SCAD: Response Spectra

Response Spectrum of Absolute Response Accelerations of a Linear Oscillator Installed in the Middle of the Span of a Simply Supported Beam with a Distributed Mass Subjected to a Kinematic Excitation of Supports (Seismic Action)

Objective: Determination of the response spectrum of response accelerations of a linear oscillator installed in the middle of the span of a simply supported beam with a distributed mass subjected to a kinematic excitation of supports.

Initial data files:

DIN_B_RS.SPR – design model
DIN_B_RS.SPC – accelerogram

Problem formulation: The simply supported beam of constant cross-section with the uniformly distributed mass μ is subjected to the kinematic excitation of the supports according to the specified accelerogram:

\[ \ddot{{z}}(t)=\ddot{{z}}_{s0} \cdot \left( {1-\frac{t}{t_{d} }} \right). \]

Determine the response spectrum of the absolute response accelerations of the linear oscillator installed in the middle of the span.

References: John M. Biggs, Introduction to Structural Dynamics, McGraw-Hill Book Companies, New York, 1964, p.256-263;

Kiselev V.A., Structural Mechanics. Special Course. Dynamics and Stability of Structures. Moscow, Stroyizdat, 1980, p. 65-67.

Initial data:

E = 3.0·107 psi = 2.1092·107 tf/m2 - elastic modulus;
I = 333.333 in4 = 138.7448·10-6 m4 - cross-sectional moment of inertia of the beam.
h = 14 in = 0.3556 m - height of the cross-section of the beam;
L = 240 in = 6.0960 m - beam span length;
μ = 0.2 lb·sec2/in2 = 0.1406 tf·s2/m2 - value of the uniformly distributed mass of the beam;
\( \ddot{{z}}_{s0} \) = ±386.2200 in/sec2 = ±9.81 m/s2 - amplitude values of the acceleration of the supports according to the accelerogram;
td = 0.10 sec = 0.10 s - half-interval of the kinematic excitation of supports;
g = 386.2200 in/sec2 =9.81 m/s2 - gravitational acceleration;
 

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 kinematic excitation of supports is described by the graph of the acceleration variation with time (accelerogram) and is given in the form of the action along the Z axis of the global coordinate system (direction cosines to the X, Y, Z axes: 0.00, 0.00, 1.00) with the scale factor to the values of the accelerogram equal to 1.00. The height of the beam structure in the model is directed along the Z axis of the global coordinate system. The dissipation factor (energy absorption factor) is taken with the minimum value ξ = 0.000001 for the oscillator and for the structure. The intervals between the time points of the graph of the acceleration variation with time are equal to Δt = 0.01 s. When plotting the graph the acceleration is taken with the values \( \ddot{{z}}(t)=\ddot{{z}}_{s0} \cdot \left( {1-n \cdot \Delta t} / {t_{d} } \right). \) at the time points n·Δt. The conversion factor for the added static loading is equal to k = 1.000 (mass generation). Number of nodes in the design model – 33.

 

Results in SCAD


Design model and the given accelerogram

 

Comparison of solutions:

The comparison was performed with the solution of the problem obtained in Abaqus (the solution was provided by A.I. Popov — Atomproekt).

 

Frequency

Acceleration

Hz

g

Abaqus

SCAD

0

0,0000

0,0000

0,05

0,0000

0,0007

0,1

0,0000

0,0029

0,15

0,0000

0,0064

0,2

0,0000

0,0114

0,25

0,0038

0,0178

0,3

0,0027

0,0256

0,35

0,0216

0,0347

0,4

0,0200

0,0452

0,45

0,0490

0,0569

0,5

0,0503

0,0700

0,55

0,0832

0,0842

0,6

0,0881

0,0997

0,65

0,1218

0,1163

0,7

0,1312

0,1340

0,75

0,1642

0,1528

0,8

0,1799

0,1726

0,85

0,2096

0,1934

0,9

0,2310

0,2152

0,95

0,2565

0,2378

1

0,2824

0,2613

1,05

0,3045

0,2855

1,1

0,3338

0,3105

1,15

0,3625

0,3362

1,2

0,3876

0,3626

1,25

0,4182

0,3895

1,3

0,4481

0,4171

1,35

0,4758

0,4453

1,4

0,5043

0,4739

1,45

0,5395

0,5030

1,5

0,5690

0,5325

1,55

0,5964

0,5625

1,6

0,6324

0,5928

1,65

0,6656

0,6235

1,7

0,6953

0,6545

1,75

0,7270

0,6857

1,8

0,7628

0,7171

1,85

0,7932

0,7487

1,9

0,8267

0,7804

1,95

0,8572

0,8121

2

0,8939

0,8441

2,05

0,9265

0,8760

2,1

0,9559

0,9079

2,15

0,9913

0,9398

2,2

1,0234

0,9717

2,25

1,0561

1,0035

2,3

1,0887

1,0353

2,35

1,1193

1,0669

2,4

1,1498

1,0984

2,45

1,1855

1,1298

2,5

1,2171

1,1611

2,55

1,2467

1,1923

2,6

1,2762

1,2234

2,65

1,3048

1,2544

2,7

1,3405

1,2853

2,75

1,3721

1,3160

2,8

1,4027

1,3465

2,85

1,4312

1,3769

2,9

1,4597

1,4071

2,95

1,4862

1,4370

3

1,5158

1,4667

3,05

1,5454

1,4963

3,1

1,5749

1,5255

3,15

1,6045

1,5546

3,2

1,6320

1,5832

3,25

1,6595

1,6115

3,3

1,6860

1,6395

3,35

1,7115

1,6671

3,4

1,7370

1,6943

3,45

1,7604

1,7211

3,5

1,7829

1,7476

3,55

1,8084

1,7736

3,6

1,8318

1,7994

3,65

1,8583

1,8247

3,7

1,8838

1,8499

3,75

1,9093

1,8744

3,8

1,9337

1,8989

3,85

1,9541

1,9226

3,9

1,9776

1,9629

3,95

2,0000

2,0807

4

2,0224

2,1999

4,05

2,0438

2,3202

4,1

2,1244

2,4415

4,15

2,1713

2,5635

4,2

2,2895

2,6862

4,25

2,4088

2,8092

4,3

2,5291

2,9324

4,35

2,6493

3,0555

4,4

2,7696

3,1784

4,45

2,8899

3,3009

4,5

2,7768

3,4226

4,55

2,8960

3,5434

4,6

3,0143

3,6631

4,65

3,1325

3,7815

4,7

3,2497

3,8982

4,75

3,3660

4,0132

4,8

3,4811

4,1262

4,85

3,5953

4,2370

4,9

3,8267

4,3453

4,95

3,9368

4,4509

5

4,0449

4,5537

5,05

4,1519

4,6535

5,15

4,3568

4,8429

5,25

4,5515

5,0178

5,35

4,7339

5,1765

5,45

5,1580

5,3179

5,55

5,2915

5,4406

5,65

5,4057

5,5436

5,75

5,5025

5,6259

5,85

5,5800

5,6867

5,95

5,6371

5,7255

6,05

5,6799

5,7418

6,15

5,6922

5,7467

6,25

5,6840

5,7459

6,35

5,6667

5,7410

6,45

5,6616

5,7305

6,55

5,6381

5,7172

6,65

5,6106

5,7002

6,75

5,5933

5,6823

6,85

5,5596

5,6593

6,95

5,5260

5,6326

7,05

5,4760

5,6019

7,15

5,4475

5,5663

7,25

5,4027

5,5263

7,35

5,3435

5,4817

7,45

5,3058

5,4330

7,55

5,2548

5,3803

7,65

5,1906

5,3244

7,75

5,1366

5,2659

7,85

5,0856

5,2063

7,95

5,0214

5,1456

8,05

4,9541

5,0832

8,15

4,8970

5,0199

8,25

4,8298

4,9568

8,35

4,7533

4,8934

8,45

4,6942

4,8276

8,55

4,6259

4,7590

8,65

4,5484

4,6880

8,75

4,4791

4,6150

8,85

4,4108

4,5400

8,95

4,3354

4,4637

9,05

4,2538

4,3856

9,15

4,1876

4,3056

9,25

4,1121

4,2230

9,35

4,0306

4,1386

9,45

3,9602

4,0539

9,55

3,8858

3,9697

9,65

3,8063

3,8864

9,75

3,7278

3,8045

9,85

3,6565

3,7235

9,95

3,5800

3,6425

10,05

3,5005

3,5625

10,25

3,3517

3,4065

10,45

3,1978

3,2517

10,65

3,0510

3,0950

10,85

2,9021

2,9342

11,05

2,7554

2,8493

11,25

2,6320

2,8338

11,45

2,6188

2,8165

11,65

2,6045

2,7987

11,85

2,5851

2,7780

12,05

2,5668

2,7555

12,25

2,5525

2,7311

12,45

2,5352

2,7038

12,65

2,5138

2,6768

12,85

2,4954

2,6496

13,05

2,4791

2,6235

13,25

2,4608

2,6008

13,45

2,4393

2,5794

13,65

2,4190

2,5580

13,85

2,4200

2,5829

14,05

2,4669

2,6253

14,25

2,5025

2,6754

14,45

2,5321

2,7204

14,65

2,5545

2,7508

14,85

2,5668

2,7680

15,05

2,5770

2,7730

15,25

2,5780

2,7656

15,45

2,5708

2,7468

15,65

2,5627

2,7165

15,85

2,5433

2,6732

16,05

2,5270

2,6209

16,25

2,4995

2,5750

16,45

2,4730

2,5384

16,65

2,4383

2,5071

16,85

2,4037

2,4806

17,05

2,3629

2,4496

17,25

2,3221

2,4087

17,45

2,2742

2,3604

17,65

2,2294

2,3022

17,85

2,1774

2,2363

18,05

2,1284

2,1646

18,25

2,0724

2,0879

18,45

2,0204

2,0594

18,65

1,9602

2,1352

18,85

1,9215

2,1977

19,05

1,9541

2,2423

19,25

2,0071

2,2704

19,45

2,0530

2,2876

19,65

2,0968

2,2987

19,85

2,1356

2,3222

20,05

2,1672

2,3682

20,55

2,2365

2,5018

21,05

2,2783

2,5673

21,55

2,3028

2,5700

22,05

2,3007

2,5106

22,55

2,2854

2,3830

23,05

2,2528

2,3421

23,55

2,2039

2,3615

24,05

2,1427

2,3313

24,55

2,0734

2,2381

25,05

1,9949

2,2560

25,55

1,9888

2,2237

26,05

2,0601

2,3334

26,55

2,1142

2,4863

27,05

2,1580

2,5792

27,55

2,1865

2,6055

28,05

2,1988

2,5678

28,55

2,1978

2,4566

29,05

2,1876

2,2866

29,55

2,1702

2,3166

30,05

2,1386

2,4101

30,55

2,0989

2,4521

31,05

2,0520

2,4243

31,55

1,9980

2,3600

32,05

2,0071

2,3904

32,55

2,0520

2,5854

33,05

2,0907

2,7152

 


Response Spectra

 

 

Abaqus

SCAD

Deviation

Frequency at which the maximum acceleration occurs (Hz)

6.15

6.15

0 %

Maximum acceleration (g)

5,6921

5.7467

0.95 %

Spectra correlation coefficient

0.995