Page 268 - Handbook of Civil Engineering Calculations, Second Edition

PRESTRESSED CONCRETE 2.53

inclines downward to the right. A load is positive if it acts downward. The vertical shear
at a given section is positive if the portion of the beam to the left of this section exerts an
upward force on the concrete. A bending moment is positive if it induces compression
above the centroidal axis and tension below it. A compressive stress is positive; a tensile
stress, negative.
The notational system is as follows. Cross-sectional properties: A gross area of
2
2
section, sq.in. (cm ); A s area of prestressing steel, sq.in. (cm ); d effective depth of
section at ultimate strength, in. (mm); h total depth of section, in. (mm); I moment of
4
4
inertia of gross area, in (cm ); y b distance from centroidal axis to bottom fiber, in.
3
3
(mm); S b section modulus with respect to bottom fiber I/y b , in (cm ); k b distance
from centroidal axis to lower kern point, in. (mm); k t distance from centroidal axis to
upper kern point, in. (mm). Forces and moments: F i initial prestressing force, lb (N); F f
final prestressing force, lb (N); F f /F i ; e eccentricity of prestressing force, in.
(mm); e con eccentricity of prestressing force having concordant trajectory; angle
between trajectory (or tangent to trajectory) and horizontal line; m slope of trajectory;
w vertical load exerted by curved tendons on concrete in unit distance; w w unit beam
weight; w s unit superimposed load; w DL unit dead load; w LL unit live load; w u
unit ultimate load; V p prestress shear; M p prestress moment; M w bending moment
due to beam weight; M s bending moment due to superimposed load; C u resultant
compressive force at ultimate load; T u resultant tensile force at ultimate load. Stresses:
f c
ultimate compressive strength of concrete, lb/sq.in. (kPa); f ci
compressive strength
of concrete at transfer; f s
ultimate strength of prestressing steel; f su stress in pre-
stressing steel at ultimate load; f bp stress in bottom fiber due to initial prestressing
force; f bw bending stress in bottom fiber due to beam weight; f bs bending stress in
bottom fiber due to superimposed loads; f bi stress in bottom fiber at initial state f bp
f bw ; f bf stress in bottom fiber at final state f bp f bw f bs ; f cai initial stress at cen-
troidal axis. Camber: p camber due to initial prestressing force, in. (mm); w cam-
ber due to beam weight; i camber at initial state; f camber at final state.
The symbols that refer to the bottom fiber are transformed to their counterparts for the
top fiber by replacing the subscript b with t. For example, f ti denotes the stress in the top
fiber at the initial state.

DETERMINATION OF PRESTRESS SHEAR
AND MOMENT

The beam in Fig. 31a is simply supported at its ends and prestressed with an initial force
of 300 kips (1334.4 kN). At section C, the eccentricity of this force is 8 in. (203.2 mm),
and the slope of the trajectory is 0.014. (In the drawing, vertical distances are exaggerated
in relation to horizontal distances.) Find the prestress shear and prestress moment at C.

Calculation Procedure:

1. Analyze the prestressing forces
If the composite concrete-and-steel member is regarded as a unit, the prestressing forces
that the steel exerts on the concrete are purely internal. Therefore, if a beam is simply sup-
ported, the prestressing force alone does not induce any reactions at the supports.
Refer to Fig. 31b, and consider the forces acting on the beam segment GB solely as a
result of F i . The left portion of the beam exerts a tensile force F i on the tendons. Since GB