Unit 14: Structural Mechanics in onstruction and ivil Engineering hapter 9 Point of ontra-flexure 9.1 What else can Shear Force and ending Moment iagrams Tell us? Look at the beam shown in Fig. 1 (a). It is supported at and and experiences a point load at and at the free end. y examining the beam and deducing the way in which it might bend (in the same way as we did with the examples at the very beginning of this chapter), we can deduce that: The beam is sagging at point ; The beam is hogging at support ; The beam is hogging at point. learly, somewhere between points and, the nature of the beam s deflection switches from sagging to hogging. This point is termed the point of contra-flexure. ut where, exactly, does the point of contraflexure occur? y now you should be able to calculate the reactions and draw the shear force and bending moment diagrams. These are shown in Figs. 1(b) and (c) respectively. Now, earlier in this chapter you were introduced to a convention which stated that the bending moment diagram is always drawn on the tension side of the zero line. This suggests that: If the bending moment profile is below the zero line, tension occurs in the bottom face of the beam, which suggests it is sagging; If the bending moment profile is above the zero line, tension occurs in the top face of the beam, which suggests it is hogging. It follows from this that where the bending moment diagram crosses the zero line, the nature of deflection of the beam switches from sagging to hogging (or vice versa). Therefore a point of contra-flexure occurs wherever the bending moment profile crosses the zero line. In the current hapter 9 Page 1
Unit 14: Structural Mechanics in onstruction and ivil Engineering example, that point is 2.5 metres from the left-hand end of the beam. This is determined by recognising that the two (hatched) triangles that constitute the bending moment diagram are similar (in the mathematical sense of the word). The deflected profile of the beam is shown in Fig. 1(d). 25 kn 10 kn 2 m 2 m 3 m R = 5 kn R = 30 kn (a) eam iagram 5 0 5 25 30 10 10 0 20 20 (b) Shear Force iagram 30 0 0 10 (c) ending Moment iagram Sagging Hogging (d) eflected Form Figure 1: eflected forms and ontra-flexure hapter 9 Page 2
Unit 14: Structural Mechanics in onstruction and ivil Engineering Practical Example 1 raw the shear force and bending moment diagrams and sketch the deflected form for the beam shown in figure 2. Identify the position of the points of contra-flexure. (The solution is given during the lesson). 20 kn 80 kn 10 kn E 2 m 2 m 2 m 3 m R R Figure 2 hapter 9 Page 3
Unit 14: Structural Mechanics in onstruction and ivil Engineering 9.2 What you should remember from the last three chapters Shear is a cutting or slicing action which causes a beam to break or snap. If a beam is subjected to a load it will bend. If the loading is increased, the bending will increase and eventually the beam will break ( if it doesn t fail in shear first). shear force is the force tending to produce a shear failure at a given point in a beam. The value of shear force at any point in a beam = the algebraic sum of all upward and downward forces to the left of the point. beam will fail in either bending or shear. Which occurs first can only be determined by calculation. The bending moment is the magnitude of the bending effect at any point in a beam. The value of bending moment at any point on a beam = the sum of all bending moments to the left of the point. Shear force and bending moment diagrams are graphically representations of shear force and bending moment and their variation along a beam. The bending moment diagram is drawn either above or below the zero line, dependent on whether the beam experiences tension in the top or bottom at the point concerned (top: above the line, bottom: below the line). Where the shear force is zero, the bending moment is either a local maximum, a local minimum or zero. It follows from this that the position of maximum bending moment can be determined from drawing the shear force diagram first. If a beam experiences point loads only, the shear force diagram will be a series of steps and the bending moment diagram will contain only straight lines (usually sloping). Where a beam experiences uniformly distributed loads, the shear force diagram will comprise sloping straight lines and the bending moment diagram will be curved. The point of contra-flexure is where the deflected form of a beam switches between hogging and sagging. The bending moment diagram will cross the zero line at this point. nd don t forget! hapter 9 Page 4
Unit 14: Structural Mechanics in onstruction and ivil Engineering 9.3 More Examples raw the shear force and bending moment diagrams for each of the beams shown in Fig. 3. The solutions are given during the lesson. 16 kn/m a 5 m 3 m R R 30 kn 20 kn b 1 m 4 m 3 m R R 10 kn/m 50 kn c 4 m 2 m 2 m R R hapter 9 Page 5
Unit 14: Structural Mechanics in onstruction and ivil Engineering 40 kn/m 30 kn d 3 m 3 m 3 m R R Figure 3 hapter 9 Page 6