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This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) P flexibility of ___________________ spring = 4- flexibility of beam = 4- (0) (b) Analysis of skeletal structures 305 With pin-jointed frames, be they two- or three-dimensional, there is only one stress resultant per member, viz. its axial load. When forces act on an elastic structure, it undergoes deformations, causing displacements at every point within the structure. The solution of forces in the frames is accomplished by relating the stress resultants to the displacements. The number of equations needed is governed by the degrees of freedom, i.e. the number of possible component displacements. At one end of the member of a pin-jointed plane frame, the member displacement has translational components in the x and y directions only, and no rotational displacement. The number of degrees of freedom is two. By similar reasoning it will be apparent that the number of degrees of freedom for a rigid-jointed plane frame member is three. For a member of a three-dimensional pin-jointed frame it is also three, and for a similar rigid-jointed frame it is six. 9.5.1 Stiffness and flexibility Forces and displacements have a vital and interrelated role in the analysis of structures. Forces cause displacements and the occurrence of displacements implies the existence of forces. The relationship between forces and displacements is defined in one of two ways, viz. flexibility and stiffness. Flexibility gives a measure of displacements associated with a given set of forces acting on the structure. This concept will be illustrated by considering the example of a spring loaded at one end by a static load P (see Fig. 9.18). As the spring is linearly elastic, the extension, D, produced is directly proportional to the applied load, P. The deflection produced by a unit load (defined as the flexibility of the spring) is obviously D/P. Figure 9.18(b) illustrates the deflection response of a beam to an applied load P. Once again the flexibility of the beam is D/P. In the simple cases considered above, flexibility simply gives the load–displacement response at a point. A more generalized definition applicable to the displace- Fig. 9.18 Flexibility
This material is copyright - all rights reserved. Reproduced under licence from The Steel Construction Institute on 12/2/2007 To buy a hardcopy version of this document call 01344 872775 or go to http://shop.steelbiz.org/ Steel Designers' Manual - 6th Edition (2003) 306 Introduction to manual and computer analysis Fig. 9.19 Flexibility coefficients for a loaded beam ment response at a number of locations will now be obtained by considering the beam sketched in Fig. 9.19. Considering a unit load acting at point 1 (Fig. 9.19(b)), the corresponding deflections at points 1, 2 and 3 are denoted as f11, f21 and f31 (the first subscript denotes the point at which the deflection is measured; the second subscript refers to the point at which the unit load is applied).The terms f11, f21, f31 are called flexibility coefficients. Figure 9.19(c) and (d) give the corresponding flexibility coefficients for load positions 2 and 3 respectively. By the principle of superposition, the total deflections at points 1, 2 and 3 due to P1,P2 and P3 can be written as D1 = P1f11 + P2f12 + P3f13 D2 = P1f21 + P2f22 + P3f23 D3 = P1f31 + P2f32 + P3f33 Written in matrix form, this becomes ÏD1 ¸ Èf f f Ô Ô ÌD 2 ˝ = Í f f f Í Ô ÓD Ô 3 ˛ ÎÍ f f f 11 12 13 21 22 23 31 32 33 ˘ ÏP1 ¸ ˙ Ô Ô 2 ˙ ÌP ˝ ˚˙ Ô Ô ÓP3 ˛ (9.14)
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