107x

2026-01-19

VE0074 | Frame Structure Subjected to Earthquake Loading

Description

A planar two-story single-bay frame structure is subjected to earthquake loading. Beams representing the floors are considered as rigid. Determine the natural frequencies of the structure while neglecting the self-weight and assuming the lumped masses are at the floor levels. For each obtained frequency, specify the displacements of the floors as well as the equivalent forces generated using the response spectrum analysis.

Material	Elastic	Modulus of Elasticity	E	48000.000	MPa
Material	Elastic	Poisson's Ratio	ν	0.500	-
Geometry	Columns	Width	w	0.500	m
		Height	h	0.500	m
		Length	L	5.000	m
Mass	Floor	First	m₁	5e5	kg
Mass	Floor	Second	m₂	5e5	kg

Frame Structure Subjected to Earthquake Loading

Response Spectrum

Analytical Solution

Modal Analysis

Due to the rigid beams and lumped masses, the structure can be modeled as a shear building with two degrees of freedom. Its undamped free vibration is described by the differential equation

M \ddot{u} + K u = 0

Differential Equation of Free Vibration

where u is the displacement vector, M is the mass matrix,

M = [\begin{matrix} m_{1} & 0 \\ 0 & m_{2} \end{matrix}]

Mass Matrix

and K is the stiffness matrix

K = [\begin{matrix} 4 k & - 2 k \\ - 2 k & 2 k \end{matrix}]

Stiffness Matrix

The flexural stiffness of a column in the X-direction can be defined as follows:

k = \frac{E w h^{3}}{L^{3}}

Flexural Stiffness of Column in X-Direction

The eigenvalues ω (and natural frequencies f) and eigenvectors Φ can be determined using the eigenvalue analysis.

|K - ω^{2} M| = 0

Eigenvalue Analysis

f_{1} = 0.964 Hz; f_{2} = 2.523 Hz

Natural Frequencies

ϕ_{1} = [\begin{matrix} 0.618 \\ 1.000 \end{matrix}]; ϕ_{2} = [\begin{matrix} 1.000 \\ - 0.618 \end{matrix}]

Eigenvectors

Response Spectrum Analysis

After calculation of the eigenvalues and eigenvectors, the displacement and equivalent forces can be calculated according to the Response Spectrum Analysis (RSA). The values of the spectral acceleration S_a are determined from the given accelerogram and corresponding period.

T_{1} = \frac{1}{f_{1}} = 1.038 s \to S_{a} = 0.578 m \cdot s^{- 2}

T_{2} = \frac{1}{f_{2}} = 0.396 s \to S_{a} = 1.500 m \cdot s^{- 2}

Response Spectrum

u_{k (j)} = \frac{ϕ_{k (j)} \sum_{i} m_{i} ϕ_{i (j)}}{ω_{j}^{2} \sum_{i} m_{i} ϕ_{i (j)}^{2}} S_{a}

k	Point of the structure
j	Mode shape
i	Mass
S_a	Spectral acceleration

RSA - Displacements

F_{k (j)} = \frac{ϕ_{k (j)} \sum_{i} m_{i} ϕ_{i (j)}}{\sum_{i} m_{i} ϕ_{i (j)}^{2}} m_{k} S_{a}

k	Point of the structure
j	Mode shape
i	Mass
S_a	Spectral acceleration

RSA - Equivalent Forces

Calculated displacements and equivalent forces follow. The columns in the matrices correspond to the mode shape; the rows correspond to the mass point.

u = [\begin{matrix} 11.410 & 1.650 \\ 18.462 & - 1.020 \end{matrix}] mm

Result Displacements

F = [\begin{matrix} 209.195 & 207.295 \\ 338.484 & - 128.115 \end{matrix}] kN

Result Equivalent Forces

RFEM and RSTAB Settings

Modeled in RFEM 6.11, RSTAB 9.11, and RFEM 5.39 (RF-DYNAM Pro), RSTAB 8.39 (DYNAM Pro)
Geometrically linear analysis is considered
Mass is considered in the X-direction
Diagonal mass matrix is generated
Root of characteristic polynomial is used as solving method

Results

Quantity	Analytical Solution	RFEM 6	Ratio	RFEM 5 RF-DYNAM Pro	Ratio
u_1,1 [mm]	11.410	11.411	1.000	-	-
u_2,1 [mm]	18.462	18.464	1.000	-	-
u_1,2 [mm]	1.650	1.650	1.000	-	-
u_2,2 [mm]	-1.020	-1.020	1.000	-	-
F_1,1 [kN]	209.195	209.160	1.000	209.177	1.000
F_2,1 [kN]	338.484	338.440	1.000	338.456	1.000
F_1,2 [kN]	207.295	207.300	1.000	207.295	1.000
F_2,2 [kN]	-128.115	-128.120	1.000	-128.115	1.000

Remark: Equivalent forces in RFEM 6 are calculated from rigid beam internal forces.

Quantity	Analytical Solution	RSTAB 9	Ratio	RSTAB 8 DYNAM Pro	Ratio
u_1,1 [mm]	11.410	11.410	1.000	-	-
u_2,1 [mm]	18.462	18.462	1.000	-	-
u_1,2 [mm]	1.650	1.650	1.000	-	-
u_2,2 [mm]	-1.020	-1.020	1.000	-	-
F_1,1 [kN]	209.195	209.180	1.000	209.214	1.000
F_2,1 [kN]	338.484	338.460	1.000	338513	1.000
F_1,2 [kN]	207.295	207.300	1.000	207.292	1.000
F_2,2 [kN]	-128.115	-128.120	1.000	-128.114	1.000

Remark: Equivalent forces in RSTAB 9 are calculated from rigid beam internal forces.

Model 000159 | Verification Example 000074 | 1

804x

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Verification Example 000074 | 1

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