Page 185 - Bridge and Highway Structure Rehabilitation and Repair

160 SECTION 2 STRENGTHENING AND REPAIR WORK

behavior of steel and concrete structures. An approximate upper bound linear analysis can fi rst
be carried out and then compared to results from nonlinear analysis to estimate the degree of
nonlinearity and its effects. An example of nonlinear material behavior is shown by the C. S.
Whitney stress-strain curve for reinforced concrete sections.
At higher loads as cracking develops, most materials exhibit a nonlinear behavior. Stiffness
properties of concrete and composite members are based on uncracked sections or on cracked
sections consistent with anticipated behavior. Progressive cracking based on experimental ob-
servations is included in the mathematical model.
A load deflection behavior is usually more accurately defined than stress-strain behavior

since in the latter case physical changes such as crack formations would occur.

Principle of superposition: Large deflection analysis is inherently nonlinear, and loads are

not proportional to displacements. Superposition of deflections and stresses cannot be used.

From an analytical point of view, deflections, moments, and forces from dead load analysis can-
not be directly added to those from large deflection live load analysis. In other words, live load

analysis results cannot be “superposed” on dead load analysis results, and a unique solution will
be required for the combined dead and live loads acting together.
4.10.2 Simplifi ed Nonlinear Analysis
P-Delta method: Under concentric compressive axial forces, a slender column is subjected
to out-of-plane deflection “Delta” giving rise to additional moment of P-Delta. Additional mo-

ment causes an equivalent eccentricity of axial load.
Synergism may be defined as the interaction of moments and forces that when combined

produces a total effect which is greater than the sum of individual moment or force. Hence, syn-
ergistic effect of interaction results in apparent softening of the column, which can be expressed
as a loss of stiffness. Quantitatively this can be classified as a second order effect. When axial

compressive stress is high and reaches Euler bucking stress, failure by buckling takes place.
Approximate method can be used by selecting a moment correction factor. LRFD load factors
are used for analysis rather than applying the load factors after analysis is complete. Changes in

lateral deflection for each incremental load are incorporated in equilibrium equations. Iterative

adjustment of deflections is carried out until convergence is reached.

Moment magnification of long columns (AASHTO 4.5.3.2.2b):
M 3 M 4 M 2s (4.21)
2b
c
b
s
where M 3 Moment on compression member, due to factored gravity loads (K-ft)
2b
M 3 Moment on compression member, due to factored gravity or lateral loads
2s
that result in side sway 9 l /1500
u
f 3 f 4 f (4.22)
c
b 2b
s 2s
where f 3 Stress corresponding to M (ksi), f 3 Stress corresponding to M (ksi)
2b 2b 2s 2s
3 C /(1 6 X ) 9 1 (4.23)
m
1
b
where X 3 P /() P ) where P 3 Factored axial load (Kips), ) 3
1 u k e u k
Stiffness reduction factor (1 for steel and 0.75 for concrete)
2
Euler buckling load P 3 ! EI/(K l ) 2 (4.24)
e u
where K 3 Effective length factor,
l 3 Unsupported length of compression member
u
3 1/(1 6 X );
s
2
where X 3 $P /() $P )
2 u k e