1、Unit34The Concept of Statical DeterminacyText1.IntroductionA structure which can be fully solved from the equations of equilibrium in this way is said to be statically determinate.The structure in Fig.1,which has four external reactions,cannot be solved by this method because the number of unknown r
2、eactions is greater than the number of equations which can be derived by considering the equilibrium of the external force system.The structure in Fig.2 is also insoluble by equilibrium due to the fact that the number of internal forces which it contains is greater than the number of independent equ
3、ations which can be derived by considering only the equilibrium of all possible free-body-diagrams.These structures are said to be statically indeterminate.Fig.34.1The framework(a)is statically determinate.Framework(b)is statically indeterminate.Fig34.2Framework is statically indeterminateTextStruct
4、ures can therefore be subdivided into two categories,those which are statically determinate and those which are statically indeterminate.The two types behave in significantly different ways in response to load and the decision as to which should be adopted in a particular situation is an important a
5、spect of structural design.Most 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.2.T
6、he characteristics of statically determinate and statically indeterminate structures2.1 Internal forcesIn Fig.3 two independent statically determinate structures,ABC and ADC,are shown.They happen to share the same supports.A and C,but in every other respect they are independent.If horizontal loads o
7、f P and 2P are applied to joints B and D,respectively,the structures will resist these internal forces and reactions will be developed,all of which can be calculated from the equations of equilibrium,and the elements will undergo axial strain,the magnitudes of which will depend on the elasticity of
8、the material and the sizes of the element cross-sections.Both joints B and D will suffer la-Text-teral deflections but these will not affect the internal forces in the elements,which will be solely dependent on the external loads and on the geometries of the arrangement(to a first approximation).Fig
9、34.3The pattern of internal forces in a statically indeterminate structure depends on the properties of the elements as well as on the overall geometry of the arrangement.If a fifth element is added,which connects joints B and D,the system becomes statically indeterminate.The two joints are now cons
10、trained to deflect by the same amount under all load conditions and if the two loads are applied as before the extent of the resulting elongation or contraction of the elements will not be the same as occurred when the joints B and D were free to deflect independently.This means that the joint which
11、 previously deflected less will be pulled or pushed further than before and the reverse will occur to the other joint.A transfer of load will thereforeTextoccur along the element BD and this will alter the pattern of internal forces in the whole frame.The amount of load transfer,and therefore of cha
12、nge to the internal force system,will depend on the difference between the deflections which occurred to the two joints in the statically determinate forms.This is determined by the rigidity of the elements,so the distribution of internal forces in the statically indeterminate structure is therefore
13、 dependent on the properties of the elements as well as on the overall geometry of the frame and the magnitudes of the external loads.The element properties must therefore be taken into account in the analysis of this structure.This is generally true of statically indeterminate structures and is one
14、 of the important differences between statically determinate and statically indeterminate structures.The fact that element properties have to be considered in the analysis of statically indeterminate structures makes their analysis much more complicated than that of equivalent statically determinate
15、 structures;in particular,it requires that the rigidity of the elements be taken into account.As this can only be done once the element dimensions have been decided and a material selected,it means that the design calculations for statically indeterminate structures must be carried out on a trial an
16、d error basis.A set of element sizes must be selected initially to allow the analysis to be carried out.Once the internal forces have been calculated the suitability of the trial sizes can be assessed by calculating the stress which will occur in them.The element sizes must normally be alte-Text-red
17、 to suit the particular internal forces which occur and this causes a change in the pattern of the internal forces.A further analysis is then required to calculate the new internal forces,followed by a further revision of the element dimensions.The sequence must be continued until satisfactory eleme
18、nt sizes are obtained.Cycles of calculations of this type are routine in computer-aided design.By comparison,the calculations for statically determinate structures are much more straightforward.The internal forces in the elements depend solely on the external loads and on the overall geometry of the
19、 structure.They can therefore be calculated before any decision on element dimensions or a structural material has been taken.Once the internal forces are known,a material can be chosen and appropriate element dimensions selected.These will not affect the pattern of the internal forces and so a sing
20、le sequence of calculations is sufficient to complete the design.2.2 Efficiency in the use of materialThe efficiency with which structural material is used is normally greater with statically indeterminate structures because the presence of a larger number of constraints allows a more direct transmi
21、ssion of loads to the foundations and a more even sharing of load by all of the elements.The benefits of statical indeterminacy in this respect are most easily seen in relation to structures with rig-Text-id joints,in which the resulting structural continuity causes smaller bending moments to occur
22、than are present in equivalent statically determinate structures under the same load conditions.As before the differences between the two types of structure can be appreciated by studying very simple examples.2.3 The 1ack-of-fit problemWith the possible exception of in situ reinforced concrete struc
23、tures,most structures are prefabricated to some extent so that their construction on site is a process of assembly.As prefabricated components can never be produced with precisely the correct dimensions,the question of lack-of-fit and of the tolerance which must be allowed for this is a necessary co
24、nsideration in structural design.It can affect the decision on whether to use a statically determinate or indeterminate form,because the tolerance of statically determinate structures to lack-of-fit is much greater than that of statically indeterminate structures.The arrangement in Fig.7(a)is static
25、ally determinate while that in Fig.7(b)is an equivalent statically indeterminate form.It will be assumed that the frames are assembled from straight elements,that the structural material is steel and that the hinge-type joints are made by bolting.The elements would be fabricated in a steel fabricati
26、on workshop and all bolt ho-Text-les would be pre-drilled.However,it would be impossible to cut the elements to exactly the correct length,or to drill the bolt holes in exactly the correct positions；there would always be some small error no matter how much care was taken in the fabrication process.F
27、ig7The lack-of-fit problem.The initial stages of the assembly would be the same for both forms and might consist of bolting the beams to the tops of the two columns.The resulting arrangements would still be mechanisms at this stage and any discrepancies which existed between the length of the next e
28、lement to be inserted,that is the first diagonal element,and the length of the space into which it must fit could be eliminated by swaying the assembly until the distance between the joints was exactly the same as the length of the element.TextThe insertion of the first diagonal element would comple
29、te the assembly of the statically determinate form.To complete the statically indeterminate form the second diagonal must be added.If any discrepancy exists between the length of this and the distance between the joints to which it must be attached,the distance cannot now be adjusted easily by movin
30、g the partly assembled frame because it is now a structure and will resist any force which is applied to it in an attempt to alter its shape.A significant force would therefore have to be applied to distort the frame before the final element could be inserted.This would produce stress in the element
31、s,which would tend to restore the frame to its original shape when the force was released after the insertion of the final element.The presence of the second diagonal element in the frame would prevent it from returning to its original shape,however,and the result would be that all of the elements i
32、n the frame would finally carry a permanent stress as a result of the lack-of-fit.This would be additional to any stress which they had to carry as a result of the application of the frames legitimate load.The performance in respect of lack-of-fit is an important difference between statically determ
33、inate and statically indeterminate structures.Statically determinate structures can be assembled fairly easily despite the fact that it is impossible to fabricate structural components with a-Text-bsolute accuracy as any discrepancy which exists between the actual dimensions of components and their
34、intended dimensions can normally be accommodated during the construction process.This does,of course,result in a final structural geometry which is slightly different from the shape which was planned,but the level of accuracy reached in the fabrication is normally such that any discrepancy is undete
35、ctable to the naked eye despite being significant from the point of view of the introduction of lack-of-fit stresses.In the case of statically indeterminate structures even small discrepancies in the dimensions can lead to difficulties in assembly and the problem becomes more acute as the degree of
36、indeterminacy is increased.It has two aspects firstly,there is the difficulty of actually constructing the structure if the elements do not fit perfectly,and secondly,there is the possibility that lack-of-fit stresses may be developed,which will reduce its carrying capacity.The problem is dealt with
37、 by minimising the amount of lack-of-fit which occurs and also by devising means of adjusting the lengths of the elements during construction(for example by use of packing plates).Both of these require that high standards are achieved in the detailed design of the structures,in the manufacture of it
38、s components and also in the setting out of the structure on site.A consequence of the lack-of-fit problem,therefore,is that both the design and the construction of statically indeterminate structures are more difficult and therefore more expensive than those of equivalent statically determinate str
39、uctures.Text2.4 Thermal expansion and temperature stressesIt was seen in Section 2.3 that in the case of statically indeterminate structures stresses can be introduced into the elements if they do not fit perfectly when the structure is assembled.Even if perfect fit were to be achieved initially,how
40、ever,any subsequent alteration to the dimensions of elements due to thermal expansion or contraction would lead to the creation of stress.Such stress is known as temperature stress.It does not occur in statically determinate structures,in which small changes in dimensions due to thermal expansion ar
41、e accommodated by minor adjustments to the structure s shape without the introduction of stress.Thermal expansion must be considered in the design of most statically indeterminate structures and the elements made strong enough to resist the resulting additional stress which will occur.This depends o
42、n the range of temperature to which the structure will be exposed and on the coefficient of thermal expansion of the material.It is a factor which obviously reduces the load carrying capacity and therefore efficiency of statically indeterminate structures.Text3.Design considerations in relation to s
43、tatical determinacyMost structural geometries can be produced in either a statically determinate or a statically indeterminate form depending on how the constituent elements are connected together.The question of which should be adopted in a particular case is one of the fundamental issues of the de
44、sign process and the decision is influenced by the factors which have been considered above.The main advantage of statically indeterminate structures is that they allow a more efficient use of material than equivalent statically determinate forms.It is therefore possible to achieve longer spans and
45、carry heavier loads than with statically determinate equivalents.The principal disadvantage of statically indeterminate structures are that they are more complex to design and more difficult to construct than statically determinate equivalents;these factors usually make them more expensive despite t
46、heir greater efficiency.Other disadvantages are the possibilities of lack-of-fit and temperature stresses and the greater susceptibility of statically indeterminate structures to damage as a result of differential settlement of foundations.These various factors are weighed against each other by the
47、designer of a structure who must decide which type is more suitable in an individual case.TextThe decision as to which material should be used for a structure is often related to the decision on determinacy.Reinforced concrete is ideal for statically indeterminate structures due to the ease with whi
48、ch continuity can be achieved without the disadvantage of the lack-of-fit problem and also to its low coefficient of thermal expansion,which results in temperature stresses being low.Most reinforced concrete structures are therefore designed to be statically indeterminate.The use of steel for static
49、ally indeterminate structures,on the other hand,can be problematical due to the lack-of-fit problem and to the relatively high coefficient of thermal expansion of the material.Steel therefore tends to be used for statically determinate structures rather than for statically indeterminate structures u
50、nless the particular advantages of indeterminacy are specifically required in conjunction with the use of steel.Steel and timber are in fact particularly suitable for statically determinate structures due to the ease with which hinge-type joints can be produced in these materials.TextUsually the cir
