Approximate Analysis of Indeterminate Structures - Theory & Concepts

Approximate Analysis of Indeterminate Structures - Theory & Concepts

Why Approximate Methods?

Before performing complex computer analysis (like Matrix Stiffness Methods), engineers often use approximate methods to quickly estimate member forces. This allows them to "size" the members preliminarily, obtaining

I

and

A

values which are strictly required as input for rigorous indeterminate analysis. They also serve as a crucial logic check against computer output.

These methods work by making explicit, educated assumptions about the deformed shape of the structure to force it to become statically determinate.

Portal vs. Cantilever Method Applicability

For building frames subjected to lateral loads (wind or earthquake), the choice of approximate method depends entirely on the building's height-to-width ratio.

Selecting the Method

The Portal Method is most accurate for low-rise to medium-rise building frames (where the height is generally equal to or less than the width). These frames deform primarily via "shear racking"—where the lateral drift is caused by the individual bending of the columns in each story, acting somewhat like a series of stacked portal frames.
The Cantilever Method is more accurate for high-rise, slender building frames (where the height is significantly greater than the width, typically $\text{height} > 4 \times \text{width}$ ). In these structures, the axial deformation of the columns (shortening on the leeward side, elongating on the windward side) becomes the dominant factor. The entire building acts like a giant vertical cantilever beam rooted in the ground.

Portal Method

For low-rise building frames subjected to lateral loads, the structure acts like a series of independent portals.

Important

Assumptions (Portal Method):

By making these three assumptions, enough unknown internal moments are "released" to make each story statically determinate.

Inflection points occur at the mid-height of each column. The column bends in double-curvature. At the inflection point, the internal bending moment is exactly zero ( $M=0$ ), effectively acting as an internal hinge.
Inflection points occur at the mid-span of each beam. Similarly, the beams are assumed to have zero moment at their center. (Note: This assumption is only truly valid if the frame has relatively uniform bay spacings and uniform lateral loading).
Horizontal shear is distributed among the columns such that interior columns take twice the shear of exterior columns. An interior column is considered part of two adjacent portals, while an exterior column is part of only one. Therefore, an interior column ( $2V$ ) is twice as stiff laterally as an exterior column ( $V$ ).

Explore the Portal Method visually by adjusting the number of bays, loads, and dimensions.

Cantilever Method

Suitable for tall, slender building frames (high-rise) which deform primarily in global bending.

Important

Assumptions (Cantilever Method):

Inflection points occur at the mid-height of each column. (Same as Portal, moment is zero at mid-height).
Inflection points occur at the mid-span of each beam. (Same as Portal, moment is zero at mid-span. Again, assumes uniform bay spacing).
Axial stress in a column is proportional to its distance from the centroid of the frame's cross-sectional area. Because the entire building acts like a bending beam, the columns on the "windward" side go into tension, and the "leeward" side goes into compression, following a linear stress distribution $f = \frac{My}{I}$ .

Axial Stress Distribution

\sigma = \frac{My}{I}

Since stress

\sigma = P/A

, axial force

P

is proportional to distance

y

from the neutral axis (centroid).

Procedure

Procedure

STEP-BY-STEP

Locate the Neutral Axis (Centroid): Find the centroid of the column group based on their cross-sectional areas.
Calculate Column Axial Forces: Pass a horizontal section through the mid-height of the columns (inflection points). Assume the axial stress in each column is proportional to its distance from the neutral axis. Use global moment equilibrium about the centroid to solve for the axial forces.
Determine Beam Shears: Isolate individual beam-column joints. The vertical shear in a beam is equal to the difference in axial forces of the columns framing into it.
Calculate Final Moments: Use the known shears and the assumed inflection points to find the moments at the ends of beams and columns.

The Substitute Frame Method

The Substitute Frame Method (often based on the ACI Equivalent Frame Method) is an approximate analysis technique for continuous, multi-story building frames subjected strictly to vertical (gravity) loads. It is a highly practical way to analyze a single floor without solving the entire building frame simultaneously.

Method Mechanics

Instead of analyzing a 10-story frame for gravity loads on the 5th floor, we isolate a substitute frame consisting only of:

The floor beam(s) under consideration.
The columns immediately above and below that floor, which are assumed to be fixed at their far ends.

Assumptions and Procedures:

Gravity Load Isolation: Because gravity loads primarily affect the loaded floor and dampen out rapidly in adjacent stories, the far ends of the connecting columns can safely be assumed as fully fixed against rotation.
Moment Distribution: This isolated sub-frame is then quickly solved using a simplified cycle of the Moment Distribution Method to determine the design bending moments in the floor beam and the moments transferred into the upper and lower columns.

Estimating Lateral Drift

For serviceability, ensuring the building does not sway excessively (drift) under wind loads is just as important as structural strength. Once the approximate internal forces are found via the Portal or Cantilever method, the story drift can be quickly estimated.

Virtual Work for Drift Estimation

To find the lateral deflection

\Delta

at the top of a frame, we use the Virtual Work equation:

1 \cdot \Delta = \int \frac{Mm}{EI} dx

For an approximate frame analysis where bending moments vary linearly and inflection points are assumed at mid-lengths, this integral can be simplified algebraically into a summation of the contributions from the columns and the beams in each story. The calculation is often tabulated, summing the flexural deformations of the girders and columns to verify if the frame meets building code drift limits (usually

H/400

H/600

The Factor Method

An approximate method for analyzing frames, particularly useful when quick estimates of lateral drift or member forces are required without setting up full stiffness matrices or running rigorous iterative procedures like Moment Distribution. The method relies on empirically derived factors that distribute loads proportionally. It is generally superseded by computer analysis today, but retains historical and educational value.

Key Takeaways

Portal Method: Assumes shear distribution (Exterior $V$ , Interior $2V$ ). Best for low-rise frames dominated by shear deformation ( $H \le W$ ).
Cantilever Method: Assumes axial stress varies linearly with distance from centroid. Best for high-rise frames dominated by global bending deformation ( $H > 4W$ ).
Inflection Points: Both lateral methods assume zero moment at mid-height of columns and mid-span of beams.
Substitute Frame Method: Used exclusively for gravity loads, isolating a single floor and its immediate columns (assumed fixed at far ends) to simplify calculation.
Drift Estimation: Approximate moments are directly plugged into Virtual Work equations to check building sway serviceability limits.
Approximate Analysis is vital for preliminary design (sizing members) and acting as a logical sanity check against computer software results.

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