V. Time History - Rectangular Pulse Force

Determine the maximum displacement at the top and maximum bending stress in the columns of a one-story portal frame subject to a rectangular pulse force.

Reference

Chopra, Anil K. 1995. Dynamics of Structures: Theory and Applications to Earthquake Engineering. Upper Saddle River, NJ:Prentice Hall, Example 4.1, p 136.

Problem

From the reference:

A one-story building, idealized as a 12-ft high frame with two columns hinged at the base and a rigid beam, has a natural period of 0.5 sec. Each column is an America standard wide-flange steel section W8x18. Its properties for bending about its major axis are I_x = 61.9 in⁴, S = I_x/c = 15.2 in³; E = 30,000 ksi. Neglecting damping, determine the maximum response of this frame due to a rectangular pulse force of amplitude 4 kips and duration t_d = 0.2 sec. The response quantities of interest are displacement at the top of the frame and maximum bending stress in the columns.

The following time vs. force function is applied for the rectangular pulse force:

Rectangular pulse force

Calculations

The natural frequency of the structure is:

ω_n = 2π/T_n = 2π/0.5 = 12.57 rad/sec.

Also, the lateral stiffness of the frame:

k = 2 \frac{3 E I}{L^{3}} = 2 \frac{3 \times 30,000 \times 61.9}{{12 \times 12}^{3}} = 3.73 kips/in

Solving $ω_{n} = \sqrt{\frac{k}{m}}$ for m, $m = \frac{k}{{ω_{n}}^{2}} = \frac{3.73 \times 10^{3}}{{12.57}^{2}} = 23.62 lbs$ .

Thus, the force applied at the top of the frame as the mass load is W = m×g = 23.62 lbs × 32.2 ft/s² × 12 in/ft = 9,122 lbs-force = 9.122 kips.

Comparison

Result	Reference	STAAD.Pro	Difference
Max. dynamic deformation (in)	2.04	2.03965	negligible
Max. bending moment (kip·in)	547.8	547.98	negligible
Max. bending stress (ksi)	36.04	36.0	negligible

Modeling notes:

A new material with the modulus of elasticity specified is used in place of the built-in steel material.
For the generation of the STAAD.Pro model, the cross-section of the beam is take as 6" × 6" and the cross-section properties are assigned using a user-provided table for prismatic shapes. A massive (10⁶ ft⁴) moment of inertia about local z and y axes is specified to approximate an infinitely rigid beam.
Pinned supports are supplied at the base of each column. A fixed-but allowing only translation along the global X axis is assigned to the tops of the columns / beam ends.
As STAAD.Pro defaults to a damping ratio of 0.05 if no arrival time is specified (i.e., an arrival time of t = 0 seconds), a negligible damping of 1.1×(10)^-6 is assigned.

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\08 Dynamic Analysis\Time History - Rectangular Pulse Force.STD is typically installed with the program.

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 06-Mar-17
END JOB INFORMATION
INPUT WIDTH 79
SET SHEAR
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 0 12 0; 3 5 12 0; 4 5 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
START USER TABLE
TABLE 1
UNIT FEET KIP
PRISMATIC
BEAM
1e+06 1e+06 0.001 0.001 0.001 0.001 0.5 0.5
END
DEFINE MATERIAL START
ISOTROPIC XYZ
E 4.32e+06
POISSON 0.3
DENSITY 0.00283
DAMP 0.03
ALPHA 6.5e-06
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
2 UPTABLE 1 BEAM
MEMBER PROPERTY AMERICAN
1 3 TABLE ST W8X18
CONSTANTS
MATERIAL XYZ ALL
SUPPORTS
1 4 PINNED
2 3 FIXED BUT FX
DEFINE REFERENCE LOADS
LOAD R1 LOADTYPE Mass  TITLE REF LOAD CASE 1
JOINT LOAD
2 FX 9.11751
END DEFINE REFERENCE LOADS
DEFINE TIME HISTORY DT 0.0001
TYPE 1 FORCE SAVE
0 4 0.2 4 0.2 0 0.4 0 0.6 0
ARRIVAL TIME
0
DAMPING 1.1e-06
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
REFERENCE LOAD
R1 1.0 
TIME LOAD
2 FX 1 1 1.000000
PERFORM ANALYSIS
UNIT INCHES KIP
PRINT JOINT DISPLACEMENTS LIST 2 3
PRINT MEMBER FORCES LIST 1 3
PRINT MEMBER STRESSES LIST 1 3
FINISH

STAAD.Pro Output

   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = PLANE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
      2    1    2.03965   0.00000   0.00000   0.00000   0.00000   0.00000
      3    1    2.03965   0.00000   0.00000   0.00000   0.00000   0.00000
   ************** END OF LATEST ANALYSIS RESULT **************
    55. PRINT MEMBER FORCES LIST 1 3
  MEMBER   FORCES   LIST     1        
      STAAD PLANE                                              -- PAGE NO.    5
   MEMBER END FORCES    STRUCTURE TYPE = PLANE
   -----------------
   ALL UNITS ARE -- KIP  INCH     (LOCAL )
  MEMBER  LOAD  JT     AXIAL   SHEAR-Y  SHEAR-Z   TORSION     MOM-Y      MOM-Z
      1    1     1      0.00      3.81     0.00      0.00      0.00       0.00
                 2      0.00     -3.81     0.00      0.00      0.00     547.98
      3    1     3      0.00      3.81     0.00      0.00      0.00     547.98
                 4      0.00     -3.81     0.00      0.00      0.00       0.00
   ************** END OF LATEST ANALYSIS RESULT **************
    56. PRINT MEMBER STRESSES LIST 1 3
  MEMBER   STRESSES LIST     1        
      STAAD PLANE                                              -- PAGE NO.    6
  MEMBER STRESSES
  ---------------
  ALL UNITS ARE KIP /SQ INCH
 MEMB   LD  SECT    AXIAL    BEND-Y     BEND-Z   COMBINED  SHEAR-Y  SHEAR-Z
      1    1   .0      0.0        0.0       0.0       0.0      2.2      0.0
              1.0      0.0 C      0.0      36.0      36.0      2.2      0.0
      3    1   .0      0.0        0.0      36.0      36.0      2.2      0.0
              1.0      0.0 C      0.0       0.0       0.0      2.2      0.0