Example1 Column Analogy Method

☕ https://buymeacoffee.com/pankajkporwal ☕ ☕ https://buymeacoffee.com/pankajkporwal ☕ Column Analogy Method This method is used to find end moments and fixed end moments in statically indeterminate beams. If both ends of a beam are fixed, then end moments will be equal to fixed end moments. It is based on analogy between pressure at edges or corners of the base of column and end moments in beam or portal frame. Analogous column is a short column with cross section width L and breadth 1/EI, where L is the length of the beam and E is Young’s Modulus of beam material and I is second moment of area of beam cross section. M(x)/EI loading is applied on an analogous column where M(x) is bending moment diagram for statically determinate beam (simply supported beam) corresponding to given statistically indeterminate beam. Ex. Find end moments (fixed end moments) for a beam subjected to concentrated load at distance a from end A and b from end B. The length of beam is L, Young’s modulus E, and second moment of area I. Both the ends of beam are fixed. Ex. Find end moment (fixed end moments) for a beam subjected to uniformly distributed load ( udl ) w kN/m. The length of beam is L, Young’s modulus E, and second moment of area I. Both the ends of beam are fixed. Ex. Find end moment (fixed end moments) for a beam subjected to right angle triangular load with maximum value w kN/m. The length of beam is L, Young’s modulus E, and second moment of area I. Both the ends of beam are fixed. Ex4. Find end moments (fixed end moments) for a beam subjected to concentrated load W at the mid span. The second moment of area for first half of the beam is 2I and that for second half of the beam is I. Young’s modulus E is constant. Both ends of the beam are fixed. Ex5. Find end moments for a beam subjected to concentrated load P at the mid span. One end is fixed and one end is pinned / hinged (pin / hinge support). Second moment of area I and Young’s modulus E are constant. Stiffness and Carry Over Factor using Column Analogy Method Rotational stiffness is equal to the moment required to produce one radian rotation at a point. Carry over factor is the ratio of moment developed at far end of beam segment to moment applied at the near end of the beam segment. Column analogy method is used to find the stiffness and carry over factor. For this we apply a unit downward load at the point (corresponding to the near end of beam, say A) on the analogous column. Analogous column is a short column with cross section width L and breadth 1/EI, where L is the length of the beam and E is Young’s Modulus of beam material and I is second moment of area of beam cross section. The pressure developed at edge A on the base of analogous column is equal to rotational stiffness at A and minus of the ratio of pressure at edge B to pressure at edge A is equal to carry over factor.