Axial Force Biaxial Moment Calculation for Load Combinations in Modal Response Spectrum Analysis
The horizontal elastic design spectrum is used in the direction of a given earthquake while making earthquake calculations with the mode combination method. For each vibration mode considered, the largest displacements, relative floor displacements, internal forces and stresses are found. The largest modal behavior magnitudes found are combined using the Exact Quadratic Combination rule. In this analysis, it does not give information about when the said behavior magnitude occurred and its correlation with other loadings. The values found as a result of 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:
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 the 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 these internal forces are obtained by the Complete Quadratic Combination rule, they should be considered as 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 |
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 |
2) 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 |
3) 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 |
4) 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 |
5) -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 |
6) -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 |
7) -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 |
8) -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.
