Basic Energy Principles E = Elastic (or Young’s) modulus G = Shear modulus in Stiffness Analysis = Poisson ratio Stress-Strain Relations Normally, the shear modulus is The application of any theory expressed in terms of the elastic requires knowledge of the physical modulus and Poisson ratio as properties of the material(s) E G comprising the structure. We are 2(1 ) limiting our attention to linear The most widely used civil elastic structural response. engineering structural materials, Further assuming that the material steel and concrete, have uniaxial is homogenous and isotropic, we stress-strain diagrams of the types only need to know two of the shown in Fig. 1. Mild steels yield following three material constants: 1 2
offset criterion. Yield strengths for steel vary from less than 250 MPa to more than 700 MPa. For practical purposes, steel behaves as an ideal material in both tension and compression below the yield Fig. 1: Typical Stress () – Strain (e) or buckling stress. The elastic Curves for (a) Steel and (b) Concrete modulus and Poisson ratio for with a pronounced permanent steel are always close to 200,000 MPa and 0.3, respectively. elongation at a stress ym (Fig.1a). High strength steels yield gra- Concrete is less predictable, but dually, which requires an arbitrary under short-duration compressive stress not greater than u/3 – u/2, definition of its yield strength yh, its behavior is reasonably linear such as the commonly used 0.2%3 4
1 Work and Energy (Fig. 1b in which typical values for
u are: 30 MPa ≤ u ≤ 50 MPa). An The principle of conservation of elastic modulus of E = 22,000 MPa energy is fundamentally important and Poisson ratio of = 0.15 are in structural analysis. This typical for concrete. In using principle, expressed as energy or concrete for analysis, the ACI code work balance, is applicable to both specifies using the gross cross rigid and deformable structures. area properties to perform Rigid structures only require analyses to determine the force multiplying the external forces by distributions in frame structures, the respective displacements. i.e., ignore the reinforcing steel Deformable structures also require and tension cracking in calculating the summation of the internal the force distributions. stresses acting through the
5 6
respective deformations. Internal Limiting attention to gradually work is called strain energy and applied forces, i.e., ignoring inertial must be accounted for in the forces caused by dynamic loads, energy balance. and linear elastic response leads The work dW of a force F acting to through a change in displacement WFdkd 11 d in the direction of F is 00 112 dW Fd (1) kF (3) 22111 Over 1, the total work is F 1 F1 WFd (2) 0 k 1
7 8 1
2 Expanding to a vector of forces where U= strain energy for the and displacements leads to element. Equation (5) is a 1 WF{} (4) homogeneous, quadratic polyno- 2 mial in terms of the local coor- The special case shown dinate element displacements {u} in the right figure: or global coordinate element displacement {v}. 1 u 1 WF0xx Fu 2 v2 Expanding (4) for a single element ({F} = [k] {u} or {F} = [K] {v}): Wu[k]{u}1 2 1 v[K]{v}U (5) 2 9 10
Principle of Virtual Displacements to constructing stiffness equations. In prior chapters we established The principle of virtual the relationships of framework displacements can be stated as analysis directly utilizing the basic If a deformable structure is in conditions of equilibrium and equilibrium and remains in displacement continuity. Hence- equilibrium while it is subject to a forth, we will use energy principles, virtual distortion, the external specifically the principle of virtual virtual work done by the external displacements since it permits forces acting on the structure is mathematical manipulations that equal to the internal virtual work are not possible with direct done by the stress resultants. procedures. We restrict our attention to virtual displacements Recall: virtual imaginary, not since this principle is applicable 11 real, or in essence but not in fact12
3 The principle of virtual displacements Equation (6) is based on the is expressed mathematically as conservation of energy principle,
Wext = Wint (6) i.e. the work done by the external forces going through a virtual F displacement equals the work F1 done by the internal forces due to Wext the same virtual displacement. W The external virtual work can be generalized to a system of forces 1 as s Wqdx()P(7) where Wext = F1 = external ext i i virtual work (shaded blue area in i1
the figure) and Wint = internal 13 14 virtual work.
internal virtual work (dWint) is The internal virtual work (Wint) is a function of the structure type. d( u) dW Fdx (8a) Since this course focuses on frame intdx x members, only axial and bending where u = virtual axial displace- deformations will be considered. ment and Fx = real axial force. Axial Deformation Recalling from your mechanics of
Consider the axial force system materials class that axial strain ex =
shown in Fig. 2. The differential du/dx and the axial force Fx = x A (axial stress times area), (8a) can be rewritten as
dW int e x x Adx (8b)
Integrating (8b) over the length of
15 16 Fig. 2: Axial Deformation the element and substituting
4 Hooke’s law (x = Eex) leads to work is L LL W M dx EI dx WeEAedxint x x int z z z z 0 00 L L d(v)22 dv d( u) du (10) EA dx (9) EI dx dx dx dx22 dx 0 0 For the beam bending (flexure) case (Fig. 3), the internal virtual where v = virtual transverse
displacement; z = d(v)/dx =
virtual rotation; Mz = real moment 2 2 about the z-axis; z = d v/dx = curvature strain about the z-axis;
and Mz = EI kz.
Fig. 3: Bending Deformation 17 18
Analytical Solutions Using Principle of Virtual Displacements NOTE: A difficulty in applying the Consider the simple axial force principle of virtual displacements is structure shown in Fig. 4. The real that functions must be assumed or developed for the real and virtual L 2 1 F , u displacement functions in (9) and x, u x2 2 (10). Development of these Fig. 4: Axial Deformation Structure expressions will follow finite element mechanics, which is displacement u: covered in a later section. u = x/L u2 The real strain is
ex = du/dx = u2/L Imposing a virtual displacement
19 20
5 u2 results in an external virtual work We will consider various of expressions for the virtual
Wext = u2 Fx2 displacement to demonstrate the In order to calculate the internal principle of virtual displacements. virtual work First, consider L d( u) du u = (x/L) u WEAdx 2 int dx dx 0 The internal virtual work: L expressions for u and u over the uuEA WdxEAuu22 length of the axial deformation intLLL 2 2 0 structure must be assumed. We Equating the external and internal will consistently assume the real virtual works gives displacement u: u2 Fx2 = u2 (EA/L) u2 u = (x/L) u2 or
21 22
u2 = Fx2 L/EA The internal virtual work: which is exact. L uxu22 Consider next: WcosdxEAint 2L 2L L u = (x/L)2 u 0 2 EA The internal virtual work: uu 22L L 2u u EA WxdxEAuu22 Which again gives the exact int2 LL 2 2 L 0 solution:
Which again gives the exact u2 = Fx2 L/EA solution: These three virtual displacement
u2 = Fx2 L/EA expressions all resulted in an exact Lastly, consider: solution since the real displace- ment solution was exact. If the u = u2 sin(x/2L) chosen real displacements 23 24
6 A = A (1-x/2L) correspond to stresses that 1 F 1 x2 x, u identically satisfy the conditions 2 L of equilibrium, any form of Fig. 5: Nonprismatic Axial Deformation admissible virtual displacement Structure will suffice to produce the exact Consider next the nonprismatic solution. axial deformation structure of Fig. 5. We will repeat the process Notice the adjective “admissible” in considered for Fig. 4 with front of virtual displacement. reference to the geometry of Fig. Admissible means that the 5. Considering the first case: chosen function is physically u = (x/L) u continuous and satisfies all 2 L essential boundary conditions, i.e., uu WA(1)dxE22 x is appropriately zero at all intLL 1 2L 0 supports. 25 26
3EA1 Equating the external and internal Wuint 2 u 2 4L virtual works leads to Equating the external and internal 3F L u x2 virtual works leads to 2 2EA1 4Fx2 L u2 Considering the third virtual 3EA1 displacement expression: Considering the second virtual u = u2 sin(x/2L) displacement expression: leads to 2 u = (x/L) u2 L uxxu22 leads to W1cosdxEAint 1 2L 2L 2L L L 2 0 2u 22 x u WxdxEAint 1 11EA1 L2 2L L uu22 0 2L EA 1 EA1 2u22 u u (0.818) u 3L 27 22L 28
7 Again, equating the external and The principle of virtual internal virtual works leads to displacements has its greatest
1.222Fx2 L application in producing u2 EA1 approximate solutions. NOTE: None of the three solutions The standard procedure is to match. This is because neither the adopt a virtual displacement of real or virtual displacements are the same form as the real exact. However, we produced displacement. Adopting different three good approximate solutions. forms for the real and virtual The exact solution for Fig. 5 is displacements can lead to
1.387Fx2 L unsymmetric stiffness matrices. u2 EA1
29 30
Special Transformations Contragradience Principal in Analysis If one transformation is known, e.g., Congruent Transformation the local to global displacements, the force transformation will be A matrix triple product in which the transpose of the displacement pre-multiplying matrix is the transformation provided both sets transpose of the post-multiplying of forces and displacements are matrix, e.g. conjugate and vice versa. Such a [C][A][B][A]or[D][A][B][A]TTtransformation is known as Significance of the transformation is contragradient (or contragredient) that [C] and [D] will each be sym- under the stipulated conditions of metric if [B] is symmetric, which is conjugacy. Conjugate simply one of the reasons all our stiffness means that the force-displacement pair only produce work in the matrices were symmetric. 31 32 direction of the displacement.
8 For linear analysis, this is always the case when using orthogonal coordinate systems. A good example are the coordinate transformations for a truss member (17.21) in which the transformation matrices are rectangular:
{ua} = [Ta] {va} T {Faaa } [T ] {Q }
cossin00 [Ta ] 00cossin
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9