Simply Supported Beam with a Distributed Mass Subjected to a Constant Shear Force Moving along the Span of the Beam at a Constant Speed
Objective: Determination of the strain state of a simply supported beam with a distributed mass subjected to a constant shear force moving along the span of the beam at a constant speed.
Initial data files:
| File name | Description |
|---|---|
| DIN_B_ML1.spr Diagram_DIN_B_ML1.txt |
The action of the constant shear force moving along the beam span is specified in the form of forces applied in all nodes of the design model according to the following variant: The delay time for each nodal force is different. The graph describing the load variation with time is the same for all nodal forces.
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The action of the constant shear force moving along the beam span is specified in the form of forces applied in all nodes of the design model according to the following variant: The delay time is the same for all nodal forces. Each nodal force has its corresponding graph describing the load variation with time. |
Problem formulation: The constant shear force P moves at a constant speed v along the span of the simply supported beam with a uniformly distributed mass μ. Determine the natural oscillation modes and natural frequencies p of the simply supported beam, as well as the deflection η in the cross-section in the middle of the beam span with time.
References: Timoshenko S.P., Course of the Theory of Elasticity, Kiev, Naukova Dumka, 1972, p. 345.
Initial data:
| E = 3.0·106 tf/m2 | - elastic modulus; |
| ν = 0.2 | - Poisson’s ratio; |
| b = 0.4 m | - width of the rectangular cross-section of the beam; |
| h = 0.8 m | - height of the rectangular cross-section of the beam; |
| l = 8.0 m | - beam span length; |
| γ = 2.5 tf/m3 | - specific weight of the beam material; |
| P = 76.8 tf | - value of the constant force moving along the beam span; |
| g = 10.00 m/s2 | - gravitational acceleration; |
| μ = 2.5·0.4·0.8/10.0 = 0.08 tf·s2/m2 | - value of the uniformly distributed mass of the beam; |
| I = 0.4·(0.8)3/12 = 0.017067 m4 | - cross-sectional moment of inertia of the beam. |
The speed of the constant force v is taken depending on the values of the beam span and the fundamental natural period of the beam T1:
v = l / T1.
Finite element model: Design model – grade beam / plate, 32 bar elements of type 3. Boundary conditions of the simply supported ends of the beam are provided by imposing constraints in the direction of the degree of freedom Z. The dimensional stability of the design model is provided by imposing a constraint in the node of the cross-section along the symmetry axis of the beam in the direction of the degree of freedom UX. The distributed mass is specified by transforming the static load from the self-weight of the beam μ·g.
The calculation is performed in two stages: first the natural oscillation modes and natural frequencies p are determined by the modal analysis, and then the deflections η in the cross-section in the middle of the beam span are determined by the direct integration of the equations of motion method.
The action of the constant shear force moving along the beam span is specified in the form of forces applied in all nodes of the design model along the Z axis of the global coordinate system with the scale factor of 1.0 according to the following variants:
- The delay time for each nodal force is different and is determined as t0 = 2·(m-1)·Δtint, where m is the number of finite elements counted from the support node of the beam to the node considered along the load path. The graph describing the load variation with time is the same for all nodal forces. When plotting the graph the nodal force is taken with consecutive values: 0; 0.5·P; P; 0.5·P; 0 at time points: 0; Δtint; 2·Δtint; 3·Δtint; 4·Δtint; 5·Δtint, measured from the delay time t0, at subsequent time points the nodal force is equal to 0.
- The delay time is the same for all nodal forces and is equal to t0 = 0. Each nodal force has its corresponding graph describing the load variation with time. When plotting the graph the nodal force at the time points from 0 to 2·(m-1)·Δtint is equal to 0, at the time points from 2·(m-1)·Δtint to 2·(m+1)·Δtint inclusive is taken with consecutive values: 0; 0.5·P; P; 0.5·P; 0, at subsequent time points the nodal force is equal to 0, where m is the number of finite elements counted from the support node of the beam to the node considered along the load path.
In both cases the intervals between the time points of the load variation graphs are equal to the time it takes to cover half the distance between the adjacent nodes of the design model at the speed v: Δtint = L / (2·n·v) = T1 / (2·n) and correspond to the integration step, where n is the number of finite elements in the design model. The duration of the process is equal to the time it takes the load moving at the speed v to cover the beam span l: t = l/v =T1. Critical damping ratios for the 1-st and 2-nd natural frequencies are taken with the minimum value ξ = 0.0001. The conversion factor for the added static loading is equal to k = 0.981 (mass generation). Number of nodes in the design model – 33. The modal integration method is used in the calculation. The determination of the natural oscillation modes and natural frequencies is performed by the method of subspace iteration. The matrix of concentrated masses is used in the calculation.
Results in SCAD
Design model
1-st -- 16-th natural oscillation modes
Graph of the variation of the deflection η in the cross-section in the middle of the beam span with time (m)
Amplitude value of the deflection η in the cross-section in the middle of the beam span and the deformed models at the respective time point (m)
Comparison of solutions:
Natural frequencies p, rad/s
|
Oscillation mode |
Theory |
SCAD |
Deviations, % |
|---|---|---|---|
|
1 |
123.370 |
123.370 |
0.00 |
|
2 |
493.480 |
493.480 |
0.00 |
|
3 |
1110.330 |
1110.325 |
0.00 |
|
4 |
1973.921 |
1973.887 |
0.00 |
|
5 |
3084.251 |
3084.120 |
0.00 |
|
6 |
4441.322 |
4440.919 |
0.01 |
|
7 |
6045.133 |
6044.087 |
0.02 |
|
8 |
7895.684 |
7893.275 |
0.03 |
|
9 |
9992.974 |
9987.907 |
0.05 |
|
10 |
12337.005 |
12327.069 |
0.08 |
|
11 |
14927.777 |
14909.367 |
0.12 |
|
12 |
17765.288 |
17732.721 |
0.18 |
|
13 |
20849.539 |
20794.097 |
0.27 |
|
14 |
24180.531 |
24089.155 |
0.38 |
|
15 |
27758.262 |
27611.778 |
0.53 |
|
16 |
31582.734 |
31353.470 |
0.73 |
The dashed line shows the value of the static deflection
Graph of the variation of the deflection η in the cross-section in the middle of the beam span with time according to the theoretical solution (m)
Amplitude value of the deflection η in the cross-section in the middle of the beam span, m
|
Theory |
SCAD |
|||
|---|---|---|---|---|
|
Time, s |
Deflection, m |
Time, s |
Deflection, m |
Deviation, % |
|
0.0339 |
0.002842 |
0.0334 |
0.002837 |
0.18 |
Notes: In the analytical solution the natural frequencies of oscillations p of the simply supported beam are determined according to the following formula:
\[ \frac{n^{2}\cdot \pi^{2}}{l^{2}}\cdot \sqrt {\frac{E\cdot I}{\mu }} , \]
where n = 1, 2, 3, 4, … – natural mode number.
In the analytical solution the deflections η in the cross-section in the middle of the beam span with time are determined according to the following formula:
\[ \eta \left( t \right)=\frac{2\cdot P\cdot l^{3}}{\pi^{4}\cdot E\cdot I}\cdot \sum\limits_{n=1} {\left[ {\frac{\sin \left( {\frac{n\cdot \pi }{2}} \right)}{n^{4}\cdot \left( {1-\frac{\mu \cdot l^{2}\cdot v^{2}}{n^{2}\cdot \pi^{2}\cdot E\cdot I}} \right)}\cdot \left( {\sin \left( {\frac{n\cdot \pi \cdot v}{l}\cdot t} \right)-\frac{l\cdot v}{n\cdot \pi }\cdot \sqrt {\frac{\mu }{E\cdot I}} \cdot \sin \left( {\frac{n^{2}\cdot \pi ^{2}}{l^{2}}\cdot \sqrt {\frac{E\cdot I}{\mu }} \cdot t} \right)} \right)} \right]} ; \]