1

1. 120 DYNAMICS OF STRUCTURES
7-5 INTEGRATION METHODS
The other general numerical approach to step-by-step dynamic response analysis
makes use of integration to step forward from the initial to the final conditions for
each time step. The essential concept is represented by the following equations:
˙v1 = ˙v0 +
h
0
¨v(τ) dτ (7-13a)
v1 = v0 +
h
0
˙v(τ) dτ (7-13b)
which express the final velocity and displacement in terms of the initial values of
these quantities plus an integral expression. The change of velocity depends on
the integral of the acceleration history, and the change of displacement depends on
the corresponding velocity integral. In order to carry out this type of analysis, it
is necessary first to assume how the acceleration varies during the time step; this
acceleration assumption controls the variation of the velocity as well and thus makes
it possible to step forward to the next time step.
Euler-Gauss Procedure
The simplest integration method, known as the Euler-Gauss method, is based
on assuming that the acceleration has a fixed constant value during the time step.
The consequence of this assumption is that the velocity must vary linearly and the
displacement as a quadratic curve during the time step. Figure 7-3 illustrates this type
of behavior for a formulation where it is assumed that the constant acceleration is the
average of the initial and the final values attained during the step. Also shown on this
figure are expressions for acceleration, velocity, and displacement at any time τ during
the step obtained by successive integration, and for the final velocity and displacement
obtained by putting τ = h into these expressions.
To initiate this analysis for any step, it is necessary first to evaluate the initial
acceleration ¨v0, and this may be obtained by solving the dynamic equilibrium expres-
sion at time t = t0, as shown by Eq. (7-7). In addition, the final acceleration ¨v1 is
needed to apply this implicit formulation, and this value may be obtained by iteration.
Starting with an arbitrary assumption for ¨v1, values of ˙v1 and v1 are obtained from
Eqs. (a) and (b) listed in Fig. 7-3. Then an improved value of ¨v1 is calculated from the
dynamic equilibrium condition at time t1 using an expression equivalent to Eq. (7-7),
and this leads to improved values of velocity ˙v1 and displacement v1. Eventually, the
iteration converges to a fixed value of the final acceleration for this time step and the
procedure may be stepped forward to the next time step. A great advantage of this
constant average acceleration method is that it is unconditionally stable; that is, the
errors are not amplified from one step to the next no matter how long a time step is

2. RESPONSE TO GENERAL DYNAMIC LOADING: STEP-BY-STEP METHODS 121
¨v0
¨v1
v0
FIGURE 7-3
Motion based on constant average acceleration.
(a)
(b)
h
τ
t0 t1
Acceleration
(Constant)
Displacement
(Quadratic)
Velocity
(Linear)
¨vav = 
1
2
(¨v0 + ¨v1)
.
v(τ) =
.
v0 + 
τ
2
(¨v0 + ¨v1)
.
v1 =
.
v0 + 
h
2
(¨v0 + ¨v1)
.
v0
v(τ) = v0 +
.
v0τ + 
τ2
4
(¨v0 + ¨v1)
v1 = v0 +
.
v0 h + 
h2
4
(¨v0 + ¨v1)
chosen. Consequently, the time step may be selected considering only the need for
properly defining the dynamic excitation and the vibratory response characteristics of
the structure.
Newmark Beta Methods
A more general step-by-step formulation was proposed by Newmark, which
includes the preceding method as a special case, but also may be applied in several
other versions. In the Newmark formulation, the basic integration equations [Eqs. (7-
13)] for the final velocity and displacement are expressed as follows:
˙v1 = ˙v0 + (1 − γ) h ¨v0 + γ h ¨v1 (7-14a)
v1 = v0 + h ˙v0 + (
1
2
− β) h2
¨v0 + β h2
¨v1 (7-14b)
It is evident in Eq. (7-14a) that the factor γ provides a linearly varying weighting
between the influence of the initial and the final accelerations on the change of
velocity; the factor β similarly provides for weighting the contributions of these initial
and final accelerations to the change of displacement.
From study of the performance of this formulation, it was noted that the factor
γ controlled the amount of artificial damping induced by this step-by-step procedure;

3. 122 DYNAMICS OF STRUCTURES
there is no artificial damping if γ = 1/2, so it is recommended that this value be
use for standard SDOF analyses. Adopting this factor γ = 1/2 and setting β = 1/4
in Eqs. (7-14a) and (7-14b), it may be seen that this Newmark formulation reduces
directly to the expressions shown for the final velocity and displacement in Fig. 7-3.
Thus, the Newmark β = 1/4 method may also be referred to as the constant average
acceleration method.
On the other hand, if β is taken to be 1/6 (with γ = 1/2), the expressions for
the final velocity and displacement become
˙v1 = ˙v0 +
h
2
(¨v0 + ¨v1) (7-15a)
v1 = v0 + ˙v0 h +
h2
3
¨v0 +
h2
6
¨v1 (7-15b)
These results also may be derived by assuming that the acceleration varies linearly
during the time step between the initial and final values of ¨v0 and ¨v1, as shown in
Fig. 7-4; thus the Newmark β = 1/6 method is also known as the linear acceleration
method. Like the constant average acceleration procedure, this method is widely
used in practice, but in contrast to the β = 1/4 procedure, the linear acceleration
method is only conditionally stable; it will be unstable unless h/T ≤
√
3/π = 0.55.
However, as in the case of the second central difference method, this restriction has
FIGURE 7-4
Motion based on linearly varying acceleration.
(b)
(a)
Acceleration
(Linear)
Displacement
(Cubic)
Velocity
(Quadratic)
¨v0
¨v1
v0
h
t0 t1
.
v1 =
.
v0 + 
h
2
(¨v0 + ¨v1)
.
v0
v1 = v0 +
.
v0 h + ¨v0

h2
3
+ ¨v1
h2
6

v( ) = v0 +
.
v0 + ¨v0

2
2
+
¨v1− ¨v0
h

3
6
( )
.
v( ) =
.
v0 + ¨v0 +
¨v1− ¨v0
h

2
2
( )
¨v( ) = ¨v0 +
¨v1− ¨v0
h
( )

4. RESPONSE TO GENERAL DYNAMIC LOADING: STEP-BY-STEP METHODS 123
little significance in the analysis of SDOF systems because a shorter time step than
this must be used to obtain a satisfactory representation of the dynamic input and
response.
Conversion to Explicit Formulation
In general, the implicit formulations of the Beta methods are inconvenient to
use because iteration is required at each time step to determine the acceleration at
the end of the step. Accordingly, they are usually converted to an explicit form, and
the conversion procedure will be explained here for the constant average acceleration
(β = 1/4) method. The objective of the conversion is to express the final acceleration
in terms of the other response quantities; accordingly Eq. (b) of Fig. 7-3 is solved for
the final acceleration to obtain
¨v1 =
4
h2
(v1 − v0) −
4
h
˙v0 − ¨v0 (7-16a)
and this then is substituted into Eq. (a) of Fig. 7-3 to obtain an expression for the final
velocity:
˙v1 =
2
h
(v1 − v0) − ˙v0 (7-16b)
Writing the equations of dynamic equilibrium at time t1
m ¨v1 + c ˙v1 + k v1 = p1
and substituting Eqs. (7-16a) and (7-16b) leads to an expression in which the only
unknown is the displacement at the end of the time step, v1. With appropriate gathering
of terms this may be written
kc v1 = p1c (7-17)
which has the form of a static equilibrium equation involving the effective stiffness
kc = k +
2c
h
+
4m
h2
(7-17a)
and the effective loading
p1c = p1 + c
2v0
h
+ ˙v0 + m
4v0
h2
+
4
h
˙v0 + ¨v0 (7-17b)
In Eqs. (7-17) the subscript c is used to denote the constant average acceleration
method.
Using this explicit formulation, the displacement at the end of the time step, v1,
can be calculated directly by solving Eq. (7-17), using only data that was available at
the beginning of the time step. Then, the velocity at that time, ˙v1, may be calculated

5. 124 DYNAMICS OF STRUCTURES
from Eq. (7-16b). Finally, the acceleration at the end of the step, ¨v1, is derived by
solving the dynamic equilibrium equation at that time
¨v1 =
1
m
(p1 − c ˙v1 − k v1)
[rather than from Eq. (7-16a)] thus preserving the equilibrium condition.
It will be noted that the linear acceleration method can be converted to explicit
form similarly by using Eqs. (a) and (b) of Fig. 7-4 in exactly the same way. The
only differences in the formulations, then, are in the expressions for the effective
stiffness and effective loading and for the final velocity. Expressing the effective static
equilibrium equation for the linear acceleration analysis by
kd v1 = p1d (7-18)
(in which the subscript d denotes the linear acceleration method) the effective stiffness
and loading are given, respectively, by
kd = k +
3c
h
+
6m
h2
(7-18a)
p1d = p1 + m
6v0
h2
+
6
h
˙v0 + 2 ¨v0 + c
3v0
h
+ 2 ˙v0 +
h
2
¨v0 (7-18b)
When the displacement v1 has been calculated from Eq. (7-18), the velocity at the
same time is given by the following expression [equivalent to Eq. (7-16b)]:
˙v1 =
3
h
(v1 − v0) − 2 ˙v0 −
h
2
¨v0 (7-18c)
It is important to remember that the linear acceleration method is only condi-
tionally stable, but this factor is seldom important in analysis of SDOF systems as was
mentioned before. On the other hand, it is apparent that assuming a linear variation
of acceleration during each step will give a better approximation of the true behavior
than will a sequence of constant acceleration steps. In fact, numerical experiments
have demonstrated the superiority of the linear acceleration method results compared
with those obtained using constant acceleration steps, and for this reason the linear
acceleration (β = 1/6) method is recommended for analysis of SDOF systems.
7-6 INCREMENTAL FORMULATION FOR NONLINEAR
ANALYSIS
The step-by-step procedures described above are suitable for the analysis of
linear systems in which the resisting forces are expressed in terms of the entire values
of velocity and displacement that have been developed in the structure up to that time.