Solid Mechanics in Engineering

Raymond Parnes

Chapter 8 Symmetric bending of beams - basic relations and stresses - all with Video Answers

Educators

Chapter Questions

Problem 1

Sketch the shear and moment diagrams for the beams shown in Figs. (8P.1a-d) and give the values of all critical ordinates (in terms of $a, b$ and $L$ ). Indicate in which segment (if any) of the beam a state of pure bending exists.
(a)figure cant copy
(b)figure cant copy
(c)figure cant copy
(d)figure cant copy

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Problem 2

The beam shown in Fig. (8P.2) is subjected to two vertical forces and an eccentric horizontal force, each having the same magnitude $P$. (a) Sketch the shear and moment diagrams and show all critical ordinates (in terms of $a, b$ and/or c). (b) What is the required value of $e$ if segment $B C$ is to be in a state of pure bending? Sketch the resulting moment diagram.

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Problem 3

Express the shear force $V(x)$ and moment $M(x)$ as a function of $x$ for the beams shown in Figs. (8P.3a-j). Sketch the variation with $x$ showing all critical values (maxima and minima) and verify that the expressions satisfy the relations $\mathrm{d} M(x) / \mathrm{d} x=$ $V(x), d V(x) / d x=-q(x)$ and $d^2 M(x) / d x^2=-q(x)$.

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Problem 4

Express the shear force $V(x)$ and moment $M(x)$ as a function of $x$ in terms of $P_1, P_2, a_1$ and $a_2$ in the two regions $0 \leq x<a_1$ and $a_1<x \leq a_2$, for the beams shown in Fig. (8P.4) and verify that the expressions satisfy the differential relations for beams.

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Problem 5

Express the shear force $V(x)$ and moment $M(x)$ as a function of $x$ in terms of $w$ and $P$ for the beam shown in Fig. (8P.5). Sketch the variation with $x$ if $P=W L$ and verify that the expressions satisfy the differential relations for beams.

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Problem 6

Determine the axial force $F$ and express the shear force $V(x)$ and moment $M(x)$ within the span $A C$ as a function of $x$ for the beams shown in Figs. (8P.6a-e). Sketch the variation with $x$ and verify that the expressions satisfy the differential relations for beams.

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05:01

Problem 7

Beam $A B C$ is subjected to an eccentric horizontal force $P$ and a uniformly distributed load $w(\mathrm{~N} / \mathrm{m})$, as shown in Fig. (8P.7), where $0 \leq P \leq w L^2 / 8$ e. (a) Determine the expressions for $V(x)$ and $M(x)$ in segments $A B$ and $B C$. (b) Sketch the shear and moment diagrams and show all critical values. (c) What value of $P$ (in terms of $w, L$ and e) will yield the (algebraically) smallest maximum moment in segment $A B$, i.e. in $0 \leq x \leq L / 2$.
Evaluate this maximum moment and sketch the shear and moment diagrams for this value of $P$.

Eric Mockensturm

Numerade Educator

Problem 8

For the structure $A B C D$ containing a hinge at B , as shown in Fig. (8P.8), (a) determine $V(x)$ and $M(x)$ and (b) draw the shear and moment diagrams.

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02:35

Problem 9

The structure shown in fig. (8P.9) consists of two beams ABC and DEF, respectively, which are connected by means of a roller. Sketch the appropriate shear and moment diagrams for each beam and show all critical ordinates.

Surendra Kumar

Numerade Educator

02:35

Problem 10

The structure shown in Fig. (8P. 10) consists of two beams ABC and DEF, containing a hinge at B and E, respectively, and connected by means of a roller. Sketch the appropriate shear and moment diagrams for each beam and show all critical ordinates.

Surendra Kumar

Numerade Educator

Problem 11

Shear and moment diagrams are given for each of the beam structures shown in Figs. ( 8 P. 11a-c). Determine the loadings on the beam for each case.

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Problem 12

A cantilever beam $A B$, as shown in Fig. ( 8 P.12), free at $A(x=0)$ and fixed at $B(x=L)$, is subjected to a loading $q(x)=q_0(x / L)^n$, where $n \geq 0$ is an integer. (a) Prior to solving this problem, estimate whether one should expect the reactions at $B$ to increase or decrease with increasing values of $n$, (b) determine the shear $V(x)$ and moment $M(x)$, (c) find the reactions $R_{\mathrm{B}}$ and $M_{\mathrm{B}}$ and (d) sketch the shear and moment diagrams for $n=0,0.5,1$ and 2 .

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Problem 13

For the loading $q(x)=q_0 \sin (\pi \times / 2 L)$ of the simply supported beam shown in Fig. (8P.13), (a) determine the reactions at A and B , (b) determine $V(x)$ and $M(x)$ and (c) sketch the shear and moment diagrams.

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Problem 14

* A simply supported beam $A B$ is subjected to a variable loading, as shown in Fig. (8P.14), which is given by the approximating expression $q_s(x)=q_0\left(1-e^{-\alpha x / 4}\right)$. (a) Evaluate the reaction $\left(R_A\right)_2$ due to $q_0$. (b) Determine expressions for $V(x, \alpha)$ and $M(x, a)$. (c) Determine expressions for $V(x)$ and $M(x)$ as $\alpha \rightarrow \infty$. To what loading does this correspond? (d) If the loading, instead, is expressed by the approximation $q_b(x)=q_0 \tanh (\alpha x / L)$, based on an analysis of the relative magnitudes of $q_{\mathrm{b}}$ and $q_b[i . e$. without evaluating the resulting reaction $\left.\left(R_A\right)_b\right]$, estimate if $\left(R_A\right)_b$ is greater or less than $\left(R_A\right)_{\text {s }}$.

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Problem 15

A beam of depth $d$ is subjected to end couples. Upon bending, the resulting extensional strain $\epsilon_x$ at the bottom of the beam is $\epsilon_x=\epsilon_{\mathrm{t}}>0$ and $\epsilon_x=\epsilon_c<0$ at the top. (a) Determine the radius of curvature $\bar{R}$ of the middle surface (i.e., the surface midway between the bottom and top of the bearm) in terms of $\epsilon_\epsilon \epsilon_T$ and $d$. (b) What is the strain at the middle surface? Express the answer in terms of $\bar{R}$ and $R$, the radius of curvature to the neutral axis. (c) If the neutral axis lies on the middle surface, show that $\epsilon_{\mathrm{t}}=-\epsilon_{\mathrm{C}}$.

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04:04

Problem 16

A long high-strength copper wire ( $E=120 \mathrm{GPa}$ ), 3 mm in diameter, is wound about a $1.5-\mathrm{m}$ diameter drum. Determine (a) the bending moment in the wire and (b) the maximum flexural stress in the wire.

Satpal Satpal

Numerade Educator

02:23

Problem 17

Material along the lateral surfaces of a long circular cylindrical $\log$ is to be sawed off to form a rectangular cross-section $b \times d$, as shown in Fig. (8P.17). If the resulting beam is to be subjected to a moment about the $z$-axis, determine (a) the optimal ratio $d / b$ that minimises the curvature of the beam and (b) the ratio $d / b$ that minimises the maximum flexural stress.

Anand Jangid

Numerade Educator

Problem 18

A beam having a square cross-sectional area of $64 \mathrm{~mm}^2$ is subjected to a pure moment. The strain $\epsilon_x$ at the top of the cross-section is found to be $1600 \mu$. Determine (a) the radius of curvature of the deformed beam and (b) the moment acting on any cross-section if the beam is made of a high-strength steel ( $E=200 \mathrm{GPa}$ ) and (c) the maximum flexural stress.

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01:43

Problem 19

A beam, whose cross-section is as shown in Fig. (8P.19), is subjected to a moment $M$ about the $z$-axis. If the allowable stress is 120 MPa , determine the maximum permissible moment that can be applied.

Surendra Kumar

Numerade Educator

Problem 20

A beam whose cross-section consists of a semi-circle of radius $R$, as shown in Fig. (8P.20), is subjected to a moment about the horizontal axis. (a) Determine the maximum flexural stress (in absolute value), i.e. $\left|\sigma_x\right|_{\text {max }}$ in terms of $M$ and $R$. (b) What are the maximum tensile and compressive stresses if $R=2 \mathrm{~cm}$ and $M=10,000 \mathrm{~N}-\mathrm{cm}$.

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Problem 21

A beam having a rectangular cross-section (i.e, width band depth $\phi$ ) is subjected to a moment $M_2$. The beam is made of a material whose stress-strain relation is given by $\sigma=\alpha \epsilon^n$, where $\alpha>0$ is a constant and $0<n$ is an odd integer, i.e., $n=1,3,5, \ldots$ relation between the moment and the curvature $K=1 / R$ of the beam is given by if $\alpha \equiv E$ and $n=1$.

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Problem 22

A beam having a rectangular cross-section (with width $b$ and depth $d$ ) is subjected to a moment $M$. The beam is made of a material whose behaviour can be described by

$$
\sigma= \begin{cases}\alpha \epsilon^n, & \epsilon \geq 0 \\ -\alpha|\epsilon|^n, & \epsilon \leq 0\end{cases}
$$
where $\alpha \geq 0$ is a constant and $0<n \leq 1$. Show that the expressions for the flexural stress $\sigma$ and the radius of curvature $\kappa$ for this case, $0<n \leq 1$, are identical to those of Problem 8.21 for the case $1 \leq n, n$ odd.

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02:20

Problem 23

A prismatic beam is composed of two or more homogeneous materials, say ' $a$ ' and ' $b$ ', as shown in Fig. (8P.23), each having different material properties. Is it necessary to make an assumption that, when subjected to a bending moment $M$, all cross-sections remain plane and perpendicular to the deformed longitudinal axis or is this a valid conclusion, as in the case (considered in Section 6 of this chapter) of a beam consisting of a homogeneous material.

Surendra Kumar

Numerade Educator

Problem 24

A rectangular beam ( $b \times d$ ), as shown in Fig. (8P.23), is composed of two materials, ' $a$ ' and ' $b$ ', having moduli of elasticity $E_{\mathrm{a}}$ and $E_b$ respectively. Determine the location of the neutral axis as measured from the interface of the two materials.

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Problem 25

A rectangular beam ( $b \times d$ ), as shown in Fig. (8P.25), is composed of thin laminates each having a different modulus of elasticity. The resulting inhomogeneous beam can be considered as having a varying modulus of elasticity, approximated by the expression $E(\eta)=E_0[1+\beta(\eta / \sigma)]$, where $\eta$ is measured from the top of the beam and $\beta$ is a constant. Determine (a) the location $\bar{y}$ of the neutral axis (measured from the top of the beam) and (b) the radius of curvature of the neutral surface if the beam is subjected to a moment $M$ about the horizontal axis.

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01:04

Problem 26

The cross-section of a steel beam ( $E=200 \mathrm{GPa}$ ) is constructed my means of plates, which are welded together to form an l-section, as shown in Fig. (8P.26),
(a) Determine the section modulus $S=1 / c$ of the cross-section. Compare the results with a similar standard wide flange section, e.g. a W203 $\times 36$ section, as given in the tables of Appendix E (see footnote p. 269).
(b) If the maximum allowable flexural stress in the beam is 150 MPa , what is the maximum permissible moment $M$ that can be applied about the $z$-axis?
(c) What is the resultant axial force in the flange abc under this positive moment $M$ ?
(d) What is the resultant axial force in the upper part, bd, of the web under this moment?
(e) What part of the total moment $M$ is resisted by the two flanges and what part by the web?
What conclusions can be drawn from these answers?

Kratika Bhadauria

Numerade Educator

01:42

Problem 27

A beam having a trapezoidal cross-section, as shown in Fig. (8P.27), is subjected to a given moment $M$ about the horizontal axis. Determine the maximum elastic moment $M=M_E$ under which the beam behaves elastically if $\sigma_E$ is the elastic limit.

Surendra Kumar

Numerade Educator

Problem 28

The overhanging beam shown in Fig. (8P.28a) is simply supported at $B$, fixed at $E$ and contains a hinge at $C$. The cross-section and location of the neutral axis are shown in Fig. (8P.28b). The moment of inertia about the neutral axis is given as $I=720 \mathrm{~cm}^4$. (a) Draw the shear and moment diagrams and show all critical ordinates. (b) Determine the maximum flexural tensile and compressive stresses $\sigma_x$ and indicate, by means of a sketch, at which cross-sections and at which points within the cross-section they occur. (c) Determine the maximum average shear stress $\left|\tau_{x y}\right|$, which occurs along the line $\mathrm{e}-\mathrm{c}$ in the cross-section and indicate the direction of the shear stress acting on a positive $x$-plane.

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Problem 29

The wooden beam, shown in Fig. (8P.29a), is subjected to several concentrated loads. The cross-section of the beam consists of four wooden components, which are glued together, as shown in Fig. (8P.29b). Determine (a) the maximum flexural tensile stress existing in the beam, (b) the maximum compressive stress, (c) the maximum shear stress in the beam and (d) the maximum shear stress in the glue.

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Problem 30

The wooden beam, subjected to two vertical forces applied at points B and C [Fig. (8P.30a)], has a cross-section as shown in Fig. (8P.30b). Determine (a) the maximum tensile and compressive flexural stresses $\sigma_x$ in segment $B C$ and indicate, by means of a figure, where they occur in the cross-section and (b) the average shear stress along line $c-c$, which exists in segment $C D$. Indicate, by means of a figure, the direction of $\tau$ acting on a positve $x$-plane.

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07:57

Problem 31

The $$\$ 305 \times 47$$ steel beam $(E=200 \mathrm{GPa})$ shown in Fig. ( 8 P. 31$)$ is subjected to two vertical forces, $P=50 \mathrm{kN}$, which are applied at points B and C . Determine (a) the maximum tensile and compressive flexural stresses $\sigma_x$ in segment BC , (b) the average shear stress along line $c-c$, which exists in segment $A B$, (c) the average shear stress in the web at the neutral axis (i) using the expression of Eq. (8.8.4) and (ii) assuming that the shear stress is distributed uniformly over the web of the section. What is the percentage difference in the two answers? and (d) the radius of curvature of the beam within segment BC .
(a)figure cant copy
(b)figure cant copy

Ajay Singhal

Numerade Educator

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Problem 32

Determine the average maximum shear stress components along lines a-a and c-c for the beam shown in Fig. (8P.32). Indicate the directions of the shear stress acting on a positive $x$-plane by means of a sketch.
(a)figure cant copy
(b)figure cant copy

Susan Hallstrom

Numerade Educator

Problem 33

A circular pipe with inner and outer radius a and b, respectively, is used as a cantilever beam. A vertical force $P$, acting along the axis of symmetry, as shown in Fig. (8P.33), is applied at the free end of the beam. (a) Based on symmetry considerations, what conclusions can be drawn about the shear flow in the cross-section? (b) Determine the average shear stress in the pipe, $r_x$, along the line $c-c$, as a function of $\theta$. (c) If the pipe is a thin wall section with $t=b-a \ll \bar{R}$, where $\bar{R}=(b+a) / 2$ is the mean radius, show that the shear stress $\tau_{x \theta}$ is given by the approximate relation $\tau_{x \theta}=V \sin \theta / \pi \bar{R} t$. (d) Show that the resultant of the shear stresses existing in the entire cross-section is in equilibrium with the vertical load $P$.

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Problem 34

A beam having a triangular cross-section, as shown in Fig. (8P.34), is subjected to a positive shear force $V_y$. (a) Determine the maximum average shear stress $t_{x y}$ in the cross-section and the location of the line $\mathrm{c}-\mathrm{c}$ along which this stress occurs. Express the answer as $\mathrm{r}_{x y}=k \frac{V_y}{A}$, where $A$ is the cross-sectional area of the beam. (b) Are the boundary conditions satisfied at the end points of the line $c-c$ ? Explain.

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Problem 35

Repeat Problem 8.34 for a diamond-shaped cross-section shown in Fig. (8P.35).

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09:27

Problem 36

The cross-section of a beam shown in Fig. (8P.36) consists of a flange whose width is equal to the depth of the beam $(b=d)$, both having the same thickness $t$. The beam is subjected to a shear force acting in the $y$-direction. Determine the possible range of the ratio $K=\frac{(r) m,}{(r)}$, namely the ratio of the average shear stress in the web at the neutral axis to that along line $a-a$.

Rashmi Sinha

Numerade Educator

03:13

Problem 37

A beam is made up of four wooden components, which are glued together to form a cross-section, as shown in Fig. (8P.37). (a) Determine the average shear stress in the glue, $(\tau)_y$ due to a vertical shear force $V_y$ acting in the $y$-direction and $(\tau)_z$ due to a shear force $V_z$ acting in the $z$-direction. (b) If the cross-section is square (i.e., $b_0=d_0$ and $\left.b_i=d_i\right)$ and if $V_y=V_z$, show that $\frac{\left(r_y\right.}{\left(r_i\right.}=\frac{b_n}{b_j}\left(\frac{h_j^2+b_2^2}{b_j^2-b_i^2}\right)$.

Chai Santi

Numerade Educator

Problem 38

A beam made up of four wooden components, which are glued together to form a cross-section, as shown in Fig. (8P.38), is subjected to a shear force $V_y$ acting in the $y$-direction. Determine the average shear stress in the glue along the line $c-c$.

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09:27

Problem 39

A beam $A B C$, having a triangular cross-section, is simply supported at points $A$ and $B$ (where $a \leq L$ ), as shown in Fig. (8P.39), and is subjected to a uniformly distributed load $w$. (a) Determine the ratio a/L such that the largest tensile stress existing at points c of the cross-section is equal to the largest tensile stress existing at points d. (b) Based on the value $a / L$ obtained above, sketch the resulting shear and moment diagrams and show all critical ordinates. (c) Based on the same value of a/L, determine the maximum shear stress $\left|r_{x y}\right|$ existing in the beam if the cross-section is an equilateral triangle with sides $b$. Express the answer in terms of $w, L$ and $b$. [See comment (iii) following Example 8.13.]

Rashmi Sinha

Numerade Educator

02:58

Problem 40

Two C305 $\times 45$ channels are connected to two plates, each $260 \mathrm{~mm} \times 15 \mathrm{~mm}$, by means of bolts, as shown in Fig. (8P.40), to form the cross-section of a simply supported beam. If the bolts are spaced 80 cm apart, and each bolt can carry an allowable force in shear of 450 N , determine the maximum load $P$ that can be applied at the centre of the beam (assuming that this shear criterion is the governing criterion).

Vidhi Bhatt

Numerade Educator

02:45

Problem 41

Four angles L102 $\times 102 \times 9.5$ are connected to a plate 12 -mm thick by means of two $20-\mathrm{mm}$ diameter bolts (spaced 100 mm apart along the longitudinal axis) to form a cross-section, as shown in Fig. (8P,41). The section is used as a cantilever beam loaded at the free end by a force $P=240 \mathrm{kN}$. Assuming the shear stress is distributed uniformly over the cross-section of each bolt, determine the average shear stress in the bolts.

Prashant Bana

Numerade Educator

Problem 42

The cantilevered beam shown in Fig. (8P.42) has a rectangular cross-section of constant width but of varying depth $h(x)$. The beam is loaded at the free end. (a) Assuming that cross-sections remain plane and perpendicular to the deformed longitudinal axis, determine the required variation of the depth $h(x)$ in order that the maximum value of the flexure stress at all cross-sections be constant over the length of the beam, $0 \leq x<L$. (Such a beam is referred to as a beam of constant strength.) (b) Recalling that the expression $\sigma_x=M y / I(x)$ is only a good approximation if the lateral surfaces of the beam have a small slope $\theta$ with respect to the $x$-axis, say $|\theta|=5^{\circ}$, use this criterion and the answer from part (a) to determine the range of $x / L$ for which the given expression for $\sigma_x$ is a good approximation for a given beam with $h_0 / L=1 / 15$.

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Problem 43

A simply supported wooden beam whose constant width is $b$ and whose depth $h(x)$ varies with $x$ is to be used as a beam of constant strength such that the maximum flexural stress is constant over its entire length $L, 0<x<L$. The beam is subjected to a distributed load $q(x)=q_0 \sin (\pi x / L)$, as shown in Fig. (8P.43). The maximum permissible flexural stress is given as $\sigma_0$. (a) Determine the maximum required depth $h_0$ at $x=L / 2$ in terms of the given parameters and loading. (b) Determine the required variation of $h(x)$ in terms of $h_0$. (c) Referring to the limitation given in part (b) of Problem 8.42, and using the same criterion, determine the range of the loading $q_0$ that can be applied to the beam while satisfying the given criterion over its entire length. (d) Using the result of (c) above, determine the maximum loading $q_0$ and the required depth $h_0$ if $\sigma_0=10 \mathrm{MPa}, b=10 \mathrm{~cm}$ and $L=3 \mathrm{~m}$.

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02:49

Problem 44

A cantilevered steel beam whose cross-section is specified as a W254×45 section, is loaded as shown in Fig. (8P.44). Determine the total allowable load on the beam if the allowable fiexural stress and shear stress are specified as $\sigma=150 \mathrm{MPa}$ and $\mathrm{r}=80 \mathrm{MPa}$, respectively. (Assume that the shear stress in the web is uniformly distributed over the area of the web.)

Chai Santi

Numerade Educator

01:39

Problem 45

A simply supported beam of length $L$ [whose cross-section is, say, a standard ( 5 ), or channel (C) section) is to carry a load $W(\mathrm{~N})$, uniformly distributed along its length. Assume that ail properties of the cross-section are known, namely the section modulus $S$, the depth of the beam $d$ and the dimensions of the flange and web. The allowable flexural and shear stress are given as $\sigma_{\text {stilow }}$ and $\tau_{\text {allow }}$. (a) Determine, the allowable load ( $w_1$ according to the flexure criterion and ( $w_{\text {}}$, according to the shear criteria in terms of the given parameters. (Assume that the shear force is resisted by the web and that the shear stress in the web is uniformly distributed over the area of the web.) (b) Determine the ranges of $L$ for which the flexure criterion governs and for which the shear criterion governs the design. What conclusion can be drawn from this result? (c) For a steel beam of length 5 m with a cross-section $\$ 203 \times 34$, determine the allowable load $W$ if $\sigma_{\text {allow }}=200 \mathrm{MPa}$ and $\mathrm{r}_{\text {allow }}=100 \mathrm{MPa}$.

Dominador Tan

Numerade Educator

01:22

Problem 46

A wooden beam ABC, containing a hinge at B, is loaded, as shown in Fig (8P.46a), by means of a uniformly distributed load between A and B. Several wooden components are nailed together to form a cross-section, as shown in Fig. (8P.46b), with $t_{z z}=2 \times 10^6 \mathrm{~mm}^4$. The maximum shear force that can be carried by each nail is given as $f=300 \mathrm{~N}$. What is the maximum permitted spacing of (a) nails 'a' in the segment $B C$ and (b) nails ' $b$ ' in the segment $B C$ ?

Hast Aggarwal

Numerade Educator

02:54

Problem 47

A beam is made up of four component parts, which are glued together to form a hollow box section, as shown in Fig. (8P.47). The allowable flexural stress of the wood is given as 25 MPa and the allowable stress in the glue is given as 2 MPa . The beam is to be used as a simply supported beam 2 m in length, and subjected to a load $W(\mathrm{~N})$, which is uniformly distributed over its entire length. Determine the maximum load $W$ that the beam can carry.

Dominador Tan

Numerade Educator

02:02

Problem 48

A W254 $\times 67$ steel beam CD, 6 m in length, is subjected to a force of 200 kN , which is to be applied via a steel beam $A B$ of length $L$ placed symmetrically as shown in Fig. (8P.48). The allowable flexural and shear stress are given as 140 MPa and 85 MPa , respectively. (a) Determine the shortest permissible length $L$ of the beam $A B$. (b) Choose the most suitable structural W -section (namely a beam having the least weight per unit length) from the tables (see Appendix E) that may be used for the beam AB.

Hast Aggarwal

Numerade Educator

Problem 49

Consider a steel beam having either a W- or S-section with given $I_0$, section modulus $S_0$ and depth d. Assume, as in Example 8.16, that it is required to strengthen the beam in flexure by adding plates having dimensions $b \times t$ at the top and bottom flanges [see Fig. (8.10.3b)] thus increasing the section modulus to $S>S_0$. Show that if the thickness $t \ll d$, the increased modulus is given by the approximate expression $S=S_0+b t d$.

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06:27

Problem 50

A steel beam $A B C D$, containing a hinge at $C$, is simply supported at $B$ and $f i x e d$ at D, as shown in Fig. (8P.50). Using an allowable flexure stress $\sigma=180 \mathrm{MPa}_i$ an S-section of minimum weight was originally chosen to support a load $P=60 \mathrm{kN}$ at $A$, (a) What section was chosen? (b) Due to changes of the loading, it is now required to strengthen the beam to resist flexure in the segment CD in order to carry a load $P=120 \mathrm{kN}$ at $A$ by attaching plates (having the same width as the flanges) to the top and bottom flanges by means of two bolts at each flange, as shown in Fig. (8.10.3). Using the result given in Problem 8.49, determine the required thickness of the plates. Check that the resulting flexural stress is within the allowable limits. (c) Determine the maximum permissible spacing of the bolts, s, along the longitudinal axis if the thickness of the plates is 25 mm , if the bolts are 20 mm in diameter and if the allowable shear stress in the bolts is $\tau=50 \mathrm{MPa}$. (Assume the shear stress in the bolts is the average stress distributed uniformly over the cross-section.)

Rashmi Sinha

Numerade Educator

02:58

Problem 51

A beam is made up of two identical angles, which are connected by means of two bolts, as shown in Fig. (8P.51). When the beam is subjected to vertical loads acting along the $y$-axis, should one expect the bolts to undergo shear? Justify the answer (i) by physical reasoning and (ii) by analytical reasoning based on Eq. (8.8.4).

Vidhi Bhatt

Numerade Educator

07:57

Problem 52

A cantilever beam is made up of two L127 $\times 127 \times 9.5$ angles and a plate ( $254 \mathrm{~mm} \times 10 \mathrm{~mm}$ ). The angles and plate are connected by means of $20-\mathrm{mm}$ diameter bolts ' $a$ ' and ' $b$ ' as shown in Fig. (8P.52). A load $P=2000 \mathrm{~N}$ acts at the free end of the beam in the $y$-direction. Determine (a) the maximum flexural tensile and compressive stress in the beam and (b) the maximum shear stress in bolts ' $a$ ' and ' $b$ ' (assuming that the shear stress is uniformly distributed over their cross-sections) if the bolts are spaced at $40-\mathrm{cm}$ intervals along the longitudinal axis.

Ajay Singhal

Numerade Educator

03:37

Problem 53

By bonding two aluminium bars ( $E=70 \mathrm{GPa}$ ) to two brass bars ( $E=105 \mathrm{GPa}$ ), it is possible to form two different composite cross-sections of a beam, as shown in Figs. ( 8 P .53 a and b). If a moment $M=2 \mathrm{kN}-\mathrm{m}$ acts about the $z$-axis, determine the maximum flexural stress in the aluminium and the brass for each case.
(a)figure cant copy
(b)figure cant copy

Surendra Kumar

Numerade Educator

Problem 54

An aluminium bar ( $E=70 \mathrm{GPa}$ ) and a steel bar ( $E=200 \mathrm{GPa}$ ) are bonded together to form a composite cross-section, as shown in Fig. (8P.54). Determine (a) the maximum flexural stress in the steel and aluminium if the beam is subjected to a pure positive moment $M=1200 \mathrm{~N} \mathrm{~m}$ about the horizontal axis and (b) the radius of curvature of the deformed beam.

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01:04

Problem 55

A rectangular wooden beam ( $150 \mathrm{~mm} \times 250 \mathrm{~mm}$ ) with $E=12 \mathrm{GPa}$ is reinforced by means of two steel plates ( $E=200 \mathrm{GPa}$ ), as shown in Fig. (8P.55). (a) If the allowable stress in the wood and steel are given as 12 and 200 MPa , respectively, determine the maximum permissible moment that can act about the $z$-axis. (b) Determine the radius of curvature of the beam under this moment.

Kratika Bhadauria

Numerade Educator

02:44

Problem 56

A wooden beam ( $E=10 \mathrm{GPa}$ ) having a rectangular shape ( $150 \mathrm{~mm} \times 350 \mathrm{~mm}$ ) is reinforced by means of a steel plate ( $E=200 \mathrm{GPa}$ ), which is fastened securely to its top face, as shown in Fig. (8P.56). Determine the maximum flexural stress in the wood when the maximum stress in the plate is 75 MPa .

Narayan Hari

Numerade Educator

Problem 57

The cross-section of a rectangular reinforced concrete beam, reinforced by steel rods as shown in Fig. (8P.57a), can be considered as a composite of two materials: concrete and steel. As is characteristic of all brittle materials, concrete being such a material, is very weak in tension and cracks form in the tension zone when the beam undergoes flexure. It is therefore usual, when designing reinforced concrete beams according to elastic theory, to assume that the concrete can withstand compression but no tension, i.e. the concrete is assumed to withstand only flexural stresses $\sigma \leq 0$. Tension in the beam is therefore assumed to be carried only by the steel rods whose (total) cross-sectional area is denoted as $A_5$. Based on this assumption, the transformed section is as shown in Fig. (8P.57b).
(a)figure cant copy
(b)figure cant copy
(a) Denoting the modulus of elasticity of the concrete and steel as $E_c$ and $E_s$, respectively, and letting $E_s=n E_c$, show that the location of the neutral axis $y$ of the rectangular beam is given by the expression

$$
\bar{y}=\frac{n A_s}{b}\left[\left(1+\frac{2 b d}{n A_s}\right)^{1 / 2}-1\right]
$$
(b) Show that if the beam is subjected to a positive moment $M$ about the neutral axis, the maximum stress in the concrete and steel is given by

$$
\sigma_{\mathrm{c}}=-\frac{6 M}{b \bar{y}(3 d-\bar{y})^{\prime}}, \quad \sigma_{\mathrm{s}}=\frac{3 M}{A_{\mathrm{s}}(3 d-\bar{y})}
$$

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Problem 58

A reinforced concrete beam, 8 m in length, has a rectangular cross-section as shown in Fig. (8P.57a), with $b=200 \mathrm{~mm}$ and $h=350 \mathrm{~mm}$. Four steel rods, each 15 mm in diameter, are placed 75 mm from the bottom of the beam. A concentrated force $P=10 \mathrm{kN}$, located at the centre of the beam, acts in the $y$-direction. If $E_{\mathrm{s}}=200 \mathrm{GPa}$ and $E_c=15 \mathrm{GPa}$, determine the maximum flexural stress in the concrete and the steel.

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Problem 59

In designing a rectangular reinforced concrete beam, it is often desired to achieve a 'balanced design', namely one for which the maximum stress in the concrete and steel are equal to the maximum allowable stress in the two materials, $\sigma_c$ and $\sigma_s$, respectively. Show that for such a balanced design, the location $\bar{y}$ of the neutral axis is given by

$$
\bar{y}=\frac{d}{1+\sigma_{\mathrm{s}} E_{\mathrm{c}} / \sigma_{\mathrm{c}} E_{\mathrm{s}}} .
$$

where $y$ and $d$ are as shown in Fig. (8P.57b) and $E_c$ and $E_{\mathrm{s}}$ are the respective moduli of elasticity.

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Problem 60

A cantilevered beam is subjected to axial and transverse loads, as shown in Fig. (8P.60). Determine the flexural and shear stresses at points $a, b$ and $c$ due to the applied loading.

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02:37

Problem 61

An eccentric vertical load $P$ is applied (with an eccentricity e) to the free end of a cantilever beam having a circular cross-section as shown in Fig. (8P.61). Determine the maximum shear stress that exists in the beam and indicate, by means of a sketch, where it occurs in the cross-section.

Chai Santi

Numerade Educator

Problem 62

An elastic circular bar of radius $R$ and length $L$, fixed at $x=0$, is subjected to an eccentric force $P$ at its free end. The force, lying in the $x-y$ plane and inclined at an angle $\alpha$ with respect to the $x$-axis, acts at point $\mathrm{B}(x=L, y=\mathrm{e})$ as shown in fig. (8P.62a). (a) Locate the position of the neutral axis at the section $x=0$ in terms of the given parameters of the problem, i.e. determine $y_0=y_0(L, R, C, \alpha)$. (b) If $P$ acts in the $x$-direction ( $\alpha=0$ ), where must the force be applied (i.e., what must be the value of e) such that $\sigma_x=0$ at point $C$ of Fig. (8P.62a)? (c) If $P$, acting in the $x$-directions is applied at point B [as found in (b) above], and an additional torque $T=P e$ is applied, as shown in Fig. (8P.62b), what is the maximum shear stress $\tau_{\text {max }}$ at point D ? On which plane (defined by its normal $n$ with respect to the $x$-axis) does $\mathrm{r}_{\text {max }}$ act?
a.figure cant copy
b.figure cant copy

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07:57

Problem 63

A simply supported steel beam ABC is loaded as shown in Fig. (8P.63), where $P=200 \mathrm{kN}$ is inclined by an angle of $30^{\circ}$ with respect to the $x$-axis. In addition, an axial compressive force acts at the end $C$. Determine the maximum tensile and compressive stresses $\sigma_x$ that exists (i) immediately to the left of point $B$ and (ii) immediately to the right of $B$ if the cross-section is a W305 $\times 97$ section.

Ajay Singhal

Numerade Educator

05:22

Problem 64

An eccentric compressive force $P$ is applied at point $B$ in the $x$-direction with an eccentricity with respect to the centroidal longitudinal axis of a beam having a rectangular cross-section $b \times d$, as shown in Fig. (8P.64). Show that the resulting stress $\sigma_x$ is compressive (i.e., $\sigma_x \leq 0$ ) throughout the cross-section, provided that the force is applied within the 'core' of the cross-section, as shown in Fig. (8.12.6).

Chai Santi

Numerade Educator

Problem 65

A beam ABCD having a square cross-section with sides $b$ is supported at points E and $F$ and is loaded as shown in Fig. (8P.65). Determine the maximum flexural stresses and the maximum shear stress that exist in segments $B C$ and $C D$. Note: $b \ll L$.

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Problem 66

A force $P=500 \mathrm{~N}$ is applied at point A to a bent rod, as shown in Fig. (8P.66). Determine the normal and shear stresses at points $B$ and $C$.

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09:13

Problem 67

Three strain gauges are attached to an aluminium rod ( $E=70 \mathrm{GPa}$ ), $1 \mathrm{~cm} \times$ 1 cm in cross-section at points A, B and C, as shown in Fig. (8P.67). When $P$ and $F$ are applied, the following readings are obtained: $\epsilon_A=550 \mu, \epsilon_B=400 \mu$ and $\epsilon_C=-300 \mu$. Determine the magnitude of $P, F$ and the position b of $P$ as measured from $B$.

Chai Santi

Numerade Educator

04:16

Problem 68

Given a cylindrical rod $A B$ of length $L$, diameter $d$ and whose material density is $\rho\left(\mathrm{N} / \mathrm{m}^3\right)$. As shown in Fig. (8P.68), the rod is supported at A by means of a linear torsional spring having constant $\beta(\mathrm{N}-\mathrm{m} / \mathrm{rad})$ such that when the rod is inclined by an angle $\theta$ with respect to the vertical axis, the spring exerts a moment $M=\beta \theta$. Determine the minimum angle $\theta$ with respect to the vertical axis at which a tensile stress occurs in the rod.

Donald Albin

Numerade Educator

01:22

Problem 69

A circular wire having diameter $d$ is wound as a helix (with radius $R$ ) to form a coiled spring, as shown in Fig. (8P.69). If a tensile force $P$ is exerted at the two ends, determine the maximum shear stress within the wire. (Note: The average maximum shear stress in a rod of circular cross-section of area $A$ when subjected to a shear force $V$, is $\tau=\frac{4}{3} \frac{V}{A}$.)

Chai Santi

Numerade Educator

01:04

Problem 70

The cross-section of a beam, whose material behaves as an ideal elastic-plastic material with yield stress in tension and compression $\pm \sigma_0$ [see Fig. (8P.70b)], has a triangular shape as shown in Fig. (8P.70a). Determine (a) the maximum moment $M_E$ that can be applied about a horizontal axis of the cross-section for its behaviour to remain elastic, (b) the location $y_p$ of the neutral axis (measured from the apex) as the moment reaches the fully plastic moment $M_{\mathrm{p}}$ and (c) the value of $M_p$ and the ratio $M_{\mathrm{P}} / M_{\mathrm{E}}$.
(a)figure cant copy
(b)figure cant copy

Hast Aggarwal

Numerade Educator

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Problem 71

A steel beam, assumed to behave as an elastic-perfectly plastic material (with $E=200 \mathrm{GPa}$ and yield stress $\sigma_0=200 \mathrm{MPa}$ ), has a square cross-section with area $A=$ $100 \mathrm{~cm}^2$. The beam is bent by end couples, which cause strains $\epsilon=0.004$ at the top of the bearm. Determine (a) the depth of yielding, $d_y$, within the cross-section and (b) the magnitude of the applied bending couples.

Victor Salazar

Numerade Educator

03:04

Problem 72

Determine the maximum elastic and plastic moments for a W762×196 steel beam, which is assumed to behave as an elastic-perfectly plastic material with $\sigma_0=$ 250 MPa .

Surendra Kumar

Numerade Educator

03:06

Problem 73

Determine the maximum elastic and plastic moments for a beam, which is assumed to behave as an elastic-perfectly plastic material with yield point $\pm \sigma_0$ if the cross-section is a square with sides a and the moment is applied about its diagonal, as shown in Fig. (8P.73).

Surendra Kumar

Numerade Educator

02:29

Problem 74

A beam with a cross-section, shown in Fig. (8P.74), is subjected to bending about the horizontal axis. Determine the ultimate plastic moment $M_p$ if the beam material is elastic-perfectly plastic with yield stress $\pm \sigma_0$.

Surendra Kumar

Numerade Educator

02:37

Problem 75

A beam of rectangular cross-section consisting of a linearly elastic core (with modulus of elasticity E) bonded to top and bottom plates, as shown in Fig. (8P.75), is subjected to a moment about the $z$-axis. The plates consist of a rigid-plastic material with a yield stress $\pm \sigma_0$ [see Fig. (4.6.2)]. (a) What is the smallest value of the bending moment $M_{r r}$ that will cause the beam to deform? (b) Assuming that plane cross-sections remain plane and perpendicular to the deformed longitudinal axis, obtain the bending-curvature relation.

Chai Santi

Numerade Educator

Problem 76

A beam, in a state of pure bending, having a rectangular cross-section $b \times d$ is subjected to a moment about the $z$-axis, as shown in Fig. (8P.76a). The linear strainhardening behaviour of the beam is described by the stress-strain curve of Fig. (8P.76b), where $E_1$ and $E_2$ are the moduli of elasticity in the two regions.
(a) Show that if the flexural strains at the bottom and top of the cross-section are $\pm \epsilon_m$ respectively, the moment $M$ acting on the beam is given as

$$
M=\frac{b d^2}{12} \sigma_0\left\{\left[3-\left(\frac{\epsilon_0}{\epsilon_m}\right)^2\right]\left(1-\frac{E_2}{E_1}\right)+2 \frac{E_2}{E_1}\left(\frac{\epsilon_m}{\epsilon_0}\right)\right\}
$$

(b) (i) Determine $M$ if the strain $\epsilon_m= \pm 3.6 \times 10^{-3}$ and if $b=20 \mathrm{~mm}, d=30 \mathrm{~mm}, E_1=$ $100 \mathrm{GPa}, E_2=50 \mathrm{GPa}, \sigma_0=120 \mathrm{MPa}$ and (ii) sketch the stress distribution in the cross-section due to the moment.
(c) The beam specified in (b) is now unloaded, i.e. the moment is reduced to zero. Sketch the stress distribution, $\sigma_{\text {unioad }}$ acting on the cross-section due to this unloading. Determine the residual stresses $\sigma_{\mathrm{res}}$ and sketch their distribution in the cross-section.

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Problem 77

The differential relations for a beam subjected to a distributed load $q(x)$ were found to be [see Eqs. (8.3.1) (8.3.3)]

$$
\frac{\mathrm{d} M(x)}{\mathrm{d} x}=V(x), \quad \frac{\mathrm{d} V(x)}{\mathrm{d} x}=-q(x) \quad \text { and } \quad \frac{d^2 M(x)}{\mathrm{d} x^2}=-q(x) \text {. }
$$

Show, by a similar derivation that if, in addition, a distributed moment $m(x)(N-\mathrm{m} / \mathrm{m})$ also acts on the beam about the $z$-axis [see Fig. (8P.77)], the differential relations are

$$
\frac{d M(x)}{d x}=V(x)+m(x), \quad \frac{d V(x)}{d x}=-q(x) \quad \text { and } \quad \frac{d^2 M(x)}{d x^2}=-q(x)+\frac{d m(x)}{d x}
$$

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Problem 78

A composite material of infinite length is made of periodically spaced laminates having different moduli of elasticity, as shown in Fig. (8P.78a). When subjected to loading and/or temperature changes in the $x$-direction, it is found that shear stresses $\tau=\mp A \sin (2 \pi x / L)$ (where $A \geq 0$ is a constant), acting in the $x$-direction, exist along the ( $\pm$ ) interface of a typical laminate over a given length $L$ within the composite, as shown in Fig. (8P.78b). (a) If the thickness of the typical laminate is h, determine the resulting moment $M(x)$ at any cross-section of the fibres. (Note: Assume that $M=0$ at $x=0$.) (b) Determine the curvature $\kappa(x)$ of the laminate if its modulus of elasticity is $E$ and, based on the curvature, sketch the approximate deformation of the laminate assuming it undergoes no deflection in the vertical direction at $x=0$ and $x=L$.

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02:37

Problem 79

A beam having a cross-section, as shown in Fig. (8P.79), is made of an ideal elastic-plastic material such that the yield stress is $\pm \sigma_0$ in tension and compression, respectively, and $\tau_0$ in shear. Determine (a) the maximum elastic moment $M_t$ about the $z$-axis in terms of $\sigma_0$ and $R$, (b) the maximum shear $V$ that can be exerted for the beam to remain elastic and (c) the ultimate plastic moment $M_p$ in terms of $\sigma_0$ and $R$.

Chai Santi

Numerade Educator

Problem 80

A beam of length $L$, consisting of a material for which the allowable stress is $\sigma_{\text {ailow, }}$ rests on a frictionless surface and is subjected to two symmetrically applied loads $P$ at positions from the ends represented by $a>0$, as shown in Fig. (8P,80). The beam is to be designed using various available square cross-sections, $b \times b$. (a) Assuming that the reactive pressure exerted by the surface on the beam is constant over the length $L$, sketch the shear and moment diagrams and label all maxima and minima in terms of $P, L$ and $a$. (b) Segment BD is coated at the top of the beam with a thin brittle adhesive (namely one which it is assumed cannot support any tension). Based on a flexural criterion, determine the location of the loads that leads to an optimal design of the beam, i.e. which minimises the dimension $b$. What is the required dimension $b$ ?

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02:40

Problem 81

A simply supported beam, having a linearly varying depth $d(x)$, as shown in Fig. (8P.81), carries a load $P$ at the centre. Determine the cross-section at which the maximum flexural stress $\sigma_x$ occurs and find ( $\left.\sigma_x\right)_{\text {max }}$. (Assume that the taper is sufficiently small so that warping of any cross-section is negligible.)

Chai Santi

Numerade Educator

09:27

Problem 82

A wooden cantilever beam of length $L$, having a rectangular cross-section with constant width b but varying depth $h(x)$, carries a total load $W(\mathrm{~N})$, which varies linearly as shown in Fig. (8P.82). The maximum allowable flexural stress is 10 MPa . Assume that plane cross-sections remain plane. (a) Determine (i) the required depth $h_0$ at the support $B$ if $L / b=40$ and $W=5 \mathrm{kN}$ and (ii) the required variation $h(x)$ if the maximum flexural stress in the beam is to be the same at all cross-sections. (b) It is known that the expression $\sigma_{\mathrm{x}}=M y / I(x)$ is only a good approximation if the lateral surfaces of the beam have a small slope with respect to the $x$-axis, $\theta$, say $|\theta|=5^{\circ}$. Using this criterion and the answer of part (a), determine the range of $x / L$ for which the given expression for the flexural $\sigma_x$ is a good approximation when the beam is subjected to the given load $W=5 \mathrm{kN}$ if $L=2 \mathrm{~m}$. (c) For the same ratio $b / L$, allowable stress $\sigma$ and total load $W$ as given above, determine the required length $L$ of the beam if the above criterion is satisfied at all cross-sections.

Rashmi Sinha

Numerade Educator

05:01

Problem 83

Repeat part (a) of Problem 8.82 for the same loading $W$ and ratio $b / L$ if the average shear stress reaches the same maximum value $\mathrm{r}=2 \mathrm{MPa}$ at all cross-sections.

Ajay Singhal

Numerade Educator

Problem 84

A composite beam whose cross-section [as shown in Fig. (8P.84)] consists of two materials ' $A$ ' and ' $B$ ' having different moduli of elasticity, $E_{\mathrm{B}} \geq E_{\mathrm{A}}$, is subjected to a moment about the z-axis. Determine the maximum flexural stress in each of the materials.

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Problem 85

A pipe consists of an external steel ( $E=200 \mathrm{GPa}$ ) pipe bonded to an internal aluminium ( $E=70 \mathrm{GPa}$ ) pipe to form a composite cross-section, as shown in Fig. (8P.85). Determine (a) the maximum stress in the aluminium and steel if the cross-section is subjected to a moment of 2500 N -m and (b) the radius of curvature of the composite beam at the cross-section.

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Problem 86

- A beam, having a rectangular cross-section $b \times d$ [Fig. (8P.86b)] (whose second moment about its horizontal centroidal axis is I), is made of a material having different moduli of elasticity in tension and compression, namely $E_{\mathrm{t}}$ and $E_c$ respectively, as shown in Fig. (8P.86a). The beam is subjected to a positive bending moment $M$ about the $z$-axis. (a) Show that the curvature-moment relation of the deformed beam can be written as $k=\frac{M}{\left.E^2\right]}$,
where $E^*$, the equivalent modulus of elasticity, is

$$
E^*=\frac{4 E_{\mathrm{t}} E_{\mathrm{c}}}{\left[\sqrt{E_{\mathrm{t}}}+\sqrt{E_{\mathrm{c}}}\right]^2} .
$$

(b) Determine the maximum flexural tensile stress.

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03:29

Problem 87

A beam, whose cross-section consists of a semi-circle, is subjected to a vertical shear force $V$ acting in the $y$-direction. Show that the average shear stress $\tau_{x y}$ along the line a-a of the cross-section shown in Fig. (8P.87), is given by

$$
\tau_{x y}=\frac{24 \pi V}{\left(9 \pi^2-64\right) R^2 \sin \theta}\left[\sin ^3 \theta-\frac{1}{\pi}(2 \theta-\sin 2 \theta)\right] .
$$

(Note: See computer-related Problem 8.97.)

Chai Santi

Numerade Educator

06:20

Problem 88

Given an elastic beam with modulus of elasticity $E$, whose cross-section is a rectangle ( $b \times d$ ) and which has an initial radius of curvature $p$, as shown in Fig. (8P.88a). A moment is applied in order to straighten out the beam [see Fig. (8P.88b)].
(a) Noting that while the beam undergoes deformation, there exists a surface for which the strain $\epsilon=0$ (namely, the neutral surface represented by the line $\mathrm{N}-\mathrm{N}^{\prime}$ ), and letting $\rho$ be the radius of curvature to the neutral surface, show that when the beam is straightened the strain at any arbitrary fibre is $\epsilon=-\frac{\eta}{\rho+\eta}$, where $\eta$ denotes the perpendicular distance from this surface to an arbitrary fibre.
(b) Show that the neutral surface does not pass through the centroid of the crosssection; i.e. $c_1 \neq c_2$, where $c_1$ and $c_2$ are shown in the figure. (Note that $c_1+c_2=d$.)
(c) (i) Show that the location of the neutral axis (which depends on the ratio $\alpha=d / \rho$ ) is given by

$$
\frac{c_2}{d}=\frac{(1-\alpha) \mathrm{e}^\alpha-1}{\alpha\left(1-\mathrm{e}^a\right)}
$$
(ii) Show that the moment required to straighten out the beam is given by the expression

$$
M=E b \rho^2\left[\frac{d}{\rho}-\frac{d\left(c_2-c_1\right)}{2 \rho^2}-\ln \left(1+\frac{c_2}{\rho}\right)+\ln \left(1-\frac{c_1}{\rho}\right)\right] .
$$

(d) Obtain a simplified expression,

$$
\frac{c_2}{d}=\frac{1}{2}\left(1+\frac{d}{6 p}\right)
$$

for the location of the neutral axis, $c_2 / d$, if the curvature is relatively small, i.e. if $d / \rho \ll 1$.
(e) Show that if the curvature is relatively small, the moment required to straighten the beam is given by $|M|=E I / \rho$, namely the same moment as is required to cause a straight beam to be bent into a curve with radius $\rho$. (We conclude that the Euler-Bernoulli relations may be applied to a beam with initial curvature $x=1 / \rho$ for the case $d / \rho \ll 1$. Clearly, this is not so if the condition $d / \rho \ll 1$ is not satisfied.)
[Hint: For parts (d) and (e), use appropriate series expansions for the logarithmic and exponential terms appearing in (c) above.
(Note: See computer-related Problem 8.101.)

Khoobchandra Agrawal

Numerade Educator

Problem 89

Two separate thin elastic strips 'A' and ' $B$ ', having the same rectangular crosssection $b \times 2 c$, (with moment of inertia $1_0$ ) are bent by end couples $M_A$ and $M_3$, as shown in Fig. (8P.89), such that the radius of curvature of the common interface H is $R_0$ and such that no separation exists at the interface. (a) Show that the required relation between the moments is $M_A=M_B\left[1+2 c / R_0+2\left(c / R_0\right)^2+\cdots\right]$ and therefore if $c \ll R_0, M_{\mathrm{A}} \simeq M_{\mathrm{B}}$. (b) The two strips are subsequently glued together and the moments are then removed. Assuming that $c \ll R_0$, (i) determine the resulting flexural stress in each strip and (ii) sketch the stress distribution in the combined cross-section. (Note : For the case $c \ll R_0$, the Euler-Bernoulli relation, $M=E I / R$, remains valid Isee part (e) of Problem 8.88].) (c) Determine the resulting final radius of curvature of the interface, $R$, of the combined strips in terms of $R_0$.

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06:01

Problem 90

A beam of length $L$, simply supported at $A$ and at a variable point $B$ as shown in Fig. (8P.90), is subjected to a uniform load $q_0$ over its entire length. It is desired to optimise the design of the beam by minimising the (absolute) value of the maximum moment in the beam. (a) Determine the position of $B$ (i.e., find b), that yields this optimal solution. (b) Determine the maximum resulting (absolute) value of the moment in the beam. (c) Sketch the variation of $\mid M_{\text {max }}$ between A and B and that of $\left|M_{\mathrm{B}}\right|$ as a function of $b / L$ for $0<b / L \leq 1$.

Eric Mockensturm

Numerade Educator

Problem 91

An elastic beam having a square cross-section $(a \times a)$ is subjected to a moment $M \equiv M_z$, about the $z$-axis, as shown in Fig. (8P.91a). (a) Determine the maximum flexural stress $\left|\left(\sigma_x\right\rangle_0\right|$. (b) By removing material at the top and bottom corners, a cross-section as shown by solid lines in Fig. (8P.91b) is obtained. Show that the resulting maximum stress $\left(\sigma_x\right)_b$ is given as

$$
\left(\sigma_x\right)_0=\frac{3 M}{c^3}\left[\frac{1-\alpha}{1-\alpha^2\left(3 \alpha^2-8 \alpha+6\right)}\right]
$$

[where $\alpha$ represents the cut over a fraction of the depth, see Fig. (8P.91b)] and thus $R_\theta<17$ What does this imply physically?
(Note: See computer-related Problem 8.100.)

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01:04

Problem 92

A moment $M_{\mathrm{t}} \leq M \leq M_{\mathrm{p}}$ causing plastic behaviour is applied about the diagonal of the cross-section of a square beam whose diagonal length is $d=2 c$, as shown in Fig. (8P.92). The beam material is elastic-perfectly plastic with yield point $\pm \sigma_0$. (a) Determine the relation between $M$ and the location of $\ell$, the interface of the elastic and plastic zones. (b) Verify that $M_1=\frac{\sigma_2 f^f}{24}$ and $M_p=\frac{v_0 d^2}{12}$ when $\ell=d / 2$ and $\ell=0$, respectively. (c) Expressing $M / M_p$ as a function of $\ell / d$, plot the non-dimensional ratio $M / M_\rho \mathrm{vs} . \ell / d$.

Hast Aggarwal

Numerade Educator

Problem 93

In deriving the expression for the flexural stress under pure bending, $\sigma_x=\frac{M y}{T}$, the stress $\sigma_y$ is assumed to be negligible; i.e. $\sigma_y \ll \sigma_x$. Consider now, for example, a simply supported beam of length $L$, having a rectangular cross-section $b \times d$, subjected to a uniformly distributed load $q_0$, as shown in Fig. (8P.93). It is clear that near the ends $x=0$ and $x=L$, the moment $M$ is very small and therefore, as $x$ approaches the end points, $\sigma_x \rightarrow 0$. Since the maximum compressive value of $\sigma_y$ directly under the load is $\left|\sigma_y\right|=q_0 / b$, the assumption $\left|\sigma_y / \sigma_x\right| \ll 1$ is clearly contradicted in this region. (a) Show that if $L / d \gg 1$, the region where this contradiction occurs is negligible with respect to the length $L$ and may therefore be disregarded in an analysis. For example, estimate $\left|\sigma_y / \sigma_x\right|$ directly under the load $q_0$ for the cases $\Delta / L=0.05$ and 0.10 with $L / d=20$ and 100, where $\Delta$ is a small distance away from the ends. (b) Determine $\left|\sigma_y / \sigma_x\right|$ directly under the load $q_0$ at $x / L=0.5$ if $L / d=20$.

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02:20

Problem 94

In deriving the expression for the flexural stress under pure bending, $\sigma_x=\frac{M y}{1}$, the stress $\sigma_y$ is assumed to be negligible; i.e., $\left|\sigma_y\right| \ll\left|\sigma_x\right|$. Consider now, for example, a beam of rectangular cross-section, $b \times 2 c$, subjected to lateral loads $q(x)$, as shown in Fig. (8P.94a). Clearly since the stress $\sigma_y=q / b$ at the top of the bearn and $\sigma_y=0$ at the bottom, $\sigma_y=f(y)$, i.e $\sigma_y$ varies with $y$ and is not zero throughout the beam. Note that here positive $\sigma_y$ is compressive.

We first isolate an element ( $b \times 2 c \times d x$ ) [see Fig. (8P.94b)]; shear forces $V$ and $V+\Delta V$ act on the two faces as shown. Let us now isolate a portion of the element, $b \times(1-\alpha) c \times d x$, where $y=-\alpha c(0 \leq \alpha \leq 1)$, as shown in Fig. (8P.94c). Since $q(x) \neq 0$, $V=V(x)$ on any cross-section; therefore, acting on this portion of the element there exist, in addition to $q(x)$, (i) shear forces (which are due to the shear stresses $\mathrm{r}_{x y}$ ) on the left and right faces, which we denote as $\Delta V$ and $\mathrm{d}(\Delta V)$, respectively, and (ii) a stress $\sigma_y$. (Note that here ' $\Delta$ ' refers to the portion of the cross-section and ' $d$ ' refers to the difference of the left and right faces.)
(a) From equilibrium considerations, show that

$$
\sigma_y=\frac{1}{b}\left[q(x)+\frac{d(\Delta V)}{d x}\right]
$$

and hence, using the relation $\mathrm{dV}(x) / \mathrm{d} x=-q(x)$,

$$
\sigma_y=\frac{q(x)}{b}\left[1-\frac{1}{l} \int_{-c}^{-a x} Q(y) d y\right] \text {, }
$$

where $Q(y)$ is the first moment about the centroidal z-axis of the portion of the cross-section $b \times(1-\alpha) c$.
(b) By evaluating the integral, obtain an expression for $\sigma_y$, namely

$$
\sigma_y(x, \alpha)=\frac{q(x)}{4 b}\left[2+3 \alpha-\alpha^3\right]
$$

$$
\sigma_y(x, y)=\frac{q(x)}{4 b}\left[2-3 \frac{y}{c}+\frac{y^3}{c^3}\right] .
$$

(c) Plot $\sigma_y$ as a function of $y$ in the beam. What is $\sigma_y$ at the neutral axis? of a simply supported beam subjected to a uniformly distributed load $q_0$ and therefore $R_\sigma \ll 1$ if $d / L \ll 1$.

Surendra Kumar

Numerade Educator

Problem 95

The following loads, acting on a cantilever beam of length $L$, free at $x=0$ and fixed at $x=L$, were measured along its length during an experiment.
$$
\begin{array}{llll|}
\text { x/L } & \text { Load }(\mathrm{N}) & \text { x/L } & \text { Load }(\mathrm{N}) \\
\hline 0 & 0 & 0.50 & 0.4621 \\
0.05 & 0.0500 & 0.55 & 0.5005 \\
0.10 & 0.0997 & 0.60 & 0.5370 \\
0.15 & 0.1489 & 0.65 & 0.5717 \\
0.20 & 0.1974 & 0.70 & 0.6044 \\
0.25 & 0.2449 & 0.75 & 0.6351 \\
0.30 & 0.2913 & 0.80 & 0.6640 \\
0.35 & 0.3364 & 0.85 & 0.6911 \\
0.40 & 0.3799 & 0.90 & 0.7163 \\
0.45 & 0.4219 & 0.95 & 0.7398 \\
& & 1.00 & 0.7616 \\
\hline
\end{array}
$$
(a) By means of a computer, plot the load as a function of $x / L$. (b) Integrating Eqs. (8.5.2) and (8.5.7) numerically, determine the shear $V$ and moment $M$ at the points $x / L$.

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Problem 96

The following loads, acting on a simply supported beam of length $L$, were measured along its length during an experiment.

$$
\begin{array}{llll}
\hline X / L & \text { Load }(\mathrm{N}) & \text { X/L } & \text { Load }(\mathrm{N}) \\
\hline 0 & 0 & 0.50 & 0.3935 \\
0.05 & 0.0488 & 0.55 & 0.4231 \\
0.10 & 0.0952 & 0.60 & 0.4512 \\
0.15 & 0.1393 & 0.65 & 0.4780 \\
0.20 & 0.1813 & 0.70 & 0.5034 \\
0.25 & 0.2212 & 0.75 & 0.5276 \\
0.30 & 0.2592 & 0.80 & 0.5507 \\
0.35 & 0.2953 & 0.85 & 0.5753 \\
0.40 & 0.3297 & 0.90 & 0.5934 \\
0.45 & 0.3624 & 0.95 & 0.6133 \\
& & 1.00 & 0.6321 \\
\hline
\end{array}
$$
(a) By means of a computer, plot the load as a function of $x / L$. (b) Integrating Eqs. (8.5.2) and (8.5.7) numerically, determine the shear $V$ and moment $M$ at the points $x / L$. (c) Based on a numerical analysis of the curve obtained in (a) for the given load, it appears that the loading could be represented analytically by the expression $q(x)=1-\mathrm{e}^{-x / h}$. Assuming this is correct, obtain the shear and moment by integrating this expression analytically according to Eqs. (8.5.2) and (8.5.7). (d) Using a computer, plot the results of (b) and (c) on the same graph and compare the results.

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Problem 97

The average shear stress $\tau_{x y}$ due to a shear force $V$ acting in the $y$-direction on a semi-circular cross-section is given as (see Problem 8.87)

$$
\mathrm{I}_{x y}=\frac{24 \pi V}{\left(9 \pi^2-64\right) R^2 \sin \theta}\left[\sin ^3 \theta-\frac{1}{\pi}(2 \theta-\sin 2 \theta)\right],
$$
where $\theta$ is as shown in Fig. (8P.87). (a) Derive the transcendental equation whose roots determine the location of the line along which the maximum shear stress occurs and solve the equation numerically. (b) (i) Evaluate the location of this line as measured by $\hat{y}$, the distance from the top of the beam and (ii) determine numerically the value of $k$ according to the relation $\tau_{\text {max }}=k \frac{V}{A}$, where $A$ is the area of the cross-section.

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Problem 98

A solid rod having a cross-section of radius a with modulus of elasticity $E$ and yield point $\pm \sigma_0$ is assumed to behave as an elastic-perfectly plastic material. The rod is subjected to a bending moment $M$, which increases incrementally from zero to $M$. the maximum elastic moment, and subsequently to $M=M_p$, the fully plastic moment [Fig. 8P.98].

Denoting the elastic-plastic interface by $\ell$,
(a) determine analytically the expression for the moment $M$ as a function of the ratio $\ell / a$;
(b) verify that $M$, as obtained in (a) above, yields $M$ and $M$ as $\ell / a=1$ and $\ell / a \rightarrow 0$ respectively;
(c) rewrite the expression obtained in (a) above in terms of $M / M p$ and $\ell / a$;
(d) by means of a computer, solve numerically for $\ell / a$ in terms of the ratio $M / M$ for values $M_i / M_p \leq M / M_p \leq 1$;
(e) using a computer, plot a curve, $\ell / a$ vs. $M / M_p$ for values $0 \leq M / M_p \leq 1$;
(f) (i) write an expression for the curvature of the rod $k \equiv 1 / R$ in terms of $E I$, $M_p$ and $f\left(M^2 / M_p\right)$, where $\ell / a=f\left(M / M_p\right)$ is the function that was determined numerically in (d) above and (ii) using a computer, plot $M / M_p$ vs. EI/R for $0 \leq$ $M / M_p \leq 1$.

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Problem 99

In Problems 8.21 and 8.22, two beams having a rectangular cross-section (with width $b$ and depth $\sigma$ ) and subjected to a moment $M$ were considered. In the first beam, the stress-strain relation was given as

$$
\sigma=\alpha \epsilon^n, \quad n=1,3,5 \ldots
$$

where $\alpha \geq 0$ is a constant. In the second beam, the stress-strain relation was given as

$$
\sigma= \begin{cases}\alpha \epsilon^n, & \epsilon \geq 0 \\ -\alpha|\epsilon|^n, & \epsilon \leq 0,\end{cases}
$$

where $0<n \leq 1$.
For both beams the expression for the curvature $\kappa$ was found to be identical, namely

$$
\kappa=\frac{2[2(n+2)]^{1 / n}}{d\left(\alpha b \sigma^2\right)^{1 / n}} M^{1 / n}
$$

(a) Using a computer, (i) plot the stress-strain curve, i.e. $\sigma / \alpha$ vs. $\epsilon$, for the cases $n=0.5,1,2$ and (ii) plot the relation $x$ vs. $M$ for the cases $n=0.5,1,2$. (b) For which beam, governed by $n=0.5$ or $n=2$, would one expect the beam to be stiffer when (i) $M$ is very small and (ii) $M$ is relatively large? (c) For what value of $M$ would the two beams bend with the same curvature?

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Problem 100

Referring to Problem 8.91, the ratio $R_e=\frac{(a) \mid}{(a)}$ of the maximum flexural stress in the two cross-sections of Figs. (8.91a and b), respectively, when subjected to the same moment $M$ is given as $R_e=(\alpha-1)^2(3 \alpha+1)$. It is desired to optimise the design by removing material at the top and bottom corners in order to minimise the maximum stress in the beam. (a) Determine the optimal values of $\alpha$ and the resulting value of $R_q$. (b) Plot $R_\sigma$ vs. $\alpha$.

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06:20

Problem 101

Referring to Problem 8.88, the location of the neutral axis in the rectangular cross-section of a beam having depth $d$ and initial radius of curvature $\rho$, is

$$
\frac{c_2}{d}=\frac{(1-\alpha) \mathrm{e}^\alpha-1}{\alpha\left(1-\mathrm{e}^\alpha\right)}
$$

where $\alpha=d / \rho$ and where $c_2$ is as shown in Fig. (8P.88a). A simplified expression for the case $d / \rho \ll 1$ is

$$
\frac{c_2}{d}=\frac{1}{2}\left(1+\frac{d}{6 \rho}\right) .
$$

Using a computer, plot the two expressions as functions of $\alpha=d / \rho$ on the same graph and determine the range of $d / \rho$ for which the difference is within $10 \%$.

Khoobchandra Agrawal

Numerade Educator