Application of Load Combinations to Modal Response Spectrum Analysis
How does ideCAD calculate column internal forces using modal response spectrum analysis according to ACI 31819?
The horizontal elastic design spectrum is used in the direction of a given earthquake while making earthquake calculations with the mode combination method. The largest displacements, relative floor displacements, internal forces, and stresses are found for each vibration mode considered. The largest modal behavior magnitudes found are combined using the Exact Quadratic Combination rule. This analysis does not give information about when the said behavior magnitude occurred and its correlation with other loadings. The values found from the combination reveal the largest possible positive (absolute) value for a single modal behavior magnitude.
If the response spectrum analysis gives an M result for the moment value, for example, this value is actually in the range of M to + M. The same relationship is valid for the axial force (N). In this case, a total of 8 calculations are made for the extremes of the 3dimensional space in which these parameters will change in order to take into account the most unfavorable situation of an element that bends biaxially under axial force.
Internal forces in the element under the effect of an earthquake:
Internal forces occurring in the element: N, Mx, My
Internal forces occurring in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Internal forces in the element: N, Mx, My
Sample
For column S01 in the Sample Project, 1.ide10 file, design internal forces will be found under the loading combination (0.9G ' Ex  0.3Ey  0.3Ez). In order to avoid confusion, only the loading combinations in Ex (5%) and Ey (5%) eccentricity effects will be combined. The directions of internal forces consisting of G ', Q', and Ez loading are given below.
Internal Forces in the Element 




For G 'loading  N = 18.7443 tf  M2 = 0.7397 tfm  M3 = 0.8767 tfm 
For Q 'loading  N = 2.118 tf  M2 = 0.0838 tfm  M3 = 0.01037 tfm 
For Ez loading  N = 8.2375 tf  M2 = 0.3251 tfm  M3 = 0.3853 tfm 
After the earthquake calculation is made with the mode combination method, the internal force values consisting of Ex (5%) and Ey (5%) loads are shown below. Since the Complete Quadratic Combination rule obtains these internal forces, they should be considered the largest possible absolute value.
Internal Forces in the Element 




For Ex (5%) loading  N = 2.7976 tf  M2 = 0.725 tfm  M3 = 2.694 tfm 
For O ( 5%) loading  N = 3.235 tf  M2 = 1.964 tfm  M3 = 0.591 tfm 
In this case, the design internal forces for the (0.9G ' Ex  0.3Ey  0.3Ez) combination are applied for each of the following situations.
Internal Forces in the Element 




For 0.9G'0.3Ez  N = 14.399 tf  M2 = 0.568 tfm  M3 = 0.673 tfm 


N = 14.399 + 2.7976 + 0.3*3.235 = 10.631 tf 
M2 = 0.568 + 0.725 + 0.3*1.964 = 1.883 tf 
M3 = 0.673 + 2.694 + 0.3 * 0.591 = 3.544 tf 


N = 14.399 + 2.7976 + 0.3*3.235 = 10.631 tf 
M2 = 0.568  0.725  0.3*1.964 = 0.746 tf 
M3 = 0.673 + 2.694 + 0.3 * 0.591 = 3.544 tf 


N = 14.399 + 2.7976 + 0.3*3.235 = 10.631 tf 
M2 = 0.568 + 0.725 + 0.3*1.964 = 1.883 tf 
M3 = 0.673  2.694  0.3 * 0.591 = 2.198 tf 


N = 14.399 + 2.7976 + 0.3*3.235 = 10.631 tf 
M2 = 0.568  0.725  0.3*1.964 = 0.746 tf 
M3 = 0.673  2.694  0.3 * 0.591 = 2.198 tf 




N = 14.399  2.7976  0.3*3.235 = 18.167 tf 
M2 = 0.568 + 0.725 + 0.3*1.964 = 1.883 tf 
M3 = 0.673 + 2.694 + 0.3 * 0.591 = 3.544 tf 


N = 14.399  2.7976  0.3*3.235 = 18.167 tf 
M2 = 0.568  0.725  0.3*1.964 = 0.746 tf 
M3 = 0.673 + 2.694 + 0.3 * 0.591 = 3.544 tf 


N = 14.399  2.7976  0.3*3.235 = 18.167 tf 
M2 = 0.568 + 0.725 + 0.3*1.964 = 1.883 tf 
M3 = 0.673  2.694  0.3 * 0.591 = 2.198 tf 


N = 14.399  2.7976  0.3*3.235 = 18.167 tf 
M2 = 0.568  0.725  0.3*1.964 = 0.746 tf 
M3 = 0.673  2.694  0.3 * 0.591 = 2.198 tf 
For the 8 cases shown above, column design is made under the influence of axial force and biaxial bending.