# Application of Load Combinations to Modal Response Spectrum Analysis

How does ideCAD calculate column internal forces using modal response spectrum analysis according to ACI 318-19?

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 3-dimensional 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:

1. Internal forces occurring in the element:  N, Mx, My

2. Internal forces occurring in the element:  N, -Mx, My

3. Internal forces in the element:  N, Mx, -My

4. Internal forces in the element:  N, -Mx, -My

5. Internal forces in the element:  -N, Mx, My

6. Internal forces in the element:  -N, -Mx, My

7. Internal forces in the element:  -N, Mx, -My

8. 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

N = -18.7443 tf

M2 = 0.7397 tfm

M3 = 0.8767 tfm

N = -2.118 tf

M2 = 0.0838 tfm

M3 = 0.01037 tfm

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

N = 2.7976 tf

M2 = 0.725 tfm

M3 = 2.694 tfm

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

1. N, M2, M3

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

1. N, -M2, M3

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

1. N, M2, -M3

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

1. N, -M2, -M3

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

1. -N, M2, M3

1. -N, M2, M3

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

1. -N, -M2, M3

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

1. -N, M2, -M3

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

1. -N, -M2, -M3

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.

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