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h2. Why are response-spectrum base reactions different from the sum of individual reactions?

*Answer:* [Response-spectrum|kb:Response-spectrum analysis] base reactions will not match the sum of individual [joint|kb:Joint] reactions because their formulations differ. Base reactions are calculated for each [mode|kb:Modal analysis] 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:
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** 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:
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** 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:
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** 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:
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** 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:
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** Point 1 reaction = {math}R_{1,1}{math}
** Point 2 reaction = {math}R_{2,1}{math}
** Point 3 reaction = {math}R_{3,1}{math}
** Point 4 reaction = {math}R_{4,1}{math}

* Mode 2:
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** Point 1 reaction = {math}R_{1,2}{math}
** Point 2 reaction = {math}R_{2,2}{math}
** Point 3 reaction = {math}R_{3,2}{math}
** Point 4 reaction = {math}R_{4,2}{math}

* Base reaction:
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** Mode 1: {math}SUM_{mode 1}=R_{1,1}+R_{2,1}+R_{3,1}+R_{4,1}{math}
** Mode 2: {math}SUM_{mode 2}=R_{1,2}+R_{2,2}+R_{3,2}+R_{4,2}{math}
** Base Reaction = {math}sqrt (SUM_{mode 1}^2+SUM_{mode 2}^2){math}

* Point reactions:
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** Point 1 reaction = {math}sqrt (R_{1,1}^2+R_{1,2}^2){math}
** Point 2 reaction = {math}sqrt (R_{2,1}^2+R_{2,2}^2){math}
** Point 3 reaction = {math}sqrt (R_{3,1}^2+R_{3,2}^2){math}
** Point 4 reaction = {math}sqrt (R_{4,1}^2+R_{4,2}^2){math}
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