PLASTIC ANALYSIS OF STRUCTURES UNIT IV PLASTIC ANALYSIS OF STRUCTURES Presented by, G.Bairavi AP/Civil

Introduction Statically indeterminate axial problems Beams in pure bending Plastic moment of resistance Plastic modulus Shape factor – Load factor Plastic hinge and mechanism Upper and lower bound theorems analysis of indeterminate beams and frames 2

Plastic analysis A method of analysis in which the ultimate strength of a structure is computed by considering the conditions for which there are sufficient plastic hinges to transform the structure into a mechanism. 3

Plastic section A section capable of reaching and maintaining the full plastic moment until a plastic collapse mechanism is formed. 4

Plastic hinge A fully yielded cross-section of a member which allows the member portions on either side to rotate under constant moment (the plastic moment). 5

Length of Plastic Hinge 6

Determine the inelastic zone length Xp of the simply supported beam shown in Figure when the section under the load becomes fully plastic. The beam has a uniform rectangular section with a plastic moment capacity of 165 kNm. The section starts to yield at a bending moment of My = Mp/ 1.5 = 165/1.5=110 kNm. 7

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Load factor A factor used to multiply a nominal load to obtain part of the design load 9

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Mechanism Structural system with a sufficient number of frictionless and plastic hinges to allow it to deform indefinitely under constant load. 11

Shape Factor is the ratio of Plastic Moment to Yield Moment S=Mp/ M 12

Stress distribution for a fully plastic section. 13

For the rectangular section shown in Figure, the plastic section where Zs = the first moment of area about the equal area axis = plastic section modulus. For the rectangular section shown in Figure, the plastic section modulus is given by 14

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Basic Three Theorems Kinematic Theorem (Upper Bound Theorem) Static Theorem (Lower Bound Theorem) Uniqueness Theorem 17

Static Theorem (Lower Bound Theorem) If a structure is in equilibrium condition under the external load and applied moment does not exceed the plastic moment of resistance M <= Mp 18

Kinematic Theorem (Upper Bound Theorem) It states that for a structure subjected to a loading corresponding to any assumed collapse mechanism must be either greater than or equal to but can not less than true collapse load. 19

Uniqueness Theorem It is quite clear that if a structure satisfies the conditions of both static and kinematic theorems, the collapse load obtained must be true and unique. Therefore, the uniqueness theorem states that a true collapse load is obtained when the structure is under a distribution of bending moments that are in static equilibrium with the applied forces and no plastic moment capacity is exceeded at any cross section when a collapse mechanism is formed. In other words, a unique collapse load is obtained when the three conditions of static equilibrium, yield, and collapse mechanism are met. 20

Mechanism Method This method requires that all possible collapse mechanisms are identified and that the virtual work equation for each mechanism is established. The collapse load Pw (or collapse load factor ac if a set of loads are applied) is the minimum of the solutions of all possible collapse mechanisms for the structure. 21

In establishing the virtual work equation, the total internal work as sum of the products of the plastic moment, Mp, and the corresponding plastic rotation, y, at all plastic hinge locations j (j ¼ A, B, . . ., etc.) must be equal to the total external work. 22

A fixed-end beam, of length L and plastic moment capacity Mp, is subject to a point load P as shown in Figure. Determine the collapse load P = Pw. 23

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Continuous Beams and Frames When a structure with n degrees of indeterminacy collapses due to the formation of p number of plastic hinges where p = n + 1, the structure fails by complete collapse; in this case, determination of the member forces for the whole structure at collapse is always possible. 26

Determine the collapse load factor P = Pw for the continuous beam Determine the collapse load factor P = Pw for the continuous beam. Plastic moment of the beam is Mp. 27

For left span AB, the plastic hinge occurs at mid span and B as shown in Figure. The virtual work equation is 28

Similarly, for right span BC with two plastic hinges shown in figure, the virtual work is 29

UDL on End Span of a Continuous Beam The load w or maximize the bending moment Mp of the internal plastic hinge so that the value of x can be found. x = 0:414L: This is the standard solution of the collapse load for UDL acting on the end span of a continuous beam. 30

What is the maximum load factor a that the beam shown in Figure can support if Mp = 93 kNm? 31

Application to Portal Frames (a) Beam mechanism—when vertical loads are applied to beams and horizontal loads to columns to form partial collapse mechanisms as shown in Figure (b) Sway mechanism—when horizontal loads are applied to form complete collapse mechanisms as shown in Figure 32

(c) Combined mechanism—a combination of beam and sway mechanisms only if unloading occurs to one or more plastic hinges as shown in Figure 33

Problem 2). A fixed-base portal frame is subject to a vertical load of 2P and a horizontal load of P shown in Figure. The length of the rafter is 6L and of the column is 4L. Find the collapse load P = Pw: 34

The portal frame has 3 degrees of indeterminacy The portal frame has 3 degrees of indeterminacy. Therefore, a complete collapse mechanism requires four plastic hinges. (iii) Combined mechanism of (i) and (ii). 35

A fixed-base portal frame is subject to two horizontal loads of 2P and 3P as shown in Figure. Find the collapse load P= Pw. 36 36

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THANK YOU 38