114 Energy methods of structural analysis
Fig. P.4.10
P.4.11 A bracket BAC is composed of a circular tube AB, whose second moment
Problems 115
obey a non-linear elastic stress-strain law given by
E=T[l+
E( 3 1
where r is the stress corresponding to strain E . Bars 15,45 and 23 each have a cross- sectional area A , and each of the remainder has an area of
A l a .
The length of member 12 is equal to the length of member 34 = 2L.If a vertical load
Po
is applied at joint 5 as shown, show that the force in the member 23, i.e. FZ3, is given by the equationany'+'
+
3 . 5 ~+
0.8 = 0 wherex = F23/Po and (Y = Po/Ar0
Fig. P.4.12
P.4.13 Figure P.4.13 shows a plan view of two beams, AB 9150mm long and DE 6100mm long. The simply supported beam AB carries a vertical load of 100000N applied at F , a distance one-third of the span from B. This beam is supported at C on the encastrk beam DE. The beams are of uniform cross-section and have the same second moment of area 83.5 x lo6 mm4. E = 200 000 N/mm2. Calculate the deflection of C .
A m . 5.6mm.
Fig. P.4.13
116 Energy methods of structural analysis
P.4.14 The plane structure shown in Fig. P.4.14 consists of a uniform continuous beam ABC pinned to a fixture at A and supported by a framework of pin-jointed members. All members other than ABC have the same cross-sectional area A . For ABC, the area is 4A and the second moment of area for bending is A 2 / 1 6 . The material is the same throughout. Find (in terms of w, A , a and Young’s modulus E ) the vertical displacement of point D under the vertical loading shown. Ignore shearing strains in the beam ABC.
A m . 30232wa2/3AE.
1.5 w h i t length
l i l l i l l
0Fig. P.4.14
P.4.15 The fuselage frame shown in Fig. P.4.15 consists of two parts, ACB and ADB, with frictionless pin joints at A and B. The bending stiffness is constant in each part, with value EI for ACB and xEI for ADB. Find x so that the maximum bending moment in ADB will be one half of that in ACB. Assume that the deflections are due to bending strains only.
Ans. 0.092.
Fig. P.4.15
P.4.16 A transverse frame in a circular section fuel tank is of radius r and constant bending stiffness EI. The loading on the frame consists of the hydrostatic pressure due to the fuel and the vertical support reaction P, which is equal to the weight of fuel carried by the frame, shown in Fig. P.4.16.
Problems 117
t'
Fig. P.4.16
Taking into account only strains due to bending, calculate the distribution of bending moment around the frame in terms of the force P, the frame radius r and the angle 0.
Ans. M = Pr(0.160 - 0 . 0 8 0 ~ 0 ~ 0 - 0.1590sin0).
P.4.17 The frame shown in Fig. P.4.17 consists of a semi-circular arc, centre B, radius a, of constant flexural rigidity EI jointed rigidly to a beam of constant flexural rigidity 2EZ. The frame is subjected to an outward loading as shown arising from an internal pressure po.
Find the bending moment at points A, B and C and locate any points of contra- flexure.
A is the mid point of the arc. Neglect deformations of the frame due to shear and noi-mal forces.
Ans. M A = -0.057pod, M B = -0.292poa2, Mc = 0.208poa2.
Points of contraflexure: in AC, at 51.7' from horizontal; in BC, 0.764~ from B.
Fig. P.4.17
P.4.18 The rectangular frame shown in Fig. P.4.18 consists of two horizontal members 123 and 456 rigidly joined to three vertical members 16, 25 and 34. All five members have the same bending stiffness EZ.
1 18 Energy methods of structural analysis
Fig. P.4.18
The frame is loaded in its own plane by a system of point loads P which are balanced by a constant shear flow q around the outside. Determine the distribution of the bending moment in the frame and sketch the bending moment diagram. In the analysis take bending deformations only into account.
A m . Shears only at mid-points of vertical members. On the lower half of the frame S4, = 0.27P to right, SS2 = 0.69P to left, = 1.08P to left; the bending moment diagram follows.
P.4.19 A circular fuselage frame shown in Fig. P.4.19, of radius r and constant bending stiffness EI, has a straight floor beam of length r d , bending stiffness EI, rigidly fixed to the frame at either end. The frame is loaded by a couple T applied at its lowest point and a constant equilibrating shear flow q around its periphery.
Determine the distribution of the bending moment in the frame, illustrating your answer by means of a sketch.
In the analysis, deformations due to shear and end load may be considered negligible. The depth of the frame cross-section in comparison with the radius r may also be neglected.
A m . MI4 = T(0.29 sine - 0.160), = 0.30Tx/r, M4, = T(0.59sine - 0.166)
1
Fig. P.4.19
Problems 119
Fig. P.4.20
P.4.20 A thin-walled member BCD is rigidly built-in at D and simply supported at the same level at C , as shown in Fig. P.4.20.
Find the horizontal deflection at B due to the horizontal force F. Full account must be taken of deformations due to shear and direct strains, as well as to bending.
The member is of uniform cross-section, of area A, relevant second moment of area in bending Z = A?/400 and ‘reduced‘ effective area in shearing A‘ = A/4. Poisson’s ratio for the material is v = 1/3.
Give the answer in terms of F , r, A and Young’s modulus E . Ans. 448FrlEA.
P.4.21 Figure P.4.21 shows two cantilevers, the end of one being vertically above the other and connected to it by a spring AB. Initially the system is unstrained. A weight W placed at A causes a vertical deflection at A of
SI
and a vertical deflection at B of6,.
When the spring is removed the weight W at A causes a deflection at A of6,.
Find the extension of the spring when it is replaced and the weight W is transferred to B.Ans.
&(Si
- S,)/(S3 - SI).Fig. P.4.21
P.4.22 A beam 2400mm long is supported at two points A and B which are