the assumption takes account of the possibility that neither the deflection nor the slope at the ends of the beam is zero . the potential energy of the system is as follows : &amp;formula; where b is the stiffness of the supports and K is a constant which depends upon the datum of the potential energy . substituting for the D&apos;s by making use of equation ( 7.51 ) then yields : &amp;formula; and for the potential energy to be stationary : &amp;formula; whence : &amp;formula; being the deflection of x = 0 , that is , at the load . had an exact solution of this problem been carried-out there would have been seven simultaneous equations to solve in the seven unknown deflections D1 , D2 , &amp;hellip; , D7 . the loss of accuracy due to adopting an approximate procedure is usually insignificant for purposes of engineering practice , i.e a few per cent . thus the correct value of D1 is 5.4 in . the considerable saving in labour achieved is usually much more important than a small loss of accuracy . in fact , it is possible in some instances , that without recourse to an approximate solution by an energy method , solution by manual activity would be too laborious to be practicable . chapter 8 . some uses of the reciprocal theorem . 8 : 1 . introduction . one of the simplest statements of the reciprocal theorem which defines the reciprocal property of linear systems , specifies that the deflection of a point i of an elastic structure in a given direction due to the application of unit force in a given direction at another point j is equal to the deflection of j when unit force is applied at i . the deflection of j is measured in the direction of the line of action of the unit force while the unit force is applied at i in the line in which the deflection due to its presence at j was measured . this is manifest when the flexibility coefficients of linear structures are calculated , since then it is found that aij = aji as shown in chapter 2 . it is also manifest when the stiffness coefficients are calculated . further proof of the reciprocal theorem is hardly necessary . a simple statement of the theorem on these lines was made by J Clerk Maxwell in his well-known paper on the analysis of frameworks ( 1864 ) but Clebsch had actually noted the reciprocal property of stiffness coefficients in his book published some two years earlier . later Betti ( 1872 ) and Rayleigh ( 1873 ) made important general contributions to the theorem independently , which led to its coming to occupy an important place in the physics of linear systems . for the purpose of structural analysis the reciprocal theorem provides useful devices for the construction of influence lines for deflections and forces in frameworks whose elasticity is linear . 8 : 2 . influence lines for deflection by the reciprocal theorem . for the purpose of illustrating this use of the reciprocal theorem it is sufficient to consider a simply supported beam with linear elasticity . thus , if the influence line for the deflection of any point P of the beam shown in fig 8.1 is required ( that is , the curve whose ordinates represent the deflection of P as a concentrated unit load traverses the beam ) , by the reciprocal theorem it is merely necessary to consider the deflected shape of the beam due to unit load at P . the reason for this is that the deflection at any other point Q of the beam due to unit load at P is : &amp;formula; where aQP is the relevant flexibility coefficient . since this is equal to the deflection of P due to unit load at Q , i.e : &amp;formula; it follows that the deformed shape of the beam caused by unit load at P represents the variation of aQP = aPQ over the length of the beam which is the influence line for the deflection of P . by similar reasoning the influence line for the deflection of any point of an elastic linear structure , in a given direction , is represented by the deformed shape of the structure due to unit load applied in the specified direction at the point in question . a convenient means of using this principle to practical advantage is afforded by scale models . such models need not be to scale in every detail ; for plane frameworks it is merely necessary that they are made of material which obeys Hooke &apos;s law of linear elasticity , to a chosen layout scale . then , for portal frameworks whose members deform primarily in bending , it is sufficient for the ratios of the second moments of area of the members to be the same as in the actual framework . the shape of the required influence line to scale can be obtained by applying a force to the model at the point in question , in the specified direction . the scale factor for the ordinates of the influence line so obtained can be found either by scaling the force applied to the model or by calculating the deflection of the actual framework at the point in question due to unit load applied there . 8 : 3 . influence lines for forces by the reciprocal theorem . a cantilever with a rigid prop at its free end , as one of the simplest statically-indeterminate systems , is suitable for demonstrating this use of the reciprocal theorem . in order to obtain the influence line for the force exerted by the prop , suppose first of all that unit concentrated load acts at any point Q of the span , as shown in fig 8.2 ( a ) . if the prop is absent the deflection of the end of the cantilever due to this load is : &amp;formula; so that the force which the prop must exert in restoring zero deflection at this point is : &amp;formula; where the flexibility coefficients aPQ and aPP refer to the cantilever . therefore , by equations ( 8.3 ) and ( 8.4 ) : &amp;formula; . now the ratio apq/app can be obtained by considering an arbitrary small displacement D&amp;prime;p of the end of the unloaded cantilever due to an arbitrary force r&amp;prime;p , as shown in fig 8.2 ( b ) , since : &amp;formula; while the resulting deflection of any other point Q is : &amp;formula; so that : &amp;formula; . therefore , by equation ( 8.5 ) : &amp;formula; . the significance of this result is that the deflection curve of the cantilever due to an arbitrary small displacement of P represents to scale the influence line for the load on the prop at P . this is in accordance with Mu&quot;ller-Breslau &apos;s principle that the influence line for the force in a member or upon a support of a linear statically-indeterminate framework is represented to scale by the change in shape of the framework due to a small displacement within the member or at the support . for the purpose of using the principle for the influence line for the bending moment at any point , the small displacement introduced there must be of the angular kind . it can be shown by virtual work that Mu&quot;ller-Breslau &apos;s principle also applies to statically-determinate systems which are not subject to gross distortion under load . Mu&quot;ller-Breslau &apos;s principle would be of very little practical value without scale model techniques . the procedure prescribed by the principle can be applied physically to a scale model for the purpose of obtaining influence lines to scale and affords an effective method of model analysis of frameworks . such models must be made of material with linear elasticity to a definite length scale . thus , if a model of the propped cantilever is made s times smaller than the actual , a small displacement &amp;formula; at P corresponds to a small displacement &amp;formula; at P of the actual system . similarly , any other point of the model Q suffers a displacement which may be multiplied by the scale factor s to obtain the corresponding displacement of the point Q of the actual cantilever due to the displacement of P of &amp;formula; . also the deformed shape of the model represents the influence line for the load on the prop of the actual system to scale . therefore , with reference to equations ( 8.8 ) and ( 8.9 ) : &amp;formula; so that : &amp;formula; and the scale factor does not appear in the final result obtained by the model in respect of influence lines for forces , because the ratios of model displacements of the linear kind are identical to the ratios of corresponding displacements of the actual structure . it is relatively easy to construct suitable models of frameworks whose members deform primarily in bending , such as portals , because then it is merely necessary for the ratios of the second moments of area of the various members to be correct . the actual scale factor in respect of second moment of area is immaterial and so models can be cut from , say , sheet celluloid , which obeys Hooke &apos;s law . Beggs pioneered the use of this kind of model . 8 : 4 . example of model analysis . the steel portal framework shown in fig 8.3 has encastr&amp;eacute; stanchion feet and the second moments of area of AB , BC , and CD are I , 2I and I , respectively . in order to obtain the influence lines for the redundants , chosen to be the reactions R1 , R2 and R3 at the foot A , a scale model may be used . the model must be made of material which has linear elasticity in accordance with Hooke &apos;s law ( e.g , it can be cut from sheet Xylonite celluloid ) , to a layout scale factor s and the ratios of the second moments of area of the model members AB , BC and CD must be 1 : 2 : 1 . the required influence lines are found by subjecting the model , mounted to reproduce the encastr&amp;eacute; conditions at A and D , to small displacements horizontally ( for the influence line for R1 ) , vertically ( for R2 ) and rotationally ( for R3 ) at A , in turn , and recording the resulting changes in shape of the model . it is important for each displacement to be applied at A separately without movement in any other direction . suppose the influence lines so obtained are as shown in fig 8.4 and that it is desired to determine the magnitudes of the reactions at A caused by the loading shown in fig 8.3 . then using subscripts m to denote that the displacements are obtained from the model : &amp;formula; which are independent of the scale of the model . for R3 , however , the scale of the model enters into the calculations and for this reason it is desirable to refer the model displacements to the corresponding values for the actual structure . thus , if the foot A of the actual framework were rotated through th radians the resulting deflections would be s times those of the model when its foot A is rotated through the same angle . using the equivalent full-scale influence line ordinates then to obtain R3 gives : &amp;formula; since th is &amp;formula; . again , for a uniformly distributed loading of intensity w over , say , CD , the corresponding values of the reactions at A are : &amp;formula; where distance x along CD refers to the model , so that if a1 , a2 and a3 are the areas enclosed by the relevant portions of the influence lines of the model , respectively : &amp;formula; and for practical purposes it is sufficiently accurate to assume that the influence lines are straight between measured ordinates . the influence line for the bending moment at a point within a member can be obtained similarly by cutting the model at the point in question and applying an angular displacement , as indicated in fig 8.5 . the required bending moment due to particular loading is then obtained from the influence line ordinates in a manner similar to that used for finding R3 . it is particularly important to measure the influence line ordinates correctly , as , for example , in fig 8.4 with respect to the line of fq2 . accuracy can also be improved by using positive and negative displacements , as shown in figs 10.20 and 10.21 . use of scale models for the analysis of frameworks is always worth considering as an alternative to manual computation , especially for frameworks of simple form whose members are of non-uniform section for reasons of economy . accuracy of model analysis tends to lie between 5 % and 10 % in relation to values calculated exactly on the basis of the same assumptions as those used in constructing the model . 