mà từ đó sự lệch tip (x = 0) là P13/ giá trị dự đoán của lý thuyết chùm đơn giản (mục ) và không bao gồm sự đóng góp lệch của các biến dạng trượt. Điều này đã được loại bỏ khi chúng tôi giả định rằng độ dốc của mặt phẳng trung lập vào cuối được xây dựng trong không. Một cuộc kiểm tra chi tiết hơn về hiệu ứng này là hướng dẫn. Biến dạng trượt tại bất kỳ điểm nào trong chùm tia được cho bởi Eq | Bending of an end-loaded cantilever 47 and from Eq. viii Pl2 Pb2 - 2EI SIG Substitution for the constants c D F and H in Eqs ix and x now produces the equations for the components of displacement at any point in the beam. Thus Px2y vPy2 Py2 PỈ2 Pb2 u 2EI 6EI 6IG 2EI SIG y xi vPxy2 Px2 Pl2x Pl2 v 2EI ŨŨ 1ẼĨ ĨẼĨ xu The deflection curve for the neutral plane is . Px3 Pl2x Pl2 0 - 6 7 2EI 3EI xiii from which the tip deflection x 0 is Pl2 3El. This value is that predicted by simple beam theory Section and does not include the contribution to deflection of the shear strain. This was eliminated when we assumed that the slope of the neutral plane at the built-in end was zero. A more detailed examination of this effect is instructive. The shear strain at any point in the beam is given by Eq. vi xy SIG b and is obviously independent of X. Therefore at all points on the neutral plane the shear strain is constant and equal to Pb2 xy SIG which amounts to a rotation of the neutral plane as shown in Fig. . The deflection of the neutral plane due to this shear strain at any section of the beam is therefore equal to Fig. Rotation of neutral plane due to shear in end-loaded cantilever. 48 Two-dimensional problems in elasticity Pbz SIG Fig. a Distortion of cross-section due to shear b effect on distortion of rotation due to shear. xiv and Eq. xiii may be rewritten to include the effect of shear as _ Px2 Pl2x p 3 Ph2 0 6ẼĨ Z 7 3E7 8 Let us now examine the distorted shape of the beam section which the analysis assumes is free to take place. At the built-in end when .V the displacement of any point is from Eq. xi _ Ỉ P 3 Py3 Pb2y 6EI 6IG 8fG The cross-section would therefore if allowed take the shape of the shallow reversed s shown in Fig. a . We have not included in Eq. xv the previously discussed effect of rotation of the neutral plane caused by shear. However this merely rotates the beam section as indicated in Fig. b . The distortion of the cross-section .
