In the 2011 ACI Code, design is based on required strengthscomputed from combina- tions of factored loads and design strengthscomputed as where is a resistance factor, also known as a strength-reduction factor,andRnis the nominal resistance. This
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process is called strength design. In the AISC Specifications for steel design, the same design process is known as LRFD (Load and Resistance Factor Design). Strength design and LRFD are methods of limit-states design, except that primary attention is placed on the ultimate limit states, with the serviceability limit states being checked after the origi- nal design is completed.
ACI Code Sections 9.1.1 and 9.1.2 present the basic limit-states design philosophy of that code.
9.1.1—Structures and structural members shall be designed to have design strengths at all sections at least equal to the required strengths calculated for the factored loads and forces in such combinations as are stipulated in this code.
The term design strengthrefers to and the term required strengthrefers to the load effects calculated from factored loads,
9.1.2—Members also shall meet all other requirements of this Code to insure adequate per- formance at service load levels.
This clause refers primarily to control of deflections and excessive crack widths.
Working-Stress Design
Prior to 2002, Appendix A of the ACI Code allowed design of concrete structures either by strength design or by working-stress design. In 2002, this appendix was deleted. The commentary to ACI Code Section 1.1 still allows the use of working-stress design, pro- vided that the local jurisdiction adopts an exception to the ACI Code allowing the use of working-stress design. Chapter 9 on serviceability presents some concepts from working-stress design. Here, design is based on working loads, also referred to as service loadsorunfactored loads. In flexure, the maximum elastically computed stresses cannot exceed allowable stressesorworking stressesof 0.4 to 0.5 times the concrete and steel strengths.
Plastic Design
Plastic design, also referred to as limit design(not to be confused with limit-states design) orcapacity design, is a design process that considers the redistribution of moments as suc- cessive cross sections yield, thereby forming plastic hingesthat lead to a plastic mecha- nism. These concepts are of considerable importance in seismic design, where the amount of ductility expected from a specific structural system leads to a decrease in the forces that must be resisted by the structure.
Plasticity Theorems
Several aspects of the design of statically indeterminate concrete structures are justified, in part, by using the theory of plasticity. These include the ultimate strength design of continuous frames and two-way slabs for elastically computed loads and moments, and the use of strut-and-tie models for concrete design. Before the theorems of plasticity are presented, several definitions are required:
• A distribution of internal forces (moments, axial forces, and shears) or corre- sponding stresses is said to be statically admissibleif it is in equilibrium with the applied loads and reactions.
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• A distribution of cross-sectional strengths that equals or exceeds the statically admissible forces, moments, or stresses at every cross section in the structure is said to be a safedistribution of strengths.
• A structure is said to be a collapse mechanismif there is one more hinge, or plastic hinge, than required for stable equilibrium.
• A distribution of applied loads, forces, and moments that results in a sufficient number and distribution of plastic hinges to produce a collapse mechanism is said to bekinematically admissible.
The theory of plasticity is expressed in terms of the following three theorems:
1. Lower-bound theorem. If a structure is subjected to a statically admissible dis- tribution of internal forces and if the member cross sections are chosen to provide a safe distribution of strength for the given structure and loading, the structure either will not col- lapse or will be just at the point of collapsing. The resulting distribution of internal forces and moments corresponds to a failure load that is a lower bound to the load at failure. This is called a lower boundbecause the computed failure load is less than or equal to the actual collapse load.
2. Upper-bound theorem. A structure will collapse if there is a kinematically admissible set of plastic hinges that results in a plastic collapse mechanism. For any kine- matically admissible plastic collapse mechanism, a collapse load can be calculated by equating external and internal work. The load calculated by this method will be greater than or equal to the actual collapse load. Thus, the calculated load is an upper boundto the failure load.
3. Uniqueness theorem. If the lower-bound theorem involves the same forces, hinges, and displacements as the upper-bound solution, the resulting failure load is the true oruniquecollapse load.
For the upper- and lower-bound solutions to occur, the structure must have enough ductility to allow the moments and other internal forces from the original loads to redis- tribute to those corresponding to the bounds of plasticity solutions.
Reinforced concrete design is usually based on elastic analyses. Cross sections are proportioned to have factored nominal strengths, and greater than or equal to the and from an elastic analysis. Because the elastic moments and forces are a statically admissible distribution of forces, and because the resisting-moment diagram is chosen by the designer to be a safe distribution, the strength of the resulting structure is a lower bound.
Similarly, the strut-and-tie models presented in Chapter 17 (ACI Appendix A) give lower-bound estimates of the capacity of concrete structures if
(a) the strut-and-tie model of the structure represents a statically admissible dis- tribution of forces,
(b) the strengths of the struts, ties, and nodal zones are chosen to be safe, rela- tive to the computed forces in the strut-and-tie model, and
(c) the members and joint regions have enough ductility to allow the internal forces, moments, and stresses to make the transition from the strut-and-tie forces and moments to the final force and moment distribution.
Thus, if adequate ductility is provided the strut-and-tie model will give a so-called safe estimate, which is a lower-bound estimate of the strength of the strut-and-tie model. Plas- ticity solutions are used to develop the yield-line method of analysis for slabs, presented in Chapter 14.
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