Approximate Analysis of Frames - Theory & Concepts

Approximate Analysis of Frames

$Learning Objectives$

Understand the purpose and limitations of approximate methods for analyzing highly indeterminate building frames.
Apply the Inflection Point and Substitute Frame methods to analyze frames for vertical gravity loads.
Use the Portal Method for analyzing low-rise frames under lateral loads.
Employ the Cantilever Method for evaluating tall, slender frames under lateral loads.

Simplified methods for analyzing complex building frames subjected to lateral and vertical loads provide an essential tool for preliminary design and verification of computer-based analyses.

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).

$Inflection Point (0.1L) Method$

STEP-BY-STEP

Assume inflection points (hinges) occur at a distance of $0.1L$ from each end of the girder, where $L$ is the clear span.
Isolate the center portion of the girder (length $0.8L$ ). It acts as a simply supported beam suspended between the hinges. Calculate its reactions at the hinges.
Isolate the end portions (length $0.1L$ ). They act as cantilever beams projecting from the columns, carrying the loads from the central beam plus any direct loads on the cantilever.
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.

$Substitute Frame Method$

STEP-BY-STEP

Isolate the substitute frame for the member of interest.
Assume all far ends of the connected members (columns above/below, beams left/right) are completely fixed.
Apply the appropriate live load patterns (e.g., checkerboard loading or adjacent span loading) to produce the maximum positive or negative moments.
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.

$Portal Method Assumptions$

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.

$Portal Method Procedure$

STEP-BY-STEP

Calculate the total story shear at each level.
Distribute this shear to the columns based on the third assumption (e.g., if there are 3 columns, the exterior take $V$ , the interior takes $2V$ . Total $= 4V$ . Solve for $V$ ).
Calculate column end moments: $M = V \times (h/2)$ where $h$ is story height.
Apply joint equilibrium ( $\sum M = 0$ ) to find the girder end moments.
Calculate girder shears from the girder end moments: $V_g = 2M_g / L$ .
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.

$Cantilever Method Assumptions$

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.

Axial Force in Cantilever Method

Determines the axial force in a column using an analog to the flexure formula, assuming the frame bends like a solid cantilever.

P_i = \frac{M_{ot} \cdot A_i \cdot x_i}{\sum (A \cdot x^2)}

Variables

Symbol	Description	Unit
$P_i$	Axial force in column i	-
$M_{ot}$	Total overturning moment at the mid-height of the story columns	-
$A_i$	Cross-sectional area of column i	-
$x_i$	Distance from the centroid of column areas to column i	-
$A$	Cross-sectional area of any column	-
$x$	Distance from centroid to any column	-

$Cantilever Method Procedure$

STEP-BY-STEP

Locate the centroid of the column areas for the story.
Calculate the total overturning moment $M_{ot}$ at the mid-height of the story columns.
Determine the axial force in each column using the flexure formula analog.
Calculate the girder shears from the column axial forces using vertical equilibrium at the joints.
Calculate the girder end moments from their shears: $M_g = V_g \times (L/2)$ .
Apply joint equilibrium ( $\sum M = 0$ ) to find the column end moments.
Calculate column shears from their end moments: $V_c = 2M_c / h$ .

Approximate Frame-Analysis Method Selection Workflow

Screen the approximation limits, separate gravity from lateral loading, then choose a method and follow its assumed hinge and equilibrium sequence.

Approximate results are intended for preliminary sizing or a reasonableness check. The lesson warns that highly irregular geometry, strongly varying stiffness, and complex dynamic loading may violate the simplifying assumptions. Portal analysis is for low-rise, shear-dominant behavior and distributes interior-column story shear at twice the exterior-column value. Cantilever analysis is for tall, slender frames where column axial deformation contributes materially; column axial force is proportional to column area and distance from the area centroid.

Interactive Tool: Portal Method Simulator

$Interactive Simulation$

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

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.

Key Takeaways

Approximate methods convert highly indeterminate frames into statically determinate structures by inserting assumed hinges (inflection points).
For vertical loads, hinges are typically assumed at $0.1L$ 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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