ASSIGNMENT 2 – TIME-HISTORY ANALYSIS, RESPONSE SPECTRUM
CALCULATION
NOTES ON SOLUTION
Stiffness calculation
Various assumptions were made about the trussed frame structure:
Assumption
Accurate
6m portal
5m portal
7m portal
6m, rigid beam
Resulting Stiffness
19,200kN/m
18,640kN/m
31,405kN/m
11,960kN/m
22,222kN/m
Period
0.7854s
0.7971s
0.6141s
0.9951s
0.73s
Analysis Method
(stiffness method, 2m deep beam-col. connection)
(moment distribution or stiffness method)
(moment distribution or stiffness method)
(moment distribution or stiffness method)
(moment-area or similar)
The 6m simple portal, with the mass also assumed to act at a height of 6m, had an agreeable cancellation of
errors leading to a good result.
The resulting maximum displacement, u max , was then obtained using the step-by-step integration
algorithm, as implemented in spreadsheet or Matlab. Maximum base shear could then be deduced as stiffness
x maximum displacement, k u max ; or alternatively, as mass x maximum absolute acceleration
m u g  u
max
El Centro
Newhall
Sylmar Hosp
Mexico City
. This gave the following results:
T = 0.73s
max displ.
max shear
8.75cm
1944kN
29.0
6456
17.5
3881
0.8
176
T = 0.7971s
max displ.
max shear
9.5
1771
26.1
4871
18.7
3491
0.9
172
T = 0.7854s
max displ.
max shear
9.37
1798
26.8
5138
18.6
3570
0.8
158
Response Spectrum
The response spectrum was also calculated using the time-history solution – by finding the maximum
displacement for a range of single degree of freedom systems with gradually increasing periods. The Excel
Data Table facility provided a good way of doing this calculation.
The maximum displacement relative to the ground was the spectral displacement, Sd.
2
The spectral pseudo acceleration Sa, could be obtained from Sd by multiplying by 
2
 2 
or   .
 T 
A common error was to multiply Sd by a particular value of  (such as that corresponding to the portal
frame) rather than the value corresponding to each period used in the plot.
A sample spreadsheet can be seen on the SM3 web site.
The true spectral acceleration, u g  u
max
, has also been calculated and plotted. You will see that it is in
close agreement with the more usual pseudo spectral acceleration values.