THE CONCRETE BRIDGE MAGAZINE

FALL 2018

ASPIRE is a quarterly magazine published by PCI in cooperation with the associations of the National Concrete Bridge Council. The editorial content focuses on the latest technology and key issues in the Concrete Bridge Industry.

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A P R O F E S S O R ' S P E R S P E C T I V E 38 | ASPIRE Fall 2018 REDISTRIBUTION OF MOMENTS in Continuous Structures by Dr. Andrea Schokker, University of Minnesota Duluth Figure 1. Simply supported determinant beam with uniform loading. All Figures: Dr. Andrea Schokker. T his article focuses on the topic of m o m e n t re d i s t r i b u t i o n , w h i c h should not be confused with moment d i s t r i b u t i o n ( t h a t g r e a t m e t h o d developed by structural engineering professor Hardy Cross for solving indeterminant structures that continues to elicit groans in structural analysis classes). Moment redistribution refers to the ability of continuous, or statically indeterminate, structures to redistribute moment at the strength limit state due to their redundancy. During the design process, analysis in the elastic range is typically performed and does not include any additional capacity of the overall structure after the elastic limit is reached. Considering moment redistribution can allow designers to account for additional capacity at the strength limit state that may be available as moment shifts (redistributes) from the section that has reached the plastic range to sections still in the elastic range. If a continuous structure is properly detailed, it can form what is known as a "plastic hinge," which results in a shift of moment to a different area, giving a higher overall structure capacity at the strength limit state than first assumed when looking only at the section capacity. The more degrees of indeterminacy a structure has, the greater the number of potential plastic hinges required to cause failure. This concept can be illustrated by a basic example. The following assumptions will be used for the example: • T h e m e m b e r h a s a n i d e a l i z e d moment-curvature relationship, where a section is elastic until the design capacity (yield) is reached and then does not take any additional moment. • After a section reaches elastic capacity, it is idealized as a hinge (resisting no additional moment). • The section capacity is the same along the entire member, with equal capacity for positive and negative applied moment. • When a sufficient number of plastic hinges have formed to make a member or span unstable so that a collapse mechanism is formed, the member is considered to have failed. Consider a simply supported determinant beam as shown in Fig. 1. The maximum moment under this condition is at the midspan of the beam, and, thus, that location is where the plastic moment M p will be reached first. If the length of the beam l is 100 ft and the section moment capacity is 1000 kip-ft, the uniform load w required to form the midspan hinge and resulting collapse is calculated as follows: wl 2 8 = 1000 kip-ft w = = 8(1000 kip-ft) (100 ft) 2 = 0.8 kip/ft With no redundancy in this system, a collapse mechanism is formed and the member fails. This determinant beam case does not allow any redistribution of moment. Now, consider the case of a redundant single-span member fixed on both ends with a uniform loading, as shown in Fig. 2a. For illustrative purposes, let's suppose we have a section capacity of 1000 kip- ft (both positive and negative moment) and a beam length of 100 ft. In this case, the plastic moment is reached first at the ends under negative moment. The moment diagram in Fig. 2b shows the moment values at the stage at which the first hinges form. The value of the applied uniform load w at this stage is calculated as follows: w l 2 12 = 1000 kip-ft w = 12(1000 kip-ft) (100 ft) 2 = 1.2 kip/ft The midspan moment at this stage is only 500 kip-ft. The beam is still stable and can continue to take load at this point, but no more moment can be added where the plastic hinges have formed (Fig. 2c). Any additional moment is taken as if the beam were simply supported (Fig. 2d). If we continue to add load, we will reach a point where a hinge is developed at midspan, where the maximum positive moment is. This location was already taking 500 kip-ft of moment when the hinges at the ends were formed and, once an additional moment of 500 kip-ft is added, the midspan section reaches its moment capacity of 1000 kip-ft. The incremental load Δw required to form the midspan hinge and resulting collapse is calculated as follows: Δ w l 2 8 = 500 kip-ft Δ w = 8(500 kip-ft) (100 ft) 2 = 0.4 kip/ft This gives us a total uniform load of w = 1.2 + 0.4 = 1.6 kip/ft on the structure prior to reaching theoretical collapse (assuming that the hinges are formed at the elastic design capacity of the section),

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