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The slope deflection method and the moment distribution methods are the elastic analysis methods which are important in order to study the performance with regard to serviceability of a structure. A structure can be made to collapse when the load is applied until the yielding occurs at some locations, the structure undergoes elastic-plastic deformations and further increase results in the state when the structure is fully plastic. At this state a sufficient number of plastic hinges are formed to transform the structure into a mechanism. By studying the mechanism and the value of the collapse load the load factor can be determined at which the structure collapses. The load factor specified for the design should be greater than or equal to the product of the load factor and the service loading.
The plastic behaviour of the materials can be studied by stress-strain relationship and the moment-curvature relationship.
2.2 Stress strain curve
When a tensile force is applied to a long metal bar a state of pure uniaxial tension exists in the middle of the bar away from the clamps at the ends. Consider a bar as shown in the figure 2.1.
The extension of the bar is measures in this region of uniaxial tension. For a specimen of gauge length L and cross-sectional area S the extension is found to be 5.65. The applied force is expressed as stress which is force/cross-sectional area. The extension caused due to the force is expressed as strain which is the (change in length)/ (original length). A typical stress-strain curve is obtained as shown in the figure 2.2 when stress on the steel specimen is plotted against the corresponding strain.
For the small values of the strain, the stress is directly proportional to strain. This can be seen in the region OU of the graph. The material remains elastic and the slope if OU is the Young's Modulus, E. U is called the upper yield point and the stress remains proportional to strain till point U which has a correspond stress of . Then there is a sudden drop in the stress to the point L and the stress which corresponds to point L is the yield stress y. The strain at the yield point L is .12. The strain is increased beyond this value there is no corresponding increase in the stress till the point H. This region of LH is the plastic region and the behaviour exhibited by the specimen is known as plastic and the increase in the stress without the increase in the stress is known as plastic flow. The corresponding strain at point H is .14 which is typically ten times the strain at the yield point. Beyond the point H, increase in the stress is required in order to achieve the increase in the strain. This is known as strain hardening. The slope of this region is about 4% of the Young's modulus. Necking and finally a cup and cone fracture occurs when the stress reaches a maximum value known as ultimate tensile strength uts and there is a further increase in the strain. The corresponding strain at fracture is about .2.
2.3 Moment curvature
The collapse load of a section in the beam may be calculated by finding the relationship between moment and curvature for the section. Consider a simply supported beam as shown the figure 2.3 (a) which has an effective span L and carries a vertical point load W.
The load is transferred to the supports by the bending of the beam. From the statics the reaction at the supports is equal to W/2. The supports provided to this beam are such that there are no bending moments or horizontal reactions at the supports. From the bending moment diagram shown in figure 2.3 (b), it can be observed that the beam is sagging throughout its length with the maximum bending moment being under the point load which is WL/4. By the engineer's bending theory which is based on elastic behaviour of the material, there is no yield in the material and the stress-strain relationships are in straight-line throughout the depth of the section as shown in the figure 2.4.
There is a level at which the stress and strain are zero and is known as the axis of zero strain which is also known as the neutral axis of the section. Stress and strain for the section are directly proportional to the distance (z) from the axis, there is maximum compression at top and tension at bottom when there is a sagging in the beam. The maximum stress, max = M/Z, where M is the bending moment and Z is the section modulus. The beam still exhibits elastic behaviour until the maximum stress reaches the yield stress. It may be noted that only the material at the outer surface of the section is yielding. Tests show that after yield the distribution of strain stays linear over the depth of the section and the simple bending theory assuming the plane sections remaining plane is still applicable. The stress at any position may be found from the stress-strain curve, as shown in the figure 2.5 and figure 2.6.
As the bending moment increases, the yielding of the section extends towards the neutral axis. From stress distribution two constant regions where yield has occurred can be seen. Now the stress remains constant but there is an increase in the strain due to the plastic flow. The result of this can be seen in figure 2.7.
The constant stress reaches the neutral axis. After all the material has yielded the section now behaves like a hinge due to the fact that strain can increase anywhere in the section without any further increase in the stress. The section has now become a plastic hinge. It can be demonstrated from the figure 2.8.
When the moment is equal to the plastic moment of resistance also known as plastic moment the plastic hinge is formed. This plastic moment is the largest moment that the section can carry. The right-hand support and the load must move when the collapse has occurred. The beam has now become a plastic mechanism. The formation of plastic hinge is illustrated in the figure 2.9.
2.4 Spread of yield through a section in bending in a member
The spread on yield in any section in bending in a member can be studied by the moment-curvature relationship.
The ideal moment-curvature relationship can be seen in the figure 2.1.
When the member is elastic it requires an increase in the moment for the corresponding increase in the curvature. But when the plastic hinge is formed in the member as it attains the plastic moment there is an increase in the curvature without any further increase in the moment. This phenomenon is known as the plastic rotation of the hinge. It is assumed that whole of the critical section has yielded when a plastic hinge is formed. This means that the strains are equal to the yield strain just above the and below the neutral axis. The only way for this assumption and the assumption of the plane sections remaining plane is valid when the strain at the top and bottom of the section are infinitely large. This condition is practically impossible. Assuming the stress distribution as shown in figure 2.11 the spread of yield can be analysed.
It can be seen that the stress distribution has a varying size of the elastic region. By using this actual moment-curvature relationship of the section can be plotted as shown in figure 2.1 (b). The actual curves are asymptotic to the ideal relationship. It can be observed that there is a small elastic region in the middle of the section. When analysing sections having shape factors close to unity this assumption gives very small error, but the errors are large when analysing sections with higher shape factor. In the ideal relationship it appears that the plastic flow is confined to the section which has the highest bending moment. In fact even before the plastic hinge is formed yielding can occur due to the bending moments at the sections which are adjacent to the critical section. This results in the formation of regions of plastic material around the critical section as shown in the figure 2.12.
The extent of these regions depends on whether the load is distributed or concentrated. When the section is under the distributed loads the extent of the region is greater due gradual changes in the bending moment as compared to the concentrated loads. Also, the shape-factor of the section determines the extent of this region, greater the shape factor, greater is the extent of these regions. There is a gradual bending in the member due to this region and not a sharp kink which results from an ideal plastic hinge. As stated before that the errors are small when the shape factors are close to unity, the errors from the idealisations are not large as most framed structures are constructed using sections of low shape factors.
There are further more complications related to the moment-curvature theory. According to the theory plastic moment is the biggest moment that a section can resist. There is a possibility of localised buckling in the sections of highest compressive stress due to which it is impossible for a section to reach the plastic moment. This depends on the slenderness of the parts in compression. Four possibilities have been identified by the moment-curvature relation as shown in figure 2.13.
The plastic state develops the full plastic moment and undergoes large amount of rotation. Though the compact section develops the full plastic moment but fails without any significant plastic rotation due to buckling. The semi-compact section reaches the yield moment but fails before reaching the plastic moment due to buckling. The bending stresses in the material may be below the yield stress when the section buckles, so that failure may occur at a moment below the first yield.
2.5 Plastic hinge
A section which is attaining its plastic moment capacity undergoes plastic rotation without any further increase in the bending moment. It can be observed in the moment-curvature idealisation. This section when it attains its full plastic moment behaves like a real hinge. This enables us to analyse the structure continuously by inserting a plastic hinge at any section that reaches its plastic moment. This provides the basic concept for the hinge-by- hinge elastoplastic analysis. Due to the formation of plastic hinges, the flexibility of the structure increaser until the stiffness of the structure becomes very small and ultimately collapse occurs. In an elastoplastic analysis for an indeterminate structure which is under increasing loads, the value of the increased loads can be calculated by considering the attainment of plastic moments in sections.
The collapse mechanism in a fixed end beam as shown in figure 2.14 and the formation of plastic hinges can be studied.
Figure 2.14 Source: (Wong, Plastic Analysis and Design of steel Structures, 29, p. 72)
For this beam it requires the formation of three plastic hinges at points A, B and C for the collapse mechanism occur. By plotting a graph of change in the load against deflection the deterioration in the stiffness of a structure can be studied. For this beam the graph is plotted for increase in the load P against the vertical deflection of point B which is under the load as shown in figure 2.15.
The black dots in the figure 2.15 represent the plastic hinge at a section which is in a fully plastic state and the bending moment of the plastic hinge is equal to its plastic moment. The state of the beam below the first plastic hinge is the elastic state and the corresponding analysis is known as elastic analysis. The state of the beam between the first and the third plastic hinge load level is known as the elastic-plastic state and the corresponding analysis is known as the elsastoplastic analysis. The state of the beam between the formations of consecutive plastic hinges is elastic and elastic analysis can be performed on the structure. The structure collapses when a fully plastic state is attained. The relative stiffness of the structure can be calculated from the graph, the slope of the curve gives the relative stiffness of the beam. As it can be observed from the graph the relative stiffness reduces with the increase in load and as more and more sections of the beam turn into plastic hinges and finally the relative stiffness becomes zero at the collapse (Wong, Plastic Analysis and Design of Steel Structures, 29, pp. 73-73).
2.6 Theorems of plastic analysis
There are three essential theorems of plastic analysis, they are, Lower Bound Theorem, Upper Bound Theorem and Uniqueness Theorem.
2.6.1 Lower Bound Theorem
The Lower bound theorem also known as Static theorem states that, if, in a structure subjected to loading defined by a positive load factor , a bending moment distribution satisfying the equilibrium and yield conditions can be found, then is less than, or equal to, the collapse load factor c. In this case the value of is lower bound to c. (Moy, Pastic Methods for Steel and Concrete Structures, 1996, p. 44)
All the possible cases of the yield conditions of the structure have to be analysed and by choosing the largest value of the collapse loads of these conditions the true collapse load can be found. The collapse mechanism for the structure cannot necessarily be obtained from the assumed yield conditions. By using this theorem the collapse load for an indeterminate structure is calculated by usually considering the static equilibrium through a flexibility approach to plot the free and reactant bending moment diagrams. This method is usually known as the "statical method" (Wong, Plastic Analysis and Design of steel Structures, 29, p. 14).
2.6.2 Upper Bound theorem
The Upper bound theorem states that, if, in a structure subjected to loading deigned by a positive load factor , a bending moment distribution satisfying the equilibrium and mechanism condition can be found, then is greater than, or equal to, the collapse load factor c. Now the value of is upper bound to c. (Moy, Pastic Methods for Steel and Concrete Structures, 1996, pp. 44-45)
By choosing the smallest value of the collapse loads obtained from all the cases of collapse mechanisms of the structure, the true collapse load can be determined. Mechanism method is derived from this theorem by equating the external work and internal work for a particular collapse mechanism (Wong, Plastic Analysis and Design of Steel Structures, 29, p. 14)
2.6.3 Uniqueness theorem
The uniqueness theorem states that, if a structure is subjected to loading, defined by a positive load factor , and a bending moment distribution at any other load factor that satisfies the yield condition, equilibrium condition and mechanism condition simultaneously (Moy, Pastic Methods for Steel and Concrete Structures, 1996, p. 45).
If the distribution of bending moments is in equilibrium with the applied loads, this condition is known as the equilibrium condition. If the bending moments nowhere exceed the plastic moment of the members this condition is known as the equilibrium condition. If there are sufficient hinges for the structure to become a mechanism this condition is known as the mechanism condition (Moy, Plastic Methods for Steel and Concrete Structures, 1996, p. 41).