Base reactions for response-spectrum analysis

Why are response-spectrum base reactions different from the sum of individual reactions?

Answer: Response-spectrum base reactions will not match the sum of individual joint reactions because their formulations differ. Base reactions are calculated for each mode before modes are combined using the CQC or SRSS modal-combination rule. Joint reactions, on the other hand, are calculated using modal combinations which are applied to each individual joint.

For example, consider a structural system with four joints and two modes. The SRSS formulation is used. Base and point reactions are generated as follows:

Mode 1:
- Point 1 reaction = R(1,1)
- Point 2 reaction = R(2,1)
- Point 3 reaction = R(3,1)
- Point 4 reaction = R(4,1)

Mode 2:
- Point 1 reaction = R(1,2)
- Point 2 reaction = R(2,2)
- Point 3 reaction = R(3,2)
- Point 4 reaction = R(4,2)

Base reaction:
- Mode 1: SUM mode 1 = R(1,1) + R(2,1) + R(3,1) + R(4,1)
- Mode 2: SUM mode 2 = R(1,2) + R(2,2) + R(3,2) + R(4,2)
- Base Reaction = √((SUM mode 1)² + (SUM mode 2)²)

Point reactions:
- Point 1 reaction = √(R(1,1)² + R(1,2)²)
- Point 2 reaction = √(R(2,1)² + R(2,2)²)
- Point 3 reaction = √(R(3,1)² + R(3,2)²)
- Point 4 reaction = √(R(4,1)² + R(4,2)²)

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Related incident:

26870: response spectrum base reactions (by fs, tg)

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Ensure that the revised version shown is consistent with the intention of the previous script:

Mode 1:
- Point 1 reaction =
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  R_
  
  Unknown macro: {1,1}
- Point 2 reaction =
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  R_
  
  Unknown macro: {2,1}
- Point 3 reaction =
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  R_
  
  Unknown macro: {3,1}
- Point 4 reaction =
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  R_
  
  Unknown macro: {4,1}

Mode 2:
- Point 1 reaction =
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  R_
  
  Unknown macro: {1,2}
- Point 2 reaction =
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  R_
  
  Unknown macro: {2,2}
- Point 3 reaction =
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  R_
  
  Unknown macro: {3,2}
- Point 4 reaction =
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  R_
  
  Unknown macro: {4,2}

Base reaction:
- Mode 1:
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  SUM_
  
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  =R_
  
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  +R_
  
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  +R_
  
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  +R_
  
  Unknown macro: {4,1}
- Mode 2:
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  SUM_
  
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  =R_
  
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  +R_
  
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  +R_
  
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  +R_
  
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- Base Reaction =
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  sqrt (SUM_
  
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  ^2+SUM_
  
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  ^2)

Point reactions:
- Point 1 reaction =
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  sqrt (R_
  
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  ^2+R_
  
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  ^2)
- Point 2 reaction =
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  sqrt (R_
  
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  ^2+R_
  
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  ^2)
- Point 3 reaction =
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  sqrt (R_
  
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  ^2+R_
  
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  ^2)
- Point 4 reaction =
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  sqrt (R_
  
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  ^2+R_
  
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  ^2)