on a particular floor beam is primarily resisted by that beam and the columns immediately above and below it. The far ends of these columns are assumed to be fixed. This significantly reduces the size of the structure to be analyzed for gravity loads.
Key Takeaways
  • Approximate methods are valuable for preliminary design and checking exact computer analyses.
  • They work by making assumptions about the locations of inflection points (M=0M=0) and force distribution to make the structure statically determinate.
  • For vertical loads, inflection points are often assumed near 0.1L0.1L from continuous supports, and the Substitute Frame Method can isolate floor systems for analysis.

Approximate Methods for Lateral Loads

Analyzing building frames for wind and earthquake forces.
When a multi-story building is subjected to lateral loads (wind or earthquake), the frame acts as a vertical cantilever. The columns bend in reverse curvature, creating points of inflection near their mid-heights. The beams also bend in reverse curvature, with inflection points near their mid-spans.
Two classical approximate methods are used for lateral analysis: the Portal Method and the Cantilever Method. The choice between them depends on the building's height and proportions.

The Portal Method

Suitable for low-to-medium-rise buildings where shear deformation dominates.
The Portal Method is generally considered appropriate for buildings up to about 5 to 7 stories. In these lower frames, lateral deflection is primarily caused by shear distortion (bending of columns and girders), while axial deformation of columns is negligible. It assumes that the building frame acts like a series of independent single-bay "portals."

Procedure

  1. Assumption 1 (Inflection Points in Columns): An inflection point (M=0M=0) exists at the mid-height of every column. (Rationale: Columns bend in double curvature under lateral load).
  2. Assumption 2 (Inflection Points in Girders): An inflection point (M=0M=0) exists at the mid-span of every girder.
  3. Assumption 3 (Distribution of Shear): The total horizontal shear force at any story level is distributed among the columns such that interior columns carry twice as much shear as exterior columns. (i.e., Exterior Column Shear = VV; Interior Column Shear = 2V2V). This is because an interior column essentially acts as part of two adjacent "portals".

Portal Method Shear Distribution

Distribution of total horizontal shear among columns in a story.

$$ \Sigma V_{story} = V_{ext,1} + 2V_{int,1} + 2V_{int,2} + \dots + V_{ext,2} $$

The Cantilever Method

Suitable for taller, slender buildings where overall bending (axial deformation of columns) is significant.
The Cantilever Method is generally more accurate for high-rise buildings (taller than 7 stories) or very slender frames. In these structures, lateral deflection is significantly influenced by the overall bending of the building acting as a giant cantilever beam. This means the axial stretching (tension) and shortening (compression) of the columns cannot be ignored.

Procedure

  1. Assumption 1: An inflection point (M=0M=0) exists at the mid-height of every column.
  2. Assumption 2: An inflection point (M=0M=0) exists at the mid-span of every girder.
  3. Assumption 3 (Axial Stress Distribution): The axial stress in any column is directly proportional to its horizontal distance from the centroidal axis of the frame's cross-section. (Assuming all columns have the same area, the axial force FyF_y is proportional to the distance xx from the centroid: Fyi/xi=constantF_{yi} / x_i = \text{constant}). This reflects the linear strain distribution typical of bending in a deep beam or "cantilever".

Important Detail

The calculation involves finding the centroid of the column areas, determining the overturning moment at the mid-height of a story, and using the formula σ=My/I\sigma = My/I (or Pi=Moverturning⋅xi⋅Ai/∑(A⋅x2)P_i = M_{overturning} \cdot x_i \cdot A_i / \sum(A \cdot x^2)) to find the axial forces in the columns.
Key Takeaways
  • The Portal Method is for low-to-medium-rise buildings (shear dominant) and assumes interior columns carry twice the shear of exterior columns.
  • The Cantilever Method is for high-rise, slender buildings (bending/axial dominant) and assumes column axial forces are proportional to their distance from the building's centroid.
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