Appendix 3 The concept of statical determinacy A3.1 Introduction the number of equations which can be derived by considering the equilibrium of the external It has been shown that the conditions for force system. The structure in Fig. A3.2 is also equilibrium of a set of coplanar forces can be insoluble by equilibrium due to the fact that summarised in the three equations of the number of internal forces which it contains equilibrium (see Appendix 1). These equations is greater than the number of independent can be solved as a simultaneous set for the equations which can be derived by considering forces in a force system which are unknown as only the equilibrium of all possible ‘free-body- was shown in connection with Fig. A1.9. diagrams’. These structures are said to be A structure which can be fully solved from statically indeterminate. the equations of equilibrium in this way is said Structures can therefore be subdivided into to be statically determinate. The structure in two categories, those which are statically Fig. A3.1, which has four external reactions, determinate and those which are statically cannot be solved by this method because the indeterminate. The two types behave in number of unknown reactions is greater than significantly different ways in response to load and the decision as to which should be adopted in a particular situation is an important aspect of structural design. Most (a) (b) structural geometries can be produced in either form and the designer of a structure must take a conscious decision as to which type is appropriate. The choice affects the detailed geometry of the structure and can influence the selection of the structural material. Fig. A3.1 The framework (a) is statically determinate. Framework (b) is statically indeterminate because the four external reactions cannot be solved from the three equations of equilibrium which can be derived. A3.2 The characteristics of statically determinate and statically Fig. A3.2 Although the external indeterminate structures force system of this structure is statically determinate the A3.2.1 Internal forces framework is statically In Fig. A3.3 two independent statically indeterminate because it determinate structures, ABC and ADC, are contains more elements than are shown. They happen to share the same required for internal stability. It will not be possible to solve the supports, A and C, but in every other respect structure for all of the internal they are independent. If horizontal loads of P forces by considering static and 2P are applied to joints B and D, equilibrium only. 140 respectively, the structures will resist these; Appendix 3: The concept of statical determinacy (a) (b) (c) Fig. A3.3 The pattern of internal forces in a statically indeterminate. Joints B and D must undergo the same indeterminate structure depends on the properties of the deflection; internal force, dependent on the relative elements as well as on the overall geometry of the magnitudes of S1 and S2, occurs in BD and this alters the arrangement. (a) ABC and ADC are independent statically whole pattern of internal forces. The final distribution of determinate structures. (b) The two structures are free to internal force depends on the elasticity of the elements as deflect independently in response to load. (c) The well as the overall geometry of the structure. presence of element BD renders the arrangement statically internal forces and reactions will be developed, statically indeterminate structure is therefore all of which can be calculated from the dependent on the properties of the elements equations of equilibrium, and the elements as well as on the overall geometry of the frame will undergo axial strain, the magnitudes of and the magnitudes of the external loads. The which will depend on the elasticity of the element properties must therefore be taken material and the sizes of the element cross- into account in the analysis of this structure. sections. Both joints B and D will suffer lateral This is generally true of statically deflections but these will not affect the indeterminate structures and is one of the internal forces in the elements, which will be important differences between statically solely dependent on the external loads and on determinate and statically indeterminate the geometries of the arrangement (to a first structures. approximation). The fact that element properties have to be If a fifth element is added, which connects considered in the analysis of statically joints B and D, the system becomes statically indeterminate structures makes their analysis indeterminate. The two joints are now much more complicated than that of constrained to deflect by the same amount equivalent statically determinate structures; in under all load conditions and if the two loads particular, it requires that the rigidity of the are applied as before the extent of the elements be taken into account. As this can resulting elongation or contraction of the only be done once the element dimensions elements will not be the same as occurred have been decided and a material selected, it when the joints B and D were free to deflect means that the design calculations for independently. This means that the joint which statically indeterminate structures must be previously deflected less will be pulled or carried out on a trial and error basis. A set of pushed further than before and the reverse will element sizes must be selected initially to occur to the other joint. A transfer of load will allow the analysis to be carried out. Once the therefore occur along the element BD and this internal forces have been calculated the will alter the pattern of internal forces in the suitability of the trial sizes can be assessed by whole frame. The amount of load transfer, and calculating the stress which will occur in them. therefore of change to the internal force The element sizes must normally be altered to system, will depend on the difference between suit the particular internal forces which occur the deflections which occurred to the two and this causes a change in the pattern of the joints in the statically determinate forms. This internal forces. A further analysis is then is determined by the rigidity of the elements, required to calculate the new internal forces, so the distribution of internal forces in the followed by a further revision of the element 141 Structure and Architecture dimensions. The sequence must be continued seen in relation to structures with rigid joints, until satisfactory element sizes are obtained. in which the resulting structural continuity Cycles of calculations of this type are routine causes smaller bending moments to occur in computer-aided design. than are present in equivalent statically By comparison, the calculations for determinate structures under the same load statically determinate structures are much conditions. As before the differences between more straightforward. The internal forces in the the two types of structure can be appreciated elements depend solely on the external loads by studying very simple examples. and on the overall geometry of the structure. The simply supported beam (Fig. A3.4), They can therefore be calculated before any whose supports offer no restraint against decision on element dimensions or a structural rotation of the beam ends, is a statically material has been taken. Once the internal determinate structure. The deflected shape of forces are known, a material can be chosen and this, in response to a uniformly distributed appropriate element dimensions selected. load, is a sagging curve in which, as in all These will not affect the pattern of the internal structures which are subjected to bending, the forces and so a single sequence of calculations intensity of the curvature at every cross-section is sufficient to complete the design. is directly proportional to the magnitude of the bending moment at that cross-section. The A3.2.2 Efficiency in the use of material curvature is greatest at mid-span and The efficiency with which structural material is decreases to zero at the supports where the used is normally greater with statically beam ends tilt but remain straight. indeterminate structures because the presence A beam whose ends are restrained against of a larger number of constraints allows a more rotation is a statically indeterminate structure direct transmission of loads to the foundations (Fig. A3.5). The fixed-end supports are each and a more even sharing of load by all of the capable of producing three external reactions elements. The benefits of statical and the total of six reactions makes the indeterminacy in this respect are most easily solution of the external force system Fig. A3.4 Load, deflection and bending moment diagrams for a statically determinate simply supported beam. Fig. A3.5 Load, deflection and bending moment diagrams for a statically indeterminate beam subjected to the same load pattern as in Fig. A3.4. The effect of the restraint at the supports, which are the cause of the statical indeterminacy, is to reduce the value of the 142 maximum bending moment. Appendix 3: The concept of statical determinacy impossible from the three equations of allows a more efficient use to be made of the equilibrium which can be derived. Another structural material. As with most gains, there is consequence of the end fixities, and of the a cost, which in this case arises from the moment reactions which result from them, is difficulty of providing fixed-ended support that the ends of the beam remain horizontal conditions. when a load is applied. The mid-span portion In more complicated structures, where many still adopts a sagging curve, but the amount of elements are present, the benefits of end fixity sag is less than in the simply supported case, are achieved by making the joints between because a reversal in the direction of the them rigid.