Approximate Methods for Analysis of Multi-Storey Frames

Why it matters

Approximate methods for analysis of multi-storey frames are crucial in structural engineering as they provide simplified techniques to evaluate the behavior of complex structures. These methods are particularly useful in preliminary design stages and for quick assessments, saving time and resources.

Key ideas

• Multi-storey Frames: Structures with multiple levels, typically used in buildings, where the load is transferred through beams and columns.
• Approximate Methods: Techniques that simplify the analysis by making assumptions about the structure's behavior, such as assuming certain points of inflection or using simplified load distributions.
• Common Methods:
  • Portal Method: Assumes that the points of inflection are at mid-height of columns and mid-span of beams, typically used for low-rise buildings.
  • Cantilever Method: Assumes that the structure behaves like a cantilever, with column axial stress varying approximately linearly with distance from the frame’s centroidal axis, suitable for tall buildings.
  • Factor Method: Uses empirical factors to estimate moments and forces based on the geometry and loading of the frame.

Portal-method assumptions

For a regular frame under lateral loads, assume zero moments near beam midspans and column midheights. At a storey with equal bays, assign each interior column twice the horizontal shear of an exterior column. These are approximations, not exact properties or general gravity-load formulas.

Worked example

A regular two-bay frame has three columns at a storey of height 3 m. The total horizontal storey shear is 80 kN. Assume portal-method conditions.

Let each exterior column shear be V_e. The interior shear is 2V_e, giving V_e + 2V_e + V_e = 80 and V_e = 20 kN. The interior column carries 40 kN.

With zero moment at each column’s midheight, the end-moment magnitudes are V_col h/2: 30 kN·m for each exterior column and 60 kN·m for the interior column. Determine moment signs and beam forces from joint and member equilibrium.

Answer: Column shear magnitudes 20, 40, 20 kN and end-moment magnitudes 30, 60, 30 kN·m for the stated approximation. A simply supported UDL result wL²/8 is not a portal-method frame solution.

Reference: University of Asia Pacific structural engineering notes.

Common mistakes

• Incorrect Assumptions: Misapplying the method to inappropriate structures, such as using the Portal Method for very tall buildings.
• Calculation Errors: Mistakes in arithmetic or unit conversions.
• Ignoring Load Combinations: Not considering all possible load cases and combinations as per IS codes.

For GATE CE

Questions often involve selecting the appropriate method for a given frame and calculating moments and shear forces. Practice problems that require identifying points of inflection and applying the correct assumptions for different building heights.

Quick check

1. What is the primary assumption of the Portal Method?
2. How does the Cantilever Method differ from the Portal Method?
3. How is 80 kN storey shear shared by three columns under the stated portal assumption?

Answers: 1. Points of inflection at mid-height of columns and mid-span of beams. 2. Uses a cantilever-like axial-stress distribution across columns rather than the portal shear-sharing assumption. 3. 20, 40, 20 kN.