Roark's Formulas: Table 8.1-4

Shear, Moment, Slope, and Deflection Formulas for Elastic Straight Beams.

TOC

Source: Roark's Formulas for Stress and Strain, 9th edition, Table 8.1, pp. 208-209, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

4. Intermediate externally created angular deformation

The beam has constant $E$ and $I$. Coordinate $x$ runs from the left end $A$ to the right end $B$, with $0\leq x\leq l$. Reactions and deflections are positive upward; slope is positive upward to the right; positive moment compresses the upper fibers. A guided end can translate vertically but cannot rotate.

Macaulay brackets satisfy ${\langle x-a\rangle}^n=0$ for $x<a$ and ${(x-a)}^n$ for $x>a$; the zeroth power is a unit step. Both one-sided limits are included when finding extrema at a jump. Cases with $a=0$ or $a=l$ are interpreted as limits from within the span.

	Transverse shear: $V=R_A$
	Bending moment: $M=M_A+R_Ax$
	Slope: $\theta=\theta_A+\frac{M_Ax}{EI}+\frac{R_Ax^2}{2EI}+\theta_0{\langle x-a\rangle}^0$
	Deflection: $y=y_A+\theta_Ax+\frac{M_Ax^2}{2EI}+\frac{R_Ax^3}{6EI}+\theta_0\langle x-a\rangle$

Selected maximum formulas below assume the positive load, couple, imposed displacement, or thermal curvature shown. Signed values are retained. The current-input tables report the minimum, maximum, and greatest absolute value over the span; for negative inputs the signs reverse. A reported position is one location attaining the extremum. The formulas labeled "maximum possible" vary $a$ while holding the other inputs fixed.

4a. Left end free, right end fixed (cantilever)

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\theta_0 =$ -0.001 rad	$\theta_B=0 =$ 0 rad
$y_A=\theta_0a =$ 0.5 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	-0.001 rad at $x=$ 0 mm	0 rad at $x=$ 500 mm	-0.001 rad at $x=$ 0 mm
Deflection	0 mm at $x=$ 500 mm	0.5 mm at $x=$ 0 mm	0.5 mm at $x=$ 0 mm

• $y_{\max}=y_A$; maximum possible value $\theta_0l$ at $a=l$.
• The elastic moment is zero throughout.

4b. Left end guided, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=-\frac{EI\theta_0}{l} =$ -200000 N$\cdot$mm	$M_B=-\frac{EI\theta_0}{l} =$ -200000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=\theta_0\left(a-\frac l2\right) =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-200000 N$\cdot$mm at $x=$ 0 mm	-200000 N$\cdot$mm at $x=$ 0 mm	-200000 N$\cdot$mm at $x=$ 0 mm
Slope	-0.0005 rad at $x=$ 500 mm	0.0005 rad at $x=$ 500 mm	0.0005 rad at $x=$ 500 mm
Deflection	-0.125 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.125 mm at $x=$ 500 mm

• Moment is constant: $M=M_A$.
• Maximum positive deflection $y_A$ when $a>l/2$; maximum possible value $\theta_0l/2$ at $a=l$.
• Maximum negative deflection $-\theta_0{(l-a)}^2/(2l)$ at $x=a$; maximum possible negative magnitude has signed value $-\theta_0l/2$ at $a=0$.

4c. Left end simply supported, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=-\frac{3EIa\theta_0}{l^3} =$ -300 N	$R_B=-R_A =$ 300 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=-\frac{3EIa\theta_0}{l^2} =$ -300000 N$\cdot$mm
$\theta_A=-\theta_0\left(1-\frac{3a}{2l}\right) =$ -0.00025 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-300000 N$\cdot$mm at $x=$ 1000 mm	0 N$\cdot$mm at $x=$ 0 mm	-300000 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0004375 rad at $x=$ 500 mm	0.0005625 rad at $x=$ 500 mm	0.0005625 rad at $x=$ 500 mm
Deflection	-0.15625 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.15625 mm at $x=$ 500 mm

• $M_{\max}=M_B$; maximum possible magnitude has signed value $-3EI\theta_0/l$ at $a=l$.
• For $a\geq2l/3$, maximum positive deflection $\theta_0a\left(1-\frac{2l}{3a}\right)^{3/2}$ at $x=l\sqrt{1-\frac{2l}{3a}}$; maximum possible value $\theta_0l/(3\sqrt{3})\approx0.19245\theta_0l$ at $x=l/\sqrt{3}$, $a=l$. No positive deflection exists for $a<2l/3$.
• Maximum negative deflection $-\theta_0a\left(1-\frac{3a}{2l}+\frac{a^3}{2l^3}\right)$ at $x=a$; maximum possible negative magnitude has signed value $-0.17404\theta_0l$ at $x=a=0.366l$.

Source correction: the printed positive-deflection coefficient $0.1926$ is replaced by the exact expression above. The printed negative-deflection coefficient is also inconsistent with its formula; substitution at $a=(\sqrt{3}-1)l/2$ gives $-0.17404\theta_0l$.

4d. Left end fixed, right end fixed

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=\frac{6EI\theta_0}{l^3}(l-2a) =$ 0 N	$R_B=-R_A =$ 0 N
$M_A=\frac{2EI\theta_0}{l^2}(3a-2l) =$ -200000 N$\cdot$mm	$M_B=\frac{2EI\theta_0}{l^2}(l-3a) =$ -200000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	-200000 N$\cdot$mm at $x=$ 0 mm	-200000 N$\cdot$mm at $x=$ 0 mm	-200000 N$\cdot$mm at $x=$ 0 mm
Slope	-0.0005 rad at $x=$ 500 mm	0.0005 rad at $x=$ 500 mm	0.0005 rad at $x=$ 500 mm
Deflection	-0.125 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.125 mm at $x=$ 500 mm

• For $a<l/2$, the greatest end moment is $M_B$ and the least is $M_A$; reflect the beam for $a>l/2$.
• Maximum possible positive moment $2EI\theta_0/l$ at $a=0$ or $a=l$.
• Maximum possible negative moment magnitude has signed value $-4EI\theta_0/l$ at $a=0$ or $a=l$.
• For $a<l/3$, positive deflection has an extremum at $x=l^2/[3(l-2a)]$; reflect for $a>2l/3$. Maximum possible positive deflection $4l\theta_0/27$ at $a=0$ or $a=l$. No positive deflection exists for $l/3<a<2l/3$.
• Maximum negative deflection $-2\theta_0a^2{(l-a)}^2/l^3$ at $x=a$; maximum possible negative magnitude has signed value $-\theta_0l/8$ at $a=l/2$.

4e. Left end simply supported, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=-\frac{\theta_0}{l}(l-a) =$ -0.0005 rad	$\theta_B=\frac{\theta_0a}{l} =$ 0.0005 rad
$y_A=0 =$ 0 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	-0.0005 rad at $x=$ 0 mm	0.0005 rad at $x=$ 500 mm	0.0005 rad at $x=$ 500 mm
Deflection	-0.25 mm at $x=$ 500 mm	0 mm at $x=$ 0 mm	-0.25 mm at $x=$ 500 mm

• $y_{\max}=-\theta_0a(l-a)/l$ at $x=a$; maximum possible magnitude has signed value $-\theta_0l/4$ at $a=l/2$.
• The elastic moment is zero throughout.

Source correction: $\theta_B=\theta_0a/l$; the scanned boundary value omits $a$. This follows from the imposed slope jump $\theta_B-\theta_A=\theta_0$.

4f. Left end guided, right end simply supported

Details

Inputs

$l =$ 1000 mm	Beam length, positive
$I =$ 1000000 mm$^4$	Second moment of area about the centroidal bending axis, positive
$E =$ 200000 MPa	Modulus of elasticity, positive
$a =$ 500 mm	Position from the left end, $0\leq a\leq l$
$\theta_0 =$ 0.001 rad	Imposed positive slope jump

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=0 =$ 0 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=\theta_0 =$ 0.001 rad
$y_A=-\theta_0(l-a) =$ -0.5 mm	$y_B=0 =$ 0 mm

Maximum moments and deformations

Current inputs; positions are measured from $A$. Values at an imposed jump are one-sided limits.

Quantity	Minimum (signed)	Maximum (signed)	Greatest absolute value (signed)
Moment	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm	0 N$\cdot$mm at $x=$ 0 mm
Slope	0 rad at $x=$ 0 mm	0.001 rad at $x=$ 500 mm	0.001 rad at $x=$ 500 mm
Deflection	-0.5 mm at $x=$ 0 mm	0 mm at $x=$ 1000 mm	-0.5 mm at $x=$ 0 mm

• $y_{\max}=y_A$; maximum possible magnitude has signed value $-\theta_0l$ at $a=0$.
• The elastic moment is zero throughout.

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