## 349 General Material Nonlinear Effects

Most structural steels can undergo large deformations before rupturing. For example, yielding in ASTM A36 steel begins at a strain of about 0.0012 in per in and continues until strain hardening occurs at a strain of about 0.014 in per in. At rupture, strains can be on the order of 0.25 in per in. These material characteristics affect the behavior of steel members strained into the yielding range and form the basis for the plastic theory of analysis and design.

The plastic capacity of members is defined by the amount of axial force and bending moment required to completely yield a member's cross section. In the absence of bending, the plastic capacity of a section is represented by the axial yield load

where A = cross-sectional area

Fy = yield stress of the material

For the case of flexure and no axial force, the plastic capacity of the section is defined by the plastic moment

where Z is the plastic section modulus (Art. 3.16). The plastic moment of a section can be significantly greater than the moment required to develop first yielding in the section, defined as the yield moment

where S is the elastic section modulus (Art. 3.16). The ratio of the plastic modulus to the elastic section modulus is defined as a section's shape factor

The shape factor indicates the additional moment beyond initial yielding that a section can develop before becoming completely yielded. The shape factor ranges from about 1.1 for wide-flange sections to 1.5 for rectangular shapes and 1.7 for round sections.

For members subjected to a combination of axial force and bending, the plastic capacity of the section is a function of the section geometry. For example, one estimate of the plastic capacity of a wide-flange section subjected to an axial force P and a major-axis bending moment Mxx is defined by the interaction equation

Py Mpx where Mpx = major-axis plastic moment capacity = ZxxFy . An estimate of the minor-axis plastic capacity of wide-flange section is

y/ py where Myy = minor-axis bending moment, and Mpy = minor-axis plastic moment capacity = ZyFy.

When one section of a member develops its plastic capacity, an increase in load can produce a large rotation or axial deformation or both, at this location. When a large rotation occurs, the fully yielded section forms a plastic hinge. It differs from a true hinge in that some deformation remains in a plastic hinge after it is unloaded.

The plastic capacity of a section may differ from the ultimate strength of the member or the structure in which it exists. First, if the member is part of a redundant system (Art. 3.28), the structure can sustain additional load by distributing the corresponding effects away from the plastic hinge and to the remaining unyielded portions of the structure. Means for accounting for this behavior are incorporated into inelastic methods of analysis.

Secondly, there is a range of strain hardening beyond Fy that corresponds to large strains but in which a steel member can develop an increased resistance to additional loads. This assumes, however, that the section is adequately braced and proportioned so that local or lateral buckling does not occur.

Material nonlinear behavior can be demonstrated by considering a simply supported beam with span L = 400 in and subjected to a uniform load w (Fig. 3.95a). The maximum moment at midspan is Mmax = wL2/8 (Fig. 3.95b). If the beam is made of a W24 x 103 wide-flange t M u u M u rq

FIGURE 3.95 (a) Uniformly loaded simple beam. (b) Moment diagram. (c) Development of a plastic hinge at midspan.

section with a yield stress Fy = 36 ksi and a section modulus Sxx = 245 in3, the beam will begin to yield at a bending moment of My = FySxx = 36 X 245 = 8820 in-kips. Hence, when beam weight is ignored, the beam carries a uniform load w = 8My/L2 = 8 X 8820/ 4002 = 0.44 kips/in.

A W24 X 103 shape, however; has a plastic section modulus Zxx = 280 in3. Consequently, the plastic moment equals Mp = FyZxx = 36 X 280 = 10,080 in-kips. When beam weight is ignored, this moment is produced by a uniform load w = 8Mp/L2 = 8 X 10,080/4002 = 0.50 kips/in, an increase of 14% over the load at initiation of yield. The load developing the plastic moment is often called the limit, or ultimate load. It is under this load that the beam, with hinges at each of its supports, develops a plastic hinge at midspan (Fig. 3.95c) and becomes unstable. If strain-hardening effects are neglected, a kinematic mechanism has formed, and no further loading can be resisted.

If the ends of a beam are fixed as shown in Fig. 3.96a, the midspan moment is Mmid = wL2/24. The maximum moment occur at the ends, Mend = wL2/12 (Fig. 3.96b). If the beam has the same dimensions as the one in Fig. 3.95a, the beam begins to yield at uniform load w = 12My/L2 = 12 X 8820/4002 = 0.66 kips/in. If additional load is applied to the beam, plastic hinges eventually form at the ends of the beam at load w = 12Mp/L2 = 12 X 10,080/ 4002 = 0.76 kips/in. Although plastic hinges exist at the supports, the beam is still stable at this load. Under additional loading, it behaves as a simply supported beam with moments Mp at each end (Fig. 3.96c) and a maximum moment Mmid = wL2/8 - Mp at midspan (Fig. 3.96d). The limit load of the beam is reached when a plastic hinge forms at midspan, Mmid = Mp, thus creating a mechanism (Fig. 3.96e). The uniform load at which this occurs

FIGURE 3.96 (a) Uniformly loaded fixed-end beam. (b) Moment diagram. (c) Beam with plastic hinges at both ends. (d) Moment diagram for the plastic condition. (e) Beam becomes unstable when plastic hinge also develops in the interior.

is w = 2Mp x 8/L2 = 2 x 10,080 x 8/4002 = 1.01 kips/in, a load that is 53% greater than the load at which initiation of yield occurs and 33% greater than the load that produces the first plastic hinges.

In continuous structural systems with many members, there are several ways that mechanisms can develop. The limit load, or load creating a mechanism, lies between the loads computed from upper-bound and lower-bound theorems. The upper-bound theorem states that a load computed on the basis of an assumed mechanism will be greater than, or at best equal to, the true limit load. The lower-bound theorem states that a load computed on the basis of an assumed bending-moment distribution satisfying equilibrium conditions, with bending moments nowhere exceeding the plastic moment Mp , is less than, or at best equal to, the true limit load. The plastic moment is Mp = ZFy, where Z = plastic section modulus and Fy = yield stress. If both theorems yield the same load, it is the true ultimate load.

In the application of either theorem, the following conditions must be satisfied at the limit load: External forces must be in equilibrium with internal forces; there must be enough plastic hinges to form a mechanism; and the plastic moment must not be exceeded anywhere in the structure.

The process of investigating mechanism failure loads to determine the maximum load a continuous structure can sustain is called plastic analysis.

The statical or equilibrium method is based on the lower-bound theorem. It is convenient for continuous structures with few members. The steps are

• Select and remove redundants to make the structure statically determinate.

• Draw the moment diagram for the given loads on the statically determinate structure.

• Sketch the moment diagram that results when an arbitrary value of each redundant is applied to the statically determinate structure.

• Superimpose the moment diagrams in such a way that the structure becomes a mechanism because there are a sufficient number of the peak moments that can be set equal to the plastic moment Mp.

• Compute the ultimate load from equilibrium equations.

• Check to see that Mp is not exceeded anywhere.

To demonstrate the method, a plastic analysis will be made for the two-span continuous beam shown in Fig. 3.97a. The moment at C is chosen as the redundant. Figure 3.97b shows the bending-moment diagram for a simple support at C and the moment diagram for an assumed redundant moment at C. Figure 3.97c shows the combined moment diagram. Since the moment at D appears to exceed the moment at B, the combined moment diagram may be adjusted so that the right span becomes a mechanism when the peak moments at C and D equal the plastic moment Mp (Fig. 3.97d).

If MC = MD = Mp , then equilibrium of span CE requires that at D,

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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