Approximate Analysis of Frames

Simplified methods for analyzing complex building frames subjected to lateral and vertical loads.

Why Use Approximate Methods?

Approximate Methods

Simplified methods for analyzing highly statically indeterminate structures (like multi-story building frames) by making reasonable assumptions about the structure's behavior. These assumptions reduce the structure to a statically determinate system, allowing for quick preliminary design, verification of computer output, and understanding of load paths.

Approximate Methods for Vertical Loads

When a building frame is subjected to gravity (vertical) loads, the primary deformations are bending in the girders.

1. Inflection Point (0.1L) Method

A highly simplified method to make the frame statically determinate by assuming the locations of zero moment (hinges).

Procedure

  1. Assume inflection points (hinges) occur at a distance of 0.1L0.1L from each end of the girder, where LL is the clear span.
  2. Isolate the center portion of the girder (length 0.8L0.8L). It acts as a simply supported beam suspended between the hinges. Calculate its reactions at the hinges.
  3. Isolate the end portions (length 0.1L0.1L). They act as cantilever beams projecting from the columns, carrying the loads from the central beam plus any direct loads on the cantilever.
  4. Calculate the fixed-end moments of these cantilevers; these are the moments transferred to the columns.

Substitute Frame Method

Substitute Frame Method

An approximate method primarily used for analyzing multi-story building frames subjected to vertical (gravity) loads. Instead of analyzing the entire massive frame at once, the method isolates a small, relevant portion (a "sub-frame") to determine the maximum moments in specific beams or columns.

Assumptions and Scope

This method is widely accepted in design codes (like the ACI code for concrete structures) for calculating design moments.
  • For Beams: To find the maximum moment in a specific floor beam, the substitute frame consists only of the beams at that floor level and the columns immediately above and below that floor. The far ends of these columns and adjacent beams are assumed to be perfectly fixed.
  • For Columns: To find the maximum moment in a column, the sub-frame consists of the column itself and the adjacent beams on the floors immediately above and below. The far ends of these members are assumed fixed.

Procedure

  1. Isolate the substitute frame for the member of interest.
  2. Assume all far ends of the connected members (columns above/below, beams left/right) are completely fixed.
  3. Apply the appropriate live load patterns (e.g., checkerboard loading or adjacent span loading) to produce the maximum positive or negative moments.
  4. Analyze this simplified sub-frame using an exact method like Moment Distribution or Slope-Deflection. Because the sub-frame is small, the analysis is rapid.

Approximate Methods for Lateral Loads

Lateral loads (wind, earthquake) cause the frame to sway. Two common approximate methods are used depending on the relative stiffness of the columns and girders.

1. Portal Method

Portal Method

An approximate analysis method used for low-rise building frames where shear deformation dominates. It assumes the frame behaves like a series of independent portal frames.

Assumptions

  1. An inflection point (hinge) exists at the mid-height of each column.
  2. An inflection point (hinge) exists at the mid-span of each girder.
  3. The total horizontal shear at any story is divided among the columns such that interior columns carry twice as much shear as exterior columns.

Procedure

Procedure

  1. Calculate the total story shear at each level.
  2. Distribute this shear to the columns based on assumption 3 (e.g., if there are 3 columns, the exterior take VV, the interior takes 2V2V. Total =4V= 4V. Solve for VV).
  3. Calculate column end moments: M=V×(h/2)M = V \times (h/2) where hh is story height.
  4. Apply joint equilibrium (∑M=0\sum M = 0) to find the girder end moments.
  5. Calculate girder shears from the girder end moments: Vg=2Mg/LV_g = 2M_g / L.
  6. Calculate column axial forces by summing the vertical shears from the connecting girders.

2. Cantilever Method

Cantilever Method

An approximate analysis method more appropriate for tall, slender building frames where axial deformation of the columns (bending of the entire frame as a cantilever) is significant.

Assumptions

  1. An inflection point (hinge) exists at the mid-height of each column.
  2. An inflection point (hinge) exists at the mid-span of each girder.
  3. The axial stress in a column is proportional to its distance from the centroid of the cross-sectional areas of all columns at that story level. (Assuming the frame bends like a solid cantilever beam).

Procedure

Procedure

  1. Locate the centroid of the column areas for the story.
  2. Calculate the total overturning moment MotM_{ot} at the mid-height of the story columns.
  3. Determine the axial force in each column using the flexure formula analog: Pi=Mot⋅Ai⋅xi∑(A⋅x2)P_i = \frac{M_{ot} \cdot A_i \cdot x_i}{\sum (A \cdot x^2)} , where xix_i is the distance from the centroid.
  4. Calculate the girder shears from the column axial forces using vertical equilibrium at the joints.
  5. Calculate the girder end moments from their shears: Mg=Vg×(L/2)M_g = V_g \times (L/2) .
  6. Apply joint equilibrium ( ∑M=0\sum M = 0 ) to find the column end moments.
  7. Calculate column shears from their end moments: Vc=2Mc/hV_c = 2M_c / h .

Interactive Tool: Portal Method Simulator

Visualize the assumed inflection points and shear distribution in a frame subjected to lateral loads using the Portal Method.

Validity and Limitations of Approximate Methods

When to Use Approximate Analysis

Approximate methods were historically the primary tool for tall building design before the advent of computers. Today, they are used for:
  • Preliminary Design: Quickly sizing members before running a full finite element analysis.
  • Verification: Providing a fast "sanity check" to ensure computer output is reasonable and free of gross data-entry errors.
  • Limitations: They are based on sweeping assumptions (like exact locations of inflection points) that may not hold true for frames with highly irregular geometries, varying stiffnesses, or complex dynamic loadings.

Assumptions of Approximate Methods

To make statically indeterminate frames solvable by statics alone, assumptions must be made about the locations of inflection points and the distribution of internal forces.

Portal Method Assumptions

Used for low-rise building frames under lateral loads. It assumes the building acts like a series of independent portals.
  • An inflection point (zero bending moment) is located at the mid-height of each column.
  • An inflection point is located at the center of each girder.
  • The total horizontal shear at any story is distributed such that interior columns carry twice as much shear as exterior columns.

Cantilever Method Assumptions

Used for high-rise building frames under lateral loads. It assumes the building acts like a vertical cantilever beam bending under the lateral load.
  • An inflection point is located at the mid-height of each column.
  • An inflection point is located at the center of each girder.
  • The axial stress in each column is proportional to its horizontal distance from the centroid of the cross-sectional areas of the columns at that story.
Key Takeaways
  • Approximate methods reduce statically indeterminate frames to determinate ones by making behavioral assumptions (usually placing hinges at inflection points).
  • For vertical loads, hinges are often assumed at 0.1L0.1L from girder ends.
  • For lateral loads on low-rise frames, the Portal Method assumes interior columns carry twice the shear of exterior columns.
  • For lateral loads on high-rise frames, the Cantilever Method assumes column axial stress is proportional to its distance from the frame's centroid.
  • Approximate methods convert highly indeterminate frames into statically determinate structures by inserting assumed hinges (inflection points).
  • For vertical loads, hinges are typically assumed at 0.1L0.1L from the supports.
  • The Portal Method (for low-to-medium rise buildings) assumes interior columns take twice the shear of exterior columns and places hinges at the mid-height of columns and mid-span of girders.
  • The Cantilever Method (for high-rise buildings) assumes column axial stresses are proportional to their distance from the frame's centroid.
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