Roark's Formulas: Table 8.1-6

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. 211-212, supplied PDF excerpt. Apparent source errors are identified beside the affected cases.

6. Uniform temperature variation from top to bottom from a to l

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.

For compactness, $\kappa=\gamma(T_2-T_1)/t$ is the imposed thermal curvature (1/mm). Temperatures vary linearly through depth $t$ and are constant along the heated portion $a\leq x\leq l$. Substitute this definition into every formula below.

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

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.

6a. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

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
$\theta_A=-\kappa(l-a) =$ -0.0012 rad	$\theta_B=0 =$ 0 rad
$y_A=\frac{\kappa}{2}(l^2-a^2) =$ 0.9 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	-0.0012 rad at $x=$ 0 mm	0 rad at $x=$ 1000 mm	-0.0012 rad at $x=$ 0 mm
Deflection	0 mm at $x=$ 1000 mm	0.9 mm at $x=$ 0 mm	0.9 mm at $x=$ 0 mm

• $M=0$ everywhere.
• $\theta_{\max}=\theta_A$; maximum possible magnitude has signed value $-\kappa l$ at $a=0$.
• $y_{\max}=y_A$; maximum possible value $+\kappa l^2/2$ at $a=0$.

Source correction: the printed maximum-deflection sign is negative, inconsistent with the positive boundary value $y_A$ and the general deflection equation.

6b. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

Boundary values

Left end A	Right end B
$R_A=0 =$ 0 N	$R_B=0 =$ 0 N
$M_A=-\frac{EI\kappa}{l}(l-a) =$ -240000 N$\cdot$mm	$M_B=M_A =$ -240000 N$\cdot$mm
$\theta_A=0 =$ 0 rad	$\theta_B=0 =$ 0 rad
$y_A=\frac{\kappa a}{2}(l-a) =$ 0.3 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	-240000 N$\cdot$mm at $x=$ 0 mm	-240000 N$\cdot$mm at $x=$ 0 mm	-240000 N$\cdot$mm at $x=$ 0 mm
Slope	-0.0006 rad at $x=$ 500 mm	0 rad at $x=$ 0 mm	-0.0006 rad at $x=$ 500 mm
Deflection	0 mm at $x=$ 1000 mm	0.3 mm at $x=$ 0 mm	0.3 mm at $x=$ 0 mm

• $M=M_A$ everywhere; maximum possible magnitude has signed value $-EI\kappa$ at $a=0$.
• Maximum slope magnitude has signed value $-\kappa a(l-a)/l$ at $x=a$; maximum possible negative magnitude has signed value $-\kappa l/4$ at $a=l/2$.
• $y_{\max}=y_A$; maximum possible value $\kappa l^2/8$ at $a=l/2$.

6c. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

Boundary values

Left end A	Right end B
$R_A=-\frac{3EI\kappa}{2l^3}(l^2-a^2) =$ -540 N	$R_B=-R_A =$ 540 N
$M_A=0 =$ 0 N$\cdot$mm	$M_B=R_Al =$ -540000 N$\cdot$mm
$\theta_A=\frac{\kappa}{4l}(l-a)(3a-l) =$ 0.00015 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	-540000 N$\cdot$mm at $x=$ 1000 mm	0 N$\cdot$mm at $x=$ 0 mm	-540000 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.0001875 rad at $x=$ 500 mm	0.00015 rad at $x=$ 0 mm	-0.0001875 rad at $x=$ 500 mm
Deflection	-0.00246914 mm at $x=$ 777.778 mm	0.0333333 mm at $x=$ 333.333 mm	0.0333333 mm at $x=$ 333.333 mm

• $M_{\max}=M_B$; maximum possible magnitude has signed value $-3EI\kappa/2$ at $a=0$.
• For $a>l/3$, maximum positive deflection $\frac{\kappa(l-a){(3a-l)}^{3/2}}{6\sqrt{3(l+a)}}$ at $x=l\sqrt{\frac{3a-l}{3l+3a}}$; maximum possible value $0.0257\kappa l^2$ at $x=0.474l$, $a=0.721l$. No positive deflection exists for $a<l/3$.
• A negative-deflection extremum, when it lies in $a\leq x\leq l$, occurs at $x=\frac{2l^3}{3(l^2-a^2)}\left[1-\frac12\sqrt{1-6{(a/l)}^2+9{(a/l)}^4}\right]$; maximum possible negative deflection $-\kappa l^2/27$ at $x=l/3$, $a=0$. At $a=l$, the thermal interval is empty.

6d. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

Boundary values

Left end A	Right end B
$R_A=-\frac{6EI\kappa a}{l^3}(l-a) =$ -720 N	$R_B=-R_A =$ 720 N
$M_A=\frac{EI\kappa}{l^2}(l-a)(3a-l) =$ 120000 N$\cdot$mm	$M_B=-\frac{EI\kappa}{l^2}(l-a)(3a+l) =$ -600000 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	-600000 N$\cdot$mm at $x=$ 1000 mm	120000 N$\cdot$mm at $x=$ 0 mm	-600000 N$\cdot$mm at $x=$ 1000 mm
Slope	-0.00015 rad at $x=$ 500 mm	0.00005 rad at $x=$ 833.333 mm	-0.00015 rad at $x=$ 500 mm
Deflection	-0.0111111 mm at $x=$ 666.667 mm	0.0111111 mm at $x=$ 333.333 mm	0.0111111 mm at $x=$ 333.333 mm

• Maximum positive moment $M_A$; maximum possible value $EI\kappa/3$ at $a=2l/3$. There is no positive moment for $a<l/3$.
• Maximum negative moment $M_B$; maximum possible negative magnitude has signed value $-4EI\kappa/3$ at $a=l/3$.
• For $l/3<a<l$, maximum positive deflection $\frac{2M_A^3}{3R_A^2EI}$ at $x=\frac{l}{3a}(3a-l)$; maximum possible value $0.01617\kappa l^2$ at $x=0.565l$, $a=0.767l$. No positive deflection exists for $a<l/3$.

Source correction: the printed stationary-point expression uses $(3l-a)$; it must be $(3a-l)$, as follows from $\theta=0$ on the portion before $a$.

6e. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

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
$\theta_A=-\frac{\kappa}{2l}{(l-a)}^2 =$ -0.0003 rad	$\theta_B=\frac{\kappa}{2l}(l^2-a^2) =$ 0.0009 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	-0.0003 rad at $x=$ 500 mm	0.0009 rad at $x=$ 1000 mm	0.0009 rad at $x=$ 1000 mm
Deflection	-0.16875 mm at $x=$ 625 mm	0 mm at $x=$ 0 mm	-0.16875 mm at $x=$ 625 mm

• $M=0$ everywhere.
• Maximum positive slope $\theta_B$; maximum possible value $\kappa l/2$ at $a=0$.
• Maximum negative slope $\theta_A$; maximum possible negative magnitude has signed value $-\kappa l/2$ at $a=0$.
• $y_{\max}=-\frac{\kappa}{8l^2}{(l^2-a^2)}^2$ at $x=(l^2+a^2)/(2l)$; maximum possible negative magnitude has signed value $-\kappa l^2/8$ at $x=l/2$, $a=0$.

6f. 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, $0\leq a\leq l$
$\gamma =$ 0.000012 $1/{}^\circ$C	Coefficient of thermal expansion
$T_1 =$ 20 $^\circ$C	Top-surface temperature
$T_2 =$ 40 $^\circ$C	Bottom-surface temperature
$t =$ 100 mm	Beam depth, positive

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	$\theta_B=\kappa(l-a) =$ 0.0012 rad
$y_A=-\frac{\kappa}{2}{(l-a)}^2 =$ -0.3 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	0.0012 rad at $x=$ 1000 mm	0.0012 rad at $x=$ 1000 mm
Deflection	-0.3 mm at $x=$ 0 mm	0 mm at $x=$ 1000 mm	-0.3 mm at $x=$ 0 mm

• $M=0$ everywhere.
• $\theta_{\max}=\theta_B$; maximum possible value $\kappa l$ at $a=0$.
• $y_{\max}=y_A$; maximum possible negative magnitude has signed value $-\kappa l^2/2$ at $a=0$.

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