O Belce Sprężonej w Fazie Odkształceń Sprężysto-Plastycznych przed Zarysowaniem - Pietrzykowski, J. (Engineering Transactions, 1961)
On the basis of the principle of plane section and assuming perfect elastic-plastic material, constant Young’s modulus and relative yield points of concrete Qc and Qr equations are obtained for the determination of the stress distribution a in a bent prestressed clement of rectangular cross-section for classified plasticity cases, Table 2. The range of appearance of each particular elastic-plastic state in function of the resultant axial force and bending moment is represented in Fig. 16a. For each particular state the limit curves are determined graphically and mathematically by considering the limit stress distributions at the beginning of the plastic state in extreme fibres. For example, the diagram B, C, D, A’, B’, B in Fig. 16b shows the plasticity (DA’) of the tension zone with the beginning of plasticity of compression zone (BC = Qc). The other limit diagrams for point C may C be obtained by turning of CD line round point C. Limit diagrams for points A’, C’, A may be found in the same manner. These curves are denoted in the diagrams by c, a, b and c’, a’, b’. As an example, the equations of the curves c, a’ are given. They are composed of two branches: rectilinear and parabolic intersecting at the characteristic point (Fig. 16a). The coordinates of this point represent the magnitude of the axial force and moment, corresponding to the beginning of the plastic state in the cross-section in compressed and stretched fibres simultaneously. The straight segments of the limit curves a, a’, c, c’ bound the region (Fig. 16a) where the value of the axial force and moment do not pro- duce stresses exceeding the elastic range in the profile. The region bounded with the curves a’, c for n < (Qc-Qr)/2 is the plasticity region in the stretched zone and for n> (Qc-Qr)/2 it represents the plasticity region in the compressed zone. The region above the curvilinear segments a’ and c, bounded by the curve p represents the values of n, m producing plasticity on both sides of the profile. In the quadrant II and III analogous regions, for a cantilever beam for instance, are determined by the limit curves c’, a. The curve p illustrates the theoretical state of full plasticity in the entire cross-section. Considering the general case of load by an axial force and moment and assuming the continuity of the body, the limit state is considered to be that of crack formation or crushing. Assuming the coefficient of filling up of the stress solid corresponding to the limit state in the cross-section to be a 0.7 the range of validity is determined for all the elastic-plastic states. On the diagram it is determined by dashed line constituting the envelope of the curves pa, Pc, Pa’s Pe determined for each state of plasticity separately. Zones of admissible plasticity before cracking and crushing are marked by oblique shading (Fig. 16a). The safe region of the validity range depending on the way of leading may be bounded either by the curve g (parallel to pa = 0.7 - oblique shaded) or the curve ω (obtained by multiplying the ordinates Da by a constant safety factor), Fig. 24. If the element is loaded by a variable moment and axial force, it is more convenient to take the curve g. The principles of choosing the safety factors are given as well as directions for the design of a profile admitting certain elastic-plastic states, by means of the system of equations of the curve Pa in the four component ranges. This enables the determination of the best dimensions of the cross-section and the prestressing parameters and the choice of the path of the cable taking inelastic states into consideration. In conclusion, equations are given enabling us to determine the plasticity regions along the beam and the computation of the values of the rotation angles of the end section in the elastic-plastic state.