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 sec-
tion, in^2 (cm^2 ) As = area of prestressing steel, in^2 (cm^2 ); d - effective depth of section at
ultimate strength, in (mm); h = total depth of section, in (mm); / = moment of inertia of
gross area, in
4
(cm
4
); yb = distance from centroidal axis to bottom fiber, in (mm); Sb = sec-
tion modulus with respect to bottom fiber = I/yb, in
3
(cm
3
); kb = distance from centroidal
axis to lower kern point, in (mm); kt = distance from centroidal axis to upper kern point,
in (mm). Forces and moments: F 1 = initial prestressing force, Ib (N); Ff = final prestress-
ing force, Ib (N); 17 = FflFt\ e = eccentricity of prestressing force, in (mm); econ = eccen-
tricity of prestressing force having concordant trajectory; 6 = angle between trajectory (or
tangent to trajectory) and horizontal line; m = slope of trajectory; w = vertical load exert-
ed by curved tendons on concrete in unit distance; ww = unit beam weight; ws = unit su-
perimposed load; WDL = unit dead load; WLL = unit live load; wu = unit ultimate load; Vp =
prestress shear; Mp = prestress moment; Mw = bending moment due to beam weight; M 5 =
bending moment due to superimposed load; C 11 = resultant compressive force at ultimate
load; Tu = resultant tensile force at ultimate load. Stresses: f'c = ultimate compressive
strength of concrete, lb/in
2
(kPa);^/ compressive strength of concrete at transfer;/J = ul-
timate strength of prestressing steel; fsu = stress in prestressing steel at ultimate load; fbp =
stress in bottom fiber due to initial prestressing force; fbw = bending stress in bottom fiber
due to beam weight;^ = bending stress in bottom fiber due to superimposed loads; fbi =
stress in bottom fiber at initial state =fbp +fbwfbf
stress in bottom fiber at final state =
(^7) Ifbp +fbw +fbstfcai = initial stress at centroidal axis. Camber: Ap = camber due to initial
prestressing force, in (mm); Aw = camber due to beam weight; A, = camber at initial state;
Ayr = 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,// denotes the stress in the top
fiber at the initial state.
DETERMINATION OF PRESTRESS SHEAR
AND MOMENT
The beam in Fig. 3 Ia 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:
- 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. 316, and consider the forces acting on the beam segment GB solely as a
result of Ft. The left portion of the beam exerts a tensile force F 1 on the tendons. Since GB