Understanding the concepts on Degrees of Freedom

o analyze the fundamental period of vibration and its associated mode shape for a multi-storey structure, we would typically look at the modal analysis results from a structural dynamics perspective. Let's break down the process and the provided information to explain the fundamental period and mode shape:

Given:

• Structure type: Five-storey building with 5 translational Degrees of Freedom (DOF).

• Mass per storey: $m$ (mass per floor level).

• Fundamental period of vibration: $T_1=2\text{seconds}$.

1. Fundamental Period of Vibration (T₁):

The fundamental period of vibration (T₁) is the time required for the structure to complete one full cycle of oscillation in its first (lowest) mode of vibration. The first mode is usually the most significant as it typically dominates the dynamic response of the structure during seismic events.

• In this case, we are told that T₁ = 2 seconds, meaning the first mode completes a full cycle every 2 seconds.

2. Mode Shape (φ₁):

The mode shape (φ₁) corresponds to the pattern of displacement that the structure undergoes during the fundamental vibration. Mode shapes represent how different parts of the structure move relative to one another during each mode of vibration.

• For a multi-storey structure, the first mode (fundamental mode) typically exhibits a "translational" shape, where the displacement increases with height.

• For a 5-storey building, the mode shape will typically show a proportional displacement at each floor level. The displacement at the top floor will be the largest, and the displacement decreases as we move down the structure.

• The mode shape for the first mode is usually represented by a set of displacements for each floor relative to the base.

3. Mathematical Representation:

The eigenvalue problem in structural dynamics is generally solved using the equation of motion for the building, which can be represented as:

$$\mathbf{M}\ddot{\mathbf{u}}+\mathbf{K}\mathbf{u}=0$$

Where:

• $\mathbf{M}$ is the mass matrix.

• $\mathbf{K}$ is the stiffness matrix.

• $\mathbf{u}$ is the displacement vector.

• $\ddot{\mathbf{u}}$ is the acceleration vector.

The eigenvalues of this equation represent the squared natural frequencies, and their corresponding eigenvectors represent the mode shapes.

The fundamental period $T_1$ is related to the natural frequency ${\omega }_1$ as:

$$T_1=\frac{2\pi }{{\omega }_1}$$

The fundamental frequency ${\omega }_1$ is derived from the smallest eigenvalue.

4. Interpretation of the Fundamental Mode:

• The mode shape associated with the fundamental period (T₁) is usually almost uniform at the lower floors, with the displacement increasing significantly towards the top.

• The first mode shape might look something like this for a five-storey building:

$${\varphi }_1=\left[\begin{array}{c}1\\ 2\\ 3\\ 4\\ 5\end{array}\right]$$

• These values represent the relative displacement at each floor (normalized). For example, the top floor (5th storey) will experience the largest displacement, and the displacement gradually reduces towards the base.

5. Interpreting the Results:

• The fundamental period $T_1=2\text{seconds}$ means that for a seismic event, the building will oscillate back and forth every 2 seconds in the first mode.

• The mode shape associated with $T_1$ will show how the building deforms during this vibration. For a building with 5 storeys, this mode shape will have larger displacements at higher floors, with smaller displacements at lower floors.

In the context of a five-storey building with 5 translational degrees of freedom (DOF), the system undergoes multiple modes of vibration when subjected to external forces such as seismic activity. Each mode has a corresponding period of vibration (time taken for one complete oscillation) and a mode shape (displacement pattern at each storey). Here's a breakdown of the provided information for the second, third, fourth, and fifth modes of vibration:

Mode Descriptions:

Second Mode (T₂ = 0.6852 seconds)

• Period (T₂): 0.6852 seconds.

• This mode is typically associated with a higher-frequency vibration than the first mode. It usually involves a larger displacement at the upper floors with less displacement at the lower floors, although the displacement pattern is more complex compared to the fundamental (first) mode.

  The mode shape (φ₂) for the second mode will have non-uniform displacement distribution, likely showing alternating displacements between storeys (i.e., different storeys may move in opposite directions).

Third Mode (T₃ = 0.4346 seconds)

• Period (T₃): 0.4346 seconds.

• This mode represents a higher frequency of vibration compared to the first two modes. The displacement pattern may show more pronounced shifts between higher and lower floors, potentially indicating some torsional effects or more complex lateral displacements.

  The mode shape (φ₃) for this mode could involve greater movement at the top floors compared to the lower floors, with possibly less overall displacement at lower levels.

Fourth Mode (T₄ = 0.3383 seconds)

• Period (T₄): 0.3383 seconds.

• This is a shorter period indicating an even higher frequency of vibration. The displacement at the upper storeys might be much more noticeable compared to the lower storeys. This could also involve more complicated deformation patterns, possibly indicating bending effects along the height of the structure.

  The mode shape (φ₄) will likely show significant differences between upper and lower floors in terms of displacement magnitude.

Fifth Mode (T₅ = 0.2966 seconds)

• Period (T₅): 0.2966 seconds.

• This is the highest frequency vibration mode. The displacement pattern for this mode will likely show larger displacements at the upper floors, with much smaller displacements at the lower floors. This mode could represent higher-order bending modes of the structure.

  The mode shape (φ₅) for this mode may also show complicated deformation with more pronounced changes in displacement between floors.

Interpreting the Mode Shapes:

In the analysis of multi-storey structures, the mode shapes (φₙ) give us an idea of how the building deforms under different vibrational frequencies. For each mode:

• Lower modes (like T₁ and T₂) typically correspond to global lateral deformations, where the building moves in a uniform or near-uniform pattern (i.e., similar movement across floors).

• Higher modes (like T₃, T₄, and T₅) tend to show more complex deformation patterns, where the floors may move in opposing directions or exhibit significant torsion (twisting).

Typical Mode Shape Patterns:

• T₁: Large displacement at the top, smaller displacement at the bottom (simple translation).

• T₂: Alternating displacements with a higher frequency, likely involving some twisting.

• T₃ and higher: Complex patterns with floor displacement becoming more pronounced at higher floors, potentially showing bending and torsional modes.

Summary of Vibration Periods and Mode Shapes:

Practical Use:

• The fundamental mode (T₁) usually governs the seismic design because it typically involves the largest deformations.

• The higher modes (T₂, T₃, T₄, and T₅) contribute to the dynamic response during an earthquake but usually to a lesser extent than the fundamental mode unless the structure is very flexible or has irregularities.

To calculate Ln, Mn, Γn, effective modal mass (Mn*), and the modal mass participating ratio for each of the five modes, we'll follow the standard structural dynamics approach, using the mass matrix of the system and modal analysis results. Here's a step-by-step breakdown of the computations based on the given equations:

1. Mass Matrix (M)

The mass matrix M contains the mass values for each storey of the building. For a multi-storey system, it is typically diagonal, with each diagonal element representing the mass of each storey (assuming the mass is lumped at each floor level).

$$M=\left[\begin{array}{ccccc}{m}_1& 0& 0& \dots & 0\\ 0& m_2& 0& \dots & 0\\ 0& 0& m_3& \dots & 0\\ \mathrm{⋮}& \mathrm{⋮}& \mathrm{⋮}& \ddots & \mathrm{⋮}\\ 0& 0& 0& \dots & m_n\end{array}\right]$$

Where:

• $m_n$ is the mass at the $n$-th storey (in this case, each storey has mass m).

The effective modal mass (Mn*) is then computed using the following formula:

$$M_n^{\ast }=\sum _{i=1}^n{\varphi }_{n,i}^2\cdot m_i$$

Where:

• ${\varphi }_{n,i}$ is the displacement in the $i$-th storey for the $n$-th mode (from the mode shapes).

• $m_i$ is the mass at the $i$-th storey.

2. Computing $L_n$ for Each Mode:

The parameter $L_n$ is typically the normalized modal displacement for the $n$-th mode. It represents the effective displacement for that mode and can be normalized in different ways. In general, the following formula is used for modal normalization:

$$L_n=\sum _{i=1}^n{\varphi }_{n,i}^2$$

Where:

• ${\varphi }_{n,i}$ is the displacement at the $i$-th storey for the $n$-th mode.

3. Computing $M_n$ (Modal Mass)

The modal mass $M_n$ is essentially the participating mass of the $n$-th mode. It is calculated as:

$$M_n=\sum _{i=1}^n{m}_i\cdot {\varphi }_{n,i}^2$$

Where:

• ${\varphi }_{n,i}$ is the mode shape (displacement) at the $i$-th storey for the $n$-th mode.

• $m_i$ is the mass at the $i$-th storey.

4. Computing ${\mathrm{\Gamma }}_n$

The parameter ${\mathrm{\Gamma }}_n$ typically refers to the modal participation factor for each mode. This is a normalized quantity that indicates the relative participation of the mode in the overall dynamic behavior. It is computed as:

$${\mathrm{\Gamma }}_n=\frac{M_n}{M_{\text{total}}}$$

Where:

• $M_n$ is the modal mass as computed earlier.

• $M_{\text{total}}$ is the total mass of the structure, which is simply the sum of the masses at all storeys:

$$M_{\text{total}}=\sum _{i=1}^n{m}_i$$

5. Effective Modal Mass (Mn)*

The effective modal mass $M_n^{\ast }$ represents the mass that is effectively participating in the $n$-th mode. This can be computed as:

$$M_n^{\ast }=M_n\cdot {\mathrm{\Gamma }}_n$$

Where:

• $M_n$ is the modal mass for the $n$-th mode.

• ${\mathrm{\Gamma }}_n$ is the modal participation factor for the $n$-th mode.

6. Modal Mass Participating Ratio

The modal mass participating ratio is the ratio of the effective modal mass $M_n^{\ast }$ to the total mass $M_{\text{total}}$, and can be expressed as:

$$\text{Modal Mass Participating Ratio}=\frac{M_n^{\ast }}{M_{\text{total}}}$$

This ratio provides insight into how much of the total mass participates in the vibrational mode.

7. Step-by-Step Calculation:

a) Mass Matrix:

Since the structure has 5 translational DOFs (one per storey), the mass matrix would look like this:

$$M=\left[\begin{array}{ccccc}m& 0& 0& 0& 0\\ 0& m& 0& 0& 0\\ 0& 0& m& 0& 0\\ 0& 0& 0& m& 0\\ 0& 0& 0& 0& m\end{array}\right]$$

Where $m$ is the mass of each floor.

b) Mode Shape Vectors:

For each mode, the mode shape vector ${\varphi }_n$ can be obtained from modal analysis (from software such as ETABS or SAP2000), which will look like:

• ${\varphi }_1=[{\varphi }_1(1),{\varphi }_1(2),{\varphi }_1(3),{\varphi }_1(4),{\varphi }_1(5){]}^T$

• ${\varphi }_2=[{\varphi }_2(1),{\varphi }_2(2),{\varphi }_2(3),{\varphi }_2(4),{\varphi }_2(5){]}^T$

• And similarly for ${\varphi }_3,{\varphi }_4,{\varphi }_5$.

c) Calculate $L_n$, $M_n$, ${\mathrm{\Gamma }}_n$, and $M_n^{\ast }$:

For each mode:

• $L_n$: Sum of squares of the mode shape components.

$$L_n=\sum _{i=1}^5{\varphi }_{n,i}^2$$

• $M_n$: Modal mass is the sum of the mass-weighted squared mode shape components.

$$M_n=\sum _{i=1}^{5}m\cdot {\varphi }_{n,i}^2$$

• ${\mathrm{\Gamma }}_n$: Modal participation factor is calculated as:

$${\mathrm{\Gamma }}_n=\frac{M_n}{\sum _{i=1}^{5}m}$$

• $M_n^{\ast }$: Effective modal mass:

$$M_n^{\ast }=M_n\cdot {\mathrm{\Gamma }}_n$$

• Modal Mass Participating Ratio: Finally, the ratio is:

$$\text{Modal Mass Participating Ratio}=\frac{M_n^{\ast }}{\sum _{i=1}^{5}m}$$

Final Output:

Once you perform these calculations for each mode using the displacement values from the mode shapes, you will have:

• Ln for each mode.

• Mn (Modal Mass) for each mode.

• Γn (Modal Participation Factor) for each mode.

• Effective Modal Mass (Mn)* for each mode.

• Modal Mass Participating Ratio for each mode.

To calculate the base shear for each of the 5 modes of vibration and the combined base shear using the SRSS rule, we'll follow the procedure as per the seismic analysis and design criteria outlined in IS 1893 (Part 1). Here's how to approach the calculation:

1. Pseudo-Acceleration Design Spectrum (Sa/T²)

The pseudo-acceleration spectrum is used to calculate the seismic forces in each mode of vibration. This is a function of the natural period of vibration $T_n$ for each mode, and the spectral acceleration $S_a(T_n)$.

The pseudo-acceleration spectrum $S_a(T_n)$ is calculated using the following formula from IS 1893:

$$S_a(T_n)=\frac{2}{T_n^2}\cdot \left(\frac{g}{R}\right)$$

Where:

• $T_n$ is the period of vibration for the $n$-th mode,

• $g$ is the acceleration due to gravity (9.81 m/s²),

• $R$ is the response reduction factor, which is dependent on the building type (e.g., for a moment-resisting frame, $R$ is typically taken as 5).

Alternatively, you can use the design spectrum (from IS 1893) for the specific seismic zone to extract the spectral acceleration values for the corresponding periods.

2. Base Shear for Each Mode (Vn)

For each mode, the base shear $V_n$ can be calculated as:

$$V_n=M_n\cdot S_a(T_n)$$

Where:

• $M_n$ is the modal mass for the $n$-th mode (as calculated previously),

• $S_a(T_n)$ is the spectral acceleration for the $n$-th mode.

3. Combined Base Shear using SRSS Rule

The combined base shear is obtained using the Square Root of the Sum of Squares (SRSS) rule:

$$V_{\text{total}}=\sqrt{V_1^2+V_2^2+\cdots +V_n^2}$$

Where:

• $V_1,V_2,\dots ,V_n$ are the base shears for each mode.

Alternatively, if the modes are uncoupled and non-interfering, you can use this formula for combining base shears.

4. Procedure:

a) Calculate Pseudo-Acceleration Spectrum $S_a(T_n)$ for Each Mode:

• For each of the 5 modes, determine $T_n$ from the mode shape.

• Use the formula to calculate $S_a(T_n)$, either from the pseudo-acceleration formula or directly from the design spectrum (if available).

b) Calculate the Modal Base Shear $V_n$:

• For each mode, multiply the modal mass $M_n$ by the corresponding $S_a(T_n)$.

c) Calculate the Combined Base Shear $V_{\text{total}}$:

• Apply the SRSS rule to combine the individual modal base shears.

Example of Calculation for a Single Mode:

Let’s say for Mode 1 (with a period $T_1=2$ seconds):

• Modal mass $M_1=1000$ kg,

• Spectral acceleration $S_a(T_1)=0.5$ m/s² (just as an example).

Then:

$$V_1=M_1\cdot S_a(T_1)=1000\cdot 0.5=500\text{N}$$

You would repeat this for each mode to calculate the base shear for each mode, and then combine them using SRSS.

Final Output:

After performing the above calculations for all five modes, you will have:

• Base shear for each mode $V_1,V_2,V_3,V_4,V_5$,

• Combined base shear $V_{\text{total}}$.

To determine how many modes should be considered in the analysis such that the modal mass participating ratio reaches 90%, you need to perform the following steps:

1. Understand Modal Mass Participating Ratio

The modal mass participating ratio is the percentage of the total mass of the structure that is involved in each specific mode of vibration. This is calculated for each mode and summed up to determine how much of the total mass is considered in the analysis.

The modal mass participating ratio for the $n$-th mode is given by:

$${\gamma }_n=\frac{M_n}{M_{\text{total}}}$$

Where:

• $M_n$ is the modal mass for the $n$-th mode.

• $M_{\text{total}}$ is the total mass of the structure (sum of the modal masses for all modes).

2. Cumulative Sum of Modal Mass Participating Ratio

To achieve 90% of the total mass, you need to sum the modal mass ratios until you reach a cumulative value of 90%. This means that you'll need to add up the modal mass participating ratios for each mode, starting from the fundamental mode (Mode 1), and continue until the cumulative sum exceeds or equals 90%.

3. Steps to Determine the Number of Modes:

1. Calculate Modal Mass for Each Mode: You first need to calculate the modal mass for each mode. This can be obtained from the mass matrix in your modal analysis (or from the results in ETABS if you have already performed the modal analysis).

2. Compute Modal Mass Participating Ratios: For each mode, calculate the modal mass participating ratio ${\gamma }_n$.

3. Sum the Modal Mass Participating Ratios: Add up the modal mass participating ratios from the first mode (Mode 1) and continue adding the subsequent modes until the cumulative participating ratio is greater than or equal to 90%.

4. Example Calculation:

Let’s assume that the modal mass participating ratios for the first few modes are as follows:

• Mode 1: ${\gamma }_1=30\mathrm{\%}$

• Mode 2: ${\gamma }_2=25\mathrm{\%}$

• Mode 3: ${\gamma }_3=15\mathrm{\%}$

• Mode 4: ${\gamma }_4=10\mathrm{\%}$

• Mode 5: ${\gamma }_5=5\mathrm{\%}$

Now, we calculate the cumulative modal mass participating ratio:

• After Mode 1: $30\mathrm{\%}$

• After Mode 2: $30\mathrm{\%}+25\mathrm{\%}=55\mathrm{\%}$

• After Mode 3: $55\mathrm{\%}+15\mathrm{\%}=70\mathrm{\%}$

• After Mode 4: $70\mathrm{\%}+10\mathrm{\%}=80\mathrm{\%}$

• After Mode 5: $80\mathrm{\%}+5\mathrm{\%}=85\mathrm{\%}$

At this point, the total participating ratio is 85%. To reach 90%, you would need to consider Mode 6, and if the ratio for Mode 6 is large enough, it will bring the cumulative total to 90%.

5. Conclusion:

You need to consider at least the first 5 modes (or more, depending on the modal mass participation) to ensure that the modal mass participating ratio reaches 90%. If you find that the first 5 modes do not contribute enough mass, you may need to include more modes.

6. Summary:

To obtain a modal mass participating ratio of 90%, you'll need to sum the contributions of the modes until the total participating mass is greater than or equal to 90%. This often requires checking the results and ensuring that the appropriate modes are included in the dynamic analysis. Typically, the first 3 to 5 modes cover most of the mass in buildings with regular geometry. However, for highly irregular buildings, more modes may be necessary.