課程名稱︰高等材料力學 課程性質︰選修 課程教師︰陳振山 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰2010.04.22 考試時限（分鐘）：3小時 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Problem 1 Stress resultants (25%) (a) In classical plate theory, we say the model works when the plate deformation is "small". What do we mean "small"? (5%) (b) Define plane stress. (5%) (c) Define in integral form the stress resultamts Mx,My,Mxy,Qx and Qy in terms of stresses σx,σy, and τxy. (5%) (d) Also show the sign convention for positive stress resultants Mx,My,Mxy,Qx and Qy. (5%) Problem 2 Curvature (15%) (a) How many curvature parameters are needed to specify the curvature at a point of a curved surfece? (5%) (b) The middle surface of a plate is deformed into the form w(x,y)=x^2y. At the point (x,y)=(1,1) find alle the curvature parameters. (5%) (c) Draw a Mohr's circle and specify the direction and magnitude of the two principal curvatures. (5%) Problem 3 Deformation of a simply-supported rectangular plate (20%) (a) A square plate of a*a is simply supported on all four edges and is under external pressure p(x,y)=sin(πx/a)sin(πy/a). The plate deformation can be expressed as w(x,y)=Wo*sin(πx/a)sin(πy/a). We know the governing equation of the plate deformation is D(▽^4 w)=p. Find the amplitude Wo? (5%) (b) Write the expression of the reactive forces Mx,My,Mxy,Qx and Qy on the four edges x=0,a and y=0,a? (5%) Draw these reactive forces on the free body diagram.(5%) Be careful with tha direction. Recall: Mx =-D(d^2w/dx^2+μ*d^2w/dy^2), Mxy =-D(1-μ)d^2w/dxdy, Qx =-Dd(▽^2 w)/dx (c) Draw the direction of the four corner forces on the free body diagram.(5%) Problem 4 Boundary conditions of rectangular plates (10%) Consider a rectangular plate of a*b. The edges x=0 is hanged, the edge x=a is also hinged but under uniform bending moment M,the edge y=0 is clamped, and the edge y=b is free. Write the boundary conditons in terms of w on all four edges. Problem 5 Deformation of a clamped circular plate under a point force (20%) (a) A full disk is clamped on the outer rim r=R and is under a concentrated force P=1 at the center. Cut a small circle of the radius r. From the force balance write the expression of the shear force distribution Qr(r)? (5%) (b) For this case the forcing term in the governing equation D(▽^4 w)=p can be set as p=0. The general sol. is then w(r)=C1+C2*lnr+C3*r^2+C4*r^2(lnr) Use the condition (dw/dr)｜(r=0) =0 to eliminate one constant. (5%) (c) Use the relation Qr=-D*d[(1/r)*d(r*(dw/dr)/dr)]/dr to express Qr in terms of the unknown constants (C1 through C4) and compare with the result of (a) to determine the second constant. (5%) (d) Use other two boundary conditions w｜(r=R) =0 and (dw/dr)｜(r=R) =0 to determine the other two constants. (5%) Problem 6 A circular membrane under uniform pressure (10%) Consider a circular memebrane fixed on the outer rim r=R and under uniform pressure p. The membrane is stretched by uniform tension so that Nr=Nθ=N=constant. The governing equation is p+N*[(d^2w/dr^2)+(1/r)*dw/dr]=0 (a) Write general solution, including homogeneous and particular solutions.(5%) (b) From the boundary conditions at the center and the outer rim, determine the deformation of the membrane. (5%) --

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