V. Beam Subject to response spectrum

Find the maximum moment due to the time history loading and compare theoretical answers to the STAAD solution.

Reference

Biggs, John M., Introduction to Structural Dynamics, McGraw Hill, 1964, pp. 256-263
Blevins, Robert D., Formulas for Natural Frequency and Mode Shape, Van Nostrand-Reinhold, 1979.

Problem

The supports of a simply supported beam are subjected to an acceleration time history. The maximum bending moment in the beam is computed for the first mode of the structure. This problem demonstrates the capabilities of STAAD to calculate the correct modal response of a structure utilizing response spectrum data.

Simple span beam

L = 240 in.

The STAAD model consists of 11 nodes and 10 elastic beam elements. Node 1 is completely restrained with the exception of having rotational freedom in the Z direction, the remaining nodes are restrained except for X and Y displacements and Z rotations. Node 11 is additionally restrained against displacements in the Y direction to provide for the simple support condition . Only the contribution of the first mode of the structure is considered.

Finite element model

Theoretical Solution

Material Properties

E = 30 x 10⁶ lb/in²

EI = 1.0 x 10¹⁰ lb-in²

m = 0.2 lb-sec²/ in²

h = 14.0 in.

From Reference 2, Table 8-1, page 108, the fundamental frequency of the beam is:

f_{i} = \frac{λ_{i}^{2}}{2 π l^{2}} \sqrt{\frac{E I}{m}} = \frac{9.869}{2 π {(240)}^{2}} \sqrt{\frac{1.0 {(10)}^{10}}{0.2}} = 6.098 h z

The modal participation factor for the fundamental mode is:

Γ = \frac{\int_{0}^{1} m ϕ (x) ⅆ x}{\int_{0}^{1} m ϕ^{2} (x) ⅆ x}

Where the first mode shape, φ(x) = sin(πx/1)

Γ = \frac{{m \int}_{0}^{1} \sin \frac{π x}{1} ⅆ x}{{m \int}_{0}^{1} \sin^{2} \frac{π x}{1} ⅆ x} = 4 / π

The maximum relative modal displacement is given by:

A_max = Γu⁰ _max

where

u⁰ _max	=	y''_so/ω²(DLF)_max
ω	=	2π*6.098Hz
y''_so	=	1.0g
(DLF)_max	=	1.648 at 6.098 Hz

therefore:

A_max = 4(1.648)(386.4)/[π(2π)²(6.098)²] = 0.5523 in.

The bending moment

M = -EIδ²u/(δx²)

Where u for the first mode = A sin(πx/l)

δ²u/(δx²) = -π²/l²A·sin(πx/l)

M = AEI·(π²/l²)·sin(πx/l)

M_max = A_maxEI·(π²/l²)

at x = l/2

M_max = 1(10)¹⁰·(0.5523)·π²/(240²) = 946.351(10)³ lb·in

at x=l/2

Comparison

Table 1. Comparison of results
Solution	Theory	STAAD.Pro	Difference
Bending Moment (kip-inch)	946.351	947.088	negligible

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Beam Subject to response spectrum.STD is typically installed with the program.

STAAD SPACE 
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* RESPONSE OF A SIMPLY SUPPORTED BEAM TO A SHOCK SPECTRUM
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 24 0 0; 3 48 0 0; 4 72 0 0; 5 96 0 0; 6 120 0 0; 7 144 0 0;
8 168 0 0; 9 192 0 0; 10 216 0 0; 11 240 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7; 7 7 8; 8 8 9; 9 9 10;
10 10 11;
MEMBER PROPERTY AMERICAN
1 TO 10 PRIS AX 20.4082 IX 40 IY 3.6139 IZ 333.333 YD 14 ZD 1.45777
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
DENSITY 3.78672
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
CUT OFF MODE SHAPE 1
SUPPORTS
1 FIXED BUT MZ
2 TO 10 FIXED BUT FX FY MZ
11 FIXED BUT FX MZ
LOAD 1
SELFWEIGHT X 1 
SELFWEIGHT Y 1 
SPECTRUM SRSS Y 1 ACC SCALE 386.4 DAMP 0.001
0.15 1.648; 0.17 1.648;
PERFORM ANALYSIS
PRINT MEMBER FORCES LIST 5
FINISH

STAAD Output

               CALCULATED FREQUENCIES FOR LOAD CASE       1
       MODE            FREQUENCY(CYCLES/SEC)         PERIOD(SEC)
         1                       6.069                  0.16476
  RESPONSE SPECTRUM LOAD     1 
     RESPONSE LOAD CASE      1
            MODAL WEIGHT (MODAL MASS TIMES g) IN POUN         GENERALIZED
      MODE           X             Y             Z              WEIGHT
         1       0.000000E+00  1.478714E+04  0.000000E+00    9.273617E+03
     SRSS          MODAL COMBINATION METHOD USED.
     DYNAMIC WEIGHT X Y Z   1.761987E+04  1.669251E+04  0.000000E+00 POUN
     MISSING WEIGHT X Y Z  -1.761987E+04 -1.905373E+03  0.000000E+00 POUN
       MODAL WEIGHT X Y Z   0.000000E+00  1.478714E+04  0.000000E+00 POUN
           MODE                 ACCELERATION-G     DAMPING
           ----                 --------------     -------
              1                      1.64933       0.00100
 MODAL BASE ACTIONS 
  MODAL BASE ACTIONS        FORCES IN POUN LENGTH IN INCH
  -----------------------------------------------------------
                                                             MOMENTS ARE ABOUT THE ORIGIN
   MODE     PERIOD        FX          FY          FZ          MX          MY          MZ
      1      0.165        0.00    24388.86        0.00        0.00        0.00  2926662.97
      STAAD SPACE                                              -- PAGE NO.    4
 PARTICIPATION FACTORS 
           MASS  PARTICIPATION FACTORS IN PERCENT         BASE SHEAR IN POUN
           --------------------------------------         ------------------
 MODE    X     Y     Z     SUMM-X   SUMM-Y   SUMM-Z       X        Y        Z
   1    0.00 88.59  0.00    0.000   88.585    0.000      0.00 24388.86     0.00
                                                    ---------------------------
                                  TOTAL SRSS  SHEAR      0.00 24388.86     0.00
                                  TOTAL 10PCT SHEAR      0.00 24388.86     0.00
                                  TOTAL ABS   SHEAR      0.00 24388.86     0.00
    34. PRINT MEMBER FORCES LIST 5
  MEMBER   FORCES   LIST     5        
      STAAD SPACE                                              -- PAGE NO.    5
   MEMBER END FORCES    STRUCTURE TYPE = SPACE
   -----------------
   ALL UNITS ARE -- POUN INCH     (LOCAL )
  MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
      5    1     5      0.00   1931.41     0.00      0.00      0.00  900734.25
                 6      0.00   1931.41     0.00      0.00      0.00  947088.00